no. 730o
V, 1
DIALOGO DELLA MUSICA ANTICA ET DELLA MODERNA
OF VINCENZO GALILEI:
TRANSLATION AND COMMENTARY
DISSERTATION
Presented to the Graduate Council of the
North Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Robert H. Herman, M. M.
Denton, Texas
August, 1973
Herman, Robert H., Dialog dell musica antics et
della modern of Vincenso Galilei: Translation and onmmeentaryDoctor
of Philosophy (Music Theory), August,
1973, 929 pp., 43 tables, bibliography, 165 titles, index.
The purpose of this study is to provide a practical
English translation of Vincenzo Galilei's significant
treatise on ancient and modern music (1581). In spite
of the important place this work holds in the history of
music, it has never before been made available in its
entirety in any language other than the original Italian.
The Dialogo della musica antics et della modern
was, in effect, an expression of monodic tendencies in
Italian music of the sixteenth century. While the
prevailing vocal compositional style was contrapuntal,
Galilei advocated the development of homophonic pieces,
which could produce in their hearers whatever affections
or emotions the composer desired to convey. The emphasis
was on simplicity of melody, with only a few chords for
accompaniment. This style, with its sparseness of means,
was influenced by the music of the ancient Greeks, whose
philosophies and art forms were in credit at the time,
having been revived by such Renaissance humanists as
Lorenzo di Medici, Marsilio Ficino, and Angelo Poliziano.
Galilei, influenced by the erudite Girolamo Mei, who was
2
a classical philologist, drew away from his former mentor,
Gioseffo Zarlino, a firm believer in traditional counterpoint.
He included in his Dialogo many attacks upon
Zarlino's teachings, engendering a long and bitter controversy
which extended until Galilei's death in 1591.
The ideas promulgated by Galilei became the subjects
of discussion by members of the Florentine Camerata of
Count Giovanni Bardi. Experiments revealed the dramatic
possibilities of the new monodies, which in turn led to
the establishment of a new art form, the opera.
Besides the 1581 edition, the Dialogo has been
reproduced in two twentieth century facsimile editions
(Rome, Reale accademia d'Italia, 1934, and New York,
Broude Bros., 1967). These versions, along with a number
of microfilm copies, were used in preparing this dissertation.
All appeared to be from the same impression.
The original Dialogo is not divided into chapters.
Division headings have been supplied in the translation
according to the logical separation of Galilei's material.
The first section of the translation concerns itself with
acoustical speculations, while the second section is a
critical discussion of ancient and modern musical art.
The third section is a critical discussion of polyphony
and the fourth section is a history of musical instruments
3
and notation. The fifth and final section concerns itself
with a critical discussion of the instrumental music of
Galilei's time. These five headings were considered by
Galilei to be the ones most pertinent to a modern treatise
on music.
In the present translation every effort has been made
to provide a lucid rendition of Galilei's work. The long,
tedious sentences which appear in the original version
have been divided, thus removing the same obstacle to
English readers as would be presented to modern Italian
readers of the 1581 edition. It has also been necessary,
on occasion, to alter the grammatical structure of
particularly idiomatic passages as well as to remove a
number of confusing particles and conjunctions. In no
instance, however, has the original meaning been distorted.
Because of increased demands upon the time of musical
scholars, there is an even greater case for translated
versions of significant treatises. It would, moreover,
permit the free flow of important musical ideas, unhampered
by the barrier of a foreign language, thus helping to
preserve for an increasing number of readers the extensive
heritage of our musical past.
1973
ROBERT HENRY HERMAN
ALL RIGHTS RESERVED
PREFACE
In the course of the ages, many works on music have
appeared, some of which have flourished briefly before
passing into obscurity. Others, possessing more enduring
qualities, have been granted a more permanent place in
the panorama of history. There are very few treatises,
standing at the foundation of an era, which form the
cultural concepts that cleave to the core of musical
society. Vincenzo Galilei's Dalogo della musica antica
et moderna was such a work, for at its inception it
fomented a fierce controversy between the proponents of
ancient and modern music. Before the strife faded, its
tremors were felt in many lands by academicians of diverse
disciplines.
The Dialogo is significant for several reasons. It
is, initially, a valuable study, both of the Renaissance
concept of ancient Greek music and the music of the
Renaissance. Galilei's work is rich in source materials,
for he cites nearly all of the ancient authorities on
music, including many incidental references to that subject
in the classical authors. The Dialogo also contains
iv
a wealth of information on many facets of musical art
including tuning theories, comparisons of Greek music
with that of Galilei's time, sixteenth-century polyphony,
the history of musical instruments and notation, and
Renaissance instrumental music.
There are, in addition, authentic examples of
ancient Greek music, the Hymns to the Muse, to the Sun,
and to Nemesis, which appear in print in the Dialogo
for the first time, as well as portions of the tables
of Alypius, which contain the early Greek notation
symbols.
Despite the importance of this treatise, no full
English translation of it has been made until now.
There are excerpts in collections by Oliver Strunk1
and Bruno Nettl,2 but neither of these is sufficient
to realize the entire potential of the Dialoo as a
valuable reference work. The purpose of this dissertation,
therefore, has been to translate the Dialogo
into English, transcribe its musical examples into
modern notation, and provide a commentary upon its
salient features.
1Oliver Strunk, Source Readings in Music History
(New York, 1950), pp.3227
2 Bruno Nettl, The Book of Musical Documents (New
York, 1948), pp. 36~7;~T4-47.
V
The Dialogo takes the form of a long rambling discourse,
149 pages in all. It is equipped with marginal
glosses known as "postilli" which are set off from the
main body of the text. Their function is to pinpoint
the location of a name or subject in the text, to clarify
or amplify a statement, or to serve as a repository
for bibliographical information. Since there are no
chapter headings or divisions in the Dialogo, the
"postilli" more or less act in their place. To eliminate
this inconvenience, chapter divisions and headings
have been supplied in the English translation.
Numbers and captions have also been added by the translator
to all musical examples and diagrams to facilitate
reference to them. In order to expedite comparisons
with the 1581 edition of the Dialogo, the original
page numbers are given in brackets in the margin of the
translation. The traditional catchword, like the musical
"custos", stands guard at the foot of each column
except where the text artistically dwindles down to a
point, heralding the approach of a large illustration
on the succeeding page.
Except for the use of indirect discourse, occasional
archaic or interchangeable spellings ("i" for
"y", "h" either retained or dropped, and so forth),
and a few obsolete verb forms, the Italian of the
vi
Dialogo is essentially the same as that of today, and,
as such, poses no monumental problems to the translator.
Galilei generally manages to say what must be said, but
his style is rather verbose, full of redundancies and
tautologies. Only occasionally does it border upon the
elegant, being more often commonplace and sometimes
decidedly awkward. He has a pronounced tendency for
complexity of sentence structure and almost a penchant
for parenthetical expressions. Sir John Hawkins describes
his writing as "clear and nervous but negligent,"
adding that "nice judges say that it is in some instances
ungrammatical."3 In order to provide a more lucid English
translation, many of the lengthy sentences have been
divided into shorter ones, and many of the unwieldy
particles and conjunctions have been eliminated. The
references to "questa" and "quella" (this and that) and
"cosa" (thing), proper enough in Italian, but vague and
confusing in English, have been replaced, where unclear,
by the words they originally modified. While it has
been necessary to deviate in some instances from the
formal grammatical structure of the Dialogo, in no case
have Galilei's statements been materially altered.
3Sir John Hawkins, A General History of the Science
and Practice of usic, 2vols., facsimile c.mewYork,
175)I 44.
vii
There are, in addition, frequent technical errors
or inconsistencies. The marginal glosses were particularly
prone to printing problems. In several instances,
one digit of a page or chapter number is missing. Similarly,
it is not always clear who is referred to by the
citation in a marginal gloss due to the omission of his
name. If there are several successive glosses, it is
also difficult to determine the object of each one.
There is an instance (Dialogo, p. 124) in which the
final letters of several lines were cut off in printing.
The same error was apparent in comparing the 1934 and
1966 facsimile editions. A number of Galilei's musical
examples came without sharps and, in one case, even without
notes (Dialogo, pp. 37-38).
Galilei's recalcitrant Venetian printers are probably
to blame for some of the errors in the Dialogo.
They held his manuscript, ostensibly at Zarlino's order,
forcing him to resort to a Florentine firm where a mere
two-thirds of the original was published under very
rushed conditions. It is not known what was done with
the other third; presumably Galilei incorporated it
later in another work.
The Dialogo contains many comparisons of Greek and
Renaissance music, which necessitates the maintenance of
several divergent nomenclatures. For reasons of clarity
viii
and in order to maintain them in the proper historical
perspective, a number of technical terms from both the
Greek and Renaissance eras have been retained in lieu
of replacement with modern equivalents (e.g. proslambanomenos,
diesis X , f fa ut, major hexachord, etc.).
The commentary to this translation takes the form
of footnotes, which serve not only as a source of bibliographical
information, but also as an amplification and
explanation of the text. The authors mentioned by Galilei
have been cited whenever possible, in both their
original form and in modern translation. The translations
of quoted materials, unless otherwise indicated,
are by this author. These often take the form of
paraphrases, rather than literal renditions, in order
to avoid unnecessary repetition of material. In
quoting the translations of other authors, their
spellings and punctuations have been preserved.
ix
TABLE OF CONTENTS
PREFACE , . . . . , . . . . , , , , ,,
LIST OF DIAGRAMS, . . , . . . . . . * *,.
LIST OF TABLES. . ., . . . . . . , . . . .
LIST OF MUSICAL EXAMPLES. . . . . . . . . ." " .
,
Chapter
I* INTRODUCTION. . . . . . * . . .
.
Vincenzo Galilei's Life and Works
The Ancient-Modern Controversy
Background for the Dialogo
TRANSLATION AND COMMENTARY
DEDICATION. . . . . . . . . . . . . . .
.
II. THE TUNING CONTROVERSY: DITONIAION
VS SYNTONON-. . . . . . . . . . .
.
III. A CRITICAL DISCUSSION OF ANCIENT AND
MODERN MUSICAL ART. . . . . . . .
,
IV. A CRITICAL DISCUSSION OF POLYPHONY.
.
V. A HISTORY OF MUSICAL INSTRUMENTS AND
NOTATION. . . . . . . . . ... . .
.
VI. A CRITICAL DISCUSSION OF CONTEMPORARY
INSTRUMENTAL MUSIC. . . . . . . .
.
INDEX . .*- . . . . . . . . . . . . . . . .,.
BIBLIOGRAPHY.. . . . 0 . . . . . . . . . . . . . 0
x
Page
iv
xi
xiii
xiv
1
29
35
o307
. 467
. 553
. 849
. 930
. 961
."
."
LIST OF DIAGRAMS
PageDiagram
I. Intervals of the Diatonic Syntonon
Applied to the Monochord. .
.
II. Interval Ratios According to the
Superparticular Genus . . . .. .
.
III. Example of the Epigoneion, Instrument
of Forty Strings, Discovered by
Epigon of Ambraciota. . . .
.
IV. Example of the Simikion, Instrument
of Thirty-five Strings. . . .
.V. Demonstration of the Thirteen Modes,
According to the Opinion of
Aristoxenus . . . . . . . .
.
VI. Demonstration of the Thirteen Modes,
According to the Opinion of
Aristoxenus, with Two Added in
the High Part by his Followers,
Which in All Make the Number of
Fifteen . . . . . . . . .. .
.
VII. Demonstration of the Eight Modes,
According to the Opinion of
Boethius. . . . . . . . . . .
.
VIII. Demonstration of the Eight Modes,
According to the Opinion of
Ptolemy . . . . . . . . . . .
.
IX. A Spherical Demonstration of the
Modes of Ptolemy. . .. . . .
.
X. A Mixed Division of the Eight Modes,
According to Gaffurio . . . . . ..
. . . . . 55
. . . .
.
96
. . . . . 259
. . . . . 260
" . . " . 310
. . . . . 334
. . . . . 336
. . . . . 373
. . . . . 384
. . . . . 423
xi
XI. Demonstration of the Eight Modes,
According to the Opinion of
Boethius, to the Strings of
Which Are Applied the Characters
According to the Usage
of the Ancient Greeks . . . . . . . . . . . . 582
XII. The Lyre of Mercury, Tuned According
to Various Opinions . . . . . . . . . . . . . 689
XIII. Heptachord Synemmenon of Terpander. . . . . . . 693
XIV. Octochord Diezeugmenon of Lycaon. . . . . . . . 706
XV. Enneachord of Prophastus Periotes . . . . . . . 710
XVI. Endecachord of Timotheus the Milesian . . .'. * 717
XVII. Decachord . . . . . . . . . . . . . . . . . . . 723
XVIII. Decatescarachord of an Unknown Author . . . . . 726
XIX. Greater Perfect System, Disjunct.'. . . . . . . 728
XX. A Composite View of All Strings Added
to the Lyre, with Their Inventors . . . . . . 732
XXI. Greater Perfect System, Conjunct. . . . . . . . 735
XXII. The Conjunction of Tetrachords
According to Ptolemy. . . . . . . . . . . . . 740
XXIII. The Lyre of Mercury Described by Homer. . . . . 769
XXIV. Form of the Ancient Lyre. . . . . . . . . . . . 789
XXV. Form of the Ancient Lyre Described
by Hyginus in Book Three of
the Constellations. . . . . . . . . . . . . . 791
XXVI. The Plectrum. . . . . . . . . . . . . . . . . . 795
XXVII. The Proportions of Various Sounding
Bodies. . . . . . . . . . . . . . . . . . . . 825
XXVIII. The Tuning of the Irish Harp. . . . . . . . . . 891
xii
LIST OF TABLES
Table Page
I. Excerpts from the Books of Alypius. . . . . . . 564
II. Description of the Hypaton Tetrachord . . . . . 664
xiii
Example
LIST OF MUSICAL EXAMPLES
Page
70
1. Location of the Tones and Semitones in
the Diatonic Syntonon of Ptolemy. . .
.
2. A Comparison of the Conjunct and Disjunct
Tetrachords of the Diatonic Syntonon.".
3. Divisions of the Major Thirteenth and
Minor Tenth . . . . . . . . . . .
.
4. The Minor Hexachord and Ditone of the
Distribution of Dydimus . . . . .
.
5. An Uncommon System of Notation. . . . .
.
6. GlCHodat Hodie . . . . . * . . . . *. . ..
7. Order of the Consonances According to
the Greeks. . . . . . . . . . . . .
.
8. Order of the Consonances According to
the Romans. . . . . . . . . . . .
.
9. Dissonant Intervals and Their Resolutions
0. Description of the Eight Ecclesiastical
Modes . . . . . . . . . . . . . . ."."..
1. Description of the Twelve Modes . . . . .
.
2. Sit Nomen Domitni. . . . . . . . . .
.
3, Greek Hymns to the Muse, to the Sun, and
t Nemesis, . . . . . . . . . . . .. ..
.
,
. . 186
* . 206
* . 234
. . 242
. . 248
. . 342
. . 344
. . 395
. . 410
. . 451
. . 584
. . 589
xiv
I
1
7I
CHAPTER I
INTRODUCTION
Vincenzo Galilei's Life and Works
The Ddaloo ella musica antics et moderna, whose
first edition bears the date of 1581, is the largest and
most important of the theoretical works of Vincenzo Galilei,
Relatively little is known concerning the life of
this noted theorist-musician. Fortunately, however, his
output is more familiar, since many of the fruits of his
genius have endured, perhaps even the greater part.
According to Fabio Fano, Vincenzo Galilei was born of
an impoverished noble Florentine family around 1520 at
S. Maria Monte in the Arno valley near Florence. His
grandfather had come there to live at the end of 1472,
since he was no longer financially able to maintain a
home in Florence. Vincenzo's parents were Michelangelo
Galilei and Maddalena di Carlo di Bergo. He had an
older sister named Lucrezia.
1
Fabio Fano in the preface to the facsimile edition
of Vincenzo Galilei, Dialogo della musica antics et moderna
(Rome, 1934).
1
2
Galilei went to Florence, probably around 1540,
in order to study music.2 His lute playing brought
him to the attention of patrons of music like Bernardetto
di Medici and Giovanni Bardi di Conti di
Vernio. Having been encouraged by Bardi and the
members of his Camerata, who supported him with his
lute playing, he undertook the study of music theory.
Aided by Bardi,3 he went to Venice in order to study
with Zarlino. The precise time of these studies is
not accurately known, but we know from Zarlino's own
statements4 that he became the successor to Cypriano
de Rore, choirmaster of San Marco in Venice, shortly
before 1565, having transferred to that city in 1541
to study music with Adrian Willaert. According to
Fano,5 Galilei's father died between 1540 and 1542,
and his mother also after 1544, leaving him to fend
for himself. After 1550, he moved to Pisa, where he
2Claude V. Palisca, "Galilei," Die Musik in Geschichte
und Gegenwart, Vol. IV (Kassel and IBasel,
1955), p.~26b.
3The dedication to the Dialogo contains Galilei's
acknowledgement of Bardi's support in those studies
to which he devoted himself since his youth. See
p. 30 below.
4 Palisca, MGG, IV, 1266; XIV, 1017-1022.
5Fabio Fano, Vincenzo Galilei, La Camerata Fiorentina,
espressione de nuove idealit'musicali, 7oT
IV of Istitugoni e monumenti dell' arte musicale
italiana (Milan, l39),~p. xxiv.
3
taught lute to distinguished merchants and foreign students
at the university.6 In 1562, he married Giulia
del fu Cosimo degli Ammannati, who bore him six, perhaps
seven, children. The first-born was the great Galileo,
born in 1564.7
Very little is known of Galilei's creative and
speculative musical efforts prior to 1574.8 Of his
works composed during those years, only the following
are extant:
(1) the first book of the Intavolature di Liuto
(1563)
(2) the dialogue, Fronimo (1568), a work on the
art of intabulation for the lute9
(3) the Primo libro di Madrigali a 4 e a 5 voci
(1574).10
In 1568, Galilei went to Venice, on account of the
6 Palisca, MGG, IV, 1266.
7 See p. 1, n. 1.
8 Fano, o. cit.
9 The second edition of this Fronimo is probably
what Galilei means by his "other work" which, in his
dedication, he promises will appear in addition to a
Latin translation of the Dialogo. See p. 33 below.
1 Only the tenor part remains. Fano, o. cit.,
p. xxv,
4
printing of the first edition of Fronimo.1 In 1572,
he moved to Florence, leaving his family under the protection
of Muzio Tedaldi, the husband of his sister-inlaw,
until they rejoined him in 1574. Beginning in 1572,
he began his correspondence with Girolamo Mei,12 a philologist
interested in Greek music. "It was Meli who
revealed to him the importance of the principle of
monody."1 3 Galilei made two journeys to Rome to see
Mei, probably between 1572 and 1577. During this time,
he was working on a didactic musical treatise, the
Compendio nella theorica della musica, which was probably
intended as a handbook for his pupils and the
members of the Camerata.1 4 It consisted, essentially,
of a shortened adaptation of Zarlino's Le Istitutioni
harmoniche. Galilei discarded this work, however,
when he determined that Zarlino's teachings were not
in conformity with Mei's views on Greek music. Mei
had formulated his opinions on that subject on the
basis of ancient manuscripts he had studied in the
1 1 Palisca, MGG, IV, 1266.
12 See Claude V. Palisca, Girolamo Mei: Letters On
Ancient And Modern Music To Vincenzo GaTTei ndiovanBniT
arcdi~(ome, 196i7ToFreprints of parts ot isue
correspondence and an exhaustive treatment of Mei's
influence on Galilei.
13Ibid., p. 1.
1 4 Palisca, MGG, IV, 1266.
5
Vatican library. The Dialogo della musica antica et
moderna (1581) was the result of Galilei's discussions
with Mei. Prior to the publication of the Dialoo,
forces not in favor of Galilei's ideas were at work
behind the scenes. Henriette Martin1 5 relates, according
to Arteaga,16 that a veritable cabal was conducted
against the author, which attempted to suppress
the edition, having begun by taking away the original
manuscript. Zarlino, in Arteaga's opinion, was the
instrument of these intrigues. The victim himself
alluded to this situation in the dedication of his
Dialo0,1 complaining of the Venetian printers who
held his manuscript to please some envious person.
Galilei took his revenge in 1583 in a new version of
Fronimo,18 where he engaged in a violent polemic with
Zarlino on the subject of the Dimostrationi harmoniche. 19
15Henriette Martin, "La 'Camerata' du Compte Bardi
et la musique florentine du XVIe siecle," Revue de
musicologie, XIII (1932), p. 65.
16 Stefano Arteaga, Rivoluzioni del Teatro musicale
italiano (Bologna, 1793).
1 7See p. 33 below.
18 Vincenzo Galilei, Fronimo, dialogo sopra l'arte
del bene intavolare et rettamente suonare a musica
(Venice,_1571) Zaln.Dmsrtoih
1Gioseffo Zarlino, Dimostrationi harmoniche
(Venice, 1571).
6
Zarlino later replied to Galilei's affronts with the
S limenti musicali. 2 0
In 1582, Galilei employed the principles he had
postulated in the Dialogo in some original compositions,
the first examples in the new monodic style.21 They
were entitled as follows:
(1) The Lamentations of Jeremiah
(2) The Responses for Holy Week
(3) and the Lament of Ugolino from Dante's Inferno.
Galilei himself sang the Lament of Ugolino, accompanied
by viols, at the home of Bardi. None of the pieces,
unfortunately, are extant. It is, however, possible to
gain certain insights into their probable structure
from other "pseudo-monodies" arranged by the same composer.
Palisca2 2 relates that Galilei left manuscript
arrangements of favorite madrigals, villanelle, and
similar pieces for solo voice and lute in a copy of
the Fronimo printed in 1568. These dated from about
1570, the time of Galilei's first interest in monody.
21Gioseffo Zarlino, Sopplimenti musicali (venice,
21 Palisca, MGG, IV, 1267.
2 2 Claude V. Palisca, "Vincenzo Galilei and SomeLinks between Pseudo-monody and Monody," Musical Quarterly,
XLVI (1960), p. 365.-
7
There are thirteen melodies for bass or baritone--this
was apparently Galilei's own voice range--twelve of
which are provided with lute reductions of the four
and five-voiced pieces from which they were taken.
Galilei evidently was accustomed to entertaining the
Camerata and other noble groups by singing the bass
part of well-known madrigals and part-songs while
accompanying himself with an intabulation of the
original composition. This would seem to establish
the fact that Galilei was already attuned to the
principle of monody, only awaiting the catalyst which
later appeared in the form of Mei's vision of reviving
ancient Greek music.
In 1584, Galilei published a contrapuntal duo,2 3
a copy of which was unearthed in a private Florentine
library by Alfred Einstein.24 "Galilei here wavers
between melodic expressiveness and scholarly treatment
once more . . . . Here again we have that 'contradictoriness'
in Galilei's historical position."2 5
2 3Vincenzo Galilei, Canto de contrappunt due
vci (Florence, 1584).
2 4 See Alfred Einstein, "Vincenzo Galilei and the
Instructive Duo," Music and Letters, XVIII (1937), 360-
368, for a discussion of" is brief, but significant
work.
25Ibid.,p. 367.
8
It should not be forgotten that this enigmatic musician
was not only a promoter of expressive music and a
pioneer of monody, but also was a master of sixteenthcentury
practice in the area of madrigals and of lute
26
music.
In 1587, Galilei produced his second book of
madrigals, It is his only collection in which all the
part-books are preserved intact. These madrigals, although
conventional in character, made the point that
the medium of expression was itself worn out and trite,
for example, the melodic and harmonic chromaticism, the
cross-relations, and the dissonant suspension chains.
Some of these madrigals, which were more inclined to
homophony and declamation, came closer to fulfilling
the ideal which Galilei approved as a theorist.2 7
Gioseffo Zarlino brought out his principal refutation
of Galilei's earlier attacks in 1588. The
Sopplimenti musicali28 consisted mainly of a systematic
series of quotations from the Dialogo or other writings
2 6 Palisca, MGG, IV, 1268.
27Se
.
28See p. 6, n. 20.
9
of Galilei, followed immediately by a none-too-subtle
set of reproofs. Galilei responded in 1589 with his
Discorso intorno all' ope di Gioseffo Zarlino, in
which he refuted Zarlino's belief that musical law
must be based on "natural" mathematical principles.
He wished to end the dogmatism and pedantry of music
theory and to replace the strict rules with simple
procedures by which physical perception and practical
musical experience are taught.29
During the last year of his life (1591), Galilei
completed a treatise on counterpoint which has been
termed his most significant achievement.30 In its
original form the work was in two parts: I primo
libro dela prattica del contrapunto intorno all' uso
delile consonanze ("The First Book of the Practice of
Counterpoint with Regard to the Use of Consonances")
and Discorso intorno all' uso delle dissonanze ("Essay
Concerning the Use of Dissonances"). He later decided
to append a third essay to his work, entitled Discorso
intorno all' uso dell' enharmonio, et di chi fusse
2 9 Palisca, MGG, IV, 1268.
30See Claude V. Palisca, "Vincenzo Galilei's Counterpoint
Treatise: A Code for the 'Seconda Pratica',"
Journal of the American Musicological Society, IX (1965),
pr 8.
10
autore del cromatico ("Essay Concerning the Use of the
Enharmonic and Concerning the Author of the Chromatic"),
and ultimately, he added a short supplement to this last
essay, called Dubbi intorno a guantoo ®io ho detto dell'
enharmonio con la solutione di essi ("Doubts Concerning
What I Have Said about the Use of the Enharmonic, with
Their Resolution"). The work, as a whole, is not titled,
but the first words of the initial essay reveal the main
subject to be "L'arte et la pratica del moderno contrapunto"
("The art and the practice of modern counterpoint").?1
All of the parts of the counterpoint treatise may be
found in the Biblioteca Nazionale of Florence.32 Galilei
set aside all the rules except the most important
ones. He both advocated and practiced the free employment
of dissonances and chromatic interweavings, which
revealed new possibilities for expression and anticipated
the essence of the seconda pratica" of the seventeenthcentury.
"The examples in his treatise on counterpoint
contain harmonic progressions whose boldness was unequaled
in the music of his time."3 3
31Ibid., p. 83.
32MSS Galileiani, Anteriori a Galileo, Vols. I-III.
3 3 Palisca, MGG, IV, 1268.
11
Before Galilei died, he conducted a number of
acoustical experiments. Rather than accept the numerical
dogmas of the Pythagoreans and Platonists, Galilei
experimented with strings of various materials, with
tubes, and other sounding bodies, to ascertain how
universal the sacred numbers were.34 His interest in
such experiments is evident from the discussion of
Bardi and Strozzi on the subject of organ pipes in
the Dialogo.35 Up to the very last, Galilei remained
a strong supporter of the theories he had proposed in
the Dialogo, seeking always to strengthen them. He was
forever mindful of current trends in music, unlike
Zarlino and other more reactionary musicians who preferred
to remain a generation behind, contenting themselves
with the dissemination of the traditions of
their learned predecessors.
The Ancient-Modern Controversy
With the coming of the Renaissance, mankind was
at last released from the spiritual subjugation of the
medieval era. The spirit of humanism, a belief in the
intrinsic worth of the individual, became prevalent
34See Palisca, "Vincenzo Galilei's Counterpoint
Treatise," p. 84.
3 5See pp. 821-829 below.
12
throughout western civilization as the Roman church
became more tolerant of secularisms. For two centuries
prior to the time of Galilei, the Italian humanists had
been reviving the glories of antiquity.? The art of
painting had been restored by artists like Giotto and
Cimabue, Dante Alighieri had written his Divina Commedia,
and Petrarch had penned his sonnets. Everywhere, artists
and musicians pursued their callings in relative freedom,
each one supported by a wealthy noble patron like Lorenzo
di Medici, or perhaps even the Pope himself. The old
Greek and Latin classics were avidly read and discussed
by learned gentlemen who habitually assembled for that
purpose in private homes. One such group in Florence
was Count Bardit s renowned Camerata,37 another the
Alterati.38 Long before Galilei, musicians had also
been probing the ancient ruins for new outlooks. Most
of the possibilities had proved futile. The return to
3 6 Warren Dwight Allen, Philosophies of Music History
(New York, 1962), p. 39.
3 7 See Manfred F. Bukofzer, Music in the Baroque
Era (New York, 1947), pp. 5-9, with~~egar o the origin
and aims of the Camerata.
38 See Claude V. Palisca, "The Alterati of Florence,
Pioneers in the Theory of Dramatic Music" in William W.
Austin, ed., New Looks at Italian Opera (New York, 1968),
pp. 9-38, for an accountTafthis lesser-known but
significant group.
13
chromaticism and enharmonicism advocated by Plutarch3 9
had been explored in vain by Don Nicola Vicentino,4 0
working on the false premise that a monophonic method
could be applied to a polyphonic composition. The
Dodecachordon of Glarean was no better, for the mild
sixteenth-century church modes could not be used to recreate
the legendary moral effects of the Greek "tonoi",
although humanists believed this possible. Another
attempt, the "vers mesures a ltantique", emulated only
one external characteristic of ancient music, which was
also insufficient to realize its effects. While a few
works of intrinsic beauty were produced, the modern
musicians had made little progress in recapturing the
lost music of the ancient Greeks.4 1
All these mistakes stemmed from misunderstanding,
poor method and inadequate knowledge. Musicians
went o the shell, rather than the core of Greek
art.
39 See Plutarch, On Music, translated by Benedict
Einarson and Phillip Tr,DeLacy (Cambridge, Massachusetts
and London, 1967), pp. 441-445.
4 0 See Nicola Vicentino, L'antica musica ridotta
alla moderna prattica (Rome, T T.V enino conAiFite
an instrument known as the archicembalo which
was designed to play in the chromatic and enharmonic
Greek scales as well as the diatonic. His experiments
were not successful and did not gain acceptance, because
it proved impractical to employ the chromatic
and enharmonic in part-music.
4 1See Palisca, Girolamo Mei, pp. 1-2.
4 2 Ibid
14
The viewpoint of the "moderns" differed greatly
from that of the "ancients". The contrapuntal style
taught by Zarlino was. founded on the teachings of his
own master, Adrian Willaert, who, in turn, came from
a long line of polyphonic composers such as Josquin
and Ockeghem. As such, Zarlino was a defender of the
"prima prattica",4 3 which incorporated a rich tradition
of discourse inherited from the Middle Ages, both
speculative and practical, with a sixteenth-century
accumulation of practical theory occasioned by particular
problems arising from contemporary musical
questions and usages. Both of these factors combined
with the growing Renaissance interest in the musical
speculation and practices of Antiquity to produce a
tradition of encyclopedic music learning. Gioseffo
Zarlino's Le Istitutioni harmoniche, published in
1558, met all the qualifications for such a comprehensive
endeavor. It gave as much attention to such
details as classical legends of the efficacy of music,
general speculations concerning the nature of mathematical
proportion, similarities between tonal patterns,
4 3 See Bukofzer, Baroque Music, pp. 1-4, concerning
the classifications of music according to styles or
practices. Ibid., p. 16, contains an apt table showing
the difference between Renaissance and Baroque styles.
15
the elements and the humors, etc., as it did to the
considerations of polyphonic composition.4 Zarlino
profited by the humanist revival of Greek texts on
music, but he was no idolator of antiquity, and what
he sifted from the mass available to him is remarkably
pertinent to his age.45 He exemplified the
mid-sixteenth-century attitude toward Greek music.
Because the conventions of "musica theorica" demanded
an explanation of the Greek musical systems and a
discussion of what they implied for modern music,
comparisons are found throughout the Istitutioni between
the Greek and modern theories. These are not
so much encouragement to musicians to emulate the
Greek models as they are compromises with the current
humanist atmosphere.46 Zarlino discussed briefly the
nature of the Greek "harmonies" and scales.4 7 He
observed that there were many conflicting opinions
about them, but decided that these "harmonies" were
of little use in the modern practice, and therefore
speaking any more about them would be futile. He
44 John Hollander, The Untuning of the S k (New
York, 1970), p. 20-21.
45See Gioseffo Zarlino, The Art of Counterpoint,
translated by Guy A. Marco anc llce. Palisca ew
Haven and London, 1968), p. xiv.
4 6 Palisca, Girolamo Mei, p. 42.
47 Zarlino, Istitutioni, pp. 293-300.,
16
considered it a vanity to try to follow the ways of the
ancients, as certain restorers of the chromatic and
enharmonic--he was alluding to Vicentino--demonstrated
clearly by their failures. It was apparent to Zarlino
that those ancient systems could not be applied to partmusic,
and since he knew of no examples of Greek music,
he was certain that the ancient practice had disappeared.
Further, Zarlino was not disturbed that modern music did
not arouse its hearers in the manner attributed to ancient
music. He thought it unnecessary to move anyone
to tears and laughter, but favored music which would
instill men with virtue. Zarlino would in no case have
been willing to abandon polyphony. To him, the object
of music was to convey honest delight while passing
one's leisure time. Because he was so imbued with
these characteristics, as well as being antagonistic
to the ideals promulgated by Galilei, Zarlino was
entirely unwilling to alter any of his conceptions of
music. Galilei, on the contrary, wished to emulate the
musical art of the Greeks since he felt it was superior
to the contemporary practice of counterpoint, which
tended to obscure the words and prevent, with its unwarranted
mixture of modes, the communication of any
desired affection or emotion to the listener. The
question uppermost in his mind was this: "Did the
17
Greeks have elaborate, many-voiced harmony, or were they
content with a single line of melody?" 4 8
In 1581, having formulated his theories with the
aid of Girolamo Mei, Galilei's reply to Zarlino was
voiced in the Dialoo della musica antica et moderna.
With this assumption of the defense of the ancients, the
strife with Zarlino had begun, a controversy which would
continue for centuries under different guises, extending
into many different localities. It was a quarrel which
would cause the name of Vincenzo Galilei to be remembered
well.
Background for the Dialogo
No study of Vincenzo Galilei would be complete without
some insight into his rather enigmatic, equivocal
personality. It has been said that he stood half in the
sixteenth and half in the seventeenth century with regard
to his musical perception. As a madrigalist and lutenist,
he was a veritable master of sixteenth-century practice.
It was, moreover, his desire to benefit the younger
generation of composers through his rich experience. On
the contrary, he promoted expressive music, and as a
pioneer of monody, he envisioned the ideals of the seventeenth
century, the era of accompanied solo music. One
4 8 Allen, 2. cite., p. 39.
49 Palisca, MGG, IV, 1267-1268.
18
finds in Galilei's Dialogo a manifestation of these
monodic tendencies, at least in theory, if not in
actual practice. Although this noble Florentine did
not show how to implement his neo-Hellenic ideas, he
did state clearly two rudimentary conceptions of the
so-called "seconda prattica":5 0
(1) A melody's emotional quality is dependent
upon the range of the voice it uses--high, medium, or
low--and upon the tempo and meter.
(2) Other independent parts should not be allowed
to interfere with voices, which express the
text.
These principles constituted a clarion call to
combat against the very musical thinking which had
itself nurtured Galilei since his early years. The
personification of such thought was the Reverend M.
Gioseffo Zarlino, with whom his relationship was
destined to deteriorate drastically with each passing
year. Zarlino, born in 1717, was not much older than
his "loving disciple", as he aptly termed his rebellious
pupil, Vincenzo. The Sopplimenti musicali5 1
5 See Palisca, "Links Between 'Pseudo-monody' And
Monody," p. 346.
51 Zarlino, Sopplimenti musicali, p. 85.
..
19
(1588), his reply to the polite attacks of the Dialogo,
was full of such sarcasms. If the irascible choirmaster
of St. Mark's in Venice was less than cordial to his
one-time scholar, Galilei's replies in the Discorso
(1589) compensated with equal aplomb. It was evident
that with the publication of the Dialogo, Galilei was
opening up a new horizon of musical endeavor. One may
well question the "display of rather pedantic erudition
and the somewhat petulant arguing over tiny points of
detail," as Pirotta52 puts it, but through such sciolisms
"we can see that the dominant feature of the book is the
unrelenting criticism of the musical language of the
time--a language based on counterpoint."
Galilei's abstruse style is that of a philologist,
and rightly so, for his association with Girolamo Mei,
a man of that profession, is well-known through a long
correspondence between them. Only Mei's letters are
extant, having been preserved in the Vatican (Vatican
MS Regina Lat. 2021), but a comparison of these letters
with Galilei's Dialogo easily manifests an unmistakable
similarity. Clearly, Mei had provided Galilei with the
bulk of material for his work, but the only acknowledgement
of indebtedness to him in the Dialogo takes the
5 2Nino Pirotta, "Temperaments and Tendencies in the
Florentine Camerata," Musical Quarterly, XL (April,
1954), 172.
20
form of a general tribute, naming Mei as a "worthy man
to whom all musicians and learned men should render
thanks and honor." 5 3 He is mentioned there once
again, merely as "one of our Florentine gentlemen".5 4
Notwithstanding, Galilei's dependence upon Mei was
generally known. Artusi, an exponent of Zarlino,
attacked the Dialogo in a pamphlet,5 5 referring to it
as "vostro non vostro Dialogo della musica antica et
moderna", and Zarlino also claimed that it was not
his work at all, but was written by several gentlemen,
friends of Galilei.56 Rising angrily to his own defense,
Galilei claimed in his Discorso (1589) that the
work was all his, saying: "he who believes differently
tells an untruth; and he who says differently tells a
lie."57 It is certain that, despite the invaluable
aid of Girolamo Mei, Galilei's work nevertheless
5 3 See p. 38 below.
5 4 See p. 588 below.
5 5 Giovanni Maria Artusi, Trattato apolo etico in
difesa dell' opere del Rdo Zarlino. This treatise i
not extE
56 Zarlino, Sopplimenti musicali, p. 131.
57 Palisca, Girolamo Mei: Letters on Ancient and
Modern Music, p 7 7
.
-
_
21
possesses a high degree of originality. Perhaps Mei
was also content in knowing that his ideas had found
fertile ground.
In conceiving the Dialogo, Galilei was concerned,
not only with the development of monody, but with the
re-grouping of contemporary music into five branches,
which were
(1) the problem of tuning,
(2) the theory of the modes,
(3) the rules of counterpoint,
(4) the history of music,
(5) and instrumental performance practice.5 8
Galilei elected to use the venerable dialogue form,
which had first been employed by Plato and the ancient
Greeks, since it suited his didactic purposes better
than discourse. He disavowed the acceptance of ideas
on sheer authority, preferring the empirical approach
of Aristotle59 by means of which.he might introduce
58 See Palisca, MGG, IV, 1268.
59Ingemar During, "Greek Music, Its Fundamental
Features and its Significance," Journal of World History,
III (1956), 318. "Aristotle was an absolute Tnovator
with his historico-critical method which enabled him to
fit his results into a wider perspective. In dealing
with a problem he regularly begins with those of his
predecessors, with special emphasis on their solutions
and the difficulties they present. His own solution of
a problem is often a compromise resting on a common
sense appraisal of different possibilities."
22
opposing views and pertinent facts for subsequent
evaluation and logical conclusion. His chosen
interlocutors, Count Bardi and Piero (or Pietro)
Strozzi, pursued their protracted polemic, presumably
in a plush parlor of Bardi's dwelling, where
the Camerata habitually "passed their days in
commendable discussions with well-bred repose. "60
It would have been somewhat incongruous, certainly,
for the noble Bardi actually to have uttered the
vast profusion of acoustical theories for which
Galilei made him responsible in the Dialogo, for
Bardi himself would have been more at home with the
adages of Aristotle or the proverbs of Plato, which
did, however, appear frequently enough in that work.
The "lofty principles" which Galilei gleefully and
glibly expounded through the mouth of Bardi often
set Strozzi in search of his "safe port" before
foundering upon the treacherous reefs of his constant
confusion. Galilei, obviously enough, enjoyed the
role of guide through the maze of clever, complex
"entanglements" which, through Bardi, he promised,
at the beginning of the Dialogo,61 to set up for
6 See p. 39 below.
61 See p. 45 below.
23
Strozzi's eventual edification and enlightenment. As
the dialogue unfolds, Strozzi seems one time to comprehend,
another to be overwhelmed with perplexity, but
he is invariably prepared with another doubt (dubbio)
which launches his "faithful pilot" Bardi onto yet
another series of sesquipedalian terminology. Under:Lying
the entire Dialogo is an unmistakable aura of
aristocratic smugness, based on the disdain of common
people as vulgar and ignorant incompetents. This
attitude seems even more appropriate for Galilei than
for Bardi, since the former, though a noble, through
the decline of his family fortune, stood uncomfortably
close to the economic apanage of the proletariat.
Bardi, however, being-affluent, could afford at least
a studied tolerance.
Galilei possessed, first of all, a love for the
popular musical idiom of his day and a desire to be of
service to the practical musician through his great
wealth of experience as a lute player and composer.
Galilei was constantly warning the "prattico" to cease
his malpractice and turn into better paths. Secondly,
he was an erudite scholar, and wished to bring about
a reform of serious vocal music. He therefore refuted
Zarlino, once his mentor, for advocating polyphony,
which Galilei felt was ruinous to compositions for
,.
24
voices, The Dialogo, in which his primary aesthetic
values and musical ideals were embodied, was the means
by which these goals could be achieved.
Galilei devotes the first section of his Dialogo
to tuning theories. He refutes Zarlino's hypothesis
that the tuning used at that time in singing is the
diatonic syntonon of Ptolemy, or "just" intonation.
He advances the opinion that the vocal tuning must be,
in practice, a compromise between the Pythagorean
tuning with its pure fifth and the Ptolemaic diatonic
syntonon. The consonant third is to be included. He
evolves a practical system of equal temperament for
the lute, using semitones in 18:17 proportion.
In the second section, Galilei examines the
distinction between the musical art of the Greeks and
that of his own time. He criticizes the systems of
Glarean and Zarlino, considering the church modes to
be ineffectual and inappropriate as representatives of
the ancient Greek tradition.
The third section is a critical discussion of
polyphony in which Galilei mocks the naive imitations
of the meaning of individual words by the contrapuntists
and their fastidious attention to the declamation of the
text. He deplores the confusion which originates in
25
church music through the independence of voices.
In the next section, Galilei gives a history of
instruments and notation, publishing for the first
time the tables of Alypius, which contain the ancient
Greek notation symbols and some authentic Greek songs.
The final section of the Dialogo is a critical
treatment of the instrumental music of Galilei's time.
IN FIORENZA M.D.LXXXI.
Appreffo Giorgio Marefcotri.
26
r
' '7
,,,,,, O ?;
W
" ,y,..
. q
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1 :, f
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TRANSLATION AND COMMENTARY
27
DIALOGUE OF VINCENZO GALILEI
NOBLE FLORENTINE
CONCERNING ANCIENT
AND MODERN
MUSIC
U6
28
To the Most Illustrious
Lord, and my Patron
The Most Worthy
Lord Giovanni Bardi,
Count of Vernio
It is truly very difficult to repay those from whom
we have received many extraordinary benefits, for it is
necessary that the one to whom we are in debt comes to
value our work. We agree that, given the inspiration,
we can fulfill these obligations. It happens that such
insights now occur less often, and therefore, many are
accused of being ungrateful. This is not the case, however,
because the giving and receiving of benefits should
not take precedence in such a noble endeavor. For if
that were so, carrying it out properly might not be in
our power--a most improper thing to think, let alone to
say. If, moreover, that is true--and it really is,
Signor Giovanni--I could still with clear conscience
sustain the debt which I owe to Your Most Illustrious
29
30
Lordship, who as a connoisseur of good literature, and
versed in the arts which are proper to a true Gentleman,
has never asked of me anything which exceeds my powers,
since they are weak. I have never been able to engage
in the service of Your Most Illustrious Lordship in a
way which leaves me any chance at all to avoid the name
of ingrate, which I just now said pertains to one who
lacks inspiration and ability. And yet, if something
done by me were of value, I would still accomplish
nothing in comparison with the many exceptional benefits
received continually from the kind hand of Your
Illustrious Lordship. How could I even in the least
repay the convenience that you have given me of being
able with calm mind to attend to those studies to which
I devoted myself since my youth, which, without your
aid, would never have reached their present conclusion.
To that is added your thoughtfulness in having procured
at my request from the most remote parts of Europe,
various books and instruments, without which it would
have been impossible to obtain that information about
music which has been obtained through their agency. In
order that this science could be shown by me to the
world much more clearly than it has been shown, perhaps
since the loss of music, has it not seemed important to
31
you to give me the convenience of travel provisions
and to bestow on me your favor in every other matter
necessary for searching many places and afterwards
deriving the most exhaustive and reliable information
possible from the customs of the inhabitants, from
ancient records, and from men intelligent in musical
science? Although some others may perform such
acts of generosity, these suppliers of riches and
protections, either through natural instinct or [iiJ
desire of glory, attach no importance to similar
expenses, and they are quite often abundantly and
willingly accustomed to making them. However, they
are so much more regarded for them to the extent
that, accompanied by equal humanity, they exceed
the common custom of such things. What greater sign
of courteous and benevolent spirit could you give me,
having left many times your more serious and important
negotiations, than to tell me enthusiastically
the obscure sentiments of the ancient and serious
writers? Although few have understood these concepts,
you stated them so accurately that one could
well believe that you were in those happy centuries
in which fullest knowledge of the art of music had
been accomplished. Music, afterwards, in the
course of many ages, has fallen into such neglect
32
that no one has been found up to today who has given
knowledge of it conforming to the truth and to that
which these ancient men of yours have left written.
This situation has moved me to try--with that talent
with which nature has endowed me and with the labors
of many years spent in that pursuit, aided also by
the favor and liberality of Your Most Illustrious
Lordship--to return this art if not into its original
state, at least into greater prominence. Up to
now, this has not been done, to my knowledge, by the
moderns that have written of it. I have judged that
it is the duty of a sincere man who is not at all
envious of the well-being of others, to give public
account of this art, since he can bring both benefit
and delight to those who will follow him, in order
that those who will read about it can ascertain if
anything there is good, and exert themselves to provide
where he has lacked completeness. I do not wish,
however, to believe that I have so dispelled the darkness
of the ancient writers, that no one else can,
having been solicited and aided by my efforts, add
to them any greater clarity. Having written it according
to my abilities and being obliged to publish
it for the reasons just now given, this labor of mine
in honor of Your Most Illustrious Lordship has just
33
been completed. I owe you this work since I have
received from you very great conveniences, which were
sought in order to complete a similar work which it
was, without doubt, my obligation to send to you.
And it has been requested that the uplifted and benefitted
one renders to the benefactor the necessary reward.
If it does not come to you with that corrected
speech which is proper, and if it does not come to you
translated into Latin--as it will soon come, together
with my other work--blame the little faith of some
Venetian printers. These men have, contrary to every
obligation, detained me many months in order to please
someone who forbade this labor of mine to be published,
either impelled by envy or to honor himself by my long
waiting. They have at last little less than denied me
the manuscripts, consigning them at the end of last
October in order that they would have to publish it
before I could copy it for you by hand. I have, for
the alleged cause, printed here two-thirds of it,
extracted from a sketch left to me. In conclusion,
I kiss your hand with every reverence, asking you to
accept with pleasure this work of mine, having no
34
regard for myself but only for your justness and your
most singular humanity. In Florence the first day of
June, 1581.
Your Most Illustrious Lordship's
Most affectionate and obligated servant,
Vincenzo Galilei
CHAPTER IIJ
CTHE TUNING CONTROVERSY:
DIATONIAION VS
SYNTONONJ
Music was numbered by the ancients [1)
among the arts that are called liberal,
that is, worthy of a free man; and
deservedly among the Greeks, its masters
and inventors--like almost all the other
sciences--it was always in much esteem.
And it was ordered by better legislators
that it must be taught, not only as delightful
to life, but also as useful to
virtue to those who were born to achieve
the perfection and the human bliss, that
is the purpose of the state. As time
went on, the Greeks lost music and the
other doctrines once more, together with
the Empire. The Romans possessed knowledge
of music, deriving it from the
Greeks; but they employed mainly that
35
36
part suitable to theaters, where tragedy
and comedy were presented, without much
valuing that which is concerned with
speculation; and continually occupying
themselves with wars, did not devote much
attention to that again, and thus they
easily forgot it. Since Italy for a long
period of time had suffered great inundations
of barbarians, every light of
science was spent; and as if all men had
been overtaken by a grave lethargy of
ignorance, they lived without any desire
to learn. As for music, they had the
same knowledge of that as of the West
Indies; and they persisted in such blindness
until first Gaffurio, then Glarean,2
1 Franchino Gaffurio was born at Lodi,
Italy in 1451 and died in 1522. His
writings exercised a powerful influence
on the musical studies of his day, and
most of his successors quote his opinions
as authoritative. His main theoretical
works are the Theorica musicae (1480) and
the Practica musicae (149>. Galilei upholds
many of is ideas in the Dialo o,
although Gaffurio had been previously
attacked by Spaturo and Aaron, on account
of some of his proportions of intervals
which fomented a long bitterly contested
strife in which many joined. For an
English translation, see Franchinus Gafurius,
Practica musicae, translated by
Irwin Young (Maison,~Milwaukee and London,
1969).
37
and after him Zarlino3--truly the princes
in this modern practice--began to investigate
what music might really be, and to
seek to extricate it from the darkness
where it had been entombed. That part
which they understood and valued, they
brought little by little to the end result
in which it is found; but it does
not seem to any men of intelligence that
2Heinrich Glarean (1488-1563), actually
Loris (Glarus was the region in
Switzerland where he was born), wrote the
important, controversial theoretical work
known as the Dodecachordon. The aim of
this treatise was to propose that there
were twelve, not eight Greek modes which
corresponded with the twelve church modes.
Galilei, in the Dialo o, takes great exception
to his s a ements, defending
Gaffurio against Glarean's criticisms.
For an English translation, see Glarean,
Dodecachordon, translated by Clement A.
Miller (Rome, 1965).
3Gioseffo Zarlino, born in 1517 at
Chioggia, died in 1599. One of the most
famous theorists of his day, he held the
post of organist at St. Mark's in Venice.
Sansovini, one of his contemporaries,
calls him "unequalled in theory and composition",
and another, Foscarini, terms
him the most famous restorer of music in
all Italy. Zarlino's most famous theoretical
works are the Istitutioni harmoniche
(1588), Dimostrationi harmonihe(1571)T,
and Sop menti mus aTi(T35 7
38
they have restored it to its ancient
state. From what can be understood from
the endless passages of the ancient histories,
from the poets and the philosophers,
they had neither attained true
nor perfect knowledge of it. Perhaps
this was caused by the coarseness of the
times, the difficulty of the subject and
the scarcity of good interpreters. Nevertheless,
these writers deserve the
highest praise, and the world owes them
a perpetual obligation, if for nothing
else, at least for having given many the
occasion of developing it with greater
dedication, in.order to see it brought
to its perfection. It appears that
this--only as much as pertains to theory--
has been attained in our times by Girolamo
Mei,4 a worthy man, to whom all musicians
4 Girolamo Mei (1519-1594), a native
of Florence, became recognized as early
as 1540 as one of the most promising
young literary men of Florence. He was
by training what today would be termed a
classical philologist. In 1561, his attention
was drawn to a book of ancient
authors on music, reawakening a former
desire to investigate the nature of Greek
music. Galilei's reading in 1572 of his
writings on that subject engendered a
39
and all scholarly men should render
thanks and honor, and after that in our
city, the Most Illustrious Lord, Giovanni
Bardi, Count of Vernio. Bardi, who has
long made a study of music and being
much delighted with it, as with all the
other sciences, has greatly ennobled it
and made it praiseworthy; having by his
example aroused the nobles to the very
same study. Many of them are accustomed
to going to his house and there passing
the time in delightful songs and in
commendable discussions with well-bred
repose. Since I am much obligated to
the courtesy of this most cultured
gentleman, and yet desirous of showing
him by some outward sign the inward wish
that I have of serving him, I have
lengthy correspondence with Mei, who then
lived in Rome. The valuable ideas gleaned
from his letters were incorporated in
Galilei's Dialogo della musica antics et
moderna (1581). F'oracomple e"testudy on
leV and his influence, see Claude V.
Palisca, Girolamo Mei: Letters On Ancient
And Modern"vusic To Tincenzo GalTTei~n
1Tovanni Roe,196J.
40
judged myself not to be able to spend
my time more profitably than to dedicate
my labors to such a subject; because in
so doing I had the hope of being able
to give him some sign of gratitude, and
to extend to the world no little aid in
leading it out of the darkness, in which,
since the aforesaid loss, it has, up to
now, been enveloped. I say this, however,
without arrogance, and with every respect
for those who from the time of Guido
d'Arezzo5 until now have written about
such matters. However, if I were to give
myself a great deal of glory for this deed,
5Guido d'Arezzo, or Aretino, was an
eleventh-century Benedictine monk, who
made important contributions to the development
of musical theory in the Middle
Ages. His pedagogical skill is evident
in his use of innovative devices such as
solmization, the Guidonian hand, and a
multi-colored musical staff. The Latin
texts of his works, the Prologs in Antiphonarium,
Micologs, and EpsoTa lii
,diotocantu,may be found inrtIn"erer
,iScri stores ecclesiastici de musica
(1784),3,vols., facsimile ed, TiTan
1931), II. Portions of his works are
translated into English in Oliver Strunk,
Source Readings in Music History (New
Yoor ,195, pp.T17-f?.
41
perhaps I would not deserve rebuke. Because
of the inclination given me by nature
for these liberal studies and the [2J
continual diligence I have exercised for
the space of many years in preparing it,
it would justify my discussing them with
good cause; but the judgement of this must
be reserved purely to intelligent men.
For this reason, other than what I said
a moment ago, and in order that I might
not defraud the world of any usefulness
it might draw. from my labors, I was
pleased to publish some of my ideas about
ancient music and about that of our times,
which until this day have been--in my
opinion--little understood by whoever has
dealt with them. This matter, without my
further testimony, is clear evidence of
the difficulty of the subject. Therefore
I desire of the reader that he prepare
himself to give judgement or make a comparison
of my writings with those of the
other moderns, with the very greatest
attention and with mind liberated from
42
every human emotion. It is clear that
whoever has a mind not entirely purged
of every passion cannot give perfect
judgement of anything that exists. Every
suggestion that may be made to me by an
intelligent man and a lover of the truth
I will receive with pleasure, and will
remain obliged to him without ever being
ashamed of learning from one who may
understand better than I. Now, since
long and continuous speaking, while
running like the gush of a torrent, does
not seem to have that power and vigor in
concluding sentences and arguments that
dialogue has, I have considered it
greatly relevant to treat my present
discourses in such a way. This, I believe,
was one of the potent causes that
prompted Plato to treat the matters of
divine philosophy in such a way. Consequently,
I have chosen the Most Illustrious
Lord Giovanni Bardi--mentioned
a little while ago--and with him, Signor
43
6Piero Strozzi, to discuss this thoroughly,
since these men are most studious of
the true music, great lovers of such
speculations as these, and suitable also
to sustain this and greater weight. On
this occasion, moreover, Signor Piero
Strozzi wishes to see which of the diatonic
species is the authentic one in
which the modern contrapuntists compose
and the singers of their songs sing.
Therefore, he speaks to Signor Giovanni
in this manner:
Strozzi: It seems an important
thing to me, Signor Giovanni, that of so
6Piero Strozzi, the scion of a very
old, established Florentine noble family,
was undoubtedly an apt choice for the
speaking partner of Bardi in the Dialogo.
There is a Strozzi chapel in the Santa
Maria Novella today, noted for its art
treasures, which attests to the affluence
and prestige of Piero's people (see p.
below), Little is known of his life,
however, but Jacopo Peri mentions him as
a learned gentleman who had heard Peri's
new style of singing with interest and
pleasure, See Strunk, Source Readings,
p. 375. Federico Ghisi7ITS oni
della monodia (Milan, 1940T)~p60,
contains a "pseudo-monody" by Piero
Strozzi, the madrigal Fuor dall humido
nido.
44
many excellent men, who have from Guido
d'Arezzo to the present written of the
art of music, not by chance, but as professors
of it, there has not been anyone,
as far as I know, who has explained the
diatonic species, in which one composes
and sings today, so that a thousand
difficulties and contradictions are not
engendered. Nonetheless, among the
principal matters, I rank this the most
principal, the most important, and the
most necessary to be known. Nor can I
succeed in being unashamed, when I
realize how little knowledge is universally
found among the materials available
to our practicing contemporaries. They
make a profession of knowing and understanding
about the power and nature of
so little, being content to be thus
esteemed by the unskilled masses. Finding
myself so encumbered, I desire greatly,
with your aid, to free myself from
such a defect.7
7Literally, "finding myself in such
pitch, I, soiled, desire to purge myself
of such a defect."
45
ardi: You continually place under
consideration the most discerning and uncommon
questions, which give everyone who
hears them an indication of your fine
talent. In order to clarify the problem
given me, I will need to develop many
intricate techniques. This, however,
since it pleases you, will not be a burden
to me,
Strozzi: If it will not be wearisome
to you to reveal them, it will give
me the greatest satisfaction to understand
them. Whenever you please, I am
ready to listen with the greatest attentiveness
you could possibly desire.
Bardi: It is necessary, on account
of this noble purpose, to examine diligently
every interval of the diatonic
species about which such a controversy
results, to see which of our diatonic
species conforms with those that are
8See Zarlino, Le Istitutioni harmoniche
(Venice, 15587 pp. 83; moT
ion harmoniche (Venice, 1571) p.-28.
What species is
the one sung today
aec rding to
Zarlino in chapter
16 of the 2nd
book of the Institutioni
harmonziche
and in the 5th
discussion at the
3rd definition in
the Dimostrationi.
46
composed and sung today. The knowledge
of this, without doubt, will lead us
into a safe port. Since all of the
practitioners of our time, under the
influence of Reverend M. Gioseffo Zarlino,
universally concur that it is important,
we will examine before every other species
that which is called the intense syntonic
of Ptolemy.9 After he examines this--if
it occurs to him--with equal care, he
will agree with what all the other moderns
like Guido d'Arezzo, Gaffurio, Glarean,
LeFevre,10 Valgulio, and other serious
9Claudius Ptolemy (2nd century A.D.),
the celebrated Greek astronomer, the
events of whose life are virtually unknown,
was the author of a three-part
treatise on music. This work gives a
clear exposition of the transposition
of the Greek modes in addition to other
important information with regard to
ancient music. The followers of Ptolemy
were very much concerned with reconciling
Pythagorean musical theory, which relied
upon numbers, with that of the Aristoxenians,
which depended upon the judgement
of the ear. There is a German translation
of Ptolemy's Harmonics in Ingemar During,
Ptolemaios und Porhrias uber die Musik
(Goteborg, T74)"
10 Jacques LeFevre d'Etaples (Jacobus
Faber Stapulensis), the author of Musica
libris quatuor demonstrata (ParisjTT3,
47
writers, have openly admitted. All of
these men of common opinion affirm that
what is sung today is the most ancient
diatonic ditoniaion, the proportions of
which were investigated with shrewd
deliberation by the strict Pythagoras
of Samos1 2 in the sixtieth olympiad. The Chronicles1 3
of Euseius.
was a sixteenth-century Pythagorean. He
made no real innovations, but his treatise
was a standard work, much in demand
long after his death, as a clear exposition
of the ancient musical number
theories,
1 1Carlo Valgulio Bresciano is known
for his translation and commentary of the
pseudo-Plutarchian treatise On Music. A
copy of this work, Plutarchi is ogu de
musica ad Titum rrF num resc a, 57),
may be Tound in theWBiitish Museum.
There is an Italian translation of both
the text of Plutarch and the commentary
in one of Vincenzo Galilei's manuscripts
(Florence, B.N.C., MSS anteriori a Galileo,
Vol. VII).
1 2Pythagoras of Samos (fl. 528 B.C.),
the noted Greek philosopher, left no
writings of his own. His teachings, disseminated
by his many followers, undoubtedly
suffered numerous modifications.
His strict number theories, which limited
the musical consonances to the octave,
fourth, and fifth, necessitated a dissonant
major third, and precluded the possibility
of modulation. Pythagoras
taught that numbers were the soul of the
universe. He related the numerical proportions
to the heavenly bodies, thus
48
Strozzi: Before your Lordship begins
to expose the heart of the proposed problem,
I desire that we always exclude in
those matters attainable by the intellect--
as Aristotle 4 says in the eighth book of Aristotle in
the 8th book
the h sics--not only the authority, but of the Phs4csat heading.22
also the prejudiced argument which would
be contrary to our desired appearance of
promulgating the familiar doctrine "music
of the spheres". The authority of
Pythagoras was invoked for hundreds of
years until equal temperament became the
standard in the eighteenth century.
1 3See Eusebius of Pamphilus, Chronici
canones, ed, by John Knight Fotheringham
(London, 1923), pp. 184-185. Eusebius
shows that Pythagoras flourished in the
sixty-second, not the sixtieth olympiad.
The olympiads were the four year intervals
between the Greek olympic games.
The sixty-second olympiad in our system
would be 528 B.C. (Christ was born in the
194th olympiad).
1 4 See Aristotle, Ph sics, in Basic
Works of Aristotle, ed. yRichard'!ceion
TNew Yx-rk,1U pp. 354-394. Galilei
is evidently referring to Aristotle's
empirical approach in examining the arguments
of others and logically eliminating
those which are not pertinent in order to
reach a logical conclusion.
truth, For it seems to me that those
who, in order to prove some conclusion
of theirs, wish it to be believed exclusively
on simple authority without
presenting any other valid arguments,
are doing a ridiculous thing, in order
not to say with the Philosopher, "like
fools". One finds that this privilege
has not been granted to others except
to the all-wise Pythagoras--just now
mentioned by you--by his followers.
Furthermore, I want you to concede that
it is lawful for me to question you
freely and answer you without any kind
of flattery, since this is truly suitable
among those who seek the truth of
things.
1 5 Calvin M. Bower, "Boethius' The
Principles of Music, An Introductio,
ransla ion,7AnTommentary," unpublished
doctoral dissertation, School of Music,
George Peabody College for Teachers,
Nashville, Tennessee, 1967, p. 100. "For
when the master Pythagoras said something,
no one thereafter dared challenge his
reasoning. The authority of the teacher
was reason enough for them."
16 See Marcus Tullius Cicero, On the
Nature of the Gods, trans. by Hube ".
oteat Thicagon.d.), p. 198.
Privilege of
Pythagoras.
Boethius, in
the 1st book
On Music,
api"33.
Cicero1 6
Concerning
tie
Nature oEhe
T078m
49
Bardi: All is conceded. First of
all, it is necessary to memorize--still
according to the syntonon of Ptolemy,
however, as I have said--the numbers in
lowest terms, between which are separately
contained each one of the intervals
which are encompassed by the queen
of the consonances. These, according
to the opinion of those who write about
such things, do not exceed the number of
fifteen. Beginning with the smallest
one, I say that the comma is contained
in its root terms by the proportion
called sesquioctogesima, between these
numbers.. .......................... 81:80
the minor semitone, between........25:24
the major semitone, between........16:15
the minor tone, between.............10:9
the major tone, between..............9:8
the minor third, between.............6:5
the major third, between.............5:4
the fourth, between...............,,4:3
the tritone, between...............45:32
the semidiapente, between..........64:45
[3)
The octave
is called
queen of
the conso-
nances.
These numbers
contain the
musical intervals
of
the syntonon.
50
the diapente, between...............3:2
the minor sixth, between.............8:5
the major sixth, between.............5:3
the minor seventh, between...........9:5
the major seventh, between..........15:8
and the queen of the consonances called
today octave, between 2:1. Perhaps the
Greeks gave it the name of diapason
since it contains in itself, as its name
implies, each of the intervals named, and
because it is valid only through each of
these other intervals. I warn you that
these intervals of the fourth, fifth,
octave, and the sesquioctava (or tone)
were not known by those names to any of
the ancient or modern musicians, except,
however, by the originators [of such
termsJ and their followers. The corruption
of these, because of the disparity
which they have with the intervals
already understood and known by such
appellations, generates no little confusion
and disturbance in the minds of
those who read their writings. They
Why the octave
is called dia-
pason.
Names of the
intervals
corrupted.
51
have taken these names quickly from the
diatonic ditoniaion and used them, in
order to substantiate some of their
schemes, to disguise the syntonon of
Ptolemy. You will understand this better
in due course. However seldom this
practice is repeated, and to whatever
extent we disagree about it, we will
relegate it sensibly to its place.
Strozzi: Why did Pythagoras construct
the fifth rather than the octave
between 3:2 thereby assigning these
numbers as the limits of the diapente,
or else between 4:3 in which he constructed
the diatessaron?
Bardi: This was not a human work
or invention, but rather one of nature.
It is quite true that Pythagoras--as I
have said--was the first to consider it.
Strozzi: What induced him to do
that, please?
Bardi: Listen! Stretch two strings
of the same length, size and quality over
a plane surface, and tune them in unison.
Method of
hearing any
interval in
its true pro-
portion.
52
53
Then divide one of them with the compass
into enough identical parts to equal the
larger number of the interval you wish to
hear. Next divide one of them by means
of an immobile bridge, in such a way that
the distance of the greater number exceeds
the smaller. Then, when the two strings
are plucked simultaneously, you will hear
from the whole--which is represented by
the larger number--and from the parts--
which are represented by the smaller
number--the dissonance or consonance
which corresponds to the proportion
which has thus been applied to these
strings. Anyone who wishes to hear some
interval on a single string can accomplish
it in another manner. First, divide the
string into enough identical parts to
equal the sum of the lowest terms of the
desired proportion. Next, place the bridge
as a divider between one term and the
other. When two parts of the whole string
which has been divided by the bridge are
plucked separately or at the same time,
Another manner,
the expected intervals will be heard.
For better understanding, here is a clear
example of the second manner. Suppose
that we want to hear, in this second
manner, the diatessaron, which is contained
in its true form by numbers in
sesquitertia proportion. Added together,
its lowest terms--which are 4:3, as I
have shown you--make seven. Next,
divide the proposed string into that
many equal parts, and place the bridge
over the point which separates the four
parts from the three. When these parts
are plucked together, or one after the
other, the consonance diatessaron will
be heard. In order to show you in this
situation some natural effects of the
proportions of numbers, I wish to put
them, as in singing, in their lowest
terms, in order to hear in both ways
each of the consonant intervals which
the diapason contains. From these simple
consonances spring the compounds.
Considering them carefully will provide
light and knowledge of many things
Example.
There are
seven
species of
diapason.
54
Fifteenth
CH
- E-H
Fourth
Twelfth
4)
)
0
Fifth
Octave
0
0
0
H
0
H
H
fa
H tf
LJO
H
0
H
H
0
H
H
H
0
To"
Octave
Nineteenth
inr
4)
a)
a)
a)
a)
0
Minor
third
Seven-
teenth
4)
a)
Ell' a)
4)
Cte4
'H
Major
third
Eleventh
0
cl
Minor
sixth
Major
tenth
IC'a CH
.H
Major
sixth
55
0
N Co
-H
56
pertinent to our discussions. You will
find, in addition, that those within the
diapason are as often consonant as dissonant--suitable,
however, for singing--
and do not exceed, rationally considered,
the number of their species.
Strozzi: I have understood very
well the cause of everything. Therefore,
you may proceed whenever you wish.
Bardi: Those who are seeking to
persuade us that the diatonic in which
they compose and sing today is the syntonon
of Ptolemy have shown, moreover,
that within the aforesaid limits is found
each singable interval. I want us to
prove to these men, with the very same
principles, that this species, so contrived,
is not in any way that which they
say it is, Instead, it consists of a
greater number and variety of intervals
than the ones proposed. They have not
wanted to mention these greater numbers,
not because they were somewhat inaccurate,
but because they were not exactly purposeful,
and hindered their schemes. First,
Other intervals,
in addition to
the first ones,
are contained in
the syntonon.
let us observe that in the syntonon of
Ptolemy, the note d la sol re is sharper
through h durum than it is through b molle
[Figure 1.]. From F fa ut and C sol fa ut
naturals to the same notes altered by this
sign [i.e.XJ, and also from b fa to
h mi [Figure 2.) is not the same interval
as it is from E la mi to its b molle [i.e.
E flat] and from G sol re ut to its diesis
[Figure 3.]. In addition, I say that one
does not find the same distance between
D sol re and E la mi as there is between
a la mi re and h mi [Compare Figures 4
and 5.]. Therefore, there is not the
same interval between C sol fa ut and
D sol re as one finds between G sol re ut
and a la mi re. When ascending, there is
not the same interval between D sol re and
F fa ut [Figure 6.] as there is between
E la mi and G sol re ut [Figure 7.]. Such
an interval does not even exist between
F fa ut and C sol fa ut [Figure 8.), when
they have been altered by the sign)x, to
a la mi re and e la mi [respectively).
Paradoxes.
57
. 58
From a la mi re, to c sol fa ut [Figure 9.J,
when it has been altered by the same sign X
,
is not the same distance as there is from
o sol fa ut to e la mi [Figure 10.]. From
a la mi re to d la sol re [Figure 11.3 there
is not the same interval as there is from
G sol re ut to C sol fa ut [Figure 12.].
The interval between F fa ut and h mi, when
considered as a fourth and a minor semitone,
is not a tritone [Figure 14.]. The dissonance
which occurs between h mi and f fa
ut [Figures 15 and 16.], when considered as
two minor thirds, is not a semidiapente.
Between D sol re and a la mi re there is
not a fifth [Figure 13.] It is not even
possible to have a similar interval by
adding together the semidiapente and the
minor semitone of the minor tone, which
is the one generally recognized up to the
present day. There is not the same interval
from Gaol re ut, altered by the sign X, and
e la mi [Figure 17.] as there is between
E la mi and c sol fa ut [Figure 18.]. Between
F La ut and d la sol re, and 0 sol fa
ut and a la mi re [Figure 19.] there is not
59
a major sixth. Between D sol re and c
sol fa ut [Figure 20.] there is not a
minor seventh. Also, that interval which
is found between E la mi and D la sol re,
considered, however, in two sesquitertias,
cannot be a similar interval [i.e. minor
seventh]. Such an interval does not occur,
even if it is considered as a fifth in the
lower portion and a minor third in the
higher, between G sol re ut and f fa ut.
Between D sol re and d la sol re, and between
C sol fa ut and c sol fa ut [Figure
21.] there is not an octave. There is
the same distance between b fa and h mi
through b molle as there is between b fa
and h mi through h durum [Figure 22.]. It
is the same distance from h mi to c sol fa
ut [Figure 23.] as it is from c sol fa ut
to b fa altered by either of these signs,
h or X. However, this h is the proper sign.
Finally, I say that each tone does not have
semitones of the same proportion and distance,
but different kinds.
60
Strozzi: Your lofty principle represents
to me, instead of a secure cove,
that part of the earth under the south
pole which is called unknown, because the
things which you just now mentioned are
so new to my ears that if I did not have
so faithful a pilot as my guide I would
abandon hope of the port.
Bardi: Be of good courage and do
not be frightened of these few reefs
which we will pass most securely with
our little craft. In order to reassure
you and ease your way, observe in practice
the example of all that has been
said. It will easily clarify everything
for you, proceeding step by step. First,
permit me to explain that in this second,
carefully constructed enumeration concerning
the disparity which the first
intervals have with the second ones, I
did not mean to say that such intervals
are not found among the notes of the
syntonon, but rather that all are not
their regular members, not even the first
61
ones, and none of. them are under consideration
by our practicing contemporaries.
I would also like to tell you that those
of which they possess knowledge are sung
only in their demonstrated proportions,
as you will soon see. I will prove to
you, therefore, according to what I have
proposed, that the two notes placed below
are not in unison, [5J
[Figure 1.)
I will prove to you also that the following
(Figure 2.] are not the same distance
apart as these are [Figure 3.J.
Et4I~I
[Figure 2.]
I
1 7 See pp. 185-186 below for Galileits
explanation of this enigma.
"
"
%NOWM IE
"
'NCO I
i
7
E[]
[Figure 3.]
These [Figure 4.J are not distant one from
the other by the very same interval as
these [Figure 5,J.
(Figure 4.)
(Figure 5.]
I say that these [Figure 6.J are less distant,
but not these [Figure 7.].
62
gob
-ILA
s
"
114
63
[Figure 6.J
[Figure 7.
I say that these two intervals are similar
to that which is found between D sol re
and F a ut; each of these is the same
as the ancient semiditone, and necessarily
dissonant,
"~~I~
[Figure 8.J
I say, in addition, that this [Figure 9.J
is a major interval, but this [Figure 1O.J
is not,
[Figure 9.J
I I. I.
I-'
64
[Figure 10.J
I say that the present [Figure 11.J is
dissonant and greater than this [Figure 12.J.
[Figure 11.3
1 1-I[Figure
12.J
I say that this [Figure 13.J is not distant
by a diapente.
Lz
[Figure 13.]
The two second notes in the following example
[Figure 14.J are not distant by a
tritone, considered, however, in the
manner previously described, that is in
a fourth in the low part and in a minor
semitone in the high part.
65
I-, I I
[Figure 14.J
These [Figure 15.J are not distant by a
semidiapente, considered in two minor thirds.
[Figure 15.J
I say that the two second notes are
not distant one from the other by a diapente
when, however, they are considered in a.
semidiapente in the high part and in a minor
semitone in the low, according to what the
example below shows.
[Figure 16.]
I say that there is not the same distance
between these [Figure 17.J notes as between
the present ones [Figure 18.].
66
-r[Figure
17.]
t --. rxI
[Figure 18.J
I say that the two intervals of the example
which follows are not major sixths.
[Figure 19.J
It is also not true that the present notes
[Figure 20.J are distant from each other
by a minor seventh, nor is anyone of these
[Figure 21.J distant by an octave.
I # I TI
;;B
[Figure 20.J
[Figure 21.]
Lod,
13
F--- -----
.
ILJL
r
r
I
67
These [Figure 22.] are equally distant.
[Figure 22.]
These [Figure 23.] are the same distance
apart.1 8
[Figure 23.]
Since you now possess a detailed under- [6J
standing of these numbers and know why
each of the first sixteen intervals
demonstrated is itself contained among
them, we may now discuss the names of
the proportions which contain them, how
these proportions are constructed, what
parts of them are more -or less remote or
near, and to what degree one proportion
exceeds the other. By means of these
principles I will come to answer the
questions just now posed, and by following
18See pp. 185-186 below.
68
the proposed order, we will see all
these particulars in the minor and
major semitone, more than in any other
interval. However, since I desire that
you understand me with the greatest
facility I can possibly engender, it
is first necessary to know--according
to the common opinion of practical
musicians and theorists of our times
who believe that what is sung today in
the diatonic genus is the syntonon of
Ptolemy--their proper position and thus
the position of major and minor tones,
and by what order they proceed through
each octave, going as much through h
durum as through b molle. The example
which follows shortly [Example 1.J will
give you this information. In this
example annotations will be made between
the pitches according to the
opinion of the same Ptolemy. Nor can
they be arranged in any other order,
because--as it appears in the distribution
of Didymus1 9 --they would be
69
more clearly displayed in convenient
signs. Moreover, the nature of the
diatonic syntonon of Ptolemy is to
proceed from low to high in each of
its tetrachords, by sesquiquindecima,
sesquioctava (or tone), and sesquinona.
In order to be better understood
by our practicing contemporaries,
we will say that each tetrachord proceeds
from low to high by major semitone,
by major tone, and by minor tone,
and on the contrary, from high to low
by minor tone, by major tone, and by
major semitone. And such, according to
what pleases Reverend M. Gioseffo Zarlino,
is the genus in which one composes, sings,
and plays today. To the reproof of his
opinion, we will dedicate the beginning
19J. Murray Barbour, Tuni and
Temperament (East Lansing,7T95T, p. 21.
"Dyimus diatonic contains the same intervals
as Ptolemy's syntonic ditoniaion,
but with the minor tone (10:9) below the
major tone (9:8) instead of the reverse."
2 0 See Zarlino, Istitutioni, p. 83.
Opinion of
Zarlino reproved
by
the author
in the Insti-
tutionihamonice,
ok 20
chapter 16.
70
Zarlino,2 1
Institutioni
hamo niche,
Took2,_.
chapter 39.
Tetrachord Meson of
the Diatonic Syntonon
of Ptolemy
Major Minor Major Major Minor Major -Major
tone tone semi- tone tone tone semitone
tone
Tetrachord Synemmenon
of the Diatonic
Syntonon of Ptolemy
Major Minor Major Major Minor Major Major
tone tone semi- tone tone tone semitone
tone
[Example 1--Location of Tones and Semitones in the
Diatonic Syntonon of Ptolemy.]
21 See Zarlino, Istitutioni, pp. 120-122.
71
[Minor Semitones of the Syntonon]2 2
linor Minor Minor Minor Minor Minor Minor
semi- semi- semi- semi- semi- semi- semitone
tone tone tone tone tone tone
Minor Minor Minor Minor Minor Minor Minor
semi- semi- semi- semi- semi- semi- semitone
tone tone tone tone tone tone
[Example 1--Continuedj
22
In the original example of minor semitones, the
sharps were omitted, probably through a printing error.
Compare Zarlino, Sopplimeni musicale, p. 137, where
the sharps are present in a quotation of the identical
example from Galilei, Dialogo, p. 6.
Semet.mp#. semst.min. SemiU.mi. Semt,4mi. Semst.min. sem.min. $smt.m5*.
$cmit. mm. emit.mm. Semi. nmih. Semt. min. Semt. min. Semut, nuld. Semit.min.
".... a"" .. + .ir.--- ' L - l ... - Li.L.0-AA- - &.-%& as
72
of our discourse, as I have promised,
in order to satisfy our first demand.
Here, I warn you that I do not understand
that there are intervals in the
syntonic other than the pure diatonic
ones and the others called chromatic
by modern contrapuntists. One sees,
according to examples of the modern
practice, that the major semitone is
indicated [by a signJ on various pitches
and the minor semitone remains on the
same pitch with some sign adjoined
[i.e. by key signatureJ. From this it
necessarily -follows that the b molle
never occurs on a minor semitone in the
low register, just as the diesis X
does not ever occur in a minor semitone
in the high register. The reverse happens
to the major semitone, which happens
to be exactly the opposite of that which
occurs in the ancient diatonic ditoniaion.
Coming at last to discuss the quality and
size of the intervals, I say that among
the different ones which the musicians
B molle never
occurs in a
low minor
semitone nor
the diesis X
in a high
minor semitone.
73
and theorists commonly keep in a safe
ratio, two of them are dissonant, having
been called semitones by those theorists.
In order to distinguish one from the
other, I accompany them, as you know,
with one of these words: major or minor.
They say that the minor semitone is that
which is contained between the numbers
25:24, This interval has, according to
its authors, a proportion of sesquivicesimusquartas.
Similarly, from the proportion
sesquiquindecima [16:15] they construct
the major semitone. On the contrary, in
the diatonic, the minor semitone, which
is also called limma, is contained between
the limits of 256:243; half of this they
then call the diaschisma. The major semitone,
also called apotome, is found between
these other limits, that is,
2187:2048,23
17)
The significance
of limma.
The meaning of
of diaschisma.
23See Willi A el, ed., Harvard Dictionary
f Music (Cambridge, Massachusetts,
196(9,p.7T0,for an explanation and an
illustration of the limma, the apotome, the
schisma, the diaschisma, and the Pythagorean
and Ptolemaic commas, These terms are also
explained in Barbo.r, aTuningdaTemperament,
pp. ix-xii.
74
Strozzi: For what reason did those
first speculative musicians construct
the major and minor semitones between
such numbers in the diatonic scale?
Bardi: Because of this, the ancient
musicians called minor semitone that
portion of the diatessaron left over
after two tones had been subtracted from
it, and because when two sesquioctaves
had been subtracted from the sesquitertia,
there remained the proportion of 256:243.
In this way they constructed such a semitone
and called it minor because two of
them added together did not fill up the
void of the tone. Therefore, they subtracted
the minor semitone from the tone
and called the remainder a major semitone;
two of these exceed the tone.
Strozzi: What is the meaning of that
term, limma?
Bardi: Limma means the same as
remainder. This was very appropriate
since the limma was nothing else, as I
have said, but that remainder of the
diatessaron after two tones had been
Why the major
and mincr semitones
are within
those
numbers.
subtracted. The Greeks also called
limma that part of a thing which taken
twice does not fill up the whole thing.
They also called the major semitone
apotome, which means in that language,
"detached". For example, when an
apotome has been taken from the ditone,
there remains the semiditone. There
have been others who have understood
as a major semitone the superquintaspartiente
76, which is the form of the
second interval of each tetrachord of
the ancient chromatic. This, in lowest
terms, comes within these numbers,
81:76. Having this knowledge, we can
understand, when we subtract the sesquivicesimusquartas
[25:24] from the sesquiquindecima
[16:15]--these are the forms
of the syntonic semitones--how much the
major semitone exceeds the minor. According
to Boethius,24 the manner of
subtracting one musical interval from
2 4 See Bower, Boethius, pp. 395-453,
with regard to Boethius musical mathematics
and Pythagorean number theories.
The meaning
of apotome.
Method of subtracting
one
interval from
another.
75
76
another is this. First, one arranges
the least terms of their proportions
placing those numbers which contain
the minor interval below, and those
which contain the major interval above.
16 15 Form of the major
semitone
x
25 24 Form of the minor
semitone
(Figure 24.]
When these numbers have been placed in
such a fashion, they represent, at first
glance, a way contrary to that used by
the arithmetician in subtracting one
number from another. Nevertheless, the
effect is the same. The music theorist,
however, does not consider the mere
value of numbers like the arithmetician,
but the quantity of sound which is included
between them. Because, most of
the time, the smaller numbers contain the
larger intervals, it happens that the
opposite of what the intellect understands
appears to the senses.
77
Strozzi: Is it not true, therefore,
that the smaller terms always contain the
larger intervals?
Bardi: No, Signor. In fact, the
larger intervals are contained by superparticular
ratios, except the dupla and
tripla, which are multiples. These smaller
intervals, besides some others which appear
as superpartient ratios, are the limma and
the apotome. When the numbers have been
set out in the demonstrated manner, we
will multiply the 16 (larger term of the
sesquiquindecima) by the 24 (smaller term
of the sesquivicesimusquartas) and the 25
(larger term of the sesquivicesimusquartas)
by the 15 (smaller term of the sesquiquindecima).
From those multiplications we
will have these products, 384:375. By
finding their common divisor we will come
to reduce them into their lowest terms
because one can more easily comprehend
the quantity of the interval and sound
when it is contained in its lowest terms.
To do that, I first measure the 384 term,
Manner of finding
the divisor of
proportions.
78
which is the larger number, by the smaller,
which was 375. The remainder is 9. Since
this is not of. common measure with them,
it cannot, consequently, be the desired
divisor. Therefore, I measure again the
smaller term, which was 375, by the 9.
The remainder is 6. Having considered
this, I find that not even this is common
measure with each term, but only with the
greater, which was 384, into which it goes
64 times, However, I measure once more
the first and greater remainder, which
was 9, by the lesser, which was 6, and 3
is left. This is necessarily the desired
divisor. Because of this, when the two
larger numbers which were first considered
are divided, the result is 128:125. Since
these numbers can in no way be reduced
into smaller numbers, they come to be in
lowest, or root terms. The interval which
is contained by these is something more
than a comma and a half. Thus, the minor
semitone is exceeded by the major by such
a quantity. We can easily verify this by
79
adding this quantity with the proportion
and numbers which contain the minor semitone,
because the product they will give
will have the same form as that which
contains the major. In order to do this,
the same order is maintained, first
arranging the numbers of their forms in
this manner.
128 125 Form of the supertripartiente
125
25 24 Form of the minor semitone
[sesquivicesimus-
quartasJ
[Figure 25.]
When one then multiplies the 128 (larger
term of the supertripartiente 125) by the
25 (larger number of the sesquivicesimusquartas)
and the 125 (smaller term of the
supertripartiente 125) by the 24 (smaller
term of the sesquivicesimusquartas), the
following products will be obtained,
3200:3000, which are related to the proportion
sesquiquindecima [16:15]. Within
these numbers is included the major
semitone, not in its root terms. If one
Method of adding
proportions
together.
[8)
80
wants to reduce them, the aforementioned
rule is to be observed.
Strozzi: I hope it will not be any
trouble for you to answer me in this case.
Bardi: I subtract from the larger
term, which is 3200, the lesser term,
which is 3000; the remainder is 200.
This excess, since it is of common
measure with each term, is also their
divisor. Therefore it is not necessary
to seek any further. In that way, when
the 3200 has been divided by 200, the
results are 16 and 15. When the 3000
is divided by the same 200, these
smallest terms (which are 16:15 as I
have said), contain virtually the same
major semitone, but between prime, not
between composite and communicant numbers,
like those larger ones. Regarding
the reduction of multiple and superparticular
intervals into lowest terms,
I could also produce the multiples
through dividing them by the smaller term
and the superparticulars by dividing them
Prime numbers
are those which
are not numbered
by other than
one. The composites
can be
measured by
other numbers.
by the difference, according to the rule.
When ascending by step toward the high
register (according to the promised
order) the proportion sesquinona follows
the sesquiquindecima. What is contained
by the sesquinona is called today the
minor tone by our practicing contemporary
musicians and theorists. After the
sesquinona comes the sesquioctava which
is called major tone to differentiate
it from the minor tone. I say that the
minor tone is contained in its true form
by the sesquinona proportion between the
numbers 10:9. The major tone is contained
by the sesquioctava proportion
between different numbers, that is 9:8,
We can easily see, with only this small
amount of information, which of these
consists precisely of the major and
minor semitone without either a deficiency
or a surplus in any of its parts.
It will be one whose form is entirely
.
similar to the root terms of the product
which results from adding them and then
Which tone
consists of
the major
and minor
semitones.
81
82
reducing them into lowest terms. Because
I have already spoken sufficiently about
the manner of adding, subtracting, and
finding the divisor of the proportions,
it will suffice for the future--when
these needs occur--that I form the example
of what I must demonstrate so that you
will understand everything easily. I
now want to demonstrate--in the manner
just now prescribed--which one of the
two tones is not suited, due to its
extremes, to contain an interval larger
than the product obtained by adding
together the terms and forms of both
semitones. We will see this in the
example below.
25 24 Form of the minor
semitone
16 15 Form of the major
semitone
400 360 Form of the minor
tone, not in
40 lowest terms
10 9 In lowest terms
[Figure 26.]
83
We have derived the minor tone from two
semitones added together. This agrees
with what Zarlino proves in his
Dimostrationi.
Strozzi: This fact does not exactly
agree with the opinion of the pure
practical musician, nor is it at all
surprising that the major tone is expected
from these Enumbers). However,
please proceed.
Bardi: Now when the minor tone has
been subtracted from the major, we will
be able to see clearly how much larger
the major is thanthe minor. This we
will understand from the next example.
9 8 Form of the major tone
x
10 9 Form of the minor tone
81 80 Form of the sesquioctagesima,
called comma today
[Figure 27.]
2 5 Gioseffo Zarlino, Dimostrationi
harmoniche (Venice, 1571), p. 173.
Zarlino,25 at
proposition 19
of the 1st discussion
and at
proposition 33
of the 3rd discussion
in the
Dimostrat oni
harmoniche].
How much the
major tone
exceeds the
minor.
84
The sesquioctagesima results from subtracting
the minor tone from the major.
The resultant quantity is called comma
by our practicing contemporaries.
Strozzi: Is this the same as the
ancient comma?
Bardi: No, Signor.
Strozzi: What is the difference?
Bardi: This! The ancient musicians
understood the comma to be the remainder
of the apotome, after the limma had been
subtracted, or, we should say, the remainder
of the tone, when two minor
semitones have been subtracted. Today
the comma is understood--as you have
heard--to be that excess by which the
tone exceeds the sesquitone. The moderns
did not wish to derive it from the
difference of the semitones like the
ancients for the reason which will be
given shortly afterward. The ancient
comma, however, is contained within these
numbers, 531441:524288. The half of this
they then called schisma.
The difference
between the ancient
comma and
that of today.
The meaning
of schisma.
85
Strozzi: Is our comma greater or is
that of the ancients greater?
Bardi: The modern one exceeds the
ancient one by the following interval,
32805: 32768.
Strozzi: How many of our commas do
the major and minor tone and the major
and minor semitone contain?
Bardi: I will tell you this briefly
for now. The sesquivicesimusquartas
consists of three commas plus more than
a fourth but less than a half of another
comma. The sesquiquindecima consists of
five commas, plus a little over an eighth
of another comma. The sesquinona exceeds
eight commas by a little less than half
a comma, and the sesquioctave exceeds
nine commas by the same amount the minor
Etonej exceeds eight commas. Boethius
tells us very well how many ancient
commas were contained in the ancient
major and minor semitones. You can also
2 6See Bower, Boethius, pp. 202-205;
205-207.
[9J
How many commas
comprise the
major tone, the
minor tone, the
major semitone,
and the minor
semitone.
Boethius, 26
book 3, chapter
14 and 15 of
De Music.
86
derive true knowledge of how much the
tone exceeds the major and minor semitones
separately from the example
given above [Figure 27.], which proves
that the minor tone contains no more
than the major and minor semitones.
From having seen it, it is understood
that the comma added to the minor tone
produces the major [tone). Thus we
know how much greater the major tone
is than the minor tone. Therefore,
concerning the intervals we have
discussed up to the present, it only
remains to inform you how much the
major tone exceeds the minor and the
major semitones. The two examples
which follow [Figures 28 and 29.] will
give you this information.
9 8 Form of the major tone
x
25 24 Form of the minor semitone
216 200 Form of the superbipartiente
25, not in lowest terms
8
27 25 In lowest terms
[Figure 28.]
87
The minor semitone of the major tone is
exceeded by the superbipartiente 25,
which consists of a major semitone plus
a comma. This is so very clear that
other proof is not necessary. However,
we will come with the example which
follows to subtract the major semitone
from the major tone in order to see
what the remainder is.
9 8 Form of the major tone
x
16 15 Form of the major semitone
135 128 Form of the super 7 partiente
128
[Figure 29.]
The major semitone of the major tone is
exceeded by the super [7J partiente 128.
This interval--as one can comprehend from
what we said earlier--contains in itself
a minor semitone plus a comma. I want--
since I am proving these truths to you
in many ways--to convince you of the true
opinion that one should have of the
intervals concerning their value, their
88
contents, and how much one exceeds the
other. We can see very well--from what
has been explained up to the present--
and comprehend the distance from b
molle, when it has been placed on
a la mi re or on e la mi, to the diesisX
when it has been placed on G sol re ut
or on d la sol re, according to what
is seen notated in this example.
b-,
The diesis X,
when placed on
G sol re ut and
d la so. re is
this much lower
than b molle
when it is
placed on a la
mi re and e la mi.
[Figure 30,]
In order that honest and upright judgement
be made, we will consider--besides
that which has been said previously--
the quality of the interval in which
such accidental signs occur. In addition,
we will consider the power which
these signs have to exercise in such a
place. Therefore, I say first that
both intervals where these effects are
d~_
L.
89
employed are in essence minor tones, each
one of which is diminished a minor semitone
in either extreme by both the
diesis X or the b molle. Now let us see
what remains of a minor tone after two
minor semitones have been extracted.
Because part of this operation happened
previously when we were examining the
quality of semitones, this single admonition
will suffice. When taking away
two minor semitones from a minor tone,
there will remain to that same interval
of a minor semitone what is left of the
major semitone when the minor semitone
has been subtracted from it. The b
molle of a la mi re and of e la mi is
sharper that the diesis A of G sol re ut
and that of D la sol re by the same
amount.
Strozzi: This has indeed been a
new and gratifying speculation which
I have many times desired to understand,
not so much for the sake of counterpoint
as for that of keyboard instruments.
90
Bardi: I want to advise you, in
addition, that when such a case arises--
although it seldom happens--in major
tones, the interval which remains will
be a comma less than a major tone,
provided, however, that its minor semitones
are considered in the form and
measure which we will say, at the
proper place, are suitable. The minor
third, or sesquitone, which has also
been called by moderns semiditone, I
said to be that interval which is
contained in its lowest terms by the
proportion sesquiquinta, between these
numbers, 6:5. This semiditone, in the
diatonic syntonon of Ptolemy--where
for the present we will mainly consider
all the intervals that it will be
necessary for us to measure in order to
culminate as much as we have taken into
consideration--contains in itself a
semitone and a tone, both of them major.
2 7 See LeFevre, Musica, p. 26.
Admonition
The meaning of
ses uit ne,
LeF vre7
book 3, chapter
24 of the
Elements of
Harmony,
91
Thus, when the numbers which include the
intervals just mentioned are added together,
the true form of the semiditone
will be obtained from their product, as
appears through the example which follows.
[10J
9 8 Form of the major tone
16 15 Form of the major semitone
144 120 Form of the minor third
not in lowest terms
6 5 In lowest terms
[Figure 31.J
Strozzi: Is our semiditone the same
as that of the ancients?
Bardi: It is in no way the same, because
ours is consonant, as you know, and
is produced in the superparticular genus
by the sesquiquinta proportion. The ancient
semiditone, as all musicians affirm,
is dissonant since it is contained in the
The meaning of
semiditone.
24
r
"
92
superpartient genus between these numbers,
32:27.
Strozzi: How have so many different
things been understood and distinctly
known by the same name?
Bardi: They are known with the
greatest difficulty and confusion, certainly,
for the one who deals with them
and for the one who listens. However, I
will proceed with the greatest distinction
and facility of which I am capable, in
order to be more clearly understood by
you, although my voice is rough in nature.
Since it is true, moreover, that the
minor third consists of a major tone and
a major semitone and, consequently, is
contained by the sesquiquinta--as
Zarlino28 particularly affirms in proposition
twenty-six of the second
discussion of his Dimostrationi--it necessarily
follows, contrary to the opinion
of the practical musician, that the
examples below [Figure 32.] are not actually
minor thirds of the same proportion
and measure as the first two shown.
The minor third
is not found between
D and F
fa ut.
93
[Figure 32.'
That only happens because each of these
contains in itself no more than a minor
tone and a major semitone. These intervals
together are not suited to produce
a minor third of the measure and proportion
of those first intervals, but
only a dissonant semitone of the ancient
diatonic ditoniaion. Although the
senses, according to what has been said,
accept it without repugnance, we will
nevertheless prove it to the intellect
in this manner. Add together the numbers
which contain both intervals, according
to the example which follows [Figure 33.J,
and then notice what product results.
When the two aforementioned intervals
have been added together, the result is
2 8 See Zarlino, Dimostrationi, p. 130.
94
the superquintapartiente 27, which is the
true form of the semiditone of the diatonic
ditoniaion.
10 9
16 15
160 135
5
32 27
[Figure 33.]
This semiditone is dissonant only because--
as has been said on a previous occasion--
it is a comma less than the minor third
of the syntonon of Ptolemy. In addition,
its form is found in the superpartient
genus, which is unsuitable, according to
Pythagoras, for a consonant interval, and
[is found] outside of the senary number.2 9
29The senary number, or six, held a
mystic significance for Renaissance musicians.
They considered six to be the
first perfect number, that is, the sum
of all the numbers of which it is a multiple
(1+2+3=1x2x3=6). Its musical
.
significance was the fact that all the
primary consonances could be expressed
as superparticular ratios, employing
only numbers from one to six. For a
fuller explanation of the ramifications
of the senary number, see Zarlino, lItutioni,
pp. 23-28.
Strozzi: Where do you suppose that
the musicians of today have obtained such
a fine consideration that between the
parts of the senary number is contained
each simple musical interval and part of
the consonant compound intervals?
Bardi: In considering the order in
which the proportions are placed in the
second genus of greater inequality called
superparticular, I firmly believe that
this occasion was brought about by having
joined the ten prime intervals together
two by two in natural order, afterwards
reducing them into their lowest terms.
Strozzi: How, may I ask?
Bardi: Here is a convenient example
which, with nothing more, will inform
you of everything I feel about it [Diagram
II]. That consideration also may have
been derived from the eighth chapter of
the third book of the Harmonics of Ptolemy
or from the fourteenth chapter of the first
book of his Quadripartite (or TetrabiblosJ
The origin of
the power of
the senary
number.
li]
95
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where he ingeniously proceeds to compare
the aspects of the planets to the forms
of the musical intervals of his times,
speaking thus:
The tetragon or square, compared
to the trine, makes sesquitertia.
Compared to the hexagon (or sextile,
as we should say), it makes
sesquialtera. Compared to the
opposite side, it makes dupla,
and with the entire circle of the
Zodiac, it makes diapason plus
diapente. All of this, again
compared to the square, makes
disdiapason. Finally, three
squares compared to two trines
have the same relation as nine
has to eight.
Strozzi: Are the forms of the imperfect
consonances not found also among
these aspects?
Bardi: No, Signor, because Heaven
does not permit or tolerate imperfection.
Whoever wishes to investigate more carefully
may also derive from each of the
two places cited the forms of all the
other musical intervals of our times.
However, enough has been said.
Strozzi: -I believed that this power
of the senary number was entirely a new
30 Compare Ptolemy, Tetrabiblos,
translated by F. E. Robbins(am ridge,
Massachusetts and London, 1940), p. 73-75.
Ptolemy3 0
compares the
aspects of
the planets
to the musical
intervals.
97
98
discovery, and I see that I was probably
not correct. This causes me to doubt
that any of the other things (concerning
inventions) which are very old are represented
as new [things] by one writersj
or another.
Bardi: Do not doubt it at all,
because the simple-minded many times, in
reading some book about any subject, believe,
through little experience, that
those things are found in no other book
but that one. In most cases, the same
things have been written in many books
a good many years before.
Strozzi: There is a division of the
square into three equal parallelograms,
the middle one of which is separated
into two equal parts, then bisected by a
line which runs from one of the angles
of the square and is placed over the
opposite side so that is is divided
into two equal parts. These different
portions of the line, compared with one
another, have power to give the forms
of a greater number of musical intervals
99
than the senario and its parts. Is this
parallelogram an ancient or, rather, a
modern invention?
Bardi: The parallelogram is question
is taken from the bulk of the second
chapter and book of the Harmonics of the
same Ptolemy.3 Concerning this, he
jokingly tells as much as his purpose
requires in order to denote the musical
intervals of those times. From what I
will now tell you, you will easily be
able--when you have an interval outside
of its true or lowest terms--to perceive
if it is enlarged or reduced from its
normal condition. I warn you, however,
that if its smaller term is diminished
from that which is appropriate to its
natural state, it is a clear indication
of being enlarged. If the smaller term
31Strozzi and Bardi are speaking of
Ptolemy's helicon, a device for illustrating
the consonances by means of
intersecting straight lines and for
showing the relative string lengths for
each pitch. See During, Ptolemaios, pp.
60-63. There is an explanation of-the
helicon in Cecil Adkins, "The Theory and
Practice of the Monochord," unpublished
doctoral dissertation, Department of Music,
State University of Iowa, Iowa City,
Iowa, 1963, pp. 53-56.
is larger than usual, such an interval
is in reduced form. How much it is
enlarged or reduced can be ascertained
by subtracting from it or adding to it
whatever amount you consider to be more
appropriate. The opposite effect would
occur in its larger term.
Strozzi: I have not completely
understood this last conception of
yours.
Bardi: Now I shall make myself
clearer. We have instead of the sesquiquinta--since
it has been assigned by
contemporary teachers of counterpoint
as the form of the minor third of the
syntonon of Ptolemy--the superquintapartiente
27. Since we now wish to
see if this proportion is larger or
smaller than the sesquiquinta, the
excess with which the 32, its larger
term surpasses the lesser term which
is 27, is able to explain it sufficiently.
This excess is.5. Now
consider how many times this 5 will go
Warning.
100
101
into the smaller term of this proportion;
it goes five times with a
remainder of 2. In the sesquiquinta,
however, the excess by which the larger
number exceeds the smaller, which is
1, goes in five times exactly. It
clearly appears that what I have said
is true, that is, since the smaller
term of the super 5 partiente 27 is
larger, when compared, however, to
that of the sesquiquinta, it is consequently
smaller by as much, then,
as has been said previously.
Strozzi: I have understood everything
very well, but tell me another
detail. Since the semiditone is not
the same thing as the minor third, with
which of these does the trihemitone
agree? I have found that this same
trihemitone has been given three different
names by various authorities.
Bardi: The trihemitone, as you
know, is the highest interval of each
tetrachord of the ancient chromatic
The meaning of
trihemitone.
102
arid does not agree with either the semiditone
or the minor third of the syntonon
because it falls under the proportion
supertripartiente 16, between these numbers,
9:16, when in lowest terms. Thus
it happens to be less than the minor
third and greater than the semiditone.
Zarlino3 2 agrees with this opinion in
chapter thirty-four of the second part
of his Institutioni harmoniche. It
remains for us to see in this case how
much the minor third exceeds the major
and minor tone. This will be explained
in the two examples which follow
(Figures 34 and 35.].
6 5 Form of the minor third
X
9 8 Form of the major tone
48 45 Form of the major semitone
not in lowest
3 terms
16 15
[Figure 34.]
3 2 Zarlino, Istitutioni, p. 113-114.
103
The minor third is larger than the major [12]
tone by a major semitone. No further
proof of this is necessary, since it has
just now been seen in combining both of
the semitones. However, let us see how
much the minor tone is exceeded by the
minor third.
6 5 Form of the minor third
x
10 9 Form of the minor tone
54 50 Form of the superbipartiente
25 not in
2 lowest terms
27 25 In lowest terms
[Figure 35.]
The minor tone is exceeded by the minor
third by the superbipartiente 25, which
consists of a major semitone, and a comma,
according to what was proved above. They
say that the major third is that imperfect
consonance which arises from the proportion
sesquiquarta and which is contained in
lowest terms by the numbers 5:4. In addition,
it consists of the major and minor
104
tone, which we will see, according to this
Next] example, by adding them together
and noting the result.
9 8
10 9
90 72
Form of the major tone
Form of the minor tone
Form of the major third
not in lowest terms
18
5 4 In lowest terms
[Figure 36.]
The major third results from adding together
both tones. From this knowledge,
together with subtracting the minor third
from the major third, we will be able to
learn how much one is exceeded by the
other (Figure 37.).
"Z ~m
105
5 4 Form of the major third
X
6 5 Form of the minor third
25 24 Form of the minor semitone
[Figure 37.]
The minor third is exceeded by the major
third by a minor semitone. Further
proof of this would be adding to it the
minor semitone in question. The simple
practical musician possesses other
species of major thirds through the use
of these signs, X and b, as accidentals
of different pitches. They are considered
by him to be of the same importance
as the first demonstrated.
This, in my opinion, is one of the
greatest abuses he perpetrates--among
many many others, besides the ones I
have already mentioned--which I have
selected to demonstrate for you.
Strozzi: I am greatly surprised
that among so many excellent men who
have written about the art of music so
cleverly and wisely, there still have
106
not been any, as far as I know, who
have, for mutual benefit, revealed and
discovered such obvious errors as these.
Bardi: That is not exactly surprising.
It happens that such a consideration
could never occur to those
who believe in singing the diatonic
ditoniaion, because the cause of that
situationsj was not found there in any
way, but only the effect. Regarding
those, then, who have said that the
species which is sung today is the
syntonon of Ptolemy, I can easily believe,
because of my many encounters
with them, that thay were very well
aware of the problem. Because revealing
it did not precisely suit
their purpose, however, as I said on
another occasion, they have passed
over it in silence since, impelled by
a vain, ambitious thought, they valued
more highly the impertinent novelty of
their whims than the convenience and
usefulness which they would have been
107
able to give to the public by explaining
the truth. In this way, they would not
have corrupted and spoiled the names of
the most important things pertaining to
the art of music. Moreover, the thirds
which I have mentioned are contained between
the notes of the present example.
[Figure 38.J
Since each one of these [intervals) is
made up of two major tones, as a result
they all happen to fall under the proportion
81:64. When it has been well
considered, one will find in this 81:64
proportion each one of those particular
accidentals which are found in the
superquintaspartiente 27. This cannot
in any way be the natural major third
of the diatonic syntonon, but only the
dissonant ditone of the diatonic.
The major third
is not found
between these
pitches.
108
ditoniaion--as Zarlino33 affirms in the
fourth discussion of his Dimostrationi
at the third definition. This proceeds
from high to low in eachof its tetrachords
by tone, tone, and limma. It
is quite true that the examples which
follow are contained by their proportion
of sesquiquarta. Consequently, they are [13J
consonances, because the interval which
has first place in the low register is
a minor tone. That which is found in
the second [place] between F fa ut and
G sol re ut is also a minor tone. In
addition, I wish to warn you that you
should consider the excess by which the
larger term of the proportion which contains
the ancient ditone, which is 17,
exceeds the smaller term.
[Figure 39.J
3 3 Zarlino, Dimostrationi, p. 268.
109
You will find--according to the rule
given you a little earlier--that because
it surpasses its smaller term, which was
64, close to four times, there is clear
indication that the aforesaid proportion
remains the true state of the sesquiquinta.
I say that the diatessaron is that middle
interval which springs from the sesquitertia
proportion, which is contained
in lowest terms by these numbers, 4:3.
It is not inappropriate that I mentioned
the fourth to you with the name of middle
interval, for it happens that among five
consonant intervals, the forms of which
are found between one and six, and the
other intervening numbers which are
placed between them in natural order, it
comes third. Two of the five consonant
intervals, the octave and the fifth, are
in the low register. Two others, the
major and minor third are in the high
register. In addition, the sound which
issues from the extremes of the fourth
is, in its simplicity of nature, not at
Diatessaron is
an interval
midway between
consonance and
dissonance.
110
all offensive to the ear as the dissonances
are, nor does it have the quality to delight
it as the other consonances do. Therefore,
deservedly, it can be stated to be midway
between dissonance and consonance; no one
should wonder about that [factJ because
Nature is not accustomed in her undertakings
to pass from one extreme to the other without
touching the middle. As the smallest
of the perfect intervals, it is less
consonant than the other, being farther
from perfection. In the same way, the
smaller imperfect consonances are more
harmonious than the larger ones, being
farther from imperfection. Exactly the
opposite happens to dissonances when
sevenths are applied to perfect consonances
and seconds are applied to imperfect consonances,
as we will clearly demonstrate
in the appropriate place. I now return
to the fourth, saying that principally
it contains in itself a major semitone,
a major tone, and a minor tone. It may be
considered as composed of the minor third
111
and the minor tone, which we will see,
according to the example, from the
product which results from adding together
the lowest terms of both of these
intervals.
6 5 Form of the minor third
10 9 Form of the minor tone
60 45 Form of the fourth, not
in lowest terms
5
4 3 In lowest terms
[Figure 40.]
The fourth may also be obtained in the
same proportion and measure from the
product of the major third and the
major semitone. We will see this in the
following example [Figure 41.] where
their forms are added together.
112
5 4 Form of the major third
16 15 Form of the major semitone
80 60 Form of the fourth, not in
lowest terms
20
4 3 In lowest terms
(Figure 41.J
Since the fourth--because it is consonant
and in its true proportion--contains the
amount I have said and proved, it necessarily
follows--contrary to the opinion
of the simple practical musician--that
the examples below are neither consonant
fourths, nor in the same proportions as
the first ones. This agrees with what
34Zarlino says in proposition twentyeight
of the second discussion of his
Dimostrationi. This happens because each
one of the fourths contains two major
34 Zarlino, Dimostrationi, p. 131.
In proposition
28 of the 2nd
discussion.
"
"
%moor
TOT
113
tones, and a major semitone which separates
one tone from the other. When the
proportions which contain these intervals
have been added together, we will have
from their product a proportion which is
a comma larger than the sesquitertia.
[Figure 42.J
The extremes of this will necessarily be
dissonant. Even the simple practical musician
can ascertain through the example
given that the tones are what I have said
they are. However, I will prove it to you
as a theorist, in the example which follows,
by means of arithmetic.
2
6 5 Form of the minor third
9 8 Form of the major tone
54 40 Form of the super 7 partiente
20, not in
lowest terms
The diatessaron
is not found
between these
pitches.
[14]
27 20
[Figure 43.]
114
From this calculation we have obtained
the super 7 partiente 20 which, as I
told you, consists of a fourth plus a
comma, This can be seen easily by
subtracting the comma from the fourth.
Besides, if this were not so, it would
follow that in descending into the low
register with a tone of the quality
which is contained between the notes
G sol re ut and a la mi re, the extremes
would not sound as a whole fifth. This
is not exactly true, as will be seen at
the appropriate place. Now, I want us
to see if the examples below [Figures
44 and 45.J, when considered with a
third uppermost and a tone below, are
whole sesquitertias or not. I also want
to know how the theorist considered it
and why. I have proposed this question
because I desire to appease each difficulty
and doubt which could possibly
arise concerning this important topic
of our discussion. A little earlier,
I proved to you in regard to the minor
115
third that when ascending between D sol re
and F fa ut, such an imperfect consonance
was not actually contained in the sesquiquinta,
but only the ancient semiditone.
(Figure 44.]35
At present, when we consider the fourth of
this example--as much through b molle as
through h durum--and each one of these
[fourths] has been composed of the interval
which I have mentioned [i.e. semiditonej
and the major tone, we will find
35The pitch "a" in the original
example [aloo, p. 14J has been emended
to an "f y an unknown hand.
116
that when the numbers which contain them
have been added together, they will give
a whole sesquitertia, as you will see
from the following example.
32 27 Form of the semiditone
9 8 Form of the major tone
288 216 Form of the fourth not
in lowest terms
72
4 3 In lowest terms
[Figure 45.]
We have obtained the fourth in its true
form by adding together two intervals.
This did not occur previously, because
the third of this next example [Figure 46.),
in comparison with the first one shown,
has been diminished by the same amount
the tone was augmented in the earlier
example [Figure 44.). The fourth in the
ancient diatonic also may be considered
as composed of such parts. This concludes
our consideration of the fourth, since I
am informing you how much it exceeds the
major and minor thirds. The two examples
[Figures 44 and 45.) given you when we
considered the fourth as composed of the
intervals just mentioned, and of the
major semitone and the minor tone, will
show this fact. Now, I come to tell you
some particulars about the tritone, which
our practicing contemporaries say is
contained by the proportion super 13
partiente 7 between these numbers, 45:32.
It contains in itself three tones and
took its name, perhaps, from those
contents. However, this only refers to
the diatonic ditoniaion where these tones
are equal. One can consider this dissonant
interval to be composed of a
major third and a major tone in the low
register as well as in the high. We can
ascertain that the aforementioned proportion
is the true form of the syntonic
tritone in many ways. Among them, the
one which is the plainest and easiest,
I will now show you. Add together the
numbers which contain the intervals
mentioned above, its propinquous parts,
which will give the same numbers which
I said contained it.
The origin
of the name
tritonne.
117
"I
5 4 Form of the major third
9 8 Form of the major tone
45 32 Form of the tritone
[Figure 46.J
The tritone in its true form and in lowest
terms is obtained from the product
of the major third and the major tone.
This interval, together with its derivative,
since their forms and extreme sounds
are very disproportionate, so to speak,
and consequently are understood with
difficulty by the intellect, they cannot
be considered nor are they found naturally
in the octave in any more than two
pitches. This is not at all surprising
since it happens that nature seldom
produces either disproportionate or
monstrous things. We can immediately
Nature seldom
produces either
perfect things
or monstrous
things.
118
119
see how much the tritone exceeds the
fourth by subtracting one from the
other.
45 32 Form of the tritone [15]
x
4 3 Form of the fourth
135 128 Form of the super 7
partiente 128
[Figure 47.]
The tritone exceeds the fourth by the
super 7 partiente 128. This proportion
consists of a minor semitone and--
contrary to the opinion of the practical
musician--a comma besides, as has been
demonstrated previously in another place.
It seems impossible to the practical
musician that when the fourth has been
subtracted from the tritone more than
an ordinary minor semitone must remain.
All the difficulty that he has in understanding
what is to him a novelty results
from not having the knowledge
which he should have of the intervals
he has continually contrived. This
120
knowledge calms those who have it perfectly.
That does not escape his reproach,
since it is impossible for those who do
not understand the propriety and virtue of
the matter, whatever it may be, to exercise
it well. And this is said by many--even
all the wise men and scholars--to be one
of the principal causes, among many others,
that the musical practice of our times
does not have that power to operate in
the minds of the hearers any of those
marvellous and virtuous effects which
the ancient practice had. Returning to
the tritone, however, I say that it cannot
be obtained by adding together the
form of the fourth and that of the minor
semitone, since a major tone must be used
as the divisor in order to accomplish
that. The major tone is capable of
containing a comma in addition to the
major and minor semitone, according to
what has been proved another time, and
as one can prove again by adding them
together. On some occasion a proportion
could occur to you between large numbers
with so little difference between them
that you would not know--in a manner of
speaking--what the difference was between
one interval and another, although you
very much wished to see this difference.
It is also not impossible in this manner
of manipulating the numbers, as in some
other cases, to subtract a very large
number from a very small one. When you
have been warned, however, by what I am
about to tell you and show you with an
example, you will ascertain which of
these contains the greater [quantityJ
and which the smaller. Therefore I say,
first, that a large interval can be
subtracted from a small one. For
example, a major tone can be subtracted
from the comma, although the comma is
much smaller than the major tone. And
in order that that may be done, we will
prove it to the senses with the present
example [Figure 48.J.
Warning.
121
122
81 80 Form of the comma
x
9 8 Form of the major tone
8 720 Form of the subsesquinona
not in lowest
terms
9 10 In lowest terms
(Figure 48.)
It seems, at first glance, from having
subtracted a minor tone from the comma,
that, consequently, the comma exceeds the
tone by a certain quantity. This offends
each one who knows about it, in regard to
the intellect and the sense. This is to
warn you, however, that the smaller
number of that interval which we have
sought to subtract from the comma has
come to the place of the greater number,
and, on the contrary, the greater to the
smaller. For this reason they do not
otherwise come to have the form of the
sesquinona, since they are contained, as
I told you another time between 10:9 and
not, on the contrary, between 9:10.
Warning.
72
64
123
Strozzi: What interval, therefore,
will be that which is contained by such
a proportion?
Bardi: That is a subsesquinona,
which manifests, in such a place, how
much the interval from which it was
derived is exceeded by the sesquioctave
and not how much the comma exceeded it.
Therefore, such proportions justly
come to be called privatives and rationals,
and those others just mentioned,
positives and reals. This will suffice
for all other such cases which may occur
to you. I come now to discuss with you
the semidiapente, which--according to
the opinion of our practicing contemporaries--is
that interval which consists
of a major tone, a minor tone, and two
major semitones. Perhaps it took such
a name because its extremes sound a
diminished diapente, having been divided,
however, into four intervals by five
3 For a fuller explanation of privatives,
rational, positives, and reals,
see Zarlino, Istitutioni, pp. 22-23.
Why the semidiapente
is
so called,
124
limits and pitches, It is contained,
moreover, in its lowest terms by such
numbers as these, 64:45, which we will
consider as composed of a fourth and
a major semitone, as much in the low
portion as in the high. From these
facts one can be capable of [doing)
it in many ways, but it is simplest
and briefest to add together the numbers
which contain the two aforementioned
intervals and see if their
product will give the form which we
have said.
.Ce.
__[16)
4 3 Form of the fourth
16 15 Form of the major semitone
64 45 Form of the semidiapente
[Figure 49.]
One may not consider or compose this
interval, as was true of the previous
interval, between natural diatonic
tones in any octave from any of its
parts except the two mentioned. If
the practical musician told me he
could compose this from two minor
thirds of the same form and of equal
proportion, I would respond and prove
that he is wrong in this, as in many
other things, since his goal is only
to appease, not the intellect or the
sense of hearing, but frequently that
of the sight. This is completely, or
more than the others, deceived with
ease, and the hearing has the same
part, or only a little greater one, in
distinguishing sounds than the ear has
in discerning the differences of colors.
The senses are particularly deceived
with regard to the smallest differences
of ordinary things and usual objects,
These, the rational intellect understands
with great difficulty. The fact
The goal of
the practical
musician.
125
that the semidiapente in its demonstrated
form cannot be composed of two minor
thirds of the same proportion is first
manifested to us by the number of its
larger term. Although it is squared,
this larger term needs to be capable of
,containingJ the geometric mean.3 The
other necessary condition is not found
in the larger term because it must also
be capable of producing the larger term
of one of those intervals into which one
will want it to divide equally. This
does not happen with the larger number
and term of the semidiapente, because
it is not related to the six which is
the largest term capable of containing
the minor third. However, we will put
all this into actual practice with a
clear example, which will consist of
adding together the terms of such intervals,
The form of the semidiapente will
in no way be derived from their products,
but only that of the super 11 partiente 25.
7Foran explanation of the geometric,
arithmetic, and harmonic means, see Bower,
pp. 128-130.
Warning.
126
127
6 5 Form of the minor third
6 5 Form of the minor third
36 25 Form of the super 11 partiente
25
[Figure 50.J
The interval which we have obtained by
adding together two minor thirds consists
of a semidiapente plus a comma. If this
is true, one may see it by subtracting
the super 19 partiente 45 (form of the
semidiapente) which will leave a comma.
This is the reason. When the theorist
considers that the semidiapente is composed
of two minor thirds, he is unable
to divide any interval of the first three
simple genera of proportions rationally
into equal parts and consequently not
the quadruple and its compounds. Since
he understands everything in terms of
arithmetic, he perceives, necessarily,
that one is larger and different from
the other. This fact I have already
shown you clearly with regard to the
128
minor third. It only remains to see,
when such diverse minor thirds are
added together, if their product will
give the true form of the semidiapente.
From this example [Figure 51.J it will
be evident that the ancient Ptolemaic
musicians had such species under consideration
also, because, if they had
considered them to be in the same
measure, they would not have assigned
to the semidiapente--which was capable
of containing both--the demonstrated
proportion. Instead, they would have
assigned the proportion which was
derived previously from the product
of the two minor thirds, each one of
which had been contained by the sesquiquinta.
The example which follows
will make it clear that the thirds
are, therefore,' different in form.
129
32 37 Form of the semiditone
6 5 Form of the minor third
192 135 Form of the semidiapente
not in lowest terms
3
64 45 In lowest terms
[Figure 51.]
Here is the semidiapente in its true form
composed of the two aforementioned intervals.
It is not necessary to see any more
of this, unless it is how much it exceeds
the triton. The example which follows
will inform us of this, in which the
tritone will be subtracted from the
semidiapente.
64 45 Form of the semidiapente
x
45 32 Form of the tritone
2048 2025 Form of the super 23 partiente
2025
[Figure 52.]
The tritone is exceeded by the semidiapente 17J
by the super 23 partiente 2025, which consists
of approximately half a comma.
130
Strozzi: Are the semidiapente and
tritone of the ancients of the same measure
and proportion as ours?
Bardi: No, Signor, because the interval
which occupied the space of the
tritone, according to the ancients, fell
under these numbers, 729:512, which exceeds
ours by a comma. Their semidiapente
fell under these other numbers,
1024:729, which, on the other hand,
happens to be less than the super 19
partiente 45, called semidiapente by
the moderns, by such an interval as
one can clearly see by subtracting one
from the other.
Strozzi: So that the greater part
of names of the musical intervals of
today, if not all, are corrupt and
spoiled?
Bardi: There is no doubt of it at
all.
Strozzi: Would it not be wise to
provide them before the thing ages and
becomes more confused in the minds of
men?
Current names
of the intervals,
131
Bardi: Yes, indeed, But this is
the province of a man of authority and
of great worth. However, we will proceed
in regard to the names of the
intervals, following, although it is
poor in quality, the common usage, if
for no other reason than to be understood.
However, let us proceed at last
to the examination of the diapente which
they say is contained in its true proportion
by the sesquialtera, called
also sesquipla, between these numbers
in its lowest terms, 3:2. I say that
it contains two major tones, a minor
tone and a major semitone, which may
be considered in many ways. The most
usual and important way, however, is
being composed of the major and minor
third as the product of the numbers
which contain it will demonstrate; this
will be seen in the example which follows
[Figure 53.].
38 See Glarean, Dodecachordon, translated
by Clement A. Miller (Rome, 1965),
P. 62.
Glarean, in
chapter 12 of
book 3, calls
the sesquialtera
sesquipla.
Form
Form
40 fo"w pe
of the major third
of the minor third
0 20 Form of the fifth not in
lowest terms
3 2 In lowest terms
[Figure 53.]
It will be in its true harmonic form
every time the major thirdj is in low
position and the minor [third) is uppermost.
I have said that the fifth is
composed of such intervals before each
of the others because those are its
nearest parts. So that you may also
understand the cause of this well, I
say that in the opinion of theorists,
the nearer parts of intervals are the
ones from whose product, when the numbers
which contain them have been added
together, can be derived not only the
same proportion of the whole, but a
The meaning of
near and far
parts of the
intervals,
132
.
r ~ : _ ___ ___
5 4
6 5
3
1C)
133
proportion which is quite close to its
same lowest terms, and sometimes--as
you have seen with the tritone and
the semidiapente--actually in them.
On the contrary, the farthest parts
Eof intervalsJ are the ones which in
their least terms all give the remotest
product. One has been derived from
their most imperfect and unequal parts
and the other from their most perfect
and equal parts. Since we are about
to consider the other parts of the
fifth and to construct it from them,
I say that whatever consists of a
fourth and of a major tone will be a
fifth, because, when the numbers of
their proportions have been added
together, its demonstrated form will
be produced by the product, although
it will be a little more remote from
its smaller terms [Figure 54.].
134
4 3 Form of the fourth
9 8 Form of the major tone
36 24 Form of the fifth not in
lowest terms
12
3 2 In lowest terms
[Figure 54.J
Since the fifth originates from those two
intervals demonstrated, as has been said,
one may argue with Zarlino39 that those
of the following example may not result
from anything except from different proportions
and genera, which is contrary
to the opinion of the practical musician
no matter how much it is proved to them
that they are dissonant.
Zarlino in the
30th proposition
of the 2nd discussion
of his
Dimostrationi.
[Figure 55.J
They are of different proportions than
those first ones because they contain a
395ee Zarlino, Dimostrationi, pp.
132-133.
"
"
fourth and a minor tone. These two intervals,
when added together, cannot
produce a fifth of the same proportion
as the two first demonstrated, but
another which will be a comma less
than these, and therefore dissonant.
In order to see how much less it is,
here is an example which will verify it.
4 3 Form of the fourth
10 9 Form of the minor tone
40 27 Form of the super 13 partiente
27
[Figure 56.]
The interval which results from adding
together the fourth and the minor tone
is the super 13 partiente 27, which
consists of a fifth lessened by a
comma. The man [i.e. Zarlino] could
assure himself more greatly of this
fact by adding it together with itself,
because he would obtain from both the
sesquialtera, form of the diapente.
From what we have said, he might wonder
The diapente
is not found
between these
pitches.
[18)
135
136
whether the following [example]--considered
as a fourth plus a tone with the fourth in
high position and the tone in low position--
is a fifth of the first or the second
species. I would respond to this saying
that it belongs to the first species and
not in any way to the second.
[Figure 57.]
If he then replied that the tone which
is in low position is minor, when it
should be major--according to the way I
have demonstrated the composition of such
intervals--I would tell him that this is
completely true. For this reason, it
happens to be consonant in that situation
because the fourth, which is in high position
in that fifthh, is greater by the
amount which is lacking to the minor tone,
which is in low position, in order to be
made equal to the major [toneJ. Therefore,
137
since it is augmented by one interval and
diminished by the other because the excess
of the first is equal to the deficiency
of the second, the fifth can be derived
from the intervals mentioned in its true
form and proportion. This can be seen
clearly by everyone by adding together
their lowest terms. The fifth can also
be obtained in its true form from the sum
of the tritone and the major semitone,
but certainly not from that of the semidiapente
and the minor semitone, as the
following example makes clear.
64 45 Form of the semidiapente
25 24 Form of the minor semi-
tone
1600 1080 Form of the super 13
partiente 27 not in
40 lowest terms
40 27 In lowest terms
[Figure 58.]
The super 13 partiente 27 is obtained from
such a sum, This consists, as has been
said, of a fifth diminished by a comma,
so that from the semidiapente and from
138
the minor semitone, one does not obtain,
as the practical musician believes, the
same product as from the major semitone
and the tritone. This does not exclude
the knowledge of how much the fifth
exceeds the tone and of how much it
exceeds the semidiapente. The two
examples which follow will give us
complete knowledge of this [Figures 59
and 60,J.
3 2 Form of the fifth
x
45 32 Form of the tritone
96 90 Form of the major semitone
not in lowest terms
6
16 15 In lowest terms
[Figure 59.J
The tritone is exceeded by the fifth
by a major semitone, additional proof
of which would be adding together the
tritone and this major semitone. By
adding these one undoubtedly should
obtain the fifth, but this is not the
case for the matter is clear in itself.
139
However, we will proceed to demonstrate
in the example which follows how much the
semidiapente is exceeded by the fifth,
3 2 Form of the fifth
x
64 45 Form of the semidiapente
135 128 Form of the super 7 partiente
128
[Figure 60.]
The semidiapente is exceeded by the fifth
by the super 7 partiente 128, which--
according to what I proved to you earlier
in examining the major tone--consists of
a minor semitone plus a comma. Therefore,
the practical musician, in addition to
being surprised, does not have to tell
me that I intend for the same interval
to contain first one form, then the other.
The minor semitone
in this
example exceeds
the comma by the
sesquivicesi-
musquartas.
[Figure 61.J
140
I say that I previously considered the
present semitone to be contained by the
sesquivicesimus quartas and finally to
be contained by the super 7 partiente 128
in which position it is actually capable
of as much as I said, I add to this that
the interval which is enclosed within the
confines of the super 7 partiente 128 is
the true form of the minor semitone of
the major tone, as I will prove to you a
little later,
Strozzi: This will be very unusual E19]
to me as well as new and agreeable.
Bardi: In this proposition I will
deal with the smallest possible case
desirable regarding the form of the minor
sixth, also known by modern practical
musicians as the minor hexachord. They
say that this interval is that imperfect
consonance which is contained by the
supertripartientequintas between these
numbers, 8:5, in its root terms. This
interval consists of two major tones, a
minor tone, and two major semitones, and
141
one may consider that, initially, it is
composed of a fourth and a minor third,
as we will see from that product which
results when the smallest numbers of
their proportions are added together.
In regard to this, there are some who
reprove Franchino [Gaffurio] because he
said that the minor sixth is composed of
the diatessaron and the larger chromatic
interval. He understood this last
interval to be the trihemitone, taking
it instead of the minor third or, to
say it better, the semiditone. These
intervals, when added together, are
not able in any way to produce the minor
sixth in the form in which it is contained
in the example which follows
shortly [Figure 62.]. They cannot even
produce it according to the diatonic
ditoniaion, in which Franchino considered
it, This happened, perhaps, because he
did not regard it as imperfect, as many
others do.
-p-
4 3 Form of the fourth
6 5 Form of the minor third
24 15 Form of the minor sixth
not in lowest terms
3
8 5 In lowest terms
[Figure 62.J
With regard to this imperfect consonance,
there is another consideration which is
what the common man believes, and which
should not be ignored. According to his
opinion, there is found in each musical
interval, a species which is not less
than the number of its pitches. This is
not true, because the species of the
minor sixth and of the major sixth are
not more than three, In addition such
a rule is not verified in any intervals
which for brevity's sake are not mentioned.
Returning now to the consid--
eration of the minor sixth, I say that
Warning.
142
143
it is also that quantity which consists
of a semidiapente and a major tone, because
when the numbers which contain
them have been added together, their
product will produce its true form. The
example which follows immediately demonstrates
this fact.
64 45 Form of the semidiapente
9, 8 Form of the major tone
576 360 Form of the minor sixth
not in lowest terms
72
8 5 In lowest terms
[Figure 63.)
Since this is true, it follows--contrary
to the opinion of the practical musician--
that none of the present examples [Figure
64.], nor any others like them are minor
sixths of the same proportions as those
first demonstrated.
Minor sixths are
not found between
these pitches.
[Figure 64.)
It therefore results that both the tone
and the semidiapente cannot produce the
minor sixth between the demonstrated
pitches as between those above, since
the tone in the latter situation is minor,
while, on the contrary, it is ma;or in
the former. Therefore, the latter
necessarily becomes less than the former
by a comma. Whatever interval results
from these, we will now see from the
product which we will obtain by adding
together the numbers of their lowest
terms,
64 45 Form of the semidiapente
10 9 Form of the minor tone
640 405 Form of the super 47 partiente
81 not in lowest
5 terms
128 81 In lowest terms
[Figure 65.J
144
145
The interval which has been obtained from
these is the super 47 partiente 81, which
contains the true minor hexachord of the
diatonic ditoniaion in its lowest terms,
which is a comma smaller than the supertripartientequintas.
The supertripartientequintas
is called today the form
of the minor sixth, although some also
call it the minor hexachord, It is
consonant in differentiation from that
ancient onewhich is dissonant, since it
is, as has been said, a comma smaller.
Anyone who takes the trouble to subtract
the ancient hexachord from the modern
one will ascertain this fact. One can
also obtain this imperfect consonance
from the fifth and the major semitone,
as we will see from the result of adding
their lowest terms together accord- [20j
ing to this example [Figure 66.J which
will indicate how much the minor sixth
exceeds the fifth.
3 2
16 15
[~J
Form of the fifth
Form of the major semitone
[8 30 Form of the minor sixth not
in lowest terms
8 5 In lowest terms
[Figure 66.]40
Where the tritone intervenes, one may not
obtain the minor sixth unless by means of
more than two intervals or through the
agency of anyone of these [just named];
however, I pass over this in silence, I
wish for us to see now the same occurrences
4 0 Galileits musical example is erroneously
a fourth instead of a major sixth.
146
6
4
147
in the major sixth, whose lowest terms
the practical musicians say are between
these numbers, 5:3, and which they say
consists of two major tones, two minor
tones, and one major semitone. It can
be considered, initially, to be composed
of a fourth and of a major third, so that
such intervals will entirely fill up the
void of its terms. I ascertain this
fact every time the result of adding together
the numbers of their proportions
is contained by the same numbers which
contain the major sixth. The example
which follows immediately clarifies this
for us,
4 3 Form of the fourth
5 4 Form of the major third
20 12 Form of the major sixth not
in lowest terms
4
5 3 In lowest terms
[Figure 67.]
148
Thus, the major sixth in its true form has
such a product, but this will not be the
case, when it is considered with similarly
appearing intervals between the pitches of
F La ut and d Ia sol re. This is because
the fourth which is found between them in
the upper portion does not fall under the
form of the first one demonstrated, but
under another form, 27:20. This, according
to what was proved in another place, was
greater by a comma. Whatever proportion
we will obtain from such intervals, you
will notice in this example where their
least terms have been added together.
The major sixth
is not found
between these
pitches.
27 20 Form of the super 7 partiente
20
5 4 Form of the major third
135 80 Form of the super 11 partiente
16 not in lowest
5 terms
27 16 In lowest terms
[Figure 68.]
s
a
149
Thus, as you see, the super 11 partiente
16 has been obtained. This is the true
form of the major hexachord of the
diatonic ditoniaion. Since it exceeds
the major sixth of the syntonic of
Ptolemy by a comma, it is necessarily
dissonant. For this reason and no other,
Zarlino4 1 stated in proposition thirtyfive
of the second discussion of his
Dimostrationi that, when adding the
major tone to the fifth, a consonance
cannot be the result. One can clearly
see that the major hexachord exceeds
the major sixth by a like quantity, as
I have said, by subtracting one from
the other. The major sixth will be
produced in very consonant form from
the product of the fifth and the minor
tone, as one can clearly see from the
next example [Figure 69.J in which the
pitches that contain them are added
together.
41 See Zarlino, Dimostrationi, pp.
138-139,
150
3 2 Form of the fifth
10 9 Form of the minor tone
30 18 Form of the major sixth not
in lowest terms
6
5 3 In lowest terms
[Figure .69.J
This imperfect consonance may also be
obtained in its true form from the minor
sixth and from the minor semitone, with
the aid, however, of the present sign,6
.
It may also be found in the following
example [Figure 70.).
I
s.p
"
" %MOW
151
8 5 Form of the minor sixth
25 24 Form of the minor semitone
200 124 Form of the major sixth not
in lowest terms
40
5 3 In lowest terms
[Figure 70,J
While one may also make judgement, not only
of how much the minor sixth is exceeded by
the major, but that these [examples) are
major hexachords, and not major sixths as
the practical musician believes, this
situation only results because the minor
semitone is derived from the major tone.
[Figure 71.]
This [major toneJ is necessarily a comma
greater than what is obtained from the minor
tone. This, as I have promised, will be
seen more clearly in its place. It is not
my intention to omit this other consideration,
for as the minor sixth could
not occur between those pitches where
the tritone intervened, one likewise may
not have the major [sixth) between those
where the semidiapente could fall. You,
yourself, will clarify the cause of this
the moment you begin to investigate.
We come at last, however, to the examination
of the minor seventh, which our
practicing contemporaries say contains
in itself three major tones, one minor,
and two major semitones. We will consider,
initially, that this interval
is composed of the fifth and of the minor
third, because its form is constituted
in the superquartapartientequintas
between these numbers, 9:5. The sesquialtera
and the sesquiquinta added together
give the abovementioned dissonance.
Here is a clear example [Figure 72.J.
Warning.
152
3 2
6 5
The minor seventh
is not found between
these pitches.
Form of the fifth
Form of the minor third
8 10 Form of the minor seventh
not in lowest terms
9 5 In lowest terms
[Figure 72.]
The minor seventh has just been derived
in its true form. This, considered between
these pitches according to the
nature of the fifth which is found between
Dfsol re and a la mi re or in a
major sixth in low position and a major
semitone in high position, will not be
of the same proportion as the first.
The same occurs in the two lower
examples [Figures 73 and 74.J because
they have, like the other example, need
of the major tone, since they need to
become octaves.
153
I
2
"
f
"
[Figure 73.J
-Il
[Figure 74,J
On the contrary, a minor tonej sufficed
that [minor seventh] of the first example
[Figure 72.J. These matters, since they
are ignored by the practical musician,
cause in him, when they are discovered,
the stupor many times mentioned. You,
yourself, will clarify that they are not
minor sevenths by adding together the
lowest terms of their designated parts.
The product of these will give the
interval 16:9, which includes between its
extremes the super 7 partiente 9, the
true form of the minor heptachord of the
diatonic ditoniaion. Nevertheless, some
154
155
of our practicing contemporaries have
called the minor seventh of the syntonon,
which is contained, as you have seen, by
the superquartapartientequintas under such
a name, without having had regard for the
difference between the two and for the
confusion it was capable of generating in
the minds of men.
5 3 Form of the major sixth
16 15 Form of the major semitone
80 45 Form of the minor heptachord
not in lowest terms
5
16 9 In lowest terms
[Figure 75.]
This heptachord is a comma less than the
minor seventh, as anyone who takes the
trouble to subtract one from the other
will clearly see. One cannot even obtain
the minor seventh from the product which
will result from adding together two sesquitertias.
This will seem impossible
to the practical musician because he
prefers to figure the case between the
pitches of the first example [Figure 72.J
156
which was the same as the below-placed
example, but divided and considered
otherwise.
[22]
4 3 Form of the fourth
4 3 Form of the fourth
16 9 Form of the minor heptachord
[Figure 76.J
The minor heptachord has been obtained in
lowest terms by adding together two sesquitertias,
I proved earlier, in this
regard, that the minor seventh was contained
between the same pitches. I said,
besides, that when the parts of any interval
are added together after they have
been divided in any manner you wish, they
must necessarily, since they have been
added together, render the whole interval
in the same form and measure which first
existed. On the contrary, the simple
157
practical musician perceives in this last
example [Figure 76.J the opposite effect
from that which I first stated, He does
not see the reason for it, because he
considers that two fourths of the same
proportion and measure are contained
between the demonstrated pitches. They
are actually different. Thus, despite
his rule, he comes to err in the beginning
and is surprised because the calculation
does not prove favorable, as it should
do upon completion. This is because the
fourth which lies in high position falls
under this proportion, 27:20, as has been
proved earlier, When this has been added
to that which has been established by the
theorist, the result will be the minor
seventh in the form which I said had been
assigned to it in the beginning of our
discussion,
Strozzi: I think I am beginning to
perceive the port, Every time you make
me se clearly, without any other authority,
that the ditone, the semiditone, the major
158
hexachord and the minor hexachord are
dissonant--in addition to the diapentes
and the diatessarons which you have previousl
y called dissonant--I will believe
absolutely that what is sung--that is as
it is customarily done today--is in no
way the diatonic syntonon of Ptolemy.
Although such intervals agree in the
instruments and voices of the men of
our times, I still do not consider, however,
that they are different in effect
and nature from those of the ancients.
Bardi: Let us continue our voyage
now that the wind is propitious and the
waves calm, because, if it pleases the
Giver of all good things, I want you to
see clearly in your mind the truth of
each smallest detail, freed from any
authoritarian influence. We should not
delay any longer because we are already
many miles away from port. I shall now
reveal another abuse of the practical
musician, who considers the fifth and
the minor third of the following example
[Figure 77.J to occupy the same space as
159
the semidiapente and the major third
combined, Consequently, when the numbers
which contain their forms have been added
together, one set will give the same
result as the other. This, according to
what we will see now, is very far from
the truth,
64 45 Form of the semidiapente
5 4 Form of the major third
320 180 Form of the minor heptachord
not in lowest
20 terms
16 9 In lowest terms
[Figure 77.]
The minor heptachord has been produced by
adding together the two demonstrated intervals,
Previously, the minor sixth
was obtained by combining the fifth and
the minor third. Therefore, it is not
true that such intervals, when combined,
160
occupy the same space. The practical
musician is greatly amazed by this, which
happens, as I have said, only because he
wishes to derive everything from one
source, since he has no knowledge of the
intervals. In order to satisfy him, I
say that when he newly examines and
understands the size of the minor third
which the minor heptachord has in its
upper part--as I last showed you--he
will accept it. All wise men of common
opinion say that nature has not made
anything in vain, I want to prove to
you with this proposition that the
demonstrated heptachord does not lack
the proper form of the minor seventh of
the. syntonon, which is the superquartapartientequintas.
The procedure will
be this. Anyone with reasonable judgement
will know from what I have proved
up to now and said, that neither the
major tone nor the minor CtoneJ, and
also the major semitone, have ever been
altered to a form different from that
which was assigned to them in the
161
beginning of our discussion, according to
the opinion of the theorist, but only the
intervals of the octave and, later, the
major seventh. This also resulted because
he had not examined them. These [examples),
when they have been considered by us carefully,
will not surpass the others in
excellence in this case. We find, moreover,
two different kinds of minor sixths
from which, by means of two different
tones, one can compose the octave in its
true form. One can compose the octave, [23)
initially, from the minor heptachord
plus the sesquioctave (or tone). It can
also be composed from the minor seventh
plus the sesquinona, which will be
clearly understood by adding together
their lowest terms and noting the result.
The many lines drawn from the center to
the circumference of a circle all remain
in the center of the circle. In the same
way, each musical interval in the octave
looks as if into a mirror like the stars
also look into the sun. Thus, each of
these receives its being and perfection,
162
according to its capacity, in the same
way. Among the minor thirds, we have
two--since we do not wish to investigate
more deeply than in the other cases--
which are contained under different
proportions. The rest of these were
found later in order to complete the
octave. These were the two major sixths,
contained under different form. Among
the former, however, we enumerated the
semiditone and among the latter, the
major hexachord, intervals which are
common to the syntonon and to the
diatonic ditonaion, as many of the
others. We also found the major third
and the ditone, and shortly afterwards
the minor sixth and the minor hexachord,
since from these the octave could be
composed far from any imperfect extreme.
We found the diatessaron contained in
different ways and afterwards the diapente
was seen to be contained by
eight different numbers. This was done,
as we will see at the proper time, only
163
in order that the octave might be constructed
in one single form from these
combined. And all this happens because
courteous Nature wishes to show, in all
her operations, her immense wisdom and
liberality. Nature does not tolerate
a vacuum, nor does she do anything or
make anything otiose or in vain. There
remains to show you, in this regard,
how much the minor seventh exceeds the
major and minor sixth, which you can
easily see by subtracting them from it.
We will now hastily consider the major
seventh, and then pass on in order to
conclude the proposed material with
that of the octave. The masters of
this modern practice of counterpoint
say, therefore, that the major seventh
is that dissonant interval which is
contained in its lowest terms by the
proportion super 7 partiente 8 between
the numbers 15:8, and that it consists
of three major tones, two minor tones,
and a major semitone. This--for the
reasons many times said--we will
164
consider to be composed, initially, of
a fifth and of a major third, according
to what appears in the following example.
3 2 Form of the fifth
5 4 Form of the major third
15 8 Form of the major seventh
[Figure 78,J
One can also obtain such an interval from
numbers which contain the form of the
tritone when it is combined with those
of the fourth, but it still will not produce
the same proportion--although this
is contrary to the opinion of the practical
musician--as the minor tone and the
major sixth when they are added together
in the manner which is seen in the following
example [Figure 79.J.
165
5 3 Form of the major sixth
10 9 Form of the minor tone
50 27 Form of the super 23 partiente
27
[Figure 79.]
The super 23 partiente 27 has been obtained
from the product of the aforesaid
two intervals. It is a comma less than
the major seventh, as you will clearly
see by subtracting it from the major
seventh. One may also obtain the major
seventh from the product of the major
sixth and the major tone. We will see
now, by subtracting the minor seventh
from the form of the major seventh, how
much one is greater than the other. The
minor seventh, according to this example
[Figure 80.J, is thus exceeded by the
major [seventh] by a semitone. From this
166
one may comprehend that the major seventh
also may be composed of the minor seventh
and a minor semitone.
15 8 Form of the major seventh
X
9 5 Form of the minor seventh
75 72 Form of the minor semitone
not in lowest terms
3
25 24 In lowest terms
[Figure 80.]
Strozzi: Is not the form of the [24]
major heptachord still the super 23 partiente
27?
Bardi: No, Signor, because that
falls under the proportion super 15 partiente
128 within these limits, 243:128,
and exceeds the super 23 partiente 27 by
the quantity 6561:6400. I come, finally,
to deal with the octave, which our practicing
contemporary contrapuntists say
is that most perfect interval which
consists of three major tones, two minor
tones, and two minor semitones and which
is contained, besides, by the proportion
167
dupla between the numbers 2:1 when in
lowest terms. We will consider this
interval, initially, composed (as its
nearest parts) of a fifth and of a
fourth. We also will verify that the
consonance diapason is entirely capable
of containing these two intervals by
adding together the numbers of their
lowest terms.
p41
3 2 Form of the fifth
4 3 Form of the fourth
12 6 Form of the octave not in
lowest terms
6
2 1 In lowest terms
[Figure 81.)
We will also, derive the octave in such a
form from the product which is obtained
by adding together the numbers which
contain the major sixth and the minor
168
third, from summing the lowest terms which
contain the major third and the minor sixth,
and also from combining those of the minor
seventh and the minor tone. It follows,
necessarily, that those of the example
below, when considered as a minor seventh
and as a major tone of the proportion
which was shown in the beginning to have
been assigned by the theorist, are not
consonant octaves, but only dissonances,
which is contrary to the opinion of the
practical musician.
-MrU - U
- -Nor
-a-t,
[Figure 82.J
For fear that they may not be as I have
said, we will experience this (according
169
to this example), by adding together the
numbers of the proportions which contain
them.
9 5 Form of the minor seventh
9 8 Form of the major tone
81 40 Form of the duplasesqui-
octagesima
[Figure 83.]
The duplasesquioctagesima, which consists
of an octave plus a comma, is obtained
from this addition, as anyone will see
clearly who takes the trouble to subtract
the octave from it, or actually to place
a number between these limits, which are
in duple proportion with the minor, in
this manner, 81:80:40. The reason that
one may not obtain the octave from the
major tone and the minor seventh is that
the intervals which we have considered
as minor sevenths in those places do
not actually fall under the proportions
which they were assigned in the beginning
of our discussion--the one concerning the
opinion of the authors that what is sung
170
today is the syntonon of Ptolemy--but only
under the minor heptachord of the diatonic
ditoniaion. This, when added together with
the sesquioctave (or tone) will produce
the dupla in its true form, as the following
example makes clear.
16 9 Form of the minor heptachord
9 8 Form of the minor tone
144 72 Form of the octave not in
lowest terms
72
2 1 In lowest terms
[Figure 84.]
The octave also may be obtained in its
true proportion from the result of adding
together the terms of the forms which
contain the major seventh and the form of
the major semitone. Thus, similarly, the
octave may be obtained from that of the
tritone and the semidiapente. Finally,
in order to know how much the octave
exceeds the major and minor seventh, one
will see this by subtracting one [seventh]
from it and then the other.
171
Strozzi: I believe that your dis- [25]
course has led me by now into a safe
port so that I no longer fear the blast
of unfriendly winds. It has so removed
from my intellect the thick fog of
ignorance that, without other reasons
and examples to influence me, I am almost
certain that in the singing of today
none of the inconveniences mentioned are
present. Nevertheless, I am also sure
that when tuning the string of an instrument
according to the demonstrated forms,
it is necessary for each one of the
demonstrated cases to be found between
these strings. From this I argue that
the intense syntonon of Ptolemy is
probably neither sung nor played.
Bardi: You are beginning to discuss
and understand it very well, but in order
better to remove each difficulty which
may recur concerning that, I want to
clarify for you two other most important
topics, and then we shall see clearly
what genus and species is sung and played
today, Therefore, I say first, that
172
between the notes of the example placed
below there are none distant from each
other by a minor semitone of the proportion
which I assigned, according to
the opinion of the theorist, in the
beginning of our discussion.
[Figure 85.]
There is also no lack of those who have
vainly doubted concerning the major
Esemitone), but they are greatly deceived.
The reason that they are not what I have
said only results because both of them
together cannot fill a space greater than
that which contains the minor tone, as
was proved in the beginning. However,
both will easily be found between the
pitches of this other example [Figure 86.),
since the place is capable of containing
173
sufficient numbers to fill the spaces
without anything lacking or anything
being left over. Therefore, since we
do not wish to proceed haphazardly and
without any reason, it will be necessary
to find new measure and form for both
those first ones, or at least for one
of them, which is not only suited to
fill the interval of the major tone, but
is actually that which the ancient musicians
had under consideration, provided
such consideration was necessary.
Since, however, there is no record or
particular example of such things being
done, these doubters, as many others,
can only be persuaded with greatest
difficulty and with powerful reasons.
One should not be dismayed by this and
permit them to resort to such plausible
and apparent means, even when there are
no others, in order that they may learn
the truth of the matter.
174
(Figure 86.]
I say--according to my ability, however--
that the ancient musicians did not observe
any difference between the major and minor
tone, unless it was the minor semitone.
This minor semitone, according to what I
indicated to you, is contained by the
super 7 partiente 128 between these numbers
in its root terms, 135:128, and this
interval consists of a minor semitone plus
a comma, as has been said previously. The
major semitone is the same in every tone.
In order that this truth also is seen
clearly in the diatonic syntonic monochord
of Ptolemy between the pitch tritesynemmenon
and the paramese, I confirm it to you, without
running afoul of authority, in yet
another manner in addition to that which I
Minor semitone
of another proportion
than
the first one
shown.
.r -4e-r
-WI--
175
have told you previously. If we wish to
obtain the fifth from the semidiapente or
the fourth from the tritone, we cannot do
so unless we augment the former and diminish
the latter by a minor semitone in
proportion to the location where it is
added and where it is subtracted, which,
as you know, is the major tone. One sees
this in the following examples [Figures
87 and 88.].
64 45 Form of the semidiapente
135 128 Form of the minor semitone
of the major tone
8640 5760 Form of the fifth not
in lowest terms
2880
3 2 In lowest terms
[Figure 87.J
45 32 Form of the tritone
x
135 128 Form of the minor semitone
of the major tone
5760 4320 Form of the fourth not
in lowest terms
1441
4 3 In lowest terms
[Figure 88.]
Or else, thus, which is the same that is
seen above in proving to you how much the
tritone is greater than the fourth:
45 32 Form of the tritone
x
4 3 Form of the fourth
135 128 Form of the minor semitone
of the major tone
[Figure 89.]
And [to confirm] that this minor semitone,
together with the major, has power to fill
entirely the void of the major tone, one
adds together the numbers of their lowest
Y
!
"
%nor
176
[26]
177
terms, so that the result one will have
from them will be certain. In addition,
if, following this example [Figure 90.J,
we wish [the interval from a la mi re to
h mi to be a major tone, as it actually is,
and if we also wish this other [interval]
to be the same distance, it is necessary
to concede a minor semitone of the
demonstrated proportion to the major tone,
provided that the place from which it is
subtracted is, like the former, capable
of containing it.
[Figure 90.J
_[ I
[Figure 91.]
178
Let us also consider that the minor semitone
which is between b fa and h mi in
the usual proportion not only will not
fill the void of the major tone, but that
note signed as an accidental on the note
h mi of the first example [Figure 90.]
will be a comma higher than that of the
second [Figure 91.] since h mi is higher
than a la mi re by a major tone. This
has been said many times. If, in that
last example [Figure 91.], one wishes in
departing from the pitch a la mi re to
ascend two semitones to h mi, one may
well consider that the distance which
is found from this a la mi re to b fa
is that of the major semitone. However,
that space which remains between b fa
and h mi is necessarily less than the
proportion which contains the major
tone, which is 135:128. From here,
perhaps, some of our contemporary contrapuntists
have had occasion to place
the b quadratum [L1J on f fa ut when it
has had to sound a fifth with h mi, and
the diesis x when it has had to form a
179
major third with D sol re or its replicate
[an octave higher) with d la sol re. This
constitutes a silent argument that the
fifth would not become "Just" with the
diesis X as with h quadratum, but a little
flat and submissive. Now, please turn
your attention from those matters to only
these two items which I wish to tell you
in addition to many others I know but omit
for brevity.
[Figure 92.]
[Figure 93.]
If between these notes [Figure 92.] is found
a major third, according to what we have
demonstrated previously, and similarly,
Whether the diesisX
or the h quadratum
should be placed
on ' fa ut.
180
a minor semitone is found between these
other notes [Figure 93.), which also
affirm it, it necessarily follows, because
of what was proved earlier, that
the extremes of the two thirds, when
sung in the manner they describe and
believe--and which actually is proper--
sounds with a very "just" fifth, since
it is contained by the sesquialtera
proportion. Therefore, if the h quadratum
has the power they say it has to
make the note on which it is placed
higher than diesis X can, the fifth,
as a result, will become larger than
its normal state. Similarly, the fourth
above it will become smaller by the
same amount the h quadratum exceeded
the diesis X in power to augment the
fifth. That appears, however, to be
not exactly true. It is quite necessary,
if one wishes the major third, its
replicate, and also the fifth to be in
their true forms and proportions, to
have a minor semitone of the proportion
181
and size which I have just now described,
but taken, however, from a place which is
capable of containing both the major semitone
and this [minor semitoneJ. Otherwise,
any actions and labors you might undertake
would be in vain, because the power to
put each one of these consonances in its
perfect genera consists principally in
the placement rather than in the capacity
of the place, and not in the difference
of the numbers, as those men believe. The
h durum was not used as a symbol unless
it was introduced after the b molle
appeared as a mode. The diesis X was
established sometime prior to both, however,
speaking always in terms of the modern
practice. Together with the b molle, it
was suitable with its [power toJ increase
and diminish to render the imperfect consonances
and the dissonant intervals in
that form [i.e. its perfect generaJ and
to bring them into the highest state of
perfection which could be desired from
their nature. At least they have been
182
introduced for such a reason, although
badly used. If I do not deceive myself,
the b molle was invented primarily as a
symbol by modern practical musicians or,
to say it better, put into use for no
other reason than because the note F fa
ut, like the others, also had a fifth in
the low register and a fourth in the
high and that of G sol re ut had a major
sixth in the low and the minor third in
the high. Also, the diesis X was introduced
in both F fa uts so that the note
h mi sounded with the high one as a
fifth and not as a semidiapente and with
the low one as a perfect fourth and not
as a "hard" fourths In addition, it made E27]
the intervals of major and minor sixth
with both d and D. After the b molle
had been placed in use, the h durum came
into consideration as a symbol by good,
modern professors of music for no other
purpose than to be able, primarily, to
transform one system into another, according
to which pleased them and was
most convenient. However, they never
183
placed this h on any note other than its
usual one. In the others, when such a
need arose, Ethey placedJ the diesis X,
which finally had been perfected in its
demonstrated form. One sees this very
well observed in their works, and if
someone answers me that it is still the
custom for everyone commonly to place the
b molle on the note a la mi re, and on
that of the e la mi which is farther
removed, I will respond that this results
from the small number of characters
which Ewe have] today in comparison with
the ancients, who had for each note an
appropriate sign to demonstrate whatever
desirable and convenient sound they
wished. But it is not, however, such a
scarcity of symbols in our times--concerning
demonstrating the difference of
the notes--which causes any deficiency or
imperfection in regard to the purpose for
which the music of today is used. If
that were so, some would have been
designed before now which would have
remedied their defect and deficiency.
184
But this has not occurred since those
Symbols) were sufficient to give any
note in the low and high registers the
form of every consonant interval which
man has been known to devise up to now,
and particularly in the two first genera
of harmony, which are the diatonic and
the chromatic. In the enharmonic, then,
the present sign, x, has been uselessly
added--because it has not been put into
practice by anyone--in order to be able
to divide the major semitone into two
parts, called enharmonic dieses by the
ancients. The moderns, however, understand
for this diesis the present symbol,
)(, as you know. This, in their opinion,
is worth two of the former. This is not
exactly true, because the ancients used
to divide the limma, or minor semitone,
of the diatonic into two enharmonic
dieses and the moderns say to divide the
sesquiquindecima, called by them major
semitone. What difference there is
between them has already been demonstrated.
185
What we have said could be restated that
there is the same distance between h mi
and b fa through h quadratum as there is
through b rotundum. In the same manner,
the identical distance is found between
c sol fa ut and h mi through h quadratum
as there is through b rotundum. And I
say, in conclusion, that d la sol re is
sharper through h quadratum than it is
through b rotundum.
Strozzi: Clarify this last topic
for me,
Bardi: I say that d la sol re is
sharper through h durum than through
b molle in comparison with this. If we
move away from a la mi re, which is a
fixed note and common to both systems,
and we want to ascend by conjunct diatonic
steps to d la sol re, we will find
by means of the disjunct system--that is
to say, through h durum--that there are
two major tones and a major semitone.
By means of the conjunct system--that isthrough
b molle--there is a major tone,
a minor tone, and a major semitone.
186
Therefore, using the conjunct system, d la
sol re necessarily becomes a comma lower
than in using the disjunct system, as the
following example shows.
Major Major Major
tone semitone tone
Major Major Minor
semitone tone tone
[Example 2--A comparison of the conjunct
and disjunct tetrachords.)
Strozsi: I have understood very well
and it is actually so.
Bardi: Because there is the same
distance from c sol fa:ut to h mi through
h durum as through b molle, it follows that
the major semitone of each tone is always
of the same quantity and proportion in one
as in the other. For the same reason,
there is the same interval from h mi to
b fa through b quadratum as there is through
b rotundum.
CThe note] d la
sol re is sharper
through h quadratum
than through b
rotundum.
187
Strossi: If it is not too much trouble,
please tell me why this 16:15, more than any
other proportion, is that which contains the
major semitone; why this 25:24 contains the
minor semitone more than another proportion;
why this 65:45 more than another proportion
contains the semidiapente; and finally, why
this other proportion 45:32, more than
another, contains the tritone? Why also,
can they not be contained by other numbers,
according to the syntonon of Ptolemy?
Bardi: I believed that I had satisfied
you with what I had told you before, but I
see that this is not so. However, I am
still under obligation to you and I want
you yourself to be the judge whether they
can be otherwise. With this, however, you
must concede to me that the true forms of
the minor sixth, of the major and the minor
third, of the fourth and of the sesquioctave
tone are actually those which were assigned,
according to the opinion of the theorist,
in the beginning of our discussion. I offer,
then, at the necessary time and place, to
188
make you clearly see and hear everything
else that you desire by means of the
monochord.
Strosi: I concede you everything
for now.
Bardi: First, I ask you how much
greater the major third is than the minor?
Strozsi: A minor semitone.
Bardi: In that way, when subtracting
the minor third from the major, those numbers
which remain necessarily contain the
minor semitone in its true form; is it
not so?
Strossi: It certainly is.
Bardi: Now we come once again with
this example to the actual experience.
5 4 Form of the major third [28J
X
6 5 Form of the minor third
25 24 Form of the minor semitone
(Figure 94.J
Here, as you see, since we have subtracted
the sesquiquinta from the sesquiquarta, the
189
remainder is the sesquivicesimusquartas,
assigned us as the form of the minor semitone.
The verification of this fact would
be adding it to the sesquiquinta and seeing
if between the two of them they gave the
form of the sesquiquarta. Since we saw
that on another occasion, however, it. would
be foolish to do it [again]. Therefore, I
ask you another question. How much greater
is the minor sixth than the semidiapente?
Strozzi: By a sesquioctave tone.
Bardi: Therefore, in subtracting one
interval from the other, must the amount
which is left consequently contain the
semidiapente?
Strozzi: Yes, it must.
Bardi: Therefore, here is a very
clear example.
8 5 Form of the minor sixth The semidiapente
is contained
within these
9 8 Form of the major tone numbers, 64:45.
64 56 Form of the semidiapente
[Figure 95.J
190
In this, the semidiapente is in its usual
lowest terms.
Strozzi: I am very much satisfied
for the present. Continue, however, and
tell me the rest.
Bardi: I ask you, in addition, how
much greater the fifth is than the tritone?
Strozzi: A major semitone.
Bardi: So that when someone subtracts
the major semitone from the fifth, will
what remains necessarily be the tritone?
Strozzi: That would necessarily
follow.
Bardi: We come, therefore, to the
example.
3 2 Form of the fifth Why the tritone
is within these
numbers, 45:32.
16 15 Form of the major semitone
45 32 Form of the tritone
[Figure 96.]
Finally, I ask you how much greater the
fourth is than the major third?
Strozzi: A major semitone.
191
Bardi: Thus, when a major third is
subtracted from the fourth, those numbers
which remain will actually contain the
major semitone in its true form, is it
not so?
Stroszi: This is true.
Bardi: We come, therefore, to demonstrate
the truth of the matter with this
example.
4 3 Form of the fourth Why the major
semitone is
within these
5 4 Form of the major third numbers, 16:15.
16 15 Form of the major semitone
[Figure 97.]
Here is the major semitone in its lowest
terms.
Strozzi: I am entirely satisfied.
Since, however, everything you have told
me and proved in many ways about the forms
of the intervals is true, what was it that
induced Plutarch42 to say in his "Table
42Plutarch, Table Talk, trans. by Paul
A. Clement and Hebetr " offleit, Vol. I
of Moralia, 16 vols. (Cambridge, MassachusetEl
and London, 1969), pp. 266-269. "Now
192
Talk" that three and one are the limits
of the diatessaron? This is such a simple
thing and yet so far from the truth.
just as the experts in the musical theory
of the lyre assert that among ratios that
of 3:2 gives the concord of the fifth,
2:1 the concord of the octave, and the
concord of the fourth (which is weakest)
consists in the ratio 4:3; so the musicologists
of Dionysus observed three concords
of wine and water, fifth, third, and
fourth, for in their song they say this:
Drink five or three, not four.
'Five,' indeed, is in the ratio 3:2,
three parts of water being mixed with two
parts of wine; 'three' is in the ratio
2:1, two parts of water being mixed with
one of wine; and four,--three parts of
water being poured into one of wine, this
is a ratio of 4:3, a drink for some group
of sensible magistrates in the prytaneion,
or logicians, their brows contracted as
they meditate upon syllogistic conversions,
a sober and a feeble mixture. Of the two
others, the mixture with ratio 2:1 brings
on that disturbing and half-drunk pitch
of intoxication
that plays upon
The inviolate strings of the mind,
for neither does it allow sobriety nor
does it completely immerse the foolish
man in strong drink. But the mixture
with a ratio of 2:3 is most harmonious,
a complete inducer of sleep and relaxer
of care, a 'protecting and soothing
governess,' in Hesiod's phrase, because
it creates a profound calm and quiet
among our lordly and disordered passions."
193
Bardi: As far as I am concerned,
Plutarch wishes, in that place, to be
considered as a drunkard and good companion
rather than as a serious mathematician.
He wishes to be considered in
keeping with the musical consonances,
and not exactly according to the proportions
of the quantity of numbers.
Above all, he intends it as a pleasant
topic for dinner conversation, which
has the same effect but not the same
result. Anyone who logically reads
the whole of that question easily
ascertains this through the words of
the writer himself. The consonance
diatessaron, therefore, as the name
implies, is contained by four notes,
which makes it the sesquitertia. Although
it is numbered among the perfects,
it is nevertheless the least perfect of
all and is more distant from the nature
of the unison than the sesquialtera and
the diapason. It is the least certain
of all to the sense, and the least
194
delightful, and if any small amount of
it is diminished, it becomes imperfect
and dissonant. This, however, is
according to the usage of the ancient
musicians. Thus, in watering the wine,
adding three out of four parts of water
is almost the least consonance and accord
which is pleasing to the taste because
now, as then, in diluting the wine almost
to the color of water, and causing it to
lose all its strength, it becomes no wine
at all. This is not agreeable to people
who are together for an enjoyable time at
the table, but for serious men who need
to remain somewhat sober while attending [29]
to some [affair] where they are the
judges and the disputants, or similar
persons. Therefore, the proverb of the
joyful men did not mean (says the
writer) that the four [parts of wine)
interfered with their task in any way.
That is one of the basic premises of the
sesquitertia, which was the last and the
most remote consonance of wine. Thus,
195
it was the sesquitertia and diatessaron
of their drinking. But it was not so
delightful to the taste, since they
were seeking the good disposition which
has as its goal the ;oy and almost the
recreation of the spirits without danger
of drunkenness, which, perhaps, can
persist long after leaving the table from
the diapason. And yet, the disputant
exalted greatly the diapente, considering
in it, as well as in the diapason, the
terms of its forms added together and
alone greater than the diatessaron. I
believe that I need to interpret the
place thus, since I want to follow the
intention of the writer and not consider
it as one compared to the three-faced
sesquitertia, because this, clearly,
is impossible. Therefore, the intention
of Plutarch in that place is to consider
only the larger terms of each proportion
of the three simple, prime consonances.
When one single unit representing wine
has been taken away, it means that the
196
others which remain are the parts of
water. Now, because the larger term of
the sesquitertia is four, from which--as
has been said--one has been subtracted
as the part of wine, that which remains
as the part of water is--as you know--
three. And therefore, the author proceeds
in regard to the sesquitertia,
comparing the one to the three. It can
also be said thus, according to what
you have heard: three out of four parts
of water. And thus we come to make
mention of proper limits for this
sesquitertia. This mode of comparison
used in the other consonances is easily
remembered. Therefore, tempering the
wine according to the sesquitertia, they
should consist--as has been said--of
three parts of water and one of wine.
This mixture makes the one who drinks
it become like its nature, which is
languid and mild. Next, according to
the sesquialtera, Eits proper limitsJ
ought to consist of three parts of
Different ways
to temper wine
with water
accord ing to
the usage of ,he
ancient musician.
197
water and one of wine. This, drunk often
in this proportion, makes the drinker
become cheerful and thus [he reflects]
the nature of the diapente. Finally,
according to the dupla, the proper limits
are in two parts, one of water and one of
wine. When this potion is drunk thus,
it makes the man become like the nature
of the diapason, which is most joyful.
But one must consider the place, the time,
the persons, and the quality of the wine
that is drunk where these effects were
caused, if one wishes to unite the strings
of this cithara perfectly. The ancient
speculators of musical proportions had
other, different methods of tempering
wine with water, but enough has been said
about it.4 Returning now to my former
topic, I say that you yourself, if you
have understood all I have discussed with
you, may discover where the proportion of
any musical interval is obtained, and not
43See Zarlino, So limenti musical,
pp. 106-110, where an en re c apter is
devoted to the refutation of Galilei's
interpretation of Plutarch's passage concerning
the tempering of wine with water.
198
only this, but [you may also clarify for
yourselfJ any other problem which may
occur to you concerning the difference in
size which exists between one interval and
another in any genus and species of harmony.
This is as much as is necessary for me to
tell you and prove to you concerning the
principal topic of the problem proposed to
me, but tell me frankly how you feel about
that.
Strozzi: I am more confused than ever
by your explanation--now that I expect
confusion less--concerning, however, the
knowledge of which species is the one which
they sing today. This is the main topic of
the discussion we have had up to now.
Bardi: What is your new difficulty?
Strozzi: Listen. If any one of our Doubt.
practicing contemporary contrapuntists
chances to use in any of his compositions
the tone and semitone in any pitch without
being, for all of that, capable of any
of the demonstrated considerations, I cannot
imagine how it can happen that so many
discrepancies are not manifested to the
199
cultivated ear. You have indeed proved to
me with convincing arguments that these
irregularities must be occurring frequently.
And if this does not happen, a still greater
inconvenience will follow. This will be
that many of the things proposed to us by
the theorist as a rule would fall under the
two already mentioned concerning the position
and difference of the location and value of
the tone and the ma;or, minor, and medial
semitone. On these two topics--as I understand
it, however--has been based, principally,
all that which you have discussed
with me up to now, unless we wish to say
that the quantity of the comma, since it
is so small, when subtracted from or added
to any interval, does not have the power
to alter the nature of its original state.
I do not believe this in any way. M. Gioseffo
Zarlino44 claims, in particular, In chapter 43
of the 2nd
that half a comma has power, when added (part of his
Ifotietini
to or taken from any consonant interval, h~amic e.
131.
See Zarlino, Istitutioni, pp. 128-
200
of making it dissonant, although he later
adds, for a joke, I believe, that the consideration
of the difference of major and
minor tones must be omitted.4 5 This consideration
when taken away bears with it
that of the various species of semitones.
When this sole speculation is carried to
the diatonic which is sung today--granting
that it is the syntonon of Ptolemy--
it becomes a different consideration
through which what is granted is perhaps
true.
Bardi: You discussed it very well
and have the greatest reason to doubt,
but here is the solution of the problem.
If the genus in which we have considered
each of its intervals with such exactness
is the syntonon of Ptolemy, I say that
the one in which they compose, sing, and
play today is not the same at all. It
In chapter 13 of
the 3rd [part]
of the same
Institutioni.
Solution of
the problem.
45Gioseffo Zarlino, The Art of Counteroint,
translated by Guy Ti~rco andl aude
Vlisca (New Haven and London, 1968), p.
29. "We are ignoring in this (the diapente]
and in the other consonances the distinction
between the large and small whole tone,
which, if taken into account, produce still
other species."
201
cannot even be the ancient diatonic
ditoniaion.
Strozzi: Are both sects therefore
deceived--so to speak--in this topic of
so much importance? Tell me this, please!
Bardi: They are fooled, certainly,
but listen! In order to clarify for you
the present status of the matter, I will
reveal to you simultaneously, using new,
authoritative examples, what species is
the one which is played on keyboard instruments,
which species is played on the lute,
and, finally, which of these approaches
nearer to perfection and why. This will
neither be useless to you nor irrelevant.
Strozzi: These will be among the
most valuable things that you could possibly
clarify in this matter.
Bardi: I find through long observation
that natural voices and instruments
made by art do not actually play
or sing in this modern music practice
any of the nine ancient diatonic species
in their simplicity. Our practicing contemporaries
inadvertently use only three
[30)
Observations
of the author.
202
of them today, mixed together in different
ways. These are the intense [diatonic] of
Aristoxenus, the ancient diatonic ditoniaion,
and the Ediatonic] syntonon of Ptolemy.
Among stringed instruments, I. consider The species of
harmony which
that the viola d'arco, the lute, and the stringed instruments
play.
fretted lyre play the intense diatonic of
Aristoxenus. Hearing and seeing in these
instruments the uniformity of tones
equally divided into two identical semitones
impels me to believe that this is
true, for the aforementioned intense [diatonic)
of Aristoxenus was distributed in
such a way, as you will understand in due
course. However, the organ, the gravi- The species which
the organ and
cembalo, and the modern harp--modern only other keyboard
instruments play,
in terms of the newly added strings and
not concerning the instrument in its
original form, which I believe is very
ancient--more or less detach themselves
from those [other speciesJ. In the division
of tones, for example, the tones
are separated into two unequal semitones.
The wind instruments like direct flutes, The species which
wind instruments
transverse flutes, cornets, and other play.
203
similar ones have power, due to the distribution
of their holes and aided by the fine
facility of their wise, expert players, to
adjust to one [species) or another according
to the need and their [player's] wishes.
This is equally true of voices, but only
when they are not willing to go against
their nature and adapt to those [species).
Moreover, concerning the composing and
singing of today, I am persuaded, because
of what I told you and am about to tell
you, that the diatonic ditoniaion is mixed
with the syntonon of Ptolemy. These are
the causes which impel me to believe this.
It is certain that if the plain syntonon
is sung, then the tones and the minor semitone--in
accordance with what I have proved
is their nature in this species--must be
unequal and different in size. Because
of this inequality, the following variety
of intervals would be sung, as you have
seen according to the different proportions:
two kinds of fifths; two kinds of fourths;
three, perhaps four, kinds of minor thirds,
The species one
composes and
sings today.
204
and as many kinds of major sixths; at least
two species of minor sixths; and two species
of major thirds. There would be the same
number of dissonances [i.e. two) and finally
[that many) octaves. The netesynemmenon is
sharper than the paranete diezeugmenon by
an amount you know. In addition, the semitone
which is found between b fa and h mi,
and all those [intervals) which are derived
as minor [semitones) from the major tone
are in a different proportion, which is
neither the sesquiquindecima [i.e. 16:15]
nor the sesquivicesimusquartas [i.e. 25:24).
We have not only told of these things on
another occasion, but one finds also--as
I realized--that we have been warned of
some of them by masters of this art.
However, there is no one at all who, in
singing these many airs together which
were introduced a hundred and fifty years
ago, has ever heard or hears now such a
confused diversity of intervals, because,
actually, [such intervals] never happened
then, nor do they happen today. This seems
to me a strong argument to persuade us that
this is true. And in order to be better
acquainted with the variability of the
intervals of this distribution, would
you tell a few things to those who believe
that Cthis distribution) is what is sung
today? Tell them to divide, using any
method they please, the ma;or thirteenth,
which is contained according to the
syntonon by the numbers 10:3, into three
sesquialteras, which are what they say
it contains. Tell them also, according
to the example which follows EExample 3.],
to divide into three sesquitertias the
dupla superbipartientequintas, form of
the minor tenth. Then, ask them how much
the second interval is exceeded by the
first. It would not be very difficult
to persuade us that the diatonic genus
which is sung today is actually the
ditoniaion--I mean the ancient diatonic--
as many have believed and still believe
today. In the first place, its ditone--
as I have said--contains two sesquioctave
tones. This interval, accompanied or
E31]
Inconvenience
which would
arise from
singing the
syntonon of
Ptolemy.
205
206
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207
alone, is dissonant. Its semiditone is of
the same nature and so is the major and
minor hexachord. In addition, the first
and lowest interval of each tetrachord of
the said species is a minor semitone (or
limma) and between the tritesynemmenon
and the paramese is the major [semitoneJ
called apotome. On the contrary, in what
is sung today, the first and lowest interval
of the tetrachord is major, which
is not what is found between the paramese
and the tritesynemmenon. One finds also
in the diatonic that the tritone exceeds
the semidiapente and, on the contrary,
in what is sung today the semidiapente
exceeds the tritone. Now, just because
none of the above-mentioned particulars
are found in our diatonic--they are even
contrary to these in a certain way--it
does not necessarily follow that this
[diatonic] is in any way that Ewhich is
sung). The diatonic of today agrees well
in some respects with the syntonon of
Ptolemy, as I am about to explain.
Initially, I do not believe I would err
208
in saying that the imperfect consonances--
omitting the consideration of dissonances
for now--almost fall under the proportions
of the diatonic ditoniaion, but I am still
not of the opinion that they are connected
with parts similar to it. For example, I
believe that the major third is contained
by an irrational proportion quite near the
sesquiquarta, but not to the point where
its sides--so to speak--are the sesquioctave
tone and the sesquinona, but only
to that point where they are two equal
parts of the said third. This is exactly
the way the third is divided in the manner
of the tetrachords of Aristoxenus, but it
is not precisely the same thing. Moreover,
I believe that the minor third is composed
of a tone of the same proportion as those
of the major and of another interval somewhat
larger than the sesquiquindecima
[16:15J. The other intervals are all
composed in such a way and of such parts.
I maintain, therefore, that none of the
other intervals except the octave is contained
in its assigned proportion; it is
The octave is
the only interval
sung today
in its true
proportion.
209
understood that I am referring to the
manner which is commonly used today in
singing. We will shortly demonstrate
this clearly. There is another point
of agreement between them, that the
tones in whose extremes our practicing
contemporary contrapuntists place the
diesis)X and the b molle as accidentals,
are minor [intervals] among the notes of
the syntonon. This tends to warn us,
not without reason, that if we used such
signs on major tones, some of these
semitones would become different in
proportion, since they would be derived
from the entirety of such a major tone.
Our diatonic also has some conformity
with the ditoniaion since each of their
intervals consistently is found in the
same measure and proportion, although
they [all] have diverse forms. In
addition, the diatessaron is found four
or five times in each species of diapason,
and the diapente at least three
or four. I do not know if this situation
can occur in another species or genus as
210
those of Aristoxenus, for instance, where
major is plentiful, since they do not
necessarily have either the tritone or
the semidiapente as all the other string
distributions do. Thus the perfect consonances
in the mode which they are sung
today happen to have greater affinity
with the diatonic ditoniaion, and the
imperfect consonances more closely relate
to the syntonon of Ptolemy, rather to
that of Dydimus--as you will learn--
but always with the same proportion and
equality of tones. One could add to this
and say that [DydimusJ agreed with neither
the syntonon nor with the diatonic, since
neither the apotome nor the super 7 partiente
128 were ever found in use in
the ditoniaion, nor the sesquivicesimusquartas
[25:24] in the syntonon, but only
the first two as a related quantity in
comparing the tritesynemmenon of the
conjunct system to the paramese of the
disjunct [system]. In addition, no interval
of any kind except the octave
211
falls--as I have said--under the proportion
and measure of the first or second species.
This applies, however, when they are sung
in the manner which is customary today.
Stroszi: Too many important things
at one time, Signor Giovanni! My mind is
so slow that I cannot follow your lofty
conceptions with the speed and promptness
with which you explain them. Therefore,
you need to slacken the pace, if you wish
me to keep up with you. However, I
clearly know, thanks to your most subtle,
ingenious, enthusiastic, true reasons
that the genera in which one sings today
in the diatonic species is not simply
the diatoniaion or the syntonon, but a
third thing mixed and composed of both.
Nevertheless--because of the little
-experience which I have in these things--
new difficulties continually occur to me
(ones of no little importance) in understanding
well what you so greatly comprehend.
However, do not let it be a
burden to you to answer me whatever is
212
necessary for me to ask you for greater
understanding of this important affair.
Bardi: Speak freely, as always.
Strozzi: In the first place, I do
not know in what manner Aristoxenus distributed
the strings of his tetrachords,
nor do I know what difference exists between
the division of the lute and that
of the keyboard instrument and consequently,
which of these approaches nearer
to perfection. I do not even know why
the thirds and the sixths of the diatonic
are dissonances or why the syntonon is by
Dydimus, rather than by Ptolemy, or even
less why the two sects were led to believe
a thing so far from the truth. I cannot
fathom how it is possible to change the
diatonic singing of the ancients into
our manner, which is so different with [32]
regard to the quantity and size of the
intervals. Finally, [I do not understand]
how, of [all] the intervals which are sung
today, there are not any, from the octave
on, which are contained in the proportions
assigned them by the theorist.
213
Bardi: In order to take away all the
difficulties you have mentioned, this is
what we will do. Let us approach a keyboard
instrument and, first of all, you
must tune its fifths as excellently as you
possibly can, but in the order which I tell
you.
Strozzi: Will it not be easy for the
ear to be deceived?
Bardi: Very easily, as much as any
other sense, and more so than [the ear]
which is unaccustomed to such speculations.
Nevertheless, the discriminating ear of one
who is well versed, accompanied also by
natural judgement and by some good standard,
is not so easily deceived. On the contrary,
there are some Ewith earsJ so perfect,
accompanied by other circumstances, who
understand each difference, however minimal,
in an instant. However, let us come to the
actual practice without undue verbosity.
First, tune A re and a la mi re in octaves.
Tune above this the octave higher than A
la mi re. Now loosen the D sol re which
is below Aa la mi re, so that it sounds
Method of ascertaining
what
discriminating
hearing is.
214
with it as a fifth in the most perfect way
desirable. Tune in the same perfection
that which is above the same a la mi re,
which will be e la mi and thus, similarly,
Tune] d la sol re below Aa la mi re.
Extract also in the same perfection the
interval which E la mi makes above A re.
We want to see now if the sense [of hearing]
has been deceived. Here is the way. Among
the consonances, there are none better
understood by the ear and [there is no
place] where it can be deceived less than
in the octave. Therefore, let us see if
the notes D sol re and E la mi respond as
a diapason with their higher counterparts.
If they are perfectly in tune in this way,
it will be evident that the ear has not
been deceived at all in the temperament
and perfection of the fifths.
Strozzi: They are excellently in tune.
Bardi: Proceed by tuning the others
according to your experience with these
[fifths).
Strozzi: It is done.
Bardi: Now play.
Strozzi: This is music which actually
would make the meek angry to hear
46
it. The manner which Timotheus used
to anger Alexander the Great and stir
him to arms must not have been any different.
Although I consider that those
imperfect (intervals) which are already
consonances are actually dissonances, I
am unable to convince myself by this
where that originates, or where the
strings are stretched according to the
species diatonic ditoniaion.
Bardi: Do you remember between
which strings the diatonic had the major
and minor semitones?
Strozzi: Yes, Signor.
Bardi: Please tell me.
4 6 Timotheus of Miletus, the most
infamous of the ancient musicians due to
his several innovations, is mentioned by
numerous ancient authorities. See Bower,
Boethius, pp. 36-39; Plutarch, On Music,
iTed by Benedict Einarson anT~illip
H. DeLacy (Cambridge, Massachusetts and
London, 1967), pp. 381, 419-423, 426; and
p. 547 below.
Effects of
the temperament
of the
ditoniaion.
215
216
Strozzi: The minor semitone in the
diatonic is found---according to what you
have told me--in each tetrachord between
the two lowest strings, and the major
semitone is found between b fa and h mi.
Bardi: Therefore, consider in such
a distribution how small the one that
formerly was major has become. On the
contrary, consider how much larger the
one that was minor [has become).
Strozzi: You are quite right. Has
this happened, perhaps, because the tones
have acquired that quantity which has
been taken away from each octave by two
major semitones?
Bardi: Actually, yes. The divine
Pythagoras could hear the ditone, the
semitone and both hexachords in this way
and no other, since this diatonic distribution
alone was in use at the time.
Those who said, after him, that these
intervals were dissonant were induced
[to say thisJ for the very same reason,
47 Zarlino, Dimostrationi, pp. 2-6.
Zarlino4 in
the preface to
his Dimostrationi.
217
although some have dreamed that [such a
thing] could happen. It is because they
have not heard [these intervals] in their
true, legitimate places--which, according
to these men, are above the octave--as if
in those times they had not had knowledge
of that quantity of strings which was
necessary to do that, or as if within the
octave, they did not make, when well
placed, a pleasanter sound than they
make outside the octave. It is also not
exactly surprising, since Pythagoras
heard those [intervals] which are contained
in the syntonon and did not value
them, that the musicians of those times
did not use them in their songs. Pythagoras,
due to the inconstancy [of things
like these), refuted and banished from
the world everything mixed, impure and
variable, since he knew that in them
caprice and rashness ruled. But like the
bee, when he gathers the purest part of
the celestial dew is fed and nourished
by it, thus the divine philosopher
What things were
valued and not
valued by Pythagoras
and why.
218
likewise tolerated among his effects the
flower of the simplest and noblest things.
Strozzi: Before we dispense with
our instrument, since it is already clear
that in the ordinary distribution the
fifths become flatter and, on the other
hand, the fourths become sharper by a
little more than that which would suit
their proportions, I desire to know why
such adjustments are necessary and if,
on the contrary, the fourths could become
diminished and the fifths augmented
if the strings of this instrument could
be suitably arranged? If this is not
possible because of some inconvenience,
demonstrate it and then tell me the
reason. Show me, in addition, how much
the fifths become diminished and the
fourths become augmented and if it is [33]
possible to remedy such inconveniences
and make each of the intervals--the
consonant ones as well as the dissonant
ones--fall under its true form and
proportion.
219
Bardi: In order to clarify well your
new doubts, it has been very much to the
point to have tempered the instrument according
to the diatonic. In that distribution--as
you have heard--those intervals
which--according to our practicing contemporaries--have
the name of imperfect consonances,
become dissonant not due to the
perfection of the fifths, as countless
dare to say, but because of the largeness
of the tones and the smallness of the
semitones.
[Figure 98.]
Since I now want to temper on this instrument
the strings of the present diapason
so that each ditone and semiditone, .together
with each major and minor hexachord,
becomes consonant, it is necessary to
reduce them to their original state.
That distribution is similar to the order
and proportion of the syntonon. It is
neither the same--as some believe and
write--nor did its authors ever think such
220
a thing. Instead, it was done by accident
when they were seeking to tune the intervals
nearer to perfection than the nature
of the instrument--rather the quantity and
quality of the strings--and their experience
with the instrument and knowledge of those
devices would allow. They always avoided
the inequality of tones--as has been said
before--together with each inconvenience
which could possibly proceed from such
inequality in this modern practice. At
present, in order that we avoid such an
inconvenience, we will subtract--mainly
according to the method of Aristoxenus,
since we cannot separate any superparticular
interval into equal parts by
another method--four seventh parts of a
comma of our times from the interval which
is between the note F fa ut and G sol re
ut, thus compressing that interval by
such a quantity. And because the very
same distance which is found between F fa
ut and G sol re ut remains between G sol
re ut and a la mi re after having lowered
How much the
intervals become
altered
in the temperament
of the
ordinary keyboard
instru-
ments.
221
the first interval by the quantity which
has been mentioned, we will flatten the
said a la mi re by an entire comma plus
a seventh part of a comma. Afterwards,
we will make the note h mi lower than
its original state by a comma plus five
sevenths of a comma. Thus the very same
space will come to be contained between
this and a la mi re as is contained between
the two lower tones combined. Now,
leaving between the note h mi and that
of c sol -fa ut the entire remainder
which has been subtracted at the rate of
three tones (which concurs with the
composition of the tritone) is not suitable
at all, because not only is the
minor third which is found between a la
mi re and c sol fa ut dissonant compared
with the acquisition made--as you can
hear--but also, when one desires to
augment the limma, which originally was
between h mi and c sol fa ut, as far as
the limit of a sesquiquindecima--or a
little more, as I have said--it is made
222
a half comma larger. Because, however,
the fifth which is found between F fa ut
and c sol fa ut does not remain diminished
by such a quantity, and because the aforesaid
minor third--in coming a little nearer
to its least imperfect state--becomes less
languid and more pleasing to the ear, we
will lower c sol fa ut two seventh parts
of a comma. Each fifth necessarily will
become diminished by this amount. When
this is done it will be necessary, if
one desires the very same distance which
is found between F fa ut and G sol re ut
and the other diminished tones to remain
between this c sol fa ut and d la sol re,
to lower d sol re six seventh parts of a
comma. And because e la mi does not
exceed the latter, we will lower it a
whole comma plus three seventh parts of
a comma. This remainder we will leave
entirely to the interval which remains
between this e la mi and F fa ut. Thus
both minor semitones [limmas] will be
augmented by a comma plus three sevenths
223
of a comma. Although the major semitone
of this new distribution exceeds the sesquiquindecima
by the same amount, it is
not unsuitable, since it was derived from
a whole greater than the sesquinona. We
will then proceed according to this
order distributing all the other strings
which this diapason has below, above and
within its limits. From what has been
said, you can now comprehend clearly that
the fifth, especially, becomes lower than
the usual state of its proportion and the
fourth becomes sharp from that proportion.
In addition, no matter how much one
wishes, on the contrary, to make the diatessaron
diminished and the fifth augmented,
it is impossible, because the principal
cause of it is inherent in the quantity
of tones which these consonances contain,
in the amount by which these tones are
diminished, and in whether the semitones
which contain such intervals are augmented.
You have also been able to see how much
it is, between what pitches, why, and how
it is] distributed. But I warn you that
The source of
the aforesaid
imperfection.
224
in this temperament the pitches which first
contained the ditone now contain the major
third and those that first contained the
semiditone now contain the minor third.
Between those pitches in which the major
hexachord was found in the diatonic, one
finds there at present the major sixth, and
the minor [sixth] comes to be contained
between those which enclosed the minor
hexachord. One consequently finds that each
tone in the demonstrated temperament becomes
diminished by four seventh parts of a comma,
the ditone by one whole comma plus the
seventh [part] of a comma, the fifth by
two seventh parts of a comma, and the major
hexachord by six seventh parts of a comma.
On the contrary, the minor hexachord comes
to be augmented by a whole comma plus a
seventh of a comma, the fourth by two
seventh parts, and the semiditone by six.
From this one can argue that those who say
that the comma is not perceptible are
greatly deceived. They have confessed, for
all of this, that in keyboard instruments,
the sound of which they profess to
understand thoroughly, the fifths are found
diminished and the fourths augmented. This
I will more readily believe they have said
for politeness rather than through knowing
that the fact is actually so.
Strozzi: I want to take this opportunity
to tell you what happened to me
some time ago while arguing with one such
player. This fellow believed unequivocally
that 'sounding only those two registers
distant by a fifth--which are ordinarily
found in each church organ above the
fifteenth--made excellent listening without
having the octave and the principal
down. He did not notice that the pipes
were so arranged that the fifth and ninth
were always together and the fourth and
seventh were also paired. With all this
it was not possible to convince him by
means of reason, so that he almost forced
me to accompany him up into an organ loft.
When he had opened the two registers mentioned
and had begun to play, he strove for
excellence in all the ways which the fifths,
the fourths, the major thirds and the minor
[34]
Dissonant
registers.
225
226
thirds would unite with the fifth above or
below, without having the usual foundation
of the octave and the principal underneath.
This defect is hidden at present from the
ears of the uneducated, because of the
great noise made by the simultaneous playing
of so many different sounding pipes.
If the authority of one of his wretched
fellows, who was pumping the air for the
pipes, had not interfered, he would have
put me in a bad way with his importunity.
CThe organ pumper] spoke to him thus:
"You have come up here, I believe, to mock
me. Either play properly or I will let up
on the bellows." Still another of these
"sages" once tried to persuade me that
the said two registers were distant from
each other by a most perfect fifth but
that each one separately, between its
particular pipes, had flat fifths, according
to the custom of the other keyboard
instruments. I replied to him [and asked]
if the G sol re ut of the lowest register
was in unison with the c sol fa ut of the
227
highest register. "You know that perfectly
well," he responded, whereupon I told him
that between two things equal in length
one could not in any way be shorter than
the other and, on the contrary, if one were
larger than the other, naturally they could
not possibly be equal. After this, he shut
up and said no more about it.
Bardi: It is indeed an important
matter that most men speak so freely and
willingly of those things which they understand
the least. However, let us leave
(such matters aside and return to our
string distribution. Since we wished to
apply it to the diatonic syntonon, we will
now need to subtract from each major tone
four seventh parts of a comma and will
have augmented the sesquinona interval--
also called minor tone--by three of these.
Because of this, they will come to be
made equal. Each sesquiquarta (form of
the major third) also becomes diminished
by a seventh part of the comma and the
sesquiquinta (form of the minor third)
228
becomes reduced by the same amount. The
diapente remains reduced by two seventh
parts of the said comma, so that the
sesquiquindecima (called today major semitone)
becomes increased by three seventh
parts of the very same interval Ei.e.
comma). Consequently, the sesquivicesimusquartas
(called minor semitone) comes to
remain in its original form, exactly the
opposite of what occurs to voices. Then,
if everything I have said is true, the
superbipartientetertia (form of the major
sixth) will have been increased by as
much as the minor third has been reduced,
and the supertripartientequintas (form
of the minor sixth) becomes similarly
increased by as much as the major third
has been diminished. The fourth comes
to be augmented by two seventh parts of
the comma, [the same amount) which the
fifth is diminished. The octave, always
removed from any corrupt extreme (for the
reason which will be stated below), remains
with the dupla in its usual perfect form.
229'
There have been others who, in distributing
the same strings in the very same species,
tuned them flatter than the previous
opinion. They have wanted, instead of the
two seventh parts of the comma which have
been taken from the diapente and added to
the diatessaron, to remove the fourth part
[of a commas from the diapente in order to
bring the diapente and diatessaron to their
least imperfect state--mentioned earlier--
by a twenty-eighth [part] of this comma.
But afterwards, the minor sixth and the
major third have remained in the same
proportion as they have in the syntonon,
because they have taken a half comma from
the major tone and have given it to the
minor [tone]. Since this made them equal,
I think the matter is worthy of consideration
at that point, but since they are
in fact the same as they were at first
we will relegate it among their other
impertinences.
Strozzi: How is it, Signor Giovanni,
that when the strings of this instrument
are distributed according to the order of
230
the ditoniaion, where the consonances
called perfect today have been placed in
their true proportions, among which none
of the thirds nor the sixths agree, that
in this [situation), where the diatessarons
become augmented only into their true form,
one does not hear between the extremes of
these, either in the low register or in
the high, any of those harsh effects which
were heard in that [other distribution).
Bardi: Such a thing cannot occur
inside or outside the extremes of the
fourth, since by diminishing the tones and
augmenting the minor semitones, the cause
of the disagreeable sound is removed from
the thirds and the sixths. These cases,
and not the perfection of the fifths and
fourths--as has been said--were hindering
their concord. In addition, one can quite
easily discover in keyboard instruments
the fifths and the fourths in their true
proportions without otherwise hindering
the imperfect [intervals), as anyone
clearly can see and hear in the temperament
we have newly discovered, which brings
into practice the greatest excellence
and perfection one could possibly desire.
Strozzi: I am entirely satisfied,
but let us proceed to the distributions
of Dydimus and Aristoxenus.
Bardi: Dydimus, a most noble Pythagorean
musician, lived some years before
Ptolemy and made a new string distribution
in each one of the three genera of harmony,
Among the others, that which he made in
the diatonic proceeds in each of its
tetrachords in the manner which you see
in this one, which is the lowest one in
the system, known as hypaton.
E la mi 72
D sol re 81
E35] How the
syntonon was
distributed
by Dydimus.
Sesquioctave (or 9 difference
tone)
Sesquinona
C fa ut
9 difference
90
Sesquiquindecima 9 difference
h mi 96
[Figure 99--J Hypaton tetrachord of
Dydimus.
231
-. - 1
-
232
Afterwards, came Ptolemy, who changed the
order of the two middle intervals of each
tetrachord, placing the upper middle one
in the position of the lower middle and
vice versa, saying that it did not suit
the larger one to be placed there, but
only suited the other, which was smaller
than it, but greater than the lowest.
From this you can understand what Ptolemy's
stand is regarding the syntonon and
can know who ought to be given the honor
and the palm.
Strozzi: Why, do you suppose, did
those who have sought to persuade us that
what is sung today is wholly syntonon--
that is, diatonic syntonon--say that it
belongs to Ptolemy rather than to Dydimus?
According to what I have seen, one distribution
does not apply any greater
convenience or inconvenience to our way
of composing and singing than the other.
:Bardi: That which has failed to annoy
you and many others has perhaps spoiled
the designs of the authors of these things.
233
Strozzi: How [could that happen),
please?
Bardi: The interval which is found
between G sol re ut and h mi [i.e. B]
in the distribution of Dydimus is a ditone
and not a major third of the variety which
most people believe is sung today. That
which is found between h mi and this G sol
re ut is a minor hexachord and not a minor
sixth. Therefore, having first stated in
their writings that such intervals were
dissonances (as you have heard they really
are), they came too extremely and too
suddenly to their conclusions to provide
[anyone] opportunity to impede their
schemes. [To demonstrate] that these intervals
appeared thus, here is the first
species of the diapason, distributed according
to the intention of Dydimus
[Example 4.]. When you have examined it
carefully, you will find that what I said
is true. Although the same has occurred
in those of Ptolemy--as you have seen--
it has not been so clarified to the sense
234
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235
and judgement of the uneducated, and until
now, it was possible for such a thing to
defraud the general public more easily.
And this is the reason why they have said Why the syntonon
is attributed to
that the diatonic species which is sung Ptolemy.
today belongs more to Ptolemy than to
Dydimus, unless we still wish to say--I
do not believe this in any way--that they
have ignored the difference which is found
between these [species). Also, as a result
of this, they gave the name of comma to the
excess by which the minor tone is exceeded
by the major tone, and not, like the ancients,
to that Eexcess] by which the major
semitone surpassed the minor, in order to
obtain more kinds of these, as you have
seen.
Strozzi: What do we believe was the
reason which persuaded so many of that
first sect--if not in the beginning as I
believe, Eat least] from Ethe time of]
Guido d'Arezzo on--that what was sung was
the ancient diatonic? Among these, there
have been many learned and renowned writers
like LeFevre, Gaffurio, Glarean and Valgulio,
236
[all of whom were] highly literate and
ingenious men (and much more), for many
great men like Aristoxenus, Euclid, and.
Vitruvius have first told them expressly
that the intervals smaller than the diatessaron
and those which are found between
the diapente and the diapason were
all dissonances. I do not know how they
could say it any more clearly in their
48Henry Macran, The Harmonics of
Aristoxenus (Oxford, T2T, p.T7.
"Though many smaller intervals than the
Fourth occur in melody, they are without
exception discords." Ibid., p.
198. "We assume then eightVmagnitudes
of concords; the smallest, the Fourth--
determined as smallest by the abstract
nature of melody; for while we can produce
several smaller intervals, they are
all discords."
49 The Eis o e of Cleonides was
formerly at u to Euclid. There is
an English translation in Strunk, Source
Readings, pp. 34-46. Cleonides caTTs
the octave, fourth and fifth symphonic
intervals because they blend together.
Intervals less than the fourth are
diaphonies, where the two notes refuse
to combine, but grate harshly on the
ear. See Ibid., p. 38.
5 0 Vitruvius, De architectura, translated
by Morris HiMcy~~organ ~nairidge,
Massachusetts, 1926), p. 142. "For
there can be no consonances either in
the case of the notes of stringed instruments
or of the singing voice, between
Aristoxenus4 8
in the 1st and
2nd [books of
the flemen s.
Euclid4 9 in the
Introduction.
Vitruvius5 0 in
the 4th chapter
of the 5th
[book].
237
writings, and anyway the matter is easily
manifested to the sense of hearing.
Bardi: It was difficult for Guido
d'Arezzo to consider such a conception,
for he was a monk and his goal was to
teach his [fellowJ monks to sing chants
with facility. Since these chants were
in unison and no harmonies of any kind
were involved, he could not doubt that
[conception] in any way. Therefore, the
reason which led him to designate the notes
in his Introduction with the same numbers5 1
which Plato used in his Timaeus (which had
first been assigned by Pythagoras and
later perpetuated by Boethius) was--as far
two intervals or between three or six or
seven; but, as written above, it is only
the harmonies of the fourth, the fifth,
and so on up to the double octave, that
have boundaries naturally corresponding
to those of the voice; and these concords
are produced by the union of the notes."
5 1 By numbers, Galilei apparently
means the string lengths designed by
Aristides Quintilianus (third or second
century A.D.), a Greek writer on music,
as. a convenient method of expressing
the Pythagorean tuning. Zarlino, Istitutioni,
p. 104, cited by Galilei (see
p~Ti ~below) incorporates these Aristidean
numbers in a diagram of Guido's
gamut.
[36J Guido
'Arezzo did
not know about
the imperfect
consonance.
238
as I am concerned--in order to give the
thing reputation. He took not only the
most ancient and famous distribution
of strings that ever had been known,
but that which had been given to mortals
by Nature. For the same reason he did
not wish to use the natural sign [i.e.
h mij on the note B fa, because neither
the Greeks nor the Romans used it in
their systems. His opinion was first
put into use by others and afterwards
was greatly approved by countless others.
Strozzi: So that in the time of
Guido dtArezzo, you believe that they
sang the lowest intervals of tetrachords
through a limma and not like today
through a major semitone, and that they
did not sing in harmony?
Bardi: I firmly believe that they
did not sing in harmony because I do not
find in his Introduction any mention of
imperfect consonances, but I am definitely
convinced that they played [in harmony).
Since the lowest interval of tetrachords
The way they
sang in [Guido's)
times.
239
was then a minor or a major semitone, I
will more easily believe that the latter,
rather than the former, was sung in that
place. This is the reason which impels
me to believe that, In the time of
Guido d'Arezzo, every light of virtue--
in a manner of speaking--had gone out
once again in Italy, particularly in
regulated [mensuralj music. It was because
of this that the mode of singing
which was acquired naturally at the
beginning of the world had not been
maintained, This is the same as that
which rustic farmers use while tilling
the fields and shepherds use to drive
away the boredom [while journeying
through the forests and mountains behind
their herds. It diverts them
from continual and heavy labors. This
kind of singing has always been used
among men from the creation up to our
times, nor will it have an end, except
together with them or with the world
itself, although Athenaeus52 says with
240
the testimony of Chamaeleon of Pontus
that men have learned this art in seeking
to imitate the song of the birds. Moreover,
up to [G.uido'sJ time or a little
before, men had wept for many continuous
centuries about their miseries. Afterwards,
little by little, they began to
apply themselves to literature, to music,
and to the other fine arts. Thereupon,
the above-mentioned Arezzo, the most
learned man in the art of music in those
times, began to reorganize the method of
singing as much as the conditions of the
century and his region would allow. He
not only gave new names to the notes,
since they had come into use' a little
before him, denoting them with the very
same characters Eas used in] the Latin
alphabet in order that people could learn
to use their voices comfortably with
facility and distinction, but he also
52See Athenaeus, The Deipnoso hists,
7 vols. translated y"IresBron
Gulick (London and New York, 1928), IV,
263.
From whom
singing is
learned.
Guido d'Arezzo
adds six notes
to the system.
designated, in addition to those of the
Greater [Perfectj System, five [notesJ in
the high register and one in the low,
afterwards distributing [all the notes
in seven hexachords. This was the same
number as the letters he used, as it
appears in his Introduction. Perhaps
this was in imitation of what Demetrius
of Phaleron5 3 said with regard to the
use of vowels, using the example of the
Egyptian priests who pronounced the notes
of their sacred songs over the sounds of
these vowels, which were seven [in numberJ.
The practical musicians who lived a little
before the time of Guido d'Arezzo used,
in order to signify the pitches of their
compositions, the same characters which
had already been employed by the ancient
Greeks and also by the Romans, placing
them on seven lines in this manner [See
Example 5.], imitating perhaps the seven
strings of the ancient cithara.
5 3See W. Rhys Roberts, trans.,
Demetrius On Style (Cambridge, 1902),
pp 1 4-o07
Demetrius of
Phaleron.
241
242
[Example 5--An uncommon system of notation.J54
54See August Ambros, Geschichte der Musik, 5 vols.
(Leipzig, 1891), II, 149. The notatin depicTed in
Example 5 corresponds to no known system. Although
Galilei tries to show that it was once in general use,
Ambros points out that if this were true, the case
would have been more publicized. See also Athanasius
Kircher, Musurgia universalis (Rome, 1650), pp. 213-214.
I11.. 1 . . 1 .| ..
-r
G
3'
243
Afterwards, Guido d'Arezzo removed the
tedium of the many lines and clefs over
which, instead of the notes of our times--
which actually were discovered in Paris
by the great Doctor Jean de Muris 5 5 --he
used points, placing them also within
the space which is found between one line
and another, as the composers do today.
By using these [points], such men acquired
the name of contrapuntists. This was very
appropriate, through composing their songs
with points, which have no other being in
nature except solely in the imagination
of men. They result by chance like the
figures composed today by the superstitious
geomants56 regarding the judgements of
55See Nicola Vicentino, L'antica
musica ridotte alla moderna prattca (Rome,
T175," p9. ~VT tino was the first to
make the extravagant claim that Jean de
Muris invented musical notes. This theory
was later supported by Grassineau, but
refuted by Mersenne and Rousseau. See
J. F. R. Stainer, "Johannes de Muris,"
Groves Dictionary of Music and Musicians,
ShREcBlom,ed. ,jvTs. (London,
1954), V, 1008.
5 6Geomants were diviners by means of
dots and lines, or by clumps of dirt
scattered on the ground.
E37J Guido
was the first
to place notes
on the spaces.
Notes were
discovered by
Jean de Nuns.
The origin
of the name
contrapuntist.
244
problems given them. These men render
their judgements of such problems according
to what the aforesaid chance has
caused--according to some of their few
principles and terms, however--without
otherwise knowing beforehand the effects
which can result when one operates in one
manner rather than another. With ten or
more geomants, rather than only one,
there will be ten different opinions
about the very same problem, which they
will form into ten or more figures. In
the same way, entrusting ten or more
contrapuntists to express the same,
particular affection of mind with their
music (according to the usage of this
century), they would express it in ten
or more different ways and in various
modes. The same [thing] will happen to
one alone, if he tries to express it ten
or more times. [All this occursJ for no
other reason than the fact that both the
geomants and the composers have allowed
chance to guide them in writing their
245
compositions, or let us say that their
principles (according to what Seneca tells Seneca,57
On Benefits
us) are full of wind and woven of fog and and also
Petroniu9
air like the dresses of lascivious Roman Arbiter.
noblewomen of his times.
Strozzi: Did the musicians prior to
Guido use, for that reason, only the lines
in writing their musical compositions?
Bardi: Yes, indeed. As I said before,
they used different symbols to denote the
clefs, and a greater number of lines, at
least in the songs which were played. This
was because they had no idea of the space
which is found between one line and another,
in order to indicate to the sense--as has
57Lucius Annaeus Seneca, "On Benefits,"
in Moral Essays 3 vols., translated by
JohnT Basore ULondon and New York, 1928-
1935), III, assim. Seneca refers continually
to the a c viousness of Roman matrons
of his times.
58 Titus Petronius, Satyicon, translated
by Michael Heseltine (London and New
York, 1916), p. 99. Petronius, quoting
Publilius Syrus, a famous writer of farce,
admonishing a cuckold; "Thy bride might
as well clothe herself with a garment of
wind as stand forth publicly naked under
her clouds of muslin."
246
been said--the strings from which they
were derived. They had at that time
Greek characters which denoted their
clefs and notes as you have been able
to comprehend from those examples
[Example 5.].
Strozzi: Would it be possible
to see some of those so-called early
modern compositions?
Bardi: It certainly would.
Strozzi: I hope it is not a burden,
but please, show me one of them.
Bardi: Here is an example59 of
one [Example 6.], among others which
I have secured, which was given to me
by one of our Florentine gentlemen6 0
who had discovered them in one of his
very oldest books. It is more complete
and better preserved than any I have
59 The musical example [Example 6.]
is not pre-Guidonian notation, as
Galilei asserts. It is actually a
"Hosanna in excelsis" trope. Compare
Analecta hymnica medii aevi, Vol.
x.IV"I, ed, by ens Bume and Henry
M. Bannister (Leipzig, 1905), p. 346.
Most copies of the Dl have only
blank staves in this example. The
247
ever seen. A little later I will show you another
contemporary example still earlier than this.
Strozzi: Here a long explanation will be needed
since I want to know how to read it, not just sing it.
notes, apparently, were either omitted through the
carelessness of the printer, or were intended to be
added at a later date. See Palisca, Girolamo lei,
pp. 154--156.
"
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c tinnul t orrnoiiaCr hft name , ,l
6 0 Galilei is referring to Girolamo Mei, whom he
mentions only once by name in the Dialoo, despite the
large role Mei played in formulating that work.
248
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250
Bardi: Let it suffice you for now
[to say] that it is--as I understand it--
a composition to be sung to the [sound of]
the aulos rather than the lyre, and it is
in the Mixolydian mode. I will interpret
for you in detail each of its other particulars
at the appropriate place, but
let us proceed to clarify the problems
you first proposed to me without placing
any new ones under consideration.
Strozzi: Let it be as you please,
but returning to Guido d'Arezzo, it seems
to me that we owe him a great deal since
he added to many fine things to the music
practice, despite the roughness of the
times. Before I forget it, however, I
am forced, contrary to our agreement, to
question you about another new problem,
if you will pardon me. I do not know
for what purpose Guido d'Arezzo added
so many notes to the system, since they
still did not sing--according to your
opinion--in harmony, but only used simple
6 1 Zarlino, Istitutioni, pp. 103-105.
Zarlino6 1 in
chapter 30 of
the 2nd part
of his Insti-
tutioni.
251
chants similar to that in the composition
you showed me.
Bardi: I do not maintain that Guido
d'Arezzo added any note to the system, but
only that he gave each one a particular
name. As for those notes which are seen
in his Introduction in addition to the
fifteen of the ancient Greeks, I firmly
maintain that he derived them from the
modes of Boethius, which are eight in
number, the extremes of which have the
same number of tones and Fare] in the same
order. Furthermore, I must say that he
derived them from wind instruments like
the organ, or else from those stringed
instruments that were in use in those
times, which were similar or perhaps even
the same as the simikion and epigoneion
of the Greeks. From these notes he took--
according to the diatonic species, which
he understood was sung in those days--
that number which comfortably coordinated
with the diversity of human voices which
sang different songs with them, but [did]
not [sing them] simultaneously, assigning
Origin of the
notes added to
the system by
Guido.
252
eight to the low register, eight to that
of the middle, and the remainder to the
high, so that, in all, since the lowest
of the eight in the middle was the same
as the highest in the low register, and
the very highest of those in the middle
was the lowest in the upper register,
they came to make a total of twenty-two
tones and strings in three octaves,
divided as I have said. Others have
said that he ordered them in seven
hexachords since he wished to denote to
us with these numbers, which are six and
seven, the excellence and perfection
which is found in this distribution. It
happens that the number of six is the
first of the perfect EnumbersJ just as
the second is twenty eight, which is
62 Pier Maria Bonini was a Florentine
mathematics professor who was the author
of Acutissimae observationes nobiliss.,
discilinar, omnium musices (Florentiae,
T520). This treaise is apparently not
extant. See Robert Eitner, Biopraphischbibliographisches
uellen-Le con der
Musiker und usigeehen der chrLTlichen
Zei'rech bis urtte~T e
neunzehnten Tahrhunders7TOvsT.(TeipZ7g,
1899-190)"T119
.
Pier Maria Bonini6 2
in his [book on)
music.
Perfect numbers
[are) 6, 28, 446,
8128 and others
like them.
253
produced by the settenario and its parts.
Others have said that they were ordered
by [Guido) in major hexachords, since
each one of these disposed in such a way
contained some one of the three species
of the diatessaron. Returning to the two
stringed instruments just now named, I
say that one of these had thirty-five
[stringsJ and the other Ehad] forty.
One can argue from the quantity of
[stringsJ that the professional players
of these [instruments] played in harmony,
no longer touching them with the plectrum,
which had gone into total oblivion, but
with the fingers, like the harp. This
species of harmony is more suited for
consonances than the others, as we have
already .declared. This mode of composing
and playing so many airs together at the
same time probably had its origin from
that manner of playing, as we will show
at the proper time. The singers of
those times produced the musical intervals
according to the distribution of the
strings and not otherwise and [this
Derivation of
the singing
of today.
254
practice] has been so conducted and used
up to today, giving the name of imperfect
consonances to the major and minor thirds
and sixths. The surnames of these Eintervals]
must cause Econfusion) in the
mind of the general public and persuade
it without thinking further to believe
and say that they were the same as the
ditone and semiditone and as the major
and minor hexachord, according to the
form in which they were contained as
Pythagorean numbers in that Introduction
of Guido d'Arezzo. That name of imperfect
consonances was actually placed there with
judgement and the greatest discretion, not
only because of their indeterminate and
variable nature, but because the greatest
part of these--in order not to say all--
appear to be, rather than actually being
consonances, although they have been
ascertained by our practicing contemporary
contrapuntists as such, without seeking
further due to necessity, having discovered
that it would be a burden to them.
255
This opinion that they were the same as
the ancient Einterval) lasted in the
minds of men until the coming of Reverend
M. Gioseffo Zarlino, who has sought with
diverse arguments to demonstrate to the
sense and to the intellect that such
imperfect consonances are not in any way
those which are found among the notes
distributed according to the diatonic
ditoniaion, nor even among those same
EnotesJ in the syntonon of Ptolemy.
Because of the novelty of this thing, he
allowed himself to induce others to believe
and say that the diatonic species
which is played and sung today is wholly
the syntonon of Ptolemy, which is not
true, as you have seen. It is not valid,
however, to say that the thirds and the
sixths are consonances in the syntonon
of Ptolemy; what we sing are the other
consonances which are, moreover, the
same as Ethose of] Ptolemy, Nevertheless,
the world owes to this man, [who is so]
exemplary of customs, life, and doctrine,
a perpetual obligation, due to the many
[39]
This is what
led Zarlino to
say that the
diatonic species
which is sung
today is the
syntonon.
Praise of
Zarlino.
256
fine efforts which he has made, particularly
in music. From these efforts one
derives knowledge of [an] infinite
Number ofJ things without which easily
the greater part of mankind would be in
darkness.
Strozzi: Signor Giovanni, I desire
to understand how those two ancient
stringed instruments, which you mentioned
earlier, were made, that is, with regard
to their form and the distribution of
their strings.
Bardi: You are asking me about
something like the platage of Archytas
of which there is no record other than
the name and the number of strings.
Although the knowledge of these is
small, I do not doubt at all that those
who wish to conjecture judiciously about
them will find approximately what form
they had and in what proportion their
strings were arranged. In order to tell
you briefly what I believe about it, I
consider that the material and the form
consisted of a wooden frame similar to
257
that of the harp, within which were
stretched, that number of strings which
you have understood, stretched either in
the same manner or in a slightly different
one, and distributed among them in a
certain way. I always, however, revert
to the judgement of one who understands
better than myself. The epigoneion was
discovered by Epigon of Ambraciota, chief
of a very famous sect, either a little
before, a little after, or at the same
time as Socrates, according to what
Porphyry63 says in his commentary on the
music of Ptolemy. This Epigon, according
to the opinion of Julius Pollux,64 was
the first who used the fingers to pluck
the strings without the plectrum. This
method of touching the strings, together
with the number of these [strings] argue
that he played harmony. This method was
6 3 See Ingemar During, Porhrips
Kommentar zur Harmonielehreds olemaios
(G t ebog, T32,p
,
6 4 See Eric Bethe ed., Pollucis
Onomasticon, 3 vols. (Leipzig,19-1937),
I,29
The meaning of
epigoneion and
the name of
its inventor.
Julius Pollux.
258
also followed later (as Suetonius Tranquillus65
tells us) by Nero. Suetonius
says in describing [Nero's) life that one
time among others when he appeared
publicly in the theater to sing and play
on the lyre in competition with a number
of professional cithara players of his
times, he first made on his lyre a
beautiful prelude with his fingers and
afterwards began to sing. It is not
known who was the inventor of the simikion,
which greatly surprises me. It
is probably true that, since it had
thirty-five strings, it must have been
discovered prior to the epigoneion which
had forty [strings). Its inventor is no
less worthy of esteem by his followers
than Epigon of Ambraciota.
65See Caius Suetonius, Lives of the
Twelve Caesars, translated by" 4.TBird
(Argus, n.d.), p. 272. Suetonius makes
no mention of plucking the strings with
the fingers, but only says that Nero made
the "usual prelude" before beginning to
sing.
259
[DIAGRAM III) [40]
EXAMPLE OF THE EPIGONEION, INSTRUMENT
OF FORTY STRINGS, DISCOVERED
BY EPIGON OF AMBRACIOTA
[DIAGRAM IV)
EXAMPLE OF THE SIMIKION, INSTRUMENT
OF THIRTY-FIVE STRINGS
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3
260
[41)
vw w w
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IN 1, --~wRkWWER--Nac---
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r
261
I come to the distribution of Aristoxenus [42]
and say that Aristoxenus,6 a most noble (AristoxenusJ
in the 2nd
musician and philosopher, made six dif- (book] of the
Elements of
ferent distributions of strings, that is, Harmony,
two diatonic, three chromatic, and one
enharmonic. But we, for brevity, will
deal with only those which will be needed,
reserving the others for where they might
be necessary. This most knowledgeable
musician used to assign to each of his
tetrachords that portion and quantity of
sound of the diapason which agreed with
[each of] their diverse genera and species.
In order to do that, he first divided the
diatessaron, which consisted of two tones
and an entire half of [one of] these [tones),
which suited his purposes, into sixty tiny
equal parts. I say [that this was] with
regard to the sound, not to the length of
the line or string, although that quantity
was also considered. [Aristoxenus] then
gave twelve of these parts to the lowest
interval of each of his tetrachords,
See Macran, Aristoxenus, pp. 202-204.
262
twenty four to that of the middle and the
rest to the highest [interval). He gave
to this system, since it was so divided
and ordered, the name of intense diatonic,
calling the lowest interval of each of
these tetrachords a semitone and calling
the other two higher intervals tones. And
this is the way the tones and semitones
which he treated in different phases of
his distributions were constructed. Of the
three chromatics which he made, he named
one the tonikon, which he arranged thus.
He assigned to the lowest interval of each
of its tetrachords twelve of the sixty tiny
parts mentioned above, into which he had
divided the diatessaron. He gave the
same number to that of the middle and the
others to the high [tetrachord).
Strozzi: Would it not have been the
same to have divided the diatessaron into
five equal parts rather than into sixty,
and in the intense diatonic to have given
one of these to the lowest interval of the
6 7 See During, Ptolemaios, pp. 44-46.
Ptolemy6 7 in
the 1st bookj
of the Harmonics
at the Th
chapter.
263
tetrachord, two to that of the middle, and
the rest to the high? [Would it not also
have been the same) in the chromatic
tonikon to have given one of these to the
lowest interval of the tetrachord, another
to that of the middle, and the three which
remained to the highest?
Bardi: It would have been [the same]
in the two distributions you have mentioned,
but assuming that that division
of the diatessaron served each of the
other four, differently distributed, he
would have been unable to avoid the tedium
of smaller numbers by dividing it into
fewer parts. [This was] because one time
it would be necessary to subtract half of
the tone and another time it would be
necessary to subtract a fourth of it. In
the case of the semitone, it would not
only be necessary to subtract a half or
a fourth, but also a third. Sometimes it
was necessary to augment one [interval]
or the other by such an amount, in order
to be able to construct conveniently any
The reason why
Aristoxenus
divided the
diatessaron
into sixty
tiny parts.
264
one of the other intervals that agreed
with his other different distributions.
In order to do that he selected the number
sixty as one which was capable of
being divided in two, three, four, five,
or six parts.
Strozzi: Could it be that the ancient
musicians should have assigned
tens of thousands of those [tiny parts]
to the tones of the diatonic ditoniaion
because the same numbers served conveniently
for the chromatic and for the
enharmonic?
Bardi: This was actually the case.
One may consider it particularly in the
ancient enharmonic, between the note
tritesynemmenon, designated by the number,
4491, and the paranete of the same
tetrachord, designated by another number,
4374. I tell you also, in regard to the
distribution of Aristoxenus, that the
intense diatonic and the chromatic
tonikon--these are* commonly used on the
lute and the viola d'arco--more nearly
265
approach perfection with regard to their
consonances, assuming for now that perfection
is the fifth within the sesquialtera
[proportion] and also the fourth
in the sesquitertia. This perfection is
not attained by those keyboard instruments
which have been preached to us as [playing]
the plain syntonon by those who have
refuted the distributions of Aristoxenus
without saying why. Anyone who takes the
trouble to examine them by means of their
particular monochords will be able to see
that fact.
Strozzi: Would he not [have been
able to make me understand with ease and
very great accuracy how greatly the
perfect and imperfect consonant intervals
of the lute and the viola become augmented
or diminished from the true state in which
they are contained [in] the syntonon, in
order to avoid the various bad relationships
and the confused and inconstant
difficulties by which they are different
from those of the diatonic ditoniaion?
266
Bardi: He would have been able Eto
do this] with clarity and facility, but
not with greater exactness than one can
show by means of the harmonic ruler.
Strozzi: Go on, please.
Bardi: Initially, the octave on the
lute and on the viola (which proceed in
the same manner stepwise), always being
far from any imperfection, consists--as
you know--of six tones, that is, twelve
semitones. By conforming more than one
should with the usage of our practicing
contemporaries, we can also say truthfully
that it consists of five tones and
two semitones. We know, therefore, that
each of their tones is less than the
sesquioctave and greater than the sesquinona.
The semitone becomes less than the
sesquiquindecima and greater than the
sesquivicesimusquartas. The minor third
is surpassed Ein sizeJ by the sesquiquinta;
the major [third] exceeds the sesquiquarta;
the diatessaron surpasses the sesquitertia;
the diapente is less than the sesquialtera;
The formation of
the intervals on
the lute.
267
the minor sixth is surpassed by the supertripartientequinta;
the major [sixth]
surpasses the superbipartientetertia; the
tritone and the semidiapente are equal;
consequently the semidiapente is smaller
and the tritone greater than the ones
contained in the syntonon.
Strozzi: Although I have understood
the way of placing frets on the lute and
viol which you have mentioned on other
occasions, it seems to me, initially, that
each of their tones is [a) sesquioctava,
and that the minor third, in addition to
many other intervals, is in the same form
in which it is contained in the syntonon [43]
and not exactly contrary to the form of
the sesquiquinta.
Bardi: How can you maintain that such
a thing can occur, since the tones are
equally divided into two parts--as you
have said and proved--and since the major
third is consonant?
Strozzi: Listen, please! The tone,
initially, falls between eighteen and
sixteen since the whole is divided into
268
eighteen equal parts of which the tone contains
two. This is the same as saying that
the tone falls between nine and eight. Is
all this not so?
Bardi: Proceed further, so that I
fully comprehend your question.
Strozzi: Does the minor third contain
three semitones which are the same value as
three eighteenth parts of the whole? And
do the fifteen [partsJ which remain, compared
to the eighteen, have the same relationship
together as six has to five, which
is the true form of the minor third according
to the syntonon.
Bardi: Now, be advised that two
eighteenth parts are only equivalent to a
ninth part of the whole in this manner of
measuring, because these parts are considered
as a quantity of the line or string.
To prove that this is true, here is the
compass. By measuring with it, you yourself
will find that the two first semitones on
the lute do not fill up a ninth part of the
length of the string, since three parts
269
are not a sixth part of the whole (stringJ
but only a lesser quantity.
Strozzi: I have understood everything
extremely well, but continue your original
discussion.
Bardi: Because of the great agreement
and similarity which the diapason, more
than every other interval, has with the
unison, not only can the simplest and most
perfect of octaves be found, but one in
which the ear can be less deceived in comprehending
it. If its lowest terms are
multiplied together a thousand times and
then added together the upper pitch is
always contained by unity. This does not
happen to any of the other intervals,
because they have not been taken as many
times as that consonant one. For this
reason, and in order to avoid both difficulty
and danger of error, we will employ
its aid and that of its replicates in this
important undertaking. It is first necessary
to know that each tone of the lute
is a sixth part of the ancient comma less
270
than the sesquioctave. I will prove it to
you in this manner. It is clear that six
sesquioctave tones are greater than the
diapason by one of these commas. If, moreover,
six tones on the lute will fill [the
diapason) entirely, without anything
remaining or lacking, each one of these
[tones on the lute] becomes, consequently,
a sixth part of this [ancient] comma less
than one of those [sesquioctave tones].
I say in addition that the sesquioctave
is surpassed by each tone of the lute by
three fourths of the sesquioctagesima,
which is, according to our practicing
contemporaries, the comma of our times
because each octave is capable--as one
can clearly understand from what we have
mentioned above--of five sesquinonas,
three commas, and two major semitones of
the syntonon. These two major semitones
give us a sesquinona, and approximately a
comma and a half more, so that we can also
consider each octave to be capable of six
sesquinone tones plus four commas and a
half, approximately. When these four
271
commas have been distributed among the
aforesaid six tones, each [tone] will
receive two-thirds plus a sixth part of
that half [comma]. Now because two-thirds
plus the sixth part of a half happens to
make, when added together, three-fourths
of an entire [comma], each tone of the
lute necessarily comes to surpass the
sesquinona by this quantity. In addition,
each of the minor thirds of [the lutej,
compared to the sesquiquinta are diminished
by three-eighths of a comma, I
will prove that to you thus. The octave
of the lute consists exactly of four
minor thirds, Let us see now, according
to the example below, in which a
dupla has been subtracted from four
sesquiquintas added together, what the
remainder will be,
1296 625 Form of four sesquiquintas
added together
x
2 1 Form of the dupla
1296 1250
2
648 625 Remainder
[Figure 100.]
272
When a dupla has been subtracted from four
sesquiquintas added together, there remains
the super 23 partiente 625, which clearly
demonstrates that the minor thirds of the
lute, compared to the sesquiquinta, are
diminished, because four of those minor
thirds fill up an octave entirely. This
super 23 partiente 625 consists of approximately
a comma and a half. When the said
interval has been distributed among four
minor thirds, it will convey to each of
these the quantity of three eighths of a
comma. Each minor third of the lute, as
I said, will be diminished by such a
quantity when compared to the sesquiquinta.
There is no doubt that since the octave of
this instrument is in its true proportion
and far from any corrupted extreme, and
since each one consists of a minor third
and a major sixth, each one of these major
sixths will be augmented in [the octave]
by the same amount as each one of those
minor thirds will be diminished. Although
the intellect understands very well the
truth of this, I nevertheless wish to
273
prove it to the sense with a convenient
example, which will be this. It is
certain that four major sixths of the [44]
lute fill up three octaves entirely. Let
us see, therefore, by subtracting three
duplas from four superbipartientetertias,
what will remain.
625 81 Four superbipartientetertias
added together
X
8 1 Three duplas added
together
625 648 Remainder
[Figure 101.]
When the octupla, form of three diapasons,
is subtracted from four superbipartientetertias
added together, the remainder is
the sub 23 partiente 648. This argues
that what I said concerning the augmentation
of the major sixths is true. Each
one of these is augmented on the lute by
three eighths of a comma. In order to see
now how greatly the major thirds are
augmented on this [instrument], we will
follow this order. It is certain that
274
three major thirds of the lute fill up
entirely the space of an octave, Let
us see now according to the example
which follows what will remain to a
dupla when three sesquiquartas added
together have been subtracted.
2 1 Dupla
X
125 64 Three sesquiquartas
128 125 Remainder
[Figure 102.J
When three sesquiquartas have been subtracted
from the dupla, the remainder is
the supertripartiente 125. From this
fact one may comprehend that the major
thirds of the lute are actually augmented.
When we examine the remainder carefully,
we will see again the amount which composes
the supertripartiente 125, which,
as I proved to you in the beginning of
our discourse, was approximately a comma
and a half. This comma and a half, distributed
among the said major thirds,
conveys to each one a half comma. Any
275
major third will be augmented by half a
comma and any minor sixth will be diminished
[by the same amount]. This I will
prove to you in this other manner. There
is no one who doubts, conceding that the
minor sixths on the lute consist of
eight semitones [apiece), that two octaves,
each containing twelve [semitones), do
not amount to the same [thing) as three
minor sixths. Let us see, therefore,
according to the example, how much of it
will remain when we subtract three supertripartientequintases
from the quadrupla,
form of the bisdiapason.
4 1 Quadrupla
X
8 5 Supertripartientequintas
20 8 First remainder
4
5 2
X
8 5
25 16 Second remainder
X
8 5
125 128 Third and final remainder
[Figure 103.]
276
The remainder is the subtripartiente
128, from which it clearly appears
that whatever amount the major third
is augmented, the minor third on the
lute is found to be diminished by the
same portion. If the situation were
any different, it would follow that
the octave, when composed of this
[minor third] and that [major thirdJ,
would be dissonant, but this is not
so in any way. In addition, I will
prove to you in this manner that the
fifths of the lute are actually
diminished as I have said they are.
It is clear that twelve lute fifths
fill the void of seven entire diapasons
precisely, since four diapasons
contain seven lute fifths
plus a semitone. Let us see now what
will remain of one interval which
contains in itself neither more nor
less than seven duplas, after twelve
sesquialteras have been subtracted
from it [Figure 104.]
1 Seven duplas added
together
3 2
256 3
x
3 2
512 9
x
3 2
1024 27
x
3 2
2048 81
x
3 2
4096 243
x
3 2
8192 729
x
3 2
16384 2187
x
3 2
32768 6561
x
3 2
65536 19683
First remainder
Second remainder
Third remainder
Fourth remainder
Fifth remainder
Sixth remainder
Seventh remainder
Eighth remainder
Ninth remainder
128
x
277
[45]
19638 Ninth remainder
x
2
59583 Tenth remainder
x
2
177147 Eleventh remainder
x
2
524288 521441 Twelfth and final
remainder
[Figure 104.J
The super 2847 partiente 521441 is the
remainder. It is plentiful in branches,
but devoid of fruits, because it does
not consist of even one entire comma
as the following example [Figure 105.J
will clearly demonstrate.
524288 521441
x
81 80
65536
278
3
131072
3
262144
3
41943040 42236771
[Figure 105.J
279
The fifths on the lute therefore become [46]
flat by less than the twelfth part of a
comma; the fourths necessarily become
augmented by as much. I will prove that
in more detail by subtracting the twelfth
part of a comma from the interval which
consists of five octaves, because it
contains virtually twelve fourths, as
the present example [Figure 106.J will
show.
32 1
x
4 3
96 4
4
24 1 First remainder
X
4 3
72 4
4
18 1 Second remainder
X
4 3
54 4
2
27 2 Third remainder
x
4 3
81 8 Fourth remainder
8 Fourth remainder
x
3
243 32 Fifth remainder
x
3
729 128 Sixth remainder
x
4 3
2187 512 Seventh remainder
x
4 3
6561 2048 Eighth remainder
x
4 3
19683 8192 Ninth remainder
x
4 3
59049 32768 Tenth remainder
x
4 3
177147 131072 Eleventh remainder
x
4 3
524288 Twelfth and final
remainder
[Figure 106.]
81
4
280
4
521441
281
I proved to you above that the semidiapente
exceeded the tritone by approximately
half a comma, so that, since one
is found to be equal to the other on the
lute, the tritone on [this instrument]
is necessarily augmented by the fourth
part of a comma, and the semidiapente
[is necessarily) diminished by the
[same amount], which is half of that
remainder by which the semidiapente surpasses
the tritone. One can comprehend
from this how discreet, diligent, and
considerate courteous Nature is in each
one of her endeavors, because those intervals
which more closely approach
perfection do not depart far from their
true state. This situation has not
necessarily occurred with those farther
from [perfection] and less with the
imperfect intervals and the dissonances,
because it was not so clarified to the
sense. One can also know by what we
have said clearly up to now how much
farther removed from perfection the
keyboard instrument is than the lute
[47]
Considerations
of the author.
282
and the viola [diarco), and also. how
difficult it is (through bad relationships
and false encounters) for their
sounds to be united together agreeably,
when these sounds continually clash in
concerts where there are no others but
those [keyboard instrumentsJ which
cause the disparity of semitones. This
is not to omit without consideration
that when the fifths acquire perfection
on the lute by submitting to augmentation,
the fourths necessarily come
to lose part of their imperfection by
being relieved of their extra size.
This is the opposite of what happens
to the [keyboard] instrument, as has
been demonstrated. With this, I believe
that I have satisfied each one
of your requests.
Strozzi: Yes, that is true, I
desire, however, that you next clarify
another problem and afterwards show me
the way I must follow when I want to
construct any monochord which we have
considered.
283
Bardi: As you see, I am ready to
satisfy [your request), but tell me what
doubt has newly occurred to you.
Strozzi: The doubt is this. Since
the tuning of the lute is so much nearer
to perfection than that of the keyboard
instruments, I am very surprised that the
players of these [keyboard instruments),
after having learned of it, have not
changed to that [new) manner [of tuning).
It happens that in addition to its
excellence, [the lute] has in each fret,
high and low, any desirable interval of
the intense diatonic and of the chromatic
tonikon of Aristoxenus, as you have told
me previously. On the contrary, in the
string distribution used by the keyboard
instruments, which is different from the
one used by the lute, many [of these
intervals] are commonly lacking. Because
of this the player of this [keyboard
instrument], however practiced and skilled,
cannot transpose a composition either into
one range or another by tone or by semitone--
I say that transposition by tone is done
284
more often--while such a composition is
very easily and usefully transposed.
Bardi: This is one of those things
which I have pondered many times, and
also have sought carefully [to see] if
[the aforesaid tuning] could be applied
to the keyboard and if it would end by
being just as it is on the lute. I find,
actually, that on the [keyboard] it
probably does not end by being that way,
but many, important defects are revealed
there.
Strozzi: How can it be that the
very same thing does not possess the same
nature in any place and time? Why, since
this particular distribution is delightful
on the lute, is it unpleasant on the
keyboard instrument?
Bardi: It can happen very easily,
as practical experience demonstrates to
us. Although the cause of it is difficult
to understand well, I do not wish for you
to be dissatisfied because I have failed
to tell you a few details which are
pertinent in this case. It is possible
Why the distribution
of the
lute does not
have a good
effect on the
keyboard
instrument.
285
that this results from having accustomed
the sense to hearing the intervals always
in the manner which I have said they are
contained. And thus, likewise, we are
used to having them in another different
way from the keyboard instrument. If we
now wish this [keyboard instrument] to be
tempered according to the usage of [the
luteJ, it is impossible to accomplish
this without offending the sense in some
particular places where the discrepancy
between Lthe intervals] is very apparent.
This is because we are already inured by
time to such a temperament, and the
particular places are those where it is
necessary to hear the difference between
the major and minor semitones. It
happens--as you know--that the lute has
the tone divided into equal parts and
the keyboard instrument has them separated
into unequal parts. The most obvious
places are between the diesis X of G sol
re ut and the b molle of a la mi re. It
[occurs] between the diesis X of d la sol
re and the b molle of e la mi, because in
286
the keyboard instrument, when we have
tuned the diesis X of d la sol re by a
major tenth with h mi, and when we intend,
as in the lute, for the same diesis%( to
perform the very same function which the
b molle of e la mi performs in being used
also to [form] a minor tenth with 0 sol
faut, it becomes so languid and flat that
it is intolerable. If, on the contrary,
in sharping it as much as suits it we
tune it well in accord with C sol fa ut
as a minor tenth when it is needed, then,
to serve as a major [tenth] with h mi, it
is so greatly stretched and tightened
that it cannot be tolerated. The same
happens in the other places where such a
difference exists between them. Such
intervals as these also can more greatly
reveal their quality and nature in the
keyboard instrument than they do in the
lute and in the viola [d'arcoJ, due to
the conformity which the practice of
singing has with its temperament, in
addition to the quality and quantity of
the sound compared with the diverse nature
287
of the situations which concur with its
production. This is because the sound of
the strings of the keyboard instrument
assails the hearing with very great
violence, but those of the lute do not.
One sees clearly, therefore, that the
sound of the strings of the [keyboard]
instrument, tempered according to the
disposition of the intervals which appear
between those of the lute (including the
thirds, the tenths, and also the major
sixths), greatly offend the ear by their
great sharpness. One also sees that the
reason why the keyboard intervals manifest
their quality there more than the
lute-intervals is only that the material
of these strings and the agent which
plucks them possess more force and
efficacy, because of their activity, to
wound the ear with greater vehemence.
This does not happen to those of the
lute because of the different quality of
one subject and the other, and the same
difference and variety can be considered
288
in iron and wood when they have been
inflamed, because iron heats with more
force and quickness than wood and introduces
its new form into that object which
is fit to receive it. As a result, the
harp is tempered according to the manner
of the lute, where the strings and the
agent which plucks them are the same.
In the harp, the sounds will undoubtedly
become no less endurable than in the
same lute. They would be unbearable,
however, in the harp every time that the
agent and the material of the strings were
changed, as we have experienced many times.
The same also can be considered when the
harp is struck by a ball of ivory or by
some strong, heavy, smooth piece of wood
after it has been beaten over a hard,
polished stone, and when it is struck by
another [object] made of cork which has
been struck with the same force over a
rough plank of willow. Let us also consider
the comparison between a blow
passing through an acute angle and one
passing through an obtuse angle. Another
example of a keyboard instrument which
the Elector August, Duke of Saxony once
presented to the happy memory of Albert
the Great of Bavaria comes to my mind in
this case, which is most effective of
all. .This instrument6 8 has strings
according to the usage of those of the
lute. They are bisected like those of
the viola Ed'arcoj by an arranged skein
cleverly made from the very same coarse
hairs as the ones from which the strings
of viol bows are made. This skein is
very easily set into gyration with a
foot by the same person who plays it,
and it continuously cuts, by means of a
wheel over which it passes, that quantity
[of string] which is directed by [the
player's] fingers. I tuned this instrument
two years ago when I was at the
court, according to the usage of the lute,
and afterwards it sounded very well, not
unlike a set of dulcet-toned viols. I
68 See F. E. Kirby, A Short Histor
of Keyboard Music (New Yrk, 1966, p.
77, with regard to this type of keyboard
instrument.
A beautiful,
cleverly
constructed
keyboard
instrument.
289
290
come to the construction of monochords
and say that in having distributed the
strings of that Einstrument2, when it
occurred to me, according to the diatonic
ditoniaion, among the other possible
ways, I adhered to this order.
I first of all divided with the compass
the whole line, length, or string (as
we ought to call it) of the plank or
ruler over which I wished to adapt it
into four equal parts. I called the
extreme low point A, denoting with it
the pitch A re, and called the extreme
high point B, without any other designation.
In the point which divided the
total length into two parts (intending
them always to be equal), I then marked
a la mi re, in that which separated A re
from A la mi re, I marked D sol re, which
happened to contain the fourth part of
6 9 The basic string distribution
discussed here is diagrammed and explained
in Cecil Adkins, "The Theory And Practice
Of The Monochord," unpublished doctoral
dissertation, Department of Music, State
University of Iowa, Iowa City, Iowa, 1963,
pp. 232-233.
70See Zarlino, Istitutioni, pp. 51-54.
Way of constructing
the di onic
monochord.
Zarlino70 teaches
another way in
book 1 , chapter
40 of his Insti-
tutioni.
291
order to provide this D sol re with its
higher octave, I opened the compass and
made two parts of the interval cut in
half which was contained between this
D sol re and the extreme high point B.
At that point where one separated from
the other, I marked d la sol re. I
again divided the whole into three parts
and distinguished them with four points
as their limits. In the second of these
(like Boethius, always calling what is
in the extreme low part first), I marked
E la mi and in the third [point] I marked
e la mi. Placing one pole of the compass
on this point, I proceeded to place the
other over the line toward the high part,
and in that point where it ended, I marked
h mi. Taking one [compass pole] away from
that [point] and holding the other firm,
I moved it toward the low part and where
it rested I marked h mi. When this was
done, I divided into two parts the whole
line which extended between d la sol re
and the extreme high point, and then
placing one of the poles of the compass
292
on this d la sol re and coming with the
other toward the low part, I signed G
sol re ut at that point where it ended.
Wishing to find the higher octave of
G sol re ut, I opened the compass again
and divided the interval which was between
this G sol re ut and point B into
two parts in the center of which I
marked. g sol re ut. Without moving the
space which at that moment was included
by the two poles of the compass, I put
one of these [polesJ on G sol re ut
and came toward the low part with the
other. Where it rested I there marked
C fa ut, and found its higher octave by
dividing into two parts the distance
which is between this C fa ut and the
extreme high point B, in the middle of
which I placed C sol fa ut. When I
again divided into two parts the whole
interval which remained above it toward
that [high point BJ and placed one of
the poles of the compass on it, coming
with the other [pole] toward the low
part, I found the position of F fa ut
293
at that point where it ended. Its higher
octave was acquired by dividing into two
parts the space which extended from it to
point B. The higher octave was in the
center of that [space). Only the note B
fa was lacking to the diatonic monochord,
which I found in this way. I divided the
line which extends between B and f fa ut
into two portions. Then I placed one of
the poles of the compass on this F fa ut
and came with the other toward the low
part. Where it ended, there I marked
b fa. I found the lower octave of that
[note] by opening the compass so that the
extremes of its poles contained the entire
empty space which was found between this
b fa and the extreme high point. Taking
the pole of the compass away from this
and holding the other firm, I moved it
toward the low part. Where it rested, I
there marked B fa, and with this I had
completed the monochord of the diatonic
ditoniaion. The ancient Greek musicians
equipped this with a single string for no
other reason than to learn and understand
Why the ancients
equipped the
mono chord with
a single string.
294
more exactly the quality and quantity of
the intervals and to avoid at the same
time the deception which could result
between two or a larger number, because
of their inconstancy with regard to the
various situations which continually
threatened them. Since we now wish to
tune the said monochord according to
the ancient chromatic, it will alone
suffice, because the difference between
the diatonic and the chromatic consists
of nothing else, to arrange the third
note of each tetrachord which contains
the trihemitone in the upper part. The
proportion of this [trihemitoneJ falls,
as you have learned, between these numbers,
19:16. In order to do this, the
same order is preserved. Divide:the
line which is found between B and E la
mi, extreme high pitch of the hypaton
tetrachord, into sixteen equal parts.
When this [quantity] has been increased
by three similar parts toward the low
register, we will mark,at that extreme
point where they end, the chromatic
[49]
Way of composing
the ancient chromatic
monochord.
295
pitch D sol re, which with E la mi will
contain the trihemitone. The same order
will be preserved in order to mark the
other chromatic pitches in the other
tetrachords. There are some who reprove
Franchino in regard to this distribution
for having varied from the [way of the]
ancients the intervals of the middle
and the high of each tetrachord, constituting
[the middle intervalJ within
these limits 18:17, which is the smaller
part of the tone which should be the
larger [one] and the high interval
with the name of trihemitone between
these numbers, 153:128, which is augmented
from its true state. Since we
also wish to temper the same diatonic
monochord according to the usage of the
ancient enharmonic, it will suffice
merely to divide into two equal parts
with the compass the distance of those
two pitches which contain among them the
minor semitone (or limma). At that point
where they divide equally, the second
enharmonic pitch will occur in each
Method of constructing
the
ancient enharmonic
monochord.
296
tetrachord, and the third pitchj will
be that which was second in the diatonic
and in the chromatic. Thus, we will have
the smaller enharmonic diesis in the
lowest interval, the larger one in that
of the middle, and the ditone in the
highest Cinterval], according to the way
their authors created them. I warn you
also that in the conjunct monochord of
the chromatic genus, the three highest
pitches of the tetrachord synemmenon
become different from the disjunct, and
thus, likewise in the enharmonic, although
the extremes are always the same
in any genera of harmony. If we want,
then, to tune the monochord of the very
oldest diatonic according to the diatonic
syntonon of Ptolemy, it is necessary--in
a way of speaking--to change from their
places the third, fifth, sixth, and
eighth fret. That is, they need to be
sharpened. From this effect perhaps,
the name of intense (syntonicJ is
acquired, according to Dydimus its author.
In each octave, moreover, of the disjunct
Way of constructing
the syntonic
monochord.
The origin of the
term syntonon.
297
system, it is necessary, when we wish to
accomplish what was mentioned, to sharpen
C fa ut, D sol re, F fa ut, and G sol re
ut and in each of those of the conjunct
Eit is necessary to sharpen] b fa, D sol
re, F fa ut, and G sol re ut. In order
to do this, I maintain this order. I
divide the whole line into six parts, and
moving toward the high part where the
first [part] from A re terminates, I
mark C fa ut. I find the octave of this
[containedJ in half the space which exists
between this C fa ut and the extreme high
poing B, and I mark it thus, C sol re ut.
In the middle of these two pitches then,
I locate F fa ut and I find its higher
octave at the point where the interval
which is found between this F fa ut and
[point] B divide into two parts. After
doing this, I divide into nine parts the
line which is between C sol fa ut, and I
sign D sol re toward that part where the
first one [of the nineJ ends. I find
its higher octave by making two parts
out of the space which lies between this
298
D sol re and [point] B. In the middle
of this, I mark d la sol re. I then
find G sol re ut by dividing the entire
line which is contained between C fa ut
and [point] B into three equal parts,
and in the extreme of the first one
toward the high part, I mark G sol re
ut. The [higher] octave of this I find
at that point where the space which is
contained between G sol re ut and [pointJ
B divides into two parts. Therefore, I
sign it g sol re ut. I adhere to the
same order in finding b fa and its
replicate as in the diatonic, and I mark
them in the same manner. Thus I hastily
leave the method of composing the syntonic
monochord of Ptolemy. I come now
to show you the way which Aristoxenus
calls intense diatonic. After that [I
will come to] his chromatic tonikon
with which the distribution of frets
upon the lute closely corresponds. This
lute and also the viola d'arco, both
modern instruments, have unexpectedly
Method of constructing
the intense
EdiatonicJ monochord
of Aristoxenus and
the chromatic
tonikon.
299
been distributed in imitation of the
chromatic tonikon. On both of these
instruments the tone has been divided
into two equal parts, as has been stated
above. In this construction the square
of the second number is greatly essential,
or the double of that, which is eighteen.
We will employ this, however, to avail
ourselves of its effect with greater
clarity and facility in the distribution
we are seeking. I therefore divide the
whole line A B into eighteen parts, and
toward the high part (moving away from
the low part) where that first part
ends, I place the first fret.7 1 I divide
once more the whole remainder of the same
number of parts [i.e. eighteen and from
the very same band, I place the first fret
under the second. I now proceed to
7 1 Galilei's approximation for equal
temperament, described here, is based on
the principle of similar proportion, in
which a correct ratio is chosen for the
semitone and then applied twelve times
to a string. Galilei's method was adopted
and emulated by many makers of instruments
and was widely discussed by writers for
two hundred fifty years. See Barbour,
Tuuning and Temperament, pp. 57-64.
The second number
squared is 9, and
its duple [isJ 18.
300
distribute the space which remains beneath
the frets, always in that very same order,
up to [the number of] twelve, which brings
me exactly to [the point] where half of
the entire string terminates. The first
and lowest octave of this, I find, has
been divided into twelve equal semitones
and six tones, as Aristoxenus instructed.
In order to do that, no other number
except eighteen is suitable, which is
obvious from this [fact). It does not
agree in any way with the first seventeen,
because that would give us a smaller number
of frets than we needed, and we would have
even fewer from sixteen and from fifteen.
It also does not agree with nineteen, because,
on the contrary, we would have a
larger quantity [of frets), and a great
deal more from twenty and twenty one.
Therefore, eighteen is the proper divisor,
rather than a larger or smaller number,
although the same thing happens to this
7 2 See Zarlino, Istitutioni, p. 24,
concerning the "sesto", or compass.
Zarlino7 2 contends
in chapter 14 of
the 1st part of
his Institutioni
that the aperture
of this compass
is exact y the 6Th
part of the circle,
which is false.
Warning.
301
[eighteenJ which also happens to the
compass in wanting to measure the circumference
of the circle six times exactly
with its aperture, which extends--as you
know--from the center to its circumference.
For this reason it has become known as the
"sesto". I therefore warn the industrious
agent to remove, with his discretion and
diligent attempts, that small inconvenience
which exists between the measurers and the
measured.
Strozzi: I have understood the whole
thing very well, but please clarify another
related problem.
Bardi: Tell me!
Strozzi: What order did the ancient
musicians follow, with regard to the numbers,
in distributing the strings of that
first species which were peculiar to the
chromatic and enharmonic of the Greater
[Perfect) System? What induced them to
leave between the first and second note
of each tetrachord of the enharmonic such
a space as 512:499, and Eto leave] between
the second and third [note] of the chromatic
Why the compass,
a geometric
instrument, is
called "sesto".
[50)
302
this [space of] 256:243 rather than
another Espace]? Tell me also if these
two species of harmony have any other
relationship and agreement with the
diatonic than the superficial
appearances of numbers?
Bardi: You will recall that I
showed you with numbers that which
I just now have demonstrated with a
line. I warn you, however, that by
examining carefully the example of
the hyperbolaion tetrachord of the
ancient diatonic ditoniaion, every
one of the difficulties mentioned will
be removed. This tetrachord has been
proposed by me because it has smaller
numbers than the others of its species.
The differences of its numbers are
more easily understood by the
intellect, as you see here [Figure 107.].
A
4-:1
0
U)
0
C',
.*d F
!04
COW
140
or
p+
0
0
43
43
0
0
-d
0
ofr
cd
0
0
0
.- I
Hn
0
0
0c
4-
0
14
0
4"9
U)
Co
4-1
CH
C',
Li
-)
CHQ
N
N
0
td
r-1
0
I
N
0CJ
0
0
0
«-1
0
10
0
0
0
43
0
C',
4~00
303
O
0
N
0
K6-1td
H
0
H
r-i
.0
"1
rl0
U~
(1)
304
In the first place, the extreme notes of
any tetrachord in each genera and species
of harmony are always the very same, as
you know. They are therefore called
stable, while those which have the power
to change the diatonic system into the
chromatic are the thirds of each tetrachord.
The occasion, moreover, which
induced the ancient musicians to permit
that particular interval between the
trite and the paranete rather than some
other was taken from here. They added
to the number which denotes the paranete
of the diatonic, half the difference
which is found between it and the nete
of the same tetrachord, which is 144
since the whole (as you have been able
to see in it) is 288. This half, when
added together, according to what I have
said, with 2592, makes 2736, and with
this number they designated in that
tetrachord the chromatic paranete. They
left to the other notes the same numbers
and positions as in the diatonic, as
What strings are
stable, which are
movable and which
are not entirely
stable or entirely
movable.
305
appears in the example, and did likewise
in the other tetrachords.
Aa Netehyperbolaion 2304
g Paranetehyperbolaion 2736
f Tritehyperbolaion 2916
e Netehyperbolaion 3072
[Figure 108.]
Now because the principal difference
which exists between the diatonic and
chromatic distributions, and the
enharmonic consists of the position of
the second note of each tetrachord,
they therefore followed this procedure
in order to denote it. They added to
the number with which they marked the
trite of the diatonic [generaJ, which
is the same as the chromatic [oneJ,
half of the difference which is found
between this diatonic trite and the
netediezeugmenon, which is 78, since
the whole is 156, as you have learned.
This trite in the enharmonic, then,
was designated with this number, 2994,
and the others [were designatedJ with
306
the same [numbersJ, according to what is
seen described here [Figure 109.J.
Hyperbolaion Tetrachord
of the Ancient Enharmonic
Aa Netehyperbolaion 2304
g Paranetehyperbolaion 2916
f Tritehyperbolaion 2994
e Netediezeugmenon 3072
[Figure 109.]
And [the ancients) observed the same in
all the other tetrachords in order to
convert them from diatonic into chromatic
or enharmonic, and the very same difference
which is found between line and line of
the three first monochords is found between
number and number of these last ones.
[CHAPTER III)
[A CRITICAL DISCUSSION OF
ANCIENT AND MODERN
MUSICAL ART]
[Bardi:) I now come to deal with [51]
tones and the modes of the ancient musicians,
and omitting, in order not to be
long and tedious, the opinions of all the
other writers, which are many and diverse
with regard to their number, name, and
place, we will discuss only the three most
famous, the first of which will concern
the Aristoxenians, the second the Ptolemaics,
and for the third we will consider
that of Boethius. It was, moreover, the
opinion of Aristoxenus (according, however,
to what Aristides Quintilianus,1
Bryennius,2 and Euclid3 in the Introduction
which he makes of music--given, however,
that this introduction) is his--
1Aristides Quintilianus' work may
be found in a German translation by Rudolf
Schafke. The original Greek text
appears in Marcus Meibom, Antiquae musicae
auctores seem, 2 vols. Ti :teT-
"i,1"2.
307
308
tell us) that the modes must be thirteen
[in number) and not seven, or eight, or
another smaller number which was recognized
before him. [This was] because,
in considering those of which many honorable
writers had made mention in his
times and earlier, and which were later
mentioned by Ptolemy and Boethius in addition
to many others, and finding between
the Lydian and the Mixolydian and
[between] the Hypolydian and the Dorian
the distance of a minor semitone (or
limma), he proceeded--in my opinion--to
ponder, thinking to himself as follows:
Since from sharping and flatting
the system by a minor semitone
there results between these modes
a clear and apparent difference
of affection, or at least this is
operated more in one than the
other according to the degree of
its effectiveness, how much more
2See Manuel Bryennius, Harmonica,
in John Wallis, Opera mathematca,T
vole. (Oxford, 1Z - 69)TTIT,417-422.
2As Galilei suggests, the Introduction
is not by Euclid; the actual author
is :leonides. See Oliver Strunk, Source
Readings in Music History (New Yor,
, 5)vpp~~3 %",
There were 13
modes according
to Aristoxenus,
Aristides Quintilianus,
Bryennius
in the 1st
of his 3 books.
(Manuel Bryennius
in the 1st
book at the
th c apter),
&uclid in the
Introduction,
Consideration
of the author.
309
so would an entire half a tone
make it? And since such a variety
of harmony and affection is
found between the aforesaid modes,
for what reason will no other
pitches be distant from each other
by such an interval?
And while he was pondering these reasons,
he divided the five tones and the two minor
semitones which contain in themselves the
species of the diapason which served the
Dorian mode into twelve equal parts, and to
each term of these parts, which came to be
thirteen since the intervals which constituted
the mean of one of these tones were
twelve in number), naming them and arranging
them in the manner which is seen
in the notated example [Diagram V.] which
follows. From the words [of Aristoxenusj,
this most noble musician and philosopher,
Ptolemy4 took occasion to reprove him for
many things, among which there are three
of some consideration. Therefore, I want
to relate them to you and afterwards prove
that such libel was unjustified.
4 See Ingemar During, Ptolemaios und
Storiios ber die Musik ( tteborg, 177),
pp, - ;3739;W4 6T6 3~;79-81.
Ptolemy reproves
Aristoxenus for
many things in
the 1st [book)
at the 9th and
12th chapters,
and in the 2nd
Sbook) at the
9th and 11th
chapters.
310
[52]EDIAGRAN v)
DEMONSTRATION OF THE THIRTEEN MODES
ACCORDING TO THE OPINION
OF ARISTOXEUS
WN
N-I
II
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22
.
2d
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Hypolydian
1cjiL
441L
Ii
a )1
Hyperaeolian
or low Lydian w
Hypophrygian
Hypoiastian N
or low Lydian A
Hypodorian
Natural
system of
the voice 0
K4
J-L
Hypermixolydian,
or Hyperphrygian
Hypoiastian
-x
2ctl01iy-,ooP4
I..Ql
-Al1
J Hyperdorian,
Mixolydian, or
Locrian
Lydian
Is
2
1W.
~JPL
a)
4-11
OJdL
7t
I) e
Aeolian, or
lowest Lydian
I Phrygian
Iastian, low
Phrygian, or
Ionian
14F 5JL
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r i1
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rs-.41
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-
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311
The first [consideration I mentioned) 53)
concerns the distribution of strings,
the second concerns the division of the
tone into equal parts, and the third and
final Econsideration) concerns the number
of modes. Ptolemy, therefore, said
this:
If half of a string, for example,
which stretches over a
plane surface is divided into
twelve equal parts with the
compass, it will be clear that
when one considers] the quanity
of sound which the whole
contains with the half, which
comes to be a diapason, the
eighth and ninth spaces will
contain the greater part of it,
rather than the first and second
spaces.
With this method of measuring, one who
pursued the situation very thoroughly
would find that one of the last parts
would contain four, five, and more times
as much as the first and second. For
the same reason, the frets on the neck
of the lute and the viola d'arco are
seen to move progressively closer and
closer together, in proportion to which
the sound of the same string, as it is
shortened, becomes progressively higher
312
in pitch. Consider a string stretched
over a plane surface which is, for example,
an arm's length. The octave
above this string is obtained from its
half (which is half an arm's length)
everytime [this string) is struck together
with its whole or a similar
string. And if [this string) were two
arm lengths, the diapason above it
would count one, and the diapason below
it would count four. By means of
this demonstration which Ptolemy makes,
it appears that "he has" (as the proverb
goes) "reasons for selling". But
the fact is not so. It happens that
Aristoxenus never understood or said
that frets would have to be distributed
as, for example, has been said with
regard to the neck of the lute, after
having first divided half the total
length of the string or line into
twelve equal parts. For Aristoxenus
knew very well--as I have demonstrated--
that he had to distribute the quantity
of sound into equal parts, not the
Aristoxenus
(is) defended
by the author.
313
quantity of the line, string, or space,
since he was acting at that time as a
musician concerned with the body of
sound, and not as a simple mathematician
concerned with the continuous quantity
Cof soundJ.5 He felt, consequently,
that the first fret alone would occupy
the ninth part of [the string's] entire
half, that is to say the eighteenth part
of the whole, and the second Ewould occupy]
the ninth part of that which remained
to the same first half after the
first semitone had been extracted. The
third fret, then, would likewise occupy
the ninth part of that space which remained
of half the string after the first
and second semitones had been removed, or
to state it yet another way, the eighteenth
[part] of the whole. Thus, the
5Dividing a string into equal parts
does not insure that the same quantity of
sound will be contained in each of these
parts, for the size of the interval
sounding increases as the amount of
string used decreases. This phenomenon
becomes apparent immediately when one fingers
an ascending scale on a stringed
instrument.
Method of placing
"just" frets on
the lute and the
viola Ed'arco).
314
others [proceed] in order, always keeping
in mind the condition mentioned above,
which is that the half of the string
must be taken, not at that point where
it terminated the first time that it was
divided into two equal parts, but where
it terminated after that number of semitones
which was necessary had been removed.
And this, as I said another time,
is the rule to be observed when one
wishes to distribute the frets "justly"
on those instruments which are [being]
investigated, like the lute, the viola
d'arco, and others. Ptolemy, in addition,
reproves Aristoxenus that the tone
cannot be divided into two equal parts,
and he wishes to prove it to him demonstrably,
speaking in the following
manner:
The tone is firmly contained between
18 and 16, so that no other
number except 17 will enter in
between [them]. This 17 when considered
as the divisor of the
sesquioctava, comes to divide it
into unequal parts because that
which is contained by the sesquidecimasexta
[i.e. 17:16] is a
315
greater part than that which is
contained by the sesquidecimaseptima
[i.e. 18:17] by the following
interval, 289:288. Whereupon,
it follows necessarily that the
tone cannot be divided into two
equal parts.
There is no man, however slow-witted, who
doubts this, with regard, however, to the
science of arithmetic. But Aristoxenus
neither said such a thing, nor understood
it, except in the manner which I demonstrated
to you, particularly in placing
frets of the lute, in which one can actually
divide each musical interval into any
desired number of equal parts, [in a manner]
no different from what [is done] by
means of the monochord. The sound, in
this case, is considered by the musician
as qualitative, not quantitative, although
Daniel Barbaro6 Lin his commentary) on
Vitruvius understands it to be the contrary.
The demonstration of Ptolemy is
the very same, for he says that it is not
possible by means of numbers to accommodate
6See Vitruvius Pollio, I dieci libri
d'architettura, translated by aniefl"EFr
aro TVenice-, 1629), pp. 227-243.
(Daniel Barbaro)
in chapter 4 of
the 5th [book).
any other interval between smaller terms
of the diapason. Nevertheless, between
the hypate and the nete one finds, in
addition to the other strings, the lichanos
and the paramese which produce the third
and the fifth in the grave register and
the fourth and the sixth in the acute.
Similarly, between F fa ut and f fa ut
of the lute one also finds (besides the
tone divided into two equal parts) the
h mi, which separates it into two identical
portions, as has been said. In
addition, [one finds in considering] two
strings, each two arm lengths in span,
which are tuned in unison, that everytime,
for example, that the distance of
of an arm length and a half is taken
from one, and three-quarters of an arm
length [is taken] from the other, and
those portions are sounded together, the
result will undoubtedly be the octave.
Concerning, then, the number of modes,
[I say that] what we have discussed above
in the person of Aristoxenus is so very
clear in itself that there is not any
contradiction.
Comparisons.
316
317
Strozzi: All is well at this point,
but who is the one who persuades us to
believe that at the time of Aristoxenus
there were modes in use [which were] distant
from each other by a minor semitone,
as you have said?
Bardi: Aristotle, in consideration
of Philoxenus, who--as you will soon learn--
was the inventor of the Hypodorian mode
(the last to be discovered) and Ptolemy,7
in the tenth chapter of the second book
[of his Harmonics), show the intervals
which were used in them according to the
design of their inventors.
Strozzi: I am surprised, therefore,
that Ptolemy wrote so boldly against
Aristoxenus, knowing how many volumes he
had written on the science of arithmetic
and [knowing] how worthy he was, since,
in addition, he had not only been a pupil
of Aristotle and a contemporary of Theophrastus,8
but had [also] acquired all
7 See During, Ptolemaios, pp. 76-79.
8 Theophrastus (?-287 B.C.), a Greek
philosopher and natural scientist, belonged
to the Aristotelian school.
318
the doctrine of Pythagoras from Xenophanes,9
a noble Pythagorean. Moreover,
as you have demonstrated, [Ptolemy] was
greatly at fault.
Bardi: This is not exactly surprising,
since the same thing has happened
between many others. What more
can be said than that which Aristotle
first wrote concerning matters of philosophy
against his divine preceptor,
Plato, even while he was still alive.
It is true that this [manJ, as a friend
of truth, wanted (as was suitable to a
philosopher) to have more respect for
it than for his master, and for Socrates
as well. Ptolemy also wrote against the
opinion which the Pythagoreans had of
the diapason plus diatessaron; his
opinion has been vigorously refuted by
9Xenophanes is credited by many ancient
writers as the teacher of Aristoxenus.
He is not to be confused with the
sixth century B.C. Eleatic philosopher
of the same name.
Plato reproved
by Aristotle.
319
Jacques LeFevre d'Etaples.10
Strozzi: It is actually so, but
may we please examine another matter
concerning the distribution of modes
of Aristoxenus, which causes me to
wonder greatly.
Bardi: Tell me what it is.
Strozzi: You said before that
Aristoxenus divided the tetrachord into
sixty tiny, equal parts, twelve of which,
in his intense diatonic, he gave to the
first and lowest interval, twenty-four
to that of the middle, and the remainder
to the highest, so that the semitone did
not come to occupy an entire half of the
sesquioctave (or tone), but a lesser
distance.
Bardi: That is true.
Strozzi: Now, for example, since in
the species of diapason which is found between
E la mi and e la mi, two tetrachords
were separated by the tone of the disjunction,
each one of these came to contain
0See Jacques LeFevre d'Etaples,
Musica libris guatuor demonstrata (Paris,
33T~ p..
[Jacques LeFevre
d'Etaples in
chapter 2 of the
3rd bookj of the
Elements of Music.
320.
six of these modes, which made the number
of twelve. That Emode) which then divided
the said tone of the disjunction into two
equal parts, came to be the Ionian. This
was the reason that each of its adjoining
extremes was found farther removed from
it than any others Ewhich were] connected
together.
Bardi: I am pleased with this particular
doubt of yours, since it indicates
that you are coming closer to understanding
Ethe situation), but I must warn you that
Aristoxenus did not divide one member
and then the other in such a way, but,
as I have said, [he divided) the entire
body of the- diapason. For this reason
(if you recall the specific words which
I used previously with regard to his distributions),
I said that they, =that is
the diatessarons, were always arranged
in conformity with his designs.
Strozzi: Does one find, moreover,
in his systems that the fifths are diminished
and the fourths augmented as on
the lute? Similarly, does one find these
intervals compared to one another?
Fifths and fourths
[are] altered in
he distributions
of Ariztoxenus,
321
Bardi: Yes, Signor. They are compared,
however, with those [intervals]
that are not contained by the sesquialtera
and by the sesquitertia, which were
assigned by Pythagoras in their true form.
Strozzi: This seems to me [to be]
quite an inconvenience [which is] not
exactly worthy of Aristoxenus that he
distributed the strings of his systems
so that the fourths and the fifths continually
are obtained outside of their
true proportions.
Bardi: Do not be so easily provoked
to reprove Aristoxenus, because I
am here needing to defend him from all
those Cmen] who attempted to speak against
him, since he was truly one of
the most judicious and learned musicians
that the world has ever known. But
listen, please! How can you believe
that the consonant intervals are sung
today, [considering that they require]
more excellent singers with discriminating
hearing than can be found?
322
Strozzi: I believe that they are
sung within their true proportions, also
that some of the artificial instruments--
as you have clearly made me see--play
nearer to these [proportionsJ and others
farther away from them.
Bardi: Which Cinstrument) is the one
which has the more perfect consonant intervals,
the lute or the keyboard instrument,
given that they are perfect when
they are in their above-mentioned forms.
Strozzi: The lute.
Bardi: What would you say if I made
you realize that the singing of today--
with regard to the perfection of the intervals--is
no less imperfect than the
playing?
Strozzi: To me, this would be one
of the most novel things that I could
ever imagine, and I could not believe it
in any way.
1 1See Gioseffo Zarlino, Istitutioni
harmoniche (Venice, 1558), pp.T15-37"
The singing of
today is little
different from
the distributions
of Aristoxenus.
Zarlino in chapter
45 of the 2nd
part of the Insti-
tutioni.
323
Bardi: Now, take note! Will you
concede to me that, in the diatonic genera
which is in use today, every musical interval
is always sung in the same proportion?
[Will you also concede], for the abovementioned
reasons, that the major third
is consonant and the ditone is dissonant
and that the space and contents of the
tone are approximately one half of our
comma less than the sesquioctave?
Strozzi: The reasons which you conveyed
to me earlier in this matter Cnowj
convince me to believe and say yes.
Bardi: Since this is true, therefore,
it is necessary that every fourth always
becomes augmented when it is sung according
to the usage of the modern practice,
and similarly, every fifth becomes diminished.
I will prove that to you in this
way. Let us ascend, for example, singing
from 0 fa ut to F fa ut. I say that this
interval is presently an augmented fourth,
because when ascending, then, from there
to G sol re ut with a space smaller than
the sesquioctave by the amount previously
324
mentioned, it should follow that a fifth
is found so diminished that it is dissonant,
in order that the sesquitertia
may exceed the sesquialtera by a sesquioctave
(or tone), as you know.
Strozzi: I do not deny that the
sesquialtera does not exceed the sesquitertia
by a sesquioctava, but only that [55]
C fa ut is found in such a way when
singing away from G sol re ut by an imperfect
fifth, because however greatly
the fourth which was sung--as you have
said--from 0 fa ut to F fa ut was augmented,
the tone which is between F fa
ut and G sol re ut may well be diminished,
as you just now said in considering
the fifth in its true form [to be) composed
of an augmented fourth and a diminished
tone, and so it is possible
for a perfect fifth to be contained in
sesquialtera proportion between this
0fautandGsolreut.
Bardi: It is true that I said
what you say [I did), but in that place,
the augmentation of the fourth was equal
325
to the diminution of the tone, which does
not happen here. Besides, if what you
say is true, which it is not in any way,
it follows that when ascending from G sol
re ut to c sol fa ut with a fourth of the
same proportion as the first, a perfect,
consonant diapason will not be found between
Ec sol fa ut] and C fa ut, which
is impossible in the manner which is sung
today. Or if this is actually the case,
it will be necessary to concede .that the
diatessaron which is sung from G sol re
ut to c sol fa ut is a different proportion
and measure from that which was
originally between C fa ut and F fa ut,
which is not a suitable thing to say.
It therefore follows necessarily--contrary
to the common opinion--that the
fifths are sung today diminished from
their true states, and the fourths fare
sung] augmented. Because of this, (according
to what I said) one comes, from
the octave onward, to sing any other interval
outside of its true proportion,
and consequently Ein a manner which isJ
326
dissimilar to those which are contained
by the senario and by the syntonon, although
the general public approves them
as perfects and is entirely satisfied
(because of not having heard the true
ones) and, since every hope of being
able to improve them has been taken
away, one could also say that the intervals
which are sung today are, due
to the inequality of semitones, more
conformed and similar to those which are
found in the temperament of the keyboard
instrument than among those of the lute.
I say to you in addition that the fifth
is understood with greater discrimination
by the general public according to
the proportion which Aristoxenus gave it
than within'the sesquialtera, its original
form. This, I firmly believe was
the result of bad usage, which corrupted
the sense, because the fifth, within the
sesquialtera, not only seems to possess,
but actually possesses a small degree of
hardness due to the extreme amount it
can be augmented, which avoids my saying
(together with others of delicate hearing)
that it is harsh. In the manner of Aristoxenus,
however, it seems that the small
degree of diminution gives it grace and
causes it to become more in keeping with
the taste of today, that is, soft and
languid, and I do not believe that that
happens for any other reason except being
accustomed to hearing them continually
under that form, or a similar one. From
this is derived an important and effective
argument to persuade [us of] that which
you will soon learn at the appropriate
place, which is that this mode of singing
was learned from stringed instruments,
and particularly from those which had no
keys, like the lute and the viola d'arco.
Strozzi: This is such a valuable,
new thing that nothing like it has ever
been heard. Do go on!
Bardi: Now consider the imperfection
Which is caused] by this sole abuse in
the music of our times and how much the
general public is deceived. [Consider
Warning.
327
328
also] how difficult it is [for the general
public] to know the truth of things and
how little knowledge it has of the true
music, not even having known up to today
the size, the quality, or the nature of
the singable, audible intervals which are
the simple elements and principles of music.
These things, together with others
pertaining to the profession of music
(which are many) were understood by Aristoxenus
and most of the ancient musicians
with supreme excellence. In addition, the
perfection and imperfection of the intervals
did not mean a thing in the world to
the way of singing of those times, since
they did not employ them (as you will
learn) in the manner which we use. The
quality of these intervals can also be
considered--as I said--in relation to
highness and lowness between one and
another constitution of thirteen modes
and the other which he made--and not
fifteen, as others have said--by comparing
one pitch to another. For example,
[let us consider the] proslambanomenos
of the [Greater Perfect) System in the
Zarlino 1 2 in chapter
3 of the 4th
part of his Institutioni
says t
Arxstxenus made
15 modes,. and in
chapter 16 of the
2nd [book) he disparages
his distributions
without any
reason.
329
Phrygian mode. I say that [this Phrygian]
proslambanomenos is higher than that of
the Hypodorian by a flat fifth, that is to
say, the mese of the Hypolydian is sharper
by an augmented diatessaron than that of
the Phrygian.
Strozzi: I have understood very well
and I am now satisfied about everything,
but [firstJ tell me, why are the flat fifth
and the diminished s.] fourth tolerated
and, on the contrary, why are the sharp
fifth and the flat fourth not permitted?
Why, then, does the octave not bear alteration
in one part nor in the other?
Bardi: I have already told you that
the fifth, with regard to bad usage, when
it is in its true form represents to the
ear a slight state of sharpness--in order
not to say tedium, as others do--rather
than anything else. Now think how much
more [tedious) it would become by being
stretched more greatly than the [state in)
which it is contained by the sesquialtera,
1 2See Zarlino, Istitutioni, pp. 299-
300; 82-86.
330
its proportion. In addition, in the
playing and singing of today, it is
not possible, for the reason which I
have demonstrated to you, to proceed
in a manner other than the one which
you have seen previously. It happens,
then, that the octave is not allowed
to be diminished or augmented. This
takes place not only through its perfection,
but through being composed
of it, since a consonance of the same
quality is taken many times, and if
two of these were altered, either up
or down, and afterwards were added together,
the extremes would be intolerable.
This thing does not happen to
the other consonances since only
dissonant intervals are composed of
these, and being so constructed, the
ear does not detect that small difference
which is found among them so
easily.
Strozzi: You have satisfied [my [56J
curiosityJ to a great extent. Continue,
331
however, the discussion of modes which
you have begun, according to the opinion
of the Aristoxenians.
Bardi: The followers of the said
Aristoxenus added two additional modes
to his [original] thirteen, and placed
these new modes toward the top. The
extreme notes of their diapasons came
toward that part [which was] outside the
ordinary system. For this reason they
were later refuted by many because they
considered that the human voice was
distinguished in three parts, that is,.
low, high, and middle--Martianus Capella 3
makes particular mention of this--and
that the number of thirteen could not
easily be distributed in three equal
portions. Therefore, they extended it
13William Harris Stahl, Martianus
Capella and the Seven Liberal Tshew
York and n n", " T), p.211 Voice
production is divided into three classes:
continuous, discrete, and intermediate.
Continuous pitch variation is used in
flowing conversation; discrete pitch variation
in music; the intermediate movement
of the voice is used in the recitation
of poetry." Compare Adolf Dick,
Martianus Capella (Leipzig, 1925), p.
500.
[Martianus Capella)
In the 9th and final
book where he
deals with music.
332
up to the number of fifteen, giving the
name of principals to five of the middle,
which are the warlike Dorian, the changeable
Iastian, the religious Phrygian, the
plain Aeolian, and' the querulous Lydian,
and [giving the namesJ of plagals to the
five modes below them, denoting them with
these titles: Hypodorian, Hypoiastian,
Hypophrygian, Hypoaeolian, and Hypolydian.
They called the five modes Cwhich were]
above Ethe principals] authentics, distinguishing
them as Hyperaeolian, Hyperiastian,
Hyperphrygian, Hyperaeolian, and
Hyperlydian. Through this order, the
principals became higher than the plagals
by a diatessaron, and the authentics were
found lower by a similar interval, so that
14
See Glarean, Dodecachordon, 2 vols.,
trans. by Clement A.~Miller (Rome, 1965),
I, 128. Compare Apuleius, The Works of
Apuleiua (London, 1853), p.Y76~~nTIgenidas
was a certain flute-player, a
honeyed modulator of every tone, and a
skilled performer in every mode, whether
it were the simple Aeolian, the varied
Asian, the plaintive Lydian, the religious
Phrygian, or the warlike Dorian."
In book 2, chapter
10 of Glareanl4
with the testimony
of A uleius in the
1st [book] of the
Floor dium.
333
from the Hypodorian to the Hyperdorian
was the distance of a minor seventh,
that is to say, five tones or ten semitones.
This was very important (according
to the usage of Aristoxenus, however)
for it greatly favored their intention,
since they had found that the extremes
of the thirteen which he made responded
by octave, and not by seventh, with
those of Ptolemy. The highest of these,
Which was] called Hypermixolydian, was
none other than the replicate of the
Hypodorian. They did the same [thing),
moreover, to the Hypoiastian and to the
Hypophrygian by raising them an octave,
forming with these the Hyperaeolian and
the very highest Hyperlydian. These became
ordered and disposed in the manner
which is seen in the demonstration below
[Diagram VI.), marked clearly with their
names. After these LAristoxenians) came
Ptolemy, whose order and opinion, being
very clever and difficult to understand
well, we will reserve in order to demonstrate
it last. We will first state,
[DIAGRAM VI)]
DEMONSTRATION OF THE THIRTEEN MODES, ACCORDING TO THE
OPINION OF ARISTOXENUS, WITH TWO ADDED IN THE
HIGH PART BY HIS FOLLOWERS, WHICH IN
ALL MAKE THE NUMBER OF FIFTEEN
4
4 i dcjr ir
4~~ iaV
0)
4
)
- I - I - I 4 I I - 4- - I -
-
______
II"
'-I I
01 1
T l
V"-l
441 1
oil
L4 IJJ
Iaatian j 4 4
Dorian Q
Hypolydian
lypoaeolian
Hypophrygian c4
Hypoiastian c
Hypodorian
0d
r Lf
Hyperlydian
(added)
Hyperaeolian
-(added)
Hypermixolydian
Hyperiaetian
Hyperdorian
Lydian
Aeolian
Phrygian
rw U
.
I ldLy
vF 144
C)
-4
I pi 4o4)12
j42aC)
1
'42
14y14 K
I
r40Lto]44I
4C
4 0
AC)
. . . I. I
raj
Ai
334
[57]
01'CLJ
0
4O
:dI 0f 4 4 cb t
A
b U
co 01 041
44 w I' bl f l
pq 44
r I
IV -W
I - 1 - 1
i------
I I L-__I
.
7111 T'E""T
- I_-."
_--
-
1"
.
1 -1 - ITT
i _4 - I I I i v - I -- - 2 am am..Oft1___I_I I 1 1
I -I- - - 19 1
,
.1-_
.
jr:4
__-T --7:K7
T
I
1 "T
1
'L^
d
335
however (since it is least difficult),
that which Boethius felt about it. He
was of the opinion, although such a view
had first been refuted by Ptolemy, that
there were eight modes. He maintained,
in addition, that each one's particular
system proceeded from low to high in the
same order, with the same steps, and with
the same pitch names which were used in
the natural, common [system], which is
that which serves the Dorian mode. He
made the Phrygian higher than the Dorian
by a tone, and the Lydian higher than the
Phrygian by a similar interval. He also
insisted that their plagals corresponded
to the diatessaron below, that the Mixolydian
was a semitone higher than the
Lydian, and that the Hypermixolydian
corresponded by octave with the Hypodorian
(in imitation of those of Aristoxenus),
ordering them and displacing them, then,
in the manner in which they are seen below
in the demonstration [Diagram VII.] under
their individual names.
[DIAGRAM VIIJ
DEMONSTRATION OF THE EIGHT MODES, ACCORDING
TO THE OPINION OF BOETHIUS
I I - i I I, - I I I I -- --- a U T- w
7ol
4
t I * *+~L.r~4 -'.ION!.
K
Dorian *j
0
Hypolydian
Hypophrygian
K
0
p4
I
0
flI
0
Hypodorian
1 tr' 1 %, M
N
4
0
%a11%-%
0
4
41---. W-+q
m4
a-
0
t
Vz: ~1
mg~
I0
«1a
- mil .b~ . .. it
N
-Q
0
0
4C)
10
d4
J
0
Hypermixolydian c
l-
Mixolydian
p4
Lydian
fr
:1
M4
21IT
Phrygian
tYFirst
species
Second
species
Third
N species
Fourth
species
Pifth
species
Sixth
species
m seventh
species
Replicate
of first
t-+ species
,O P
N
'-4
0
.1
0
0
mj0
iTI
II
K- *
h whINS
336
582
C,,
(d
41I
A
I N
C
j bu!
j a
Cn
' _ _ GO
a I I m a - 0 -Wi 1 i--------- T
I
"
r, ._
IIi d i .1
'1 aVIME FongILL
1 I T
I _--I1
1
t. ^1 +" i M1 lUI moan 41 d_ a
-SS
W ol
b
f a cl1 rk- M
Q
I
I AI n v .41 m
337
In this demonstration, one sees clearly
each smallest case which is considered by
Boethius1 5 with regard to these modes.
For example, it shows clearly what the
particular species of the diapason is
in each mode, where it is located in its
particular system, and between which
pitches [it is found]. One sees also
which ones are higher than others by a
tone, called "page" by him in that place,
or by a semitone, which he there calls
"verse". One likewise sees which of
these modes is the right side and which
is the left. For example, the proslambanomenos
signed in the Hypermixolydian
with an cX is the mese of the Hypodorian
signed with the same character, and the
netehyperbolaion of the Hypodorian, designated
with a r , is, on the
1 5See Calvin M. Bower, "Boethius'
The Principles of Music, An Introduction,
ransla on,~ WCommentary," unpublished
doctoral dissertation, School of Music,
George Peabody College for Teachers,
Nashville, Tennessee, 1967, p. 267-275,
for a complete exposition of the modes.
[59)
LBoethiusJ in
chapter 14 of
the 4th [book).
338
contrary, the mese of the Hypermixolydian.
One sees, in addition, that
the system of the Dorian mode--according
to what [Boethius) says--is distant
from that of the Mixolydian by a diatessaron
toward the low register and
distant by a diapente from the Hypermixolydian,
in addition to [seeing]
many other, important particulars
which for brevity are omitted. With
all this, there is no lack of doubts,
particularly concerning the characters
with which the ancient Greeks used to
designate the pitches of their songs.
These doubts will be removed a little
later, Besides, the demonstration
which is found in the text of Boethius
does not agree in any [of its] parts,
with regard to the highness and lowness
of the modes, with the words he
[uses to] describe them. I doubt
seriously that this was one of the
potent causes which some careless, in
order not to say unwise, men have
339
boldly stated, perhaps for their [own]
convenience, that the text in that place
is corrupt; this is false! The demonstration,
however, is quite incorrect and
battered, thanks to the little accuracy
(in order not to say the little intelligence,
which is more appropriate) of those
who have taken it upon themselves to print
it in Venice in the year 1491. Boethius
easily attained that number of modes
through the advice of Alypius, 7 although
he does not mention this, finding in this
[work] (as we will show in the proper
place) the characters for signing the
pitches of each one of these eight modes
individually. We shall see, in addition,
since there are only seven Epitches), that
all fifteen pitches of the system were not
occupied, Cjust) as he said. This will
suffice for the present concerning the
understanding of modes according to the
16 See Zarlino, Istitutioni, pp. 307-
308.
1'Alypius' exposition of the ancient
Greek musical notation may be found in
Carolus Janus rKarl Von Jan , Musici 'cri-tores aeci (Leipzig, 1895J, p.~67- .V
Zarlino16 in
chapter 8 of
the 4th Lbook]
of the Insti-
tutioni
340
opinion of Boethius. Now, in order to
understand well the order and the number
of those Emodes) according to the opinion
of Ptolemy,18 I must deviate a little
from the present consideration. Therefore,
I tell you this. The ancient
Greek musicians, according to the authority
of the same Ptolemy, had later accepted
for the first of their seven
species of diapason that which is contained
by the pitches h mi and h mi.
For the first of the four species of
diapente which they possessed, they accepted
that which is found between the
pitch of E la mi and that of h mi. They
took, for the first of the three species
of diatessaron which existed, that which
is located between h mi and E la mi. For
the second species of the diapason, they
accepted the interval which contains the
pitches of C fa ut and c sol fa ut. They
18
See During, Ptolemaios, pp. 63-64.
9 See Henry Macran, The Harmonics of
Aristoxenus (Oxford, 1902T"p. 21.
Ptolemy] in the
2nd bookJ of the
Harmonics at the
3rd LchapterJ.
Aristoxenus1 9 at
the end of the
3rd Ebook] of the
Elements,
341
said that the third was between D sol re
and d la sol re; the fourth was between
E la mi and e la mi; the fifth was be--
tween F fa ut and f fa ut; the sixth was
between G sol re ut and g sol re ut. The
seventh and last necessarily came to be
contained between the pitch a la mi re
and that of Aa la mi re. The other three
species of the diapente and the two
[other species) of the diatessaron were
those which, ascending by conjunct steps
toward the high register, proceeded following
the order of the first [ones) proposed,
according to what is seen notated
in this example [Example 7.]. After
these, the Romans accepted as the first
species of the diapason that intervalsj
which is found between Aa la mi re and
a la mi re; for the first of the diapente,
[they accepted] that which is [found)
between h mi and E la mi; and for the
first of the diatessaron, they accepted
that [interval] which is [found] between
h mi and E la mi; the other species
followed [them], descending by conjunct
step. This was the order of their
342
Species of the diapason
[SpeciesJ of the diapente
1 2 3 4
[Species) of the diatessaron
1 2 3
[Example 7--Order of
according to the Greeks.)
Order of the
consonances
according to
the Gre ks.
Ptolemy 0in
chapter 5 of
the 2nd [book].
the consonances
20See iDring, Ptolomaios, pp. 66-68.
S2 3 4 5 6 7
343
intervals, if we mean those which
Boethius has written. This is the
example LExample 8.]. And although
some have believed and said that
[these species) proceed contrary to
those of the Greeks, perhaps because
they feared that Boethius had not
understood in what manner the lowest
diapason could serve the highest tone
and the lowest [tone] could serve the
highest [diapason). We will show them
that this was not so. It is quite
true that Boethius, in the order of
numbering the consonances, seemed to
some people not to have taken care of
naming the semidiapente which is found
between F fa ut and h mi as the first
species of the diapente. In this case,
it is credible and reasonable that
Boethius understood that the hypatehypaton
ought to form an octave with
the tritesynemmenon, because each one
2 1 See Gioseffo Zarlino, The Art of
Counterpoint, translated by G7A7'Marco
aid Claude.. Palisca (New Haven and London,
1968), pp. 27-29.
L60]
Zarlino2 1 is of
contrary opinion
in the same place.
344
Species of the diapason
1 2 3 4 5edie
[species] of the diapente
1 2 3 4
[SpeciesJ of the diatessaron
1 2 3
Order of the
consonances
according to
the Roman.
Boethius in
chapter 13
sic ] of the
b6ook).
Zarl no2 3
spoke to the
contrary in
chapter 13 of
the 3rd Ebookj
of his Insti-
tutioni.
Example 8 -- Order of the consonances
according to the Romans.J
2 2 Galilei's reference to Boethius IV,
13, is erroneous.. It should read Boethius
IV, 14. See Bower, Boethius, pp. 267-275.
2 3Zarlino, Ibid.
345
of his modes, as you have seen in the
demonstration, includes both systems.
In addition, he named the said species
(which we have applied to the modern
practice by proceeding from high to
low) in this order, that is: mi, mi;
la, re; sol, ut; fa, mi; and fa, fa.
He undoubtedly understood Cthe entire
situation) because of his superior
intellect. There have been others of
the moderns who have given this particular
species of the diapente the name
of second, although Boethius called it
fourth, and others have called it other
names, according to their various designs,
saying that the first species of
diapason was between A re and a la mi re;
that the first [species) of the diapente
was between D sol re and a la mi re; and
that the first species of the diatessaron
was between this D sol re and G sol re ut.
The other species followed by conjunct
steps toward the high register. Others
have felt otherwise, according to what
has turned out well for them, without
346
submitting as a testimony of the truth anything
except their opinions. And all say
that they have imitated Boethius, whose
text--it must be that Esame] edition which
used to be procured with great difficulty
from both Basel and from Paris and which
was mentioned above--has not come to our
notice, other than having seen it in manuscript
in different, famous libraries.
Strozzi: For what reason did the
ancient Greeks not want to accept as the
first species of their diapason, nor of
their diapente, nor of their diatessaron,
that which commences with the lowest
proslambanomenos, since they were (according
to some opinions) accepted by the Romans,
as you have said?
Bardi: I have never said that the
Romans accepted for the first species of
any of their consonances that which begins
with the proslambanomenos. There are also
others [who share] this opinion, but I do
not know where it [all] began. I state
emphatically, however, that if you will
observe the order of the species of
347
consonances according to the opinion of
the Greeks, you will find that they agree
with those of the Romans.
Strozzi: I do not know how this
could possibly be, since it happens that
you said that the first species of the
diapason of the Romans is that which is
found between Aa la mi re and a la mi re
and that of the Greeks between h mi and
h mi. Besides, those of the Greeks,
which begin in the low register, go by
conjunct steps when proceeding toward
the high, and those of the Romans, on
the contrary, when moving away from the
high register, proceed toward the low
register.
Bardi: All is well. Now, please
take warning. In the next demonstration,
you will see, with regard to modes, according
to the opinion of Ptolemy, that
the species of diapason which is found
between h mi and h mi, which the Greeks
had adopted for their first [speciesJ
and which serves the Mixolydian mode, is
applied to that pitch which, in the Dorian
The order of
consonances
according to
the Greeks and
the Romans is
the same.
348
mode, was likewise known to the Romans by
such a name. The same is true of the
others in order, so that they agree very
well together. That little difference
which exists between them, with regard to
the size of the interval, only arises because
of the diversity of their purposes
in applying either species of the diapason
to either system. One could also consider
that the two extreme pitches of this particular
diapason serve the Hypodorian
mode; however, they serve it between those
Same pitches) in the Dorian system. The
ancient Greek musicians were accustomed
to begin to number the pitches of systems
from the top, coming Ethen] toward the
bottom. Consequently, they said that the
first species of the diapason was the one
which served the Mixolydian mode, and the
others [went] in descending order. Their
other species of consonances were treated
by them in the same manner. The reason,
then, that they did not accept the proslambanomenos
as the first species of the diapason
and Eaccept as] the first species of
349
the other consonances those which began
with the very lowest pitch was because
it did not intervene in any genera of
harmony nor [in any] different distribution
of species, nor between the
stable pitches, nor among the movable
(pitchesJ of their tetrachords, besides
being last to be numbered. Although it
did not intervene--as has been said--
between the pitches which enclose and
which are enclosed in tetrachords, it
is not because the proslambanomenos is
not stable in each Greater Perfect System,
for the reason which will be said
below. And to tell you more, not only
was this pitch [i.e. proslambanomenosJ
the last string [to be] added to the
cithara, but [it was added] after [the
other pitches] had been ordered and
distributed in tetrachords, and the
reasons for its addition numbered two
or three of not much importance. The
first of these [reasonsJ evolved only
because the mese came in the middle of
this system, as its name implies.
350
Without the aid of the proslambanomenos
in that place, this would have been impossible.
The second reason was because
the extreme high pitch sounded as a
disdiapason (or quint, decima) with the
extreme low pitch. None of the intervals
reputed by [the Greeks) as consonances
were lacking to that Ldisdiapasonl,
and then the mese would sound as a diapason
(or octave) with the extremes.
I said [these were] causes of not much
importance because, without [the proslambanomenos),
each species of the diapason
appeared in the system in the same order
and without any impediment. Consequently
each mode (or tone), if not perfectly
enacted with regard to its entire constitution,
at least appeared to be in effect
concerning the species of the diapason.
This aspect must be the principal cause
that, while the lyre and cithara (which
were the same thing to them according to
some higher authorities) had been received
in those early times by Mercury (its inventor)
with only four strings, it had
351
been augmented in process of time up to
the number of fifteen.
Strozzi: You are therefore of the
opinion that the lyre and the cithara
were the same instrument according to
the ancient Greek and Roman musicians?
Bardi: I have very little doubt
about it, because of the many verifications
of authorities, in addition to
which Pausanias24 says that the lyre
was discovered by Mercury, and the
cithara by Apollo.
Strozzi: I judge that your opinion
of this is highly contrary to the common
one.
Bardi: I firmly believe that you
and anyone else who has patience to hear
me will concede [to my opinion).
J. G. Frazer, Pausania's Description
of Greece, 3 vols. (London and New
YdijT8 7~T, 258. "There is an altar
of Apollo and Hermes [Mercury] in common,
because there is a Greek tale about them
that Hermes was the inventor of the lyre
and Apollo of the Ecitharal."
2 5 Zarlino, Istitutioni, pp. 61-64.
C61)
Pausanias) in
he 4th book.
Zarlino25 in the
Institution at
captTer 3 (and in
chapter 4 of the
2nd book) and in
the 1st chapter of
the 4th book),
352
Strozzi: It will be most gratifying
to understand such a thing.
Bardi: I will state briefly to you
the most famous authorities which I have
gathered in favor of both sides Eof the
question=, leaving, then, the judgement
of what is more pertinent to you and to
everyone else of rational intellect. But
listen! One finds, according to the
Greeks, five names of musical instruments
(among the many others for which they
have different meanings) which, from what
is gathered from their writings seem to
mean the very same (thing). These are
lyre, chelyn, cithara, cethra, and phorminx.
Hardly anyone doubts that lyre and chelyn
are not the same instrument, and that
either one of these, similarly, is not the
very same as that which was then called
testudo by the Romans. They also believe,
in the same way, that cithara and cethra
are the same thing, the poet calling it
one name most of the time and -the orator
the other. It is clear that the phorminx
is the same instrument as the cithara. It
How many different
names the
Greek lyre has
been given.
353
happens that Suidas2 6 confirms this, Suidas,
writing about it thus: "Phorminx, that
is, cithara." Next, Homer27 recalls in Homer.
te Odyssey that when a servant at the
banquet had given the cithara to Phemius
he added these words, "Surely this one
'phormison', " while Dydimus, the interpreter
says, "'phormison', that is, to
play on the citha a." Because of this,
if we now demonstrate that the lyre is
the same thing as the phorminx, no one
will doubt that the lyre is the same as
the eithara. This is what 28is Horace
able to teach us sufficiently, because
having translated almost word for word
26
Immanuel Bekker, ed., Suidae Lexicon
(Berlin, 1854), p. 1103.
27
See Homer, Odsse translated by
S. H. Butcher and T thers(New York, 1963),
p. 208.
28James Gow, ed., . Horati Flacci
Carmina (Cambridge, 189), -. .""
virwm
aut heroa lyra vel acri tibia sumis
celebrare Clio, quem deum?"
354
from Pindar29 that song of his which Pindar
begins, "0 Clio, what man or what hero
will begin to celebrate you with your
lyre or with the shrill tibia?", while
Pindar in place of lyre says this,
"anaxiphorminges". From this it is
plainly known that Horace calls lyre
what Pindar first called phorminx.
From the words of the same Pindar,
however, this very same Lfact] can
easily be ascertained, because a little
later he writes this: "We will
take the lyre." In that place Pindar
calls lyre that same instrument which
he earlier called phorminx.
Strozzi: I concede to you that
the phorminx was the very same thing
as the lyre, but it will not follow
because of this that the lyre was the
same thing as the cithara, since it
could easily be an ambiguous name.
29Sir John Sandys, trans., The
Odes of Pindar (London and New r,
T 3)7pTT89. The Greek word
"anaxiphorminges" is rendered as "lyre".
Bardi: Tralascio, the interpreter
of Pindar, indiscriminately uses lyre for
cithara in many places. I am able, however,
to prove this same [thing] to you
more clearly with the authority of more
serious writers. First [I shall quote]
the authority of Xenophon30 who, in that
book he wrote Lcalledj Concerning the
Care of a Famy, said this: "Those who
first learn to play cithara spoil the
lyre." In addition, Socrates, according
to Plato,31 said that Alcibiades learned
to play the cithara and shortly afterwards,
that which he earlier had called
cithara, he called lyre, Now let me
gather arguments in favor of this side,
since I am able with a single testimony
worthy of faith (and this is Suidas)32
30See Xenophon, The Oeconomicus in
J. S. Watson, XenophonThVs Minor Works
(London, 1905), p. 80.
31W. R. Lamb, Plato (London and New
York, 1927), p. 108-T
32 Bekker, Suidae Lexicon, p. 599.
[Plato] in
Alcibiades I.
1355
356
to bring you into close proximity with
the truth. He uses these words. "The
cithara is a musical instrument which
otherwise is called lyre." Could he
say anything plainer and more open than
this? Now hear, oppositely, what I have
gathered in favor of the other side,
which seems in a certain way to persuade
us [to] the contrary. Plato in the,
third Ebook] of the Republic says: "The
lyre, moreover, remains, and the cithara
is useful in the city. The syrinx aids
in country pastorales." Athenaeus in
the fourth book toward the end [says]:
"The magadis is an instrument like the
cithara, the lyre, and the barbiton."
In the same place, he alludes to a certain
Anaxilas, having taken two quotes,
in addition to the others, from a work
entitled The Builder of the _re, in
which one reads in the same order works
Compare the Plato excerpt in
Oliver Strunk, Source Readings in Music
History (New York, 155), p. 6.
34 Athenaeus, Deipnosophistae, 7
vols., translated by Charles Burton
Gulick (London and New York, 1928), II,
309.
Lyre and cithara
are the
same thing.
357
which correspond to these. "But I have
strung barbitons, trichordons, pektides,
citharas, lyres, and scindaposes."
Pollux3 5 in the fourth book [says] that
"among those, moreover, which are played
are the lyre, the cithara, and the barbiton."
Plato36 in another place, which
troubles us more, speaks always of the
cithara nobly, and, on the contrary, he
deals with the lyre as an instrument for
fun and games. Therefore, it seems impossible
to many, since the cithara--as
has been shown above--is the same thing
as the lyre, that it has been discussed
by so many in the manner which you have
learned. Nevertheless, with the sole
consideration of the diversity of names
with which our own keyboard instrument
is known [today], every scruple is easily
taken away which the words of these famous,
serious writers could possibly
bring us. This, as everyone knows, is
35 Eric Bethe, ed., Pollucis Onomasticon,
3 vols. (Leipzig, 1900-1937),It18.
3 6B. Jowett, trans., The Dialo es of
Plat, 4 vols. (New York, '72 I,,'% 8T
Platoj in the
ial o e of
c es.
358
called by us with the names of clavichord,
harpsichord, clavicembalo, spinet, buonaccordo,
archicembalo, and others, only because
of the different quantity and quality
of strings and registers and Ebecause] of
the size and form of the instrument. Yet,
in essence, one instrument is the same
thing as another, and whoever plays this
one, plays that one likewise. There is no
one, also, of rational intellect who, when
comparing a soft gravicembalo to a strident
spinet, does not judge the latter, in comparison
with the former, to be a thing for
fun and games. So, also, when comparing a
sweet, sonorous, deep organ to a shrill,
raucous, harsh regal or to the noisy
sordunes, the result will be the same.
One sees, in addition, that the viola da
gamba, the harp, and many wind instruments
which we use every day in serious musicals
and important concerts are also used for
amusement at balls and dances. But among
the artifices which are employed in both
the Eserious] and [light) ways [of playing),
359
one can consider that there is the same
difference which Plato supposes between
the dignity of the cithara and the sport
of the lyre. It may very well be also,
that when Xenophon said that those who
learn to play the cithara spoil the lyre,
he wanted to infer that the young girls
of his times, in wanting to learn to play
the cithara, practiced first on the lyre,
since its shape was more convenient and
also its sound was more suitable, through
its high pitch or [some] other [attribute],
for that tender age. Thus, through their
inexperience, they came in the beginning
to spoil the strings, both with the plectrum
and with nails. It is no different
from what happens to our own [people]
every day when in wanting to learn to
play the organ (that is to say the harpsichord),
they practice most of the time on
a spinet or on a small clavichord.
Pausanias' opinion concurred also with
this sentiment without exactly misrepresenting
it when he said that Apollo invented
the cithara, and Mercury invented
360
the lyre. It happens that Mercury discovered
the lyre in his youth, as an
instrument suitable to that age, and
Apollo discovered the cithara after he
had attained manhood. The point of this
is to say that the invention of the cithara
was attributed to Apollo, and that
of the lyre to Mercury in order to distinguish
the qualities of the [two)
persons. Aristotle3 7 also agrees with
this opinion in dealing with the difference
of the harmonies--as you will learn
later, and which I will tell you at the
appropriate place--what difference there
actually was (if there was any) between
the cithara and the lyre.
Strozzi: You have greatly satisfied
[my curiosity. If it pleases you, however,
you could return to your original
topic and continue the discussion already
begun about tones (or modes) of the ancient
Greek musicians according to the
opinion of Ptolemy. Also, please pardon
me [for the inconvenience.
See p. 363 below.
oAristotleitin
he 8th [book]
of the Politics.
361
Bardi: I told you above that the
ancient musicians gradually proceeded
to augment the strings of their cithara
and lyre up to the number of fifteen.
They distributed these strings in the
manner which one sees ordinarily in the
Greater Perfect Disjunct System of the
Dorian mode. In the process of many,
many years, knowing that this number
of strings was comfortable suited to
express any human emotion effectively,
they were satisfied. I told you, in
addition, that the meaning of the name
of the lowest string, the proslambanomenos,
since it was last to be added
to the [GreaterJ Perfect System and
last also to be numbered according to
the usage of the Greeks (although first
according to the Romans), was the same
in that language as it is in ours, that
is, "added", "taken in addition", or
"surplus". This meaning--as I have
said--seems to warn us that that was
not done through any necessity but as
a proper choice. The reason, then,
The significance
of pro-
slambanomenos.
362
that it is called stable in any Greater
Perfect System results from the same
relationship which it continually has in
every genera and species of harmony with
the netehyperbolaion. Whereupon, since
the nete is stable, then such is the
nature of all the extreme strings of any
tetrachord. The proslambanomenos also
is necessarily stable, because it is
continually found [to be] distant from
the nete and from the other stable
[strings] by as much as has been said.
In addition, in the Greater Perfect
Disjunct System, even without the example
of infinite writers, the strings are
stretched--as I said before--according
to the Dorian tone (or mode). This
harmony was reputed, approved, and valued
more than any other by every ancient musician,
poet, and philosopher. This--if I
am not deceived--resulted mainly because
the strings in its system were stretched
according to the tone (or mode) in which
one normally discourses without effort.
The human voice cannot sing all the
Why the Dorian
harmony was more
reputed than the
others.
363
pitches of an entire system more comfortably
than those which are tuned according
to this Dorian mode, the excellence of
which has continually caused it to be
adopted as an example by every famous man
who has written of the art of music. This
situation has not been true of the others.
We should not wonder about that, for it
happens that the very high modes and those
which were too low were mainly refuted by
the Platonists3 8 in their well-ordered
Republic, since those [high modes) were
woeful and these low ones were lugubrious,
Only those middle modes were accepted by
these [Platonists] since numbers and
rhythms were also necessary according to
them. Their opinion was later refuted by
Aristotle,39 who said that the mild harmonies
are not to be disdained with regard
to old men who, because of their years,
are not able to sing the extended harmonies.
The high modes then, like the Lydian, he
3 8 See the excerpt from Plato's Republic
in Strunk, Source adings, pp. 4-6.
39See Aristotle, Politics (1342b20-43)
Richard McKeon, ed., Basic Works of Aristotle
(New York, 194177T".T
.
The Platonists
refuted the
harmonies
[which were
hooigh and
too low.
Rhythm means
the dancing and
movement of the
body.
[Aristotle] at
he end of the
8th [book] of
the PolitICs.
364
conceded to young girls in order (as he
said) to make them both ornamental and
disciplined at the same time. One can
argue, in the absence of another authority,
that the compositions of the ancients were
actually sung according to the sound of
pitches in which they were found written
by their composers. The contrary happens
today with our [compositions). There is
a particular example of the system of the
Dorian tone [or mode), and not of the
others, in Ptolemy, in Boethius, in the
40Introduction of Guido d'Arezzo, in the
Theorica and Practica of Franchino [Gaffurio),
in the music of Glarean, in the
Institutioni harmoniche of Zarlino, and,
finally, in the Timaeus of Plato, there
has been mentioned the larger and better
part of the species diatonic ditoniaion,
as all the other [modes) are called, in
addition to the many different places
Where they are so-called by) other
By Introduction of Guido, Galilei
alludes to ui o's ologs in antiphonarium.
See Strunk, ourceReadings, pp.
Ti-~125.
Ptolemy, Boethius,
Guido d'Arezzo,
Franchinus [Gaffurio),
Glarean,
Zarlino, Plato.
[63]
365
authorities. The species of diapason
of this mode is the fourth [species)
which is located, according to what I
said, between the pitches of E la mi
and e la mi where Boethius placed the
fourth species lower than E la mi and
e la mi by a sesquioctave interval for
no other reason than to provide room
for the Hypermixolydian mode, which
he could not do unless he assigned to
the Dorian mode that which Ptolemy first
assigned to the Phrygian. These two
modes [i.e. Dorian and Phrygian] were
continually used in speech by the characters
of comedy and tragedy. That
[procedure] was likewise followed in
satire since [these characters) used
it on the stage of the theater,. accompanied
by the sound of the tibia Caulos],
in reciting their poems. Above this
mode.[Dorian), there were three higher
modes and there were also three modes
below it. The three higher ones were
the Phrygian and Lydian, and the highest
of all was the Mixolydian. These
Why Boethius
varied the
species of the
diapason from
the [usage of)
the Greeks.
The characters
in comedy and
tragedy recited
their poems to
the sound of
the aulos.
ibmi-mmim
366
[additional modes) were then distributed
with these conditions. By raising the
ordinary disjunct system by a sesquioctave.
interval, in which system.w-as I have said--
they sang the Dorian mode which had been
discovered by the great Thamyris of Thrace,41
one obtained the Phrygian [mode), whose
42inventor was Marsyas, or Masses, son of
Hyagnis the Phrygian and, according to
the opinion of Aristotle, the teacher of
the great Olympus. This Marsyas was the
first who played the pipe with holes,
since it had not been known before him.
He also played two of them with one breath,
and he was likewise the one who first combined
high and low sounds. Elegies with
4 1 See Plinius, The Natural Histor of
Pliny, 6 vols. (London,lTp37I251o
Thamyris was also reputed to be a former
lover of Hyakinthos by Apollodorus.
4 2 Plutarch, On Music, translated by
Benedict EinarsonanWlIllip H. DeLacy
(Cambridge, Massachusetts and London, 1967),
p. 368-369, gives the alternate name of
"Masses" for Marsyas, Marsyas is credited
by Plinius with the invention of the
Phrygian mode and of the double-pipe.
Thamyris of
Thrace invented
the Dorian
harmony,.
Marsyas invented
the Phrygian
[mode).
367
fixed form were sung to the sound of that
instrument, The chorus in tragedies used
the Phrygian harmony a great deal, as well
as the Lydian, since the latter was appropriate
for the angry and for the cries of
the doleful, and the former was suitable
for one who was rejoicing with alacrity.
These Emodes) were repudiated by Socrates
because they were apt to introduce affections
in man which were inappropriate.
For this reason he repudiated also the
Lydian and the deep Iastian as weak and
proper for drunkards, not when they were
infuriated but when they were languid.
The Dorian and the Phrygian he conceded
as useful to war. I say that the species
of the diapason in the Phrygian mode is
found between D sol re and d la sol re,
and that its mese in necessarily G sol re
ut, while that of the Dorian is a la mi re.
Once more sharpening the system by a tone
4 Galilei's reference to the Politics
is erroneous. See Plato, Republic, in
Strunk, Source Readings, p. 5-6.
These are the
harmonies which
were used by
the chorus in
tragedies.
[PlatoJ in the
8th [book) of
the Politics4 3
[sis
.
368
more than that which serves the Phrygian
mode, one will obtain that of the Lydian
mode, of which Amphion4 4 was the inventor,
although others say Menalippides, and
46others, Corebos. Pindar claims that
Anthippus at the wedding of Niobe was the
first who taught the Lydian harmony.47
The Epithalmians were especially accustomed
to singing the [Lydian mode] and
this is what Plato had called the intense
Lydian in differentiation from its weak
44 Plinius, o. _ci~t a, p. 20. "Amphion
was the inventor of music." Ibid., p. 231.
"Amphion invented the Lydian measures in
music."
45Menalippides was a fifth century
B.C. dithyrambic poet from Melos. Plutarch,
On Music, p. 385. Ibid., p. 421,
alludes to a new complexity i Greek music
which superseded the more conservative
schools.
46 Ibid. Corebos, or Torebus, or
Torrhebos, was the son of Atys, king of
Lydia. It is he who, according to ancient
sources, invented the Lydian mode.
Ibid., p. 385. "Pindar says in
his Paeans (Frag. 75, ed. Turn] that the
Lydian moe was first presented at the
wedding of Niobe."
Amphion Ewas
the) inventor
of the Lydian
harmony.
[Pindar) in
the Paeans.
369
plagal. The Lydian species of diapason
is contained between C fa ut and c sol
fa ut; its mese is f fa ut. Plutarch48
relates, following Aristoxenus, that
the first use of this particular harmony
resulted from a certain mournful event,
and that, first of all, Olympus, on the
death of Python, played upon the aulos
funeral verses according to the Lydian
usage. Plato4 9 spoke thus in describing
its nature, cursing it. "The Lydian
harmony is high, raging, and strident,
and is altogether suited for laments."
Now this particular system, when transposed
a minor semitone (or limma) higher,
produces the strings strung according to
the Mixolydian mode, which is full of
48 Ibid. "For Aristoxenus in his
first book On Music says that Olympus
was the first To perform on the auloi
a lament for Python in the Lydian mode."
The cited work of Aristoxenus exists
only in fragmentary form (Frag. 80, ed.
by Wehrli, Testim. 105, ed. Da Rios).
b 9Ibid. "Hence Plato in the third
book ofe Republic shows distaste for
such music; thus he rejects the Lydian
mode, since it is high pitched and appropriate
to lamentation."
Olympus, on
the death of
Python, played
the Lydian mode
on the aulos.
370
many affections of harmony and is very
much suited to tragedy; its mese is
e la mi, and the species of its diapason
is that which is found between h mi and
h mi. The names of the three modes which
are below the Dorian are the Hypolydian,
the Hypophrygian, and the Hypodorian.
These are called plagals, since they Why plagals
modes are soare appropriate for those who are afraid called.
and who pray entreatingly. They are
created and formed in this manner. By
lowering the system of the Lydian mode
by a diatessaron, or to a semitone from
that of the Dorian, one will obtain the
Hypolydian mode, which was discovered,
together with the Orthian nome (from
which the trochaeus, giver of the sign,
was derived), by Polymnestus of Colophon.5 0
Polymnestus,
This (PolymnestusJ used more strings than thnveHolydian
harmony.
50 Plutarch, On Music, p. 365. "There
was also a poet Po!ymdilTsus, son of Nelos
of Colophon, who composed Polymnestian
nomes." bid., p. 373. "Polymnestus too
composed nomes sung to the auloi, but
whether he employed the Orthios nome in
his music as the writers on harmonics
assert, we are unable to say definitely,
as on this point the ancients are silent,"
Ibid., p. 417. "To Polymnestus is ascribed
=t"mode now called Hypolydian."
371
any other musician of his times. The
species of the Hypolydian diapason is
between F fa ut and f fa ut; its mese
is h mi. If the system of the Phrygian
mode is once again lowered by a diatessaton,
that is to say to a tone below
that of the Hypolydian (which is very
important with regard to the effect), one
will obtain the Hypophrygian mode which
Plato called mild Iastian. I have not
yet been able--although I have searched
diligently--to find out the name of its
inventor. The [Hypophrygian] diapason
is contained between the pitches of G
sol re ut and g sol re ut; its mese is
c sol fa ut. Finally, if we transpose
the system of the Dorian mode down a
diatessaron, that is, to a tone below
the Hypophrygian, we will obtain the
lowest EmodeJ of all, the Hypodorian.
The invention of this is attributed to
Phil~xenus.51 It was the last mode to
tome into use. Its diapason is contained
5 1Philoxenus of Cythera (435-379
B.C.), a dithyrambic poet, is credited
with the invention of the Hypodorian
The inventor
of the Hypophrygian
harmony
[is
unknown o
the author.
Philoxenus (was
the] inventor of
the Hypodorian
harmony (which
was] last to be
discovered.
372
between a la mi re and Aa la mi re; its
mese, consequently, is d la sol re. And
this was the actual number and measure
of the tones (or modes) of the ancient
Greek musicians, according to the opinion
of Ptolemy. These musicians were searching
among all seven modes--there were
only seven because there were no new
species of the diapason to occupy--for
twenty-one pitches and their higher and
lower replicates; for example that of
the Hypermixolydian, which was refuted
by Ptolemy, came outside of the natural,
common system. In addition, in the
demonstration of Boethius,52 the same
occurred also to the Mixolydian mode,
since it was different from Ptolemy in
its consideration of the pitches of the
diapason of these modes. This matter is
worthy of no little consideration.
mode by Julius Pollux. See Bethe, ed.,
Pollucis Onomasticon, I, 220.
52
~See Bower, Boethius, pp. 121-128.
EBoethius] in
chapters 9, 10,
and 11 of the
2nd Ebook].
373
[DIAGRAM VIII) [64]
DEMONSTRATION OF THE EIGHT MODES, ACCORDING
TO THE OPINION OF PTOLEMY
Greater Perfect
System of the
Mixolydian Mode,
Discovered by
S c w wo e MSappho
'ro Sul ft] Le ap o
Greater Perfect
System of the
m Lydian Mode,
x Discovered by
d.4 PAaM .. A Amphion
Se_o d S ecis he is
Greater Perfect
o System of the
4 _tPhrygian Mode,
z Discovered by
ab 0Aa s w A A -4C Marsyas
Lri ec es f t e Dia aso
0' Greater
000 4Perfect
-- System of
the Dorian
Mode, Dio
4 o covered by
Thamyris
- . d 0 0 a44 wP
F at S c es -f the a aso
Greater Perfect
System of the 0
Hypophrygian
Mode, Discovered
by Polymnastus , f
Ff S o e of e a o
Greater Perfect
System of the
Mode, Its Origin
Unknown to theO P
Author ix h eiesoftheDieaso
Greater Perfect
System of theI0
Hypodorian Mode,
Discovered by
Philoxenus
v.bO w b n.bi C wAi8LeP
374
The extremes of their constitution,
according to Ptolemy (the conjunct
ones as well as the disjunct ones)
sought in each genera of harmony no
more than twenty-one pitches--as I
said--and their differences were
twenty-seven [in number). Thus, the
Mixolydian, which was higher than all
the others, came to sound a semitone
above the Lydian, which in turn came
to sound a tone above the Phrygian,
and the Dorian sounded a tone below
the Phrygian and a fourth above the
Hypodorian. Below that, a semitone
and a fourth from the Lydian, there
was the Hypolydian, and descending a
tone below the Hypolydian and a fourth
below the Phrygian, there was the Hypophrygian.
Finally, the system of the
Hypodorian was obtained by lowering
that of the Dorian by a fourth, or,
actually, [by lowering] that of the
Hypophrygian by a tone. These,
53 Zarlino, Istitutioni, pp. 299-300.
E65]
Zarlino5 3 claims
that according to
Ptolemy, there
were 8 modes,
which Ptolemy
never said.
EZarlino] even
repudiated the
8th mode in chapter
3 of the 4th
(bookJ of his
Insti utioni.
375
according to Ptolemy,5 4 prince of mathematics,
were the true, legitimate intervals
by which the modes of the ancient Greeks
were lower or higher than each other and
lower or higher than the pitches of which
they were sung. Therefore, the description
is this. In order that it will be sufficient,
I have converted it into the easiest
way which I could possibly imagine. One
must consider in this, principally, that as
the mese and the species of the diapason of
each system are different, in the same way
the whole order of the scale is different
with regard to the steps in ascending and
descending from beginning to end by tones
and semitones. That trite proverb which
they used which says, "We will pass from
the Dorian harmony to the Phrygian, that
is, from serious matters to foolish ones"
not only derived from the opposing nature
of the harmonies and tones, but from their
contrary way of ascending and descending
(with regard to the steps of tones and semitones)
from one to the other, because the
54 See During, Ptolemaios, pp. 76-79.
55 See Macran, Aristoxenus, pp. 187-208.
[PtolemyJ in
chapter TO of
the 2nd [book].
Aristoxenus5 5
in the 2nd [book.
Proverb of the
ancient Greek
musicians.
376
Phrygian ascends by the same steps which
the Dorian does in descending. This can
be clearly observed in their examples.
Their constitutions were named by the
ancient Greeks, for different reasons,
with these three different titles of
sound and sense, that is, tones, tropes,
and "ethoi", "Ethoi" (were so called]
because they demonstrated the custom;
tropes, because the whole system rotated
in passing from one to the other; and
tones, with regard to highness and lowness,
"Ethoi" demonstrated the mood
because some of them, especially with
the example of those who recited them,
induced in the listeners important and
serious thoughts, and others induced
soft, effeminate ones, which happened
in listening to them, The system rotated
every time one passed from one to
the other; for example, in passing from
the Dorian to the Phrygian, because those
same intervals which were used in the
Phrygian in proceeding from low to high
were used in the Dorian--as has been
said--in moving toward the low range
Why tones modestj
were named
differently.
Why "ethoi" were
so called.
Why tropes were
so called,
away from the high. There were also
differences of highness and lowness; it
happens that the Phrygian was a tone
higher than the Dorian, and it Ei.e.
PhrygianJ was found Cto be] the same
interval lower than the Lydian. We will
add to this another consideration; they
were called tones and not semitones because
the three principal ones, which
are more ancient than the others, that
is, the Dorian, the Phrygian, and the
Lydian just now named, were distant
from each other by a tone and acquired
their names before the other four had
come to be known. It would also not
have been inappropriate to have named
them tones and not semitones after all
seven were in use, because, between the
intervals through which their systems
proceeded from low to high or high to
low, they contained a minor heptachord
in the manner which is seen notated
here CFigure 110.].
Why tones were
so called.
377
378
Leii
[Figure 110.]
In this [illustration), the semitone is
enumerated only twice, and the tone four
times, in addition to [which] the [tone]
is more natural and nearer to perfection
than [the semitone).
Strozzi: I have always understood
with regard to scientific, scholarly
men, that understanding well how the
modes of the ancient musicians stood, how
they sang them, and how many they were in
number, not to mention the many other
cases they considered regarding them, is
one of the most difficult things to know
and comprehend well; [it is more so] than
any other [matter) pertaining to the music
of those times. This I now prove through
experience and I see that it is so, for,
with all your diligence, I am still not
entirely capable of understanding what
you have told me up to now and shown me
with the example; nor am I entirely
Order of
the modes.
379
satisfied. I hope, however, that it will
not be a burden to answer whatever I ask
with regard to these, in order that I remain
fully appeased. Would you also pardon
my importune and perhaps impertinent
requests?
Bardi: Speak freely, as always, for
I do not intend to fall short, except in
that part where my knowledge does not
extend.
Strozzi: I understand, for the most
part, that the same Ptolemy says that there
is a mese particular to each mode. I am
now about to tell you that, applying his
words to the modern practice. The mese of
the Mixolydian is d la sol re; that of the
Lydian is c sol fa ut; that of the Phrygian
is h mi; that of the Dorian is a la mi re;
that of the Hypolydian is G sol re ut;
that of the Hypophrygian is F fa ut. Finally,
that of the Hypodorian is E la mi.
Bardi: You speak very well, and you
are quite correct. You will find the same
Efactsj in my description if you will consider
it more carefully.
Strozzi: I do not know how to find
them without your aid.
Bardi: Then look! When Ptolemy5 6
describes the order of the meses of modes
in the manner which you have recited them,
he considers all seven in the natural,
ordinary system. I come, then, to consider
each one of these (modesJ, in its particular
constitution, in the order in
which I have stated them to you. You will
not find in my example any contradiction
between them, because in considering that
line which represents the mese of the Mixolydian,
which is e la mi, you will find
that this mese falls on d la sol re in the
ordinary, natural system of the Dorian mode.
All the meses of the other systems do the
same thing compared to this.
Strozzi: I understand it very well
now, but Ptolemy claims, in addition, that
the systems, continuing one after the other
56See Dearing, pp. 79-80.
[Ptolemyj in
chapter 11 of
the 2nd [book].
C66]
380
381
in the order which you have demonstrated
to me, are distant by ditones and semiditones.
Instead of these, you have
mentioned tones and semitones. I do not
know how such a difference can exist between
you [and Ptolemy]. 5 7
Bardi: I made mention of those
smaller intervals because they are more
necessary to the understanding which I
tried to give you of these modes. If
you had wanted to strain your intellect
a little, you would indeed have found
also in my demonstration the ditones and
semiditones between one tone and another.
You would only have had to consider and
compare each particular pitch of one
system with that of another, as, for example,
the h mi of the Dorian with that
of the Phrygian, or the h mi of the
Phrygian with that of the Lydian. But
57See Claude V. Palisca, Girolamo
Mei: Letters On Ancient And Modernusic
T FVincenzo aT1' andGTvanniar
(Tome, 1960 ),pp.76~ ,fora oinpfensive
discussion of this discrepancy.
How ditones and
semiditones are
considered modes.
382
the whole of one compared together with
the whole of the other, or even any part
of these--let us omit, however, the consideration
of the particular names of
the pitches according to the usage of
this modern practice--is actually not
more than a tone or a semitone--as I
first said--that is, between those which
are connected. The sight of such diversity
occurs nowhere else except in the
example given you of these [modesJ, in
which the proslambanomenos, the mese,
and the extreme nete, and thus all the
other pitches of each one have different
characters. This problem did not occur
in that of Boethius, since the pitches
were marked with the same symbols, and
proceeded in the same order of steps in
one scale as in the other, so to speak.
I do not wish to suppress this other
observation which one may consider in
the demonstration of modes distributed
according to the order of Ptolemy. The
Fact] is, that those [modesJ which are
Other considerations
of the
author regarding
the modes of
Ptolemy.
383
higher than the Dorian are represented
to the sight as distant from each other
by the same steps in which the notes of
its constitution travel when moving
away from the nete hyperbolaion by
raising c sol fa ut a third. Those
[modesJ which are below the Dorian proceed
by the very same intervals which
are found in moving away from the proslambanomenos
of the same by descending
a third to h mi. This does not occur
in the demonstration which Boethius
makes. The same is true of the difference
of steps and the number of those
modes. Whereupon, those EmodesJ of
Ptolemy, when the beginning and the end
are connected, are seen going in a perpetual
spiral like the celestial spheres,
as the present wheel clearly demonstrates
[Diagram IX.]. On this [wheel], each
similar pitch is enumerated three times.
Because of this they have with excellent
reason been given the same names as those
two circles which in the globe of the
world are the limits of the longest and
shortest day of the year.
Tropos is a Greek
term which is badly
explained in that
propose 'on of
Zarlino at the
end of the 1st chapter
of the 4th part
of the Institutioni,
384
DIAGRAM Ii) 67)
LA SPHERICAL DEMONSTRATION OF THE
MODES OF PTOLEMY)
\ :
'
385
Strozzi: This has truly been a most [68]
shrewd consideration, but now tell me
something else. In considering the mese
of each tone in the natural system, still
according to the distribution of Ptolemy,
I find, particularly, that the mese of the
Phrygian mode is below that of the Lydian
by a semitone, and that of the Mixolydian
is a tone above the Lydian. I am, however,
accustomed to believe the opposite of what
you have described, that is, that the
Lydian was a semitone below the Mixolydian,
and the Phrygian was found a tone below
the Lydian.
58 See Zarlino, Istitutioni, p. 297-
298, for a discussionof the distinction
between trope, tone, mode, and harmonia.
This question has remained a controversial
one among leading scholars today. R. P.
Winnington-Ingram, Mode In Ancient Greek
Music (Cambridge, 17T,~p.75,alluis
To Me interchangeability of these terms.
See D. B. Monro, The Modes of Ancient
Greek Music (Oxfor7 T, p.727,Ronro,
sinT? the Hyperaeolian and Hyperlydian
modes, states that "the word
'harmonia' is used of these keys, and
with it, seemingly as an equivalent, the
word 'tropes'."
386
Bardi: This is actually so, It is
one of the things which has provided many
[people] of our times [with opportunityJ
for thought and speech, and many volumes
have been written about it. It has not
been understood, notwithstanding, by
these [people], as you will learn at the
proper place. Take warning, however.
You need to remember what I said above
about the meaning of trope, and to strain
your fine genius a bit, so that, with the
aid of this other consideration, you will
understand extremely well. I tell you,
therefore, that the species of the diapason
of the Dorian mode is that of E la
mi; that of the Phrygian is D sol re; that
of the Lydian is C fa ut; and that of the
Mixolydian is h mi. These are naturally
distant from one another by the intervals
which I said, but by applying them, then,
to the tones which they serve, from that
of the Dorian onward, they are transposed
5 9 Zarlino, Istitutioni, pp. 3O7--308,
contains a discusion~~ relative positions
of the modes.
Zarlino in
chapter 8 of
the 4th part
of his Insti-
tutioni.
387
up, one by a seventh, one by a fifth, and
one by a ditone. On the other hand, the
species of diapason of the Hypolydian, the
Hypophrygian, and the Hypodorian are transposed
down by the same intervals, as it
was possible to perceive in the example
[Diagram IX.). The only reasons for this
were that some of the former species were
underneath the lowest pitch of the dia
pason of the Dorian mode and some of the
latter species were above it (considered
in the natural system, however) in addition
to the fact that [people] have claimed (and
deservedly) that the first species of the
diapason serves as the prime of the highest
mode and as the final of the lowest one,
since it was last to be found among the
order and manner of numbering the strings.
I want you now to remember that having to
demonstrate a thing which in itself is
most difficult, owing to its properties and
individualities, which are not only very
different but, on the whole, contrary, is
most often not only lacking in ease of
execution, considering all the circumstances
The species of
the diapason of
modes are transom
posed down one
time and up
another time.
Warning.
pertaining to it which should agree, but
difficult, if not impossible. It is not
exactly surprising that the example of
modes which I have fabricated in order
that you might understand Ethem] does
not have in itself such an exquisite
clarity that each smallest case is understood
with the greatest ease one could
possibly desire. The ancient Greek musicians
had a character to denote each of
their particular pitches. This character
signified the proper sound of that pitch
without having any need on lines, spaces,
or clefs as Edo) these of our times.
All of these things are the worst of
hindrances which could be.removed and
largely ameliorated by scholars of the
modern music practice with the greatest
advantage Eto all]. I will tell you how
to carry that out easily before I bring
our discussion to a close.
Strozzi: This shouldindeed be a
thing which would be desired and adopted
by every judicious practical musician, so
you should not suppress it in any way.
The names of
the musical
notes could
be greatly
improved.
388
389
Returning, however, to the business of
modes, would you tell me how Ptolemy
formed them from two major superparticular
consonances?
Bardi: Ptolemy extracted from the
Dorian mode, by means of the diatessaron,
the highest and lowest modes (which
are--as you know--the Mixolydian and
the Hypodorian) by raising and lowering
them by that particular interval of this
Dorian [mode). Afterwards, ascending a
fifth above the lowest, he created the
Phrygian and descending a diatessaron
below this Phrygian, he formed the Hypophrygian.
Ascending a diapente above
this [Hypophrygian], he formed the Lydian,
and descending a fourth below this [i.e.
LydianJ, he formed the Hypolydian. Thus,.
all seven modes were derived by means of
two major superparticular intervals.
Strozzi: How will we resolve that
tritone which is found between the mese of
the Hypophrygian and that of the Phrygian
which you say, together with Ptolemy, has
been tuned as a fourth? I am referring,
The fourth and
fifth are major
superparticular
consonances.
This is how
Ptolemy formed
the tone by means
of two major
superparticular
intervals.
390
however, to the way it happens in the
ordinary, natural system.
Bardi: Neither Ptolemy nor I said
(with regard, however, to its beginnings
and endings) that the mese of the Phrygian
and [that of] the Hypophrygian were tuned
as fourths, but I only said that they
harmonized by that interval.
Strozzi: Is it not the same to be
in harmony as it is to be in tune?
Bardi: No, Signor. [It is] not the
samej according to Ptolemy, because he
says that a consonant is that interval
which in penetrating to the ear strikes
it inoffensively. Such, according to him,
are the diatessarons, which are called
symphonies in order to distinguish them.
He further said that those which struck
the ear, not with offense, but with sweetness,
are tuned. These are the diapentes
which are called parasones in order to
signify them [properlyJ. The others, then,
were those which in being heard, not only
struck the sense inoffensively and sweetly,
but were so [agreeable] that one could not
Zarlino 6 0 in
chapter 15 of
the 2nd Ebook].
To be in harmony
and to
be in tune are
not the same
[thingJ.
The meaning
of symphonic
consonances,
The meaning of
parasones.
391
want for anything more. These were the
diapasons, which were called homophonies,
or, if you wish, antiphonies. This distinction
was also made by Aristotle.62
Strozzi: This has been a new, valuable,
and subtle distinction which I had
not expected, but tell me, in this case,
another thing. Does the same precedence
exist between the dissonances when they
sound discordantly as exists between the
consonances when they harmonize to a
greater or lesser degree?
Bardi: Do not doubt it in the least!
[In order to prove] that this is true, here
is a clear, convenient example. When you
have considered it rationally, you will
find that the major second and the minor
seventh are much less dissonant when the
major second has been resolved by the
minor third and the minor seventh has
been resolved by the major sixth, which
does not happen when they are resolved
differently. You will find that the
6 1 Zarlino, Istitutioni, pp. 105-107.
6 2 Aristotle, Problems, 2 vols., translated
by W. S. Hett(am ridge, Massachusetts
and London, 1961), I, 378-415 and
passim.
Zarlino61 in
chapter 31 of
the 2nd part
of hisnstiuton
says
a ose intervals
which
are not consonances
are
necessarily
dissonances.
The meaning
of homophony
and antiphony
Eis mentioned]
In many places
in the Problems
of Harmony.
"nsiderations
of the author
concerning the
qualities of
the intervals.
[69]
392
hearing is struck less harshly by the
tritone and the semidiapente when they
are resolved by the minor sixth or by
the major third. This had not occurred
to the first intervals named, perhaps
because of the pleasantness of the
union and the contrary motion of the
major semitone, both of which effects
are created by the parts simultaneously.
In the first ones, however, it happened
only in that moment when the other
[part] proceeded by tone, When that
happens to the high part in moving away,
it is always much more graceful than it
is when this happens to the low part.
The contrary occurs when they approach
one another, perhaps because of the
motion, which becomes languid in the
low part and intense in the high part,
or because small intervals, in moving
away by contrary motion, are more
natural in high ranges than in low
ones. The opposite effect happens in
moving inward, but such an opinion can
also result in a person from having
393
accustomed the ear in such a way. It
can also happen since in that place both
sevenths have resolved to the sixth and
both seconds have resolved to the third,
the larger of these [thirds] and the
smaller of those [sixths] are less strident
since they are nearer to their normal,
safe conditions. For the same
reason the major sixth should seem
harder to us than the minor [sixth in
the passage which is made by that Emajor
sixth] to the octave and by this minor
sixth to the fifth, as most of the time
happens. I tell you also that when one
wishes to come to the diapente from the
tritone, it would be much better to make
the low part descend while the high part
stands fast. Oppositely, if the latter
were ascending, the former would stand
fast. This happens for no other reason
than from having the tritone with the
fifth which causes an agreement in that
place (that is, the low part), but I do
not know how great it is. This agreement
394
results perhaps because of the slowness
of the motion and the softness of the
interval between those pitches, because
the semitone is always more obvious to
the ear when ascending than when descending.
In these cases, which have been
related above, and for the same reason
which has been related there, the tritone
is rendered less offensive to the sense
than the semidiapente. Perhaps [this is
trueJ because this tritone is found in
the midst of two smaller consonances
which today are called perfect. Still
another reason would be that the semidiapente
is able to become a diapente,
and the tritone is able to become a
diatessaron without altering any note,
as one can clearly see and hear in the
following example [Example 9.J. This
diversity of harmony results in the consonances
for no other reason than from
the little or great conformity which
their extreme sounds have together. On
the other hand, the dissonances, being
Why a consonance
pleases the ear
and a dissonance
displeases it,
Dissonant minor
seventh
Dissonant major
second.
Much less dissoHard.
major sixth
More greatly dissonant
major seventh
-0" I,
More greatly dissonant
minor second
Less tolerable
semidiapente
Less
quid
hard, even lanminor
;ixth
LCa
[Example 9--Dissonant intervals and
their resolutions.J
395
__1 1.
396
diverse in form and contrary (to the
extreme sounds of these consonances],
strike the ear harshly, because if we
wish each one of their extreme sounds to
be preserved intact in a certain way,
and if we do not wish it to yield to
the other, they [both] come to strike
the sense harshly. The ear is offended
even worse by the seventh than by the
semidiapente, and less so by the tritone,
perhaps because the tritone has the same
number of steps as the fourth, and the
semidiapente has the same number of
steps as the fifth. Because the semidiapente
resolves into the major third
and the tritone resolves into the minor
sixth, (the situation) there is more
imperfect. The fourth is less consonant
than the fifth, since dissonances are
contained by the same quantity of pitches
in the diatonic genera. Similarly, that
which agrees with the fifth is more dissonant,
but that which agrees with the
fourth is less dissonant. That is not
397
surprising, for it happens that when the
humors are well proportioned, any smallest
case alters them greatly. This does not
happen to those which do not agree and
unite as well. One perceives this, as I
have shown above, in the first two dissonances
named, and also in the inconstant
distance which is found between them and
their adjoining consonances, the sixth and
the third, which are generally used to
resolve them. In addition, you will find
that the minor third, which is between
these pitches, and others like them, that
is, A re [to) C fa ut have a sad (quality) [70J
when descending and a happy [quality)
when ascending. You will find that those
pitches which are located between A re
and C fa ut and between the ones similarly
constructed,. that is, E la mi and G sol
re ut, are the opposite. All these considerations,
more in the latter than in
the former part, demonstrate their qualities
and operate their effects with greater
efficacy.
398
Strozzi: I [nowJ extract this Litem]
from among the other most important ones
in your discourse which is that the diesis X,
when placed in the high part makes the
composition happy, and when the b molle is
placed in the low part, it makes it sad. But
this happens) when the parts move away from
each other, proceeding by contrary motion,
and also when the low part is moved toward
the high part in a separated and disjunct
Mannerr. The very same [thing] probably
should happen in passing by contrary motion
from the minor sixth to the major third
and from the minor third to the major sixth.
When arranged in an opposite manner their
nature should be contrary. These cases
should be less easily understood by the
sense in proportion to the increase of the
number of parts and the low part should be
more greatly revealed than the high part or
that of the middle.
Bardi: That is true.
Strozzi: You may now continue, if it
pleases you, to show me the way to save that
tritone which I said was found between the
399
mese of the Hypophrygian Cmode) and that
of the Lydian.
Bardi: Our practicing, contemporary
contrapuntists, as you know, resolve that
jtritonej in their compositions with the
minor sixth, and we will resolve it in
our compositions thus. Each particular
mese needs to be considered in its usual
system, and there will be a fourth exactly,
and not a tritone.
Strozzi: Even considering them as
you now say, we will have a major interval,
because the mese of the Hypophrygian mode
is c sol fa ut, and that of the Phrygian
is g sol re ut, a fifth higher than [the
Hypophrygian meseJ.
Bardi: You have forgotten once again
the meaning of trope, and have not--as far
as I know--understood, as yet, the status
of the most important item of business.
The mese of the Phrygian EmodeJ, which is
c sol fa ut, although it has the relationship
of fifth with that of the Hypophrygian,
which is g sol re ut, considered, however,
in the manner you say, they Eboth] need
400
not be considered thus for this reason,
but in the opposite way. Accordingly,
the mese of the Phrygian has, by being
higher than that of the Hypophrygian,
the same amount this c sol fa ut has by
being lower and not higher than g sol re
ut. This is what Ptolemy told you just
now in finding the positions of modes by
means of the two larger superparticular
consonances, and thus, likewise, it must
be understood of the others also, if one
wishes the outcome to be similar to the
way in which he describes it.
Strozzi: You have produced for me
a monumental thought. Clarify another
difficulty for me, however.
Bardi: Tell me!
Strozzi: How can it happen, in the
Lydian mode, that the pitch of c sol fa
ut is--as I have learned--the extreme
and the mese of its diapason at the same
time?
Bardi: This is the case when it is
considered in its usual system and as mese
401
in that of the Dorian, but in the same
constitution this cannot happen in any
way.
Strozzi: Why, then, did Ptolemy
assign higher species of the diapason
to lower modes and lower species of the
diapason to higher modes?
Bardi: I showed you above that
what he said was not exactly true, but
only the contrary. I say, in order to
warn you again, that if Ptolemy had, for
example, assigned to the Hypodorian mode
the first species of the diapason which
is contained, as you have learned, between
h mi and h mi, and Chad assigned] the
other species to the other modes in order,
[then] this is one of the inconveniences
which would result from these. The Mixolydian
would become a tone higher than
the Lydian. Nonetheless, one reads about
this deed that Sappho,63 the illustrious
poetess, inventress of this Mixolydian
6 3 Plutarch, On Music, p. 387., Plutarch,
citing Arisoxds, credits Sappho,
the poetess of Lesbos (sixth or seventh
century B.0,), with the invention of the
Mixolydian mode.
Sappho, inventress
of
the Mixolyw
dian harmony.
402
Modee, since she was unable (according
to the opinion of many), being a lady,
to sing comfortably her own poems or
those of others in the Lydian mode,
raised the Lydian system by a semitone,
and came to create a new mode which she
called Mixolydian. This name almost
implied that, because of the proximity
of this new mode to the Lydian, they
were mixed. We will add to their opinion,
as a more solid and firmer one, this other
consideration. The fact is that Sappho,
being herself, was constrained to this
necessity not because she was a woman,
but from the very great conformity which
the melodies of her poetry had with the
effect and nature of that particular
harmony. Since she was--as I have said--
a lady of small stature, who possessed
only average looks, and whose profession
(as the writers tell us) was not very
chaste, she was therefore given occasion--
as one sees in some fragments of her poems--
to quarrel and complain most often about
403
love while she yearned for the beauties
of the youth, Phaon.64 Besides, Ethe
fact] that the people of Phrygia ordinarily
sang their airs a tone higher than
the Dorians and a tone lower than the
Lydians, is, due to the many verifications
by authority, a clear, trite thing.
Nonetheless, if anyone wishes to proceed
to distribute the species of the diapason
in the manner which we last said, the
example of it would demonstrate the
opposite effect, because, in this order
or proceeding, it would not be more than
a semitone from the Dorian mode to the
Phrygian. Nevertheless, the truth is--
as you have learned--that the Phrygians
sang a tone, not a semitone, higher than
the Dorians. Although Ptolemy wished the
modes to be as they actually had been sung
by the ancient musicians, with regard to
64 See Bekker, Suidae Lexicon, pp. 935-
936, with reference"T" o poan Phaon.
Her vain devotion to this handsome youth
is well known to readers of the ancient
classics.
404
their distance from one another, their
order and their locations between the
particular pitches, and the proper species [71]
of the diapason, neither he nor those who
ordered them before him were able to arrange
them in a manner superior to the one
which has been shown.
Strozzi: I have understood very well.
Tell me, however, who was the author of
this difference of high and low modes? Why,
also, were Ethese modes] assigned more to
one nation than another, or was this
customary?
Bardi: Nature was the author of the
three original Emodes) which are the Dorian,
the Phrygian, and the Lydian, because, when
one nation or the other uttered words naturally
while singing and speaking, or [doing
something in between the two), there existed
between their sounds (with regard to highness
and lowness) the difference which you
have learned. It is not the exact difference
that was ascribed to it, and afterwards
instituted in it by art. This thing is
seen and heard by whoever considers well
405
What] happens every day in many other
provinces, and particularly in those of
Italy, because the Lombards generally
speak and sing in a deeper tone than the
Tuscans do, and the people of Liguria
speak in a higher voice than these.65
EThis is sufficiently clear) without
resorting to the Sicilians or more remote
nations, and without leaving the confines
of our province any more than absolutely
necessary. If this, then, results from
food, water, air, or climate, we will
leave such disputes to the inhabitants
[of these places). It suffices that the
same [thing) which occurs today in Italy
occurred daily in Asia among the peoples
of Lydia, of Phrygia, and of Doris. The
same [thing] happened in modulating diatonically,
although each interval was
afterwards regulated by Pythagoras in
the same way Jubal Cain [regulated them]
65See Vitruvius, The Ten Books on
Architecture, translatTby orris6 Eiy
Morgan (Caibridge, 1926), pp. 171-173,
with reference to the supposed effect of
climate on the voice.
Tuscans speak
with deeper
voices than
Ligurians and
with higher
voices than
Lombards.
Nature is the
inventress of
singing dia-
tonically.
406
prior to the great flood, to whom the
invention of such a practice [i.e. modulation)
was afterwards attributed by the
unenlightened.
Strozzi: What authority is it that
persuades us that the Phrygians sang a
tone, rather than another interval, below
the Lydians, and that they sang the same
interval [i.e. tone] above the Dorian?
Bardi: [We learn this) particularly
[from] the description of these modes
which Aristoxenus makes, in which he
places the Ionian between the Dorian and
the Phrygian and the Aeolian between the
Phrygian and the Lydian. [We also learn
it] from what Ptolemy and Boethius say,
in addition to many other authorities
who wrote before and after them.
Strozzi: With regard to the beginning
and end of the compositions of the
ancient musicians, did they have a
determined and particular pitch?
Bardi: No, Signor. [This was] because
the difference which existed between
407
one and another of their tones (or modes)
consisted principally in the disposition
of the strings of varying tension and in
the diversity (so to speak) of long and
short sounds placed in different order in
each system, and not like that of our
practicing contemporary contrapuntists who
have placed all Etheir confidence] in the
final pitch. Even so, some Lof these contrapuntists),
in order to appear wiser and
more scholarly than the others, add there
the arithmetical and harmonic divisions
of the diapason. These divisions, with
regard to differences of harmony, affection,
or tone [mode), have less to do
with these [contrapuntists] than you have
to do with the kingdom of Peru. According
to them, there are twelve different tones
(or modes), although they continually
compose, play, and sing in the same
strange, unknown Emode) which they employ,
without exactly noticing it, indiscriminately
at weddings and funerals [alike).
66 Zarlino, Istitutioni, pp. 308-309.
Zarlino in
chapter 9 of
the 4th part
of his Insti-
tutioni.
~aaErinc and
arithmetical
divisions have
no part in the
modes of [our]
contemporaries.
408
Strozzi: Where, do we believe, did
the application of this particular consideration
of arithmetical and harmonic divisions
to the tones [modes) of our practicing
contemporary contrapuntists have its origin?
What do you believe was the factor which
led them to believe that such a difference
had power to vary one thing from another
with regard to the diversity of harmony
[melody]?
Bardi: These are indeed two of those
things which I have pondered many times,
and truthfully, I have never found a way
to resolve them entirely.
Strozzi: With all this, it will be
gratifying to understand your opinion in
order to see if the cause of my doubts
concerning this first point is in any way
consistent with the way you feel about it,
because I consider that they derived such
foolishness from that which Boethius67 says
concerning the initials of tones modestj
and the disposition of the notes of every
67 Galilei's reference to Boethius IV,
14, is incorrect. It should read Boethius
IV, 15. See Bower, Boethius, pp. 275-276.
409
mode and sound. He says this at the beginning
of the fourteenth ric.J chapter
of the fourth book of his De music.
"Therefore, those which are called modes
result from the species of consonances."
Or perhaps this thing had its origin in
the way in which Ptolemy used the diapente
and the diatessaron to form the
modes which are found between the extremes
and that of the middle.
Bardi: These considerations of yours
do not displease me Eat all], but, in order
to make you capable of [understandingJ all
that which I believe about that matter, it
is necessary, first of all, to commit to
memory the order in which the eight ecclesiastical
modes have been arranged, which
have been instituted as follows aExample
10.J among their different species of the
diapason.
410
I
10
*r H0 H - i .I
F4 o 0
A
A P4 -l H Hr
1 2 3 4 5 6 7 8
[Example1O-Descr option of the eight
ecclesiastical modes.]8
Strozzi: You have made much mention, [72]
in this description, of Dorian, Phrygian,
and Lydian, and the other names of the
ancient modes. Is there, perhaps, any
conformity between these ecclesiastical
modes and those ancient ones?
8Galilei, Dialogo, p. 71.
- 4 f _8
Bardi: Do not doubt it at all, [for
they conform] particularly with those
[modes] of the Romans which have been
described to us by Boethius. They have,
on the other hand, been derived from him,
as you will learn, although this is contrary
to the opinion of some [musicians].
Strozzi: Tell me two other things.
First of all, who was the author of these
particular modes? Secondly, what do we
believe induced their composers to call
one mode the third rather than the first,
or [say that a certain melody belonged
to the second mode rather than to the
fourth? For what reason do they not
pass beyond the eighth mode?
Bardi: The most ancient reference
which I have found to ecclesiastical modes
is in the Introduction of Guido d'Arezzo,
who flourished during the pontificate of
John XX around the year 1020. If I remember
correctly, according to what Guido
6 9 Zarlino, Istitutioni, p. 307-308.
Zarlino69in
chapter 8 of
the 4th book
of the Insti-
tutioni.
411
412
himself shows in the Introduction, Odo70
and some others dealt with [modesJ a
little before him. I have seen these
EreferencesJ in some extremely ancient
books which I have right beside me, and
this is all I know to tell you about the
origin of modes. Next, there is the
question of why is the first [named firstJ
rather than second, or the third [named
third] rather than fourth, or [why is
one] mode [named one way] and not another?
Concerning this, you will learn everything
which has been done according to how much
my feeble intellect can discern by way
of conjecture, although [this will] not
[be] without consideration and judgement.
There is, however, no book close at hand
which speaks of Esuch thingsJ.
700ooCln
t%do of Cluny (?-942) authored the
treatise Enchiridion musices, also knownas D de maTu4_, contains the
firs y emllic use of letters for pitchesin the meaning that was to become standardfor the Middle Ages, that is, the entire
gamut from A to g, with the addition of thelow gamma and the high a'. See Strunk,
Source Readings, pp. 103-116.
The question
of authorship
of the eccle-
siastical
modes..
413
Strozzi: Who, then, was the author
of considering and applying the harmonic
and arithmetical divisions to modes in
figured songs? EWho was the firstJ to
mention and introduce twelve modes, when
only eight had been in use at first?
Bardi: Let us not place under
consideration so many controversies at
one time, if you please, but let us proceed
to decide topic by topic. In order
to avoid confusion, let us not examine
the second [thing] until the first has
been determined. Gaffurio7l was the
author of considering the harmonic and
arithmetic divisions in the modes of
either plain or measured songs. Glarean72
was the author of adding four modes to
the eight which were first demonstrated.
In that place CDodecachordonj, Glarean
71See Franchinus Gafurius, The Practi-ca Muiatranslated by Irwin !o'""nFadson, Milwaukee and London, 1969), pp.39-45.
72StrunkSource Radns,Heinrich GloreanfP
the19
great sixteenth century humanists, is bestknown to the musical reader as the authorof the 1odecachordon, who favored four
Franchino (Gaffurio
was thej author ofapplying the har-monic and arithmetic
divisions to tones.
This is found inis Practice in the
1st bo , chpter 7.
Glarean added four
modes to the orinal
eight. He re-proved :Franchino,
who is defended bythe author.
414
severely reproves and attacks Franchino,
contrary to every sense of propriety, for
two things of no little importance. I do
not wish to suppress these matters in any
way. Instead, I wish to remove such a
stain from the name of Gaffurio with the
fewest words possible. Initially, Glarean
was surprised, since Gaffurio had had
knowledge of the harmonic and arithmetic
divisions of the diapason and had considered
them, besides, in the eight original
modes, that the same consideration had not
occurred to him as that which had afterwards
dawned upon Glarean himself, with
regard to the other four modestj and
bringing Lthe total number of modes] up to
twelve. He then reproved him, largely
from ignorance. He admonished, secondly,
that EGaffurioj had not understood the
additions to the existing eight ecclesiastical
modes. The book exerted much
influence on the changing concept of the
modal system.
73See Glarean, Dodecachordon, pp.
139-140.
Glarean7 in the
2nd book at chapter
15 did not
understand the
matter of modes
according to
Aristoxenus.
415
order of modes of Aristoxenus. Both of
these calumnies are as unjust as anything
one could possibly imagine. The fact that
Gaffurio, initially, understood excellently
the order of modes of Aristoxenus is known
openly from the demonstration which he
made of them and by the words with which
he describes it. They are, after all, the
same as those which Bryennius and Aristides
Quintilianus used, according to what I
said above in describing the modes conforming
to the opinion of Aristoxenus,
having drawn them from their writings,
Glarean, on the contrary (and others more
recent than he) did not understand it, or
perhaps--as will be said later--they did
not wish to understand it because of their
particular interests. This Efact] is
clearly manifested here. Glarean claims
that the Lydian mode is, according to
Zarlino in the
8th [chapterJ of
the th part of
his Institutioni.
Another error of
Glarean. Second
book, chapter 9.75
Glarean7 6 in book
1, chapters 21
and 22 Eand in'
book 2, chapter 7.
74See Zarlino, Istitutioni, pp. 307Glarean,
Dodecachordon, I, 125?6,bid.
,I, 97-103; I, 114-115.
308.
127.
Aristoxenus, a semitone below the Mixolydian
and thus, similarly, that the
Hypophrygian is below the Hypolydian
by the same interval. Aristoxenus, on
the contrary, says that there is a tone
Ein those places]. He also says that
.the Lydian is a third below the Aeolian,
and Aristoxenus puts the Aeolian above
the Lydian by a semitone. Glarean claims,
in addition, that the Dorian is a fourth
below the Hypoionian, while Aristoxenus
says that there is a third in that place).
Let us say this in conclusion, however,
that Glarean distorts all the modes according
to Aristoxenus, from the Dorian
and Phrygian and their plagals, onward.
He does all this, moreover, because he
wishes to harmonize those Emodes] with
the ones of today, which is just as possible
as making an Ethiopian white. ENeither]
Glarean nor any other more recent
writer could ever understand how the
7 Zarlino, Istitutioni, pp. 307-308.
Zarlino? in
chapter 8, of
the 4th part
of the Insti-.
416
417
Lydian mode could be higher than the
Phrygian by the space of a tone. It
happens that the species of diapason of
the Phrygian mode was B la mi (according
to the text of Boethius, which these men
had brought for authority, seeking--as
I have said--to imitate it), and the
species of the diapason of the Lydian
was F fa ut. These were [both] distant
by a semitone, and the same difficulty
arose in their plagals, because the species
of the diapason of the Hypophrygian
was 0 fa ut. Ignoring this Efact), they
proceeded to say (rather than confess
that they did not understand the situation)
that the texts of Boethius, of
Franchino EGaffurio], of Giorgio Valla, 7 8
and of others were incorrect with regard
7 8 Galilei is probably alluding to
the treatise by Giorgio Valla entitled
Musicae, which constitutes five books out
?TlaTrger collection of forty-nine
called De +s tend s fet
(Venice,
,7
to this topic. I cannot possibly imagine
how such a thought could [ever] occur to
them. It happens that Aristoxenus, according
to the opinion of the above-mentioned
interpreters, placed the Aeolian between
the Phrygian and the Lydian, and placed
the Hypoaeolian between the Hypophrygian
and the Hypolydian. Both of these were
placed at the distance of a semitone, so
that there was necessarily a tone between
the Phrygian and the Lydian and a tone
between their plagals, as has been said
many times. Moreover, Ptolemy places between
these Emodesj the very same intervals
used by Aristoxenus and Boethius,
although he attributes to them various
species of the diapason which are considered
differently and with conditions
other than the ones which Boethius imposed.
Would it not be a fine thing, however, in
this modern practice, to follow the singing
of Ancorche col a i of Cypriano Eda
Rore2 in four parts by immediately singing,
a tone highr,ci 'ta yt
418
C73J
419
the same composer, although the bass of
the latter piece begins and ends in F fa
ut and the former begins and ends in E la
mi, which is the distance of a semitone?
Would it not also be a fine thing, following
the singing of the same Ancorcheol
partire, to sing, a semitone lower, the
Giustizia immortale of the very same
author, also in four parts, although it
ends on D sol re, a tone away from E la
mi? I am not [at all] certain [about
thisJ. Coming however to the second
topic, I say that as much as Franchino
[GaffurioJ had considered the four
authentics of the eight original modes
divided harmonically and the four plagals
divided arithmetically, and since such a
consideration as that was indeed suited,
with nothing added, to make him aware of
the other four Lmodes] which Glarean later
added, it is unfair (for the reasons which
will be revealed soon) to charge him with
ignorance (contrary to the way those men
feel) for not having introduced them. It
is only right to commend his discretion
420
and great knowledge, because he found
that each one of the seven species of
diapason had been occupied with the
seven original modes. He considered,
in addition, that the highest pitch
of the seven, together with the lowest
of the second, included all fifteen
strings of the Greater [Perfecti System.
tHe perceived, also] that if
others were added above or below,
Ethe system] would exceed its natural
limits, the same [pitchesJ would be
repeated, and nothing new would be
produced. Ptolemy also had this opin
ion, although Zarlino7 9 in chapter
seven of the fourth part of the Insutioni
harmoniche says the reverse,
In addition, he makes a case of this
matter, [saying] that it was the cause
of making jaJvariation between the
first and second modes, not on account
of the different division of the
various species, as Glarean contended,
but because the second mode extends a
79Zarlinaco, Isti.tutionip.3637
421
diatessaron lower than the first [mode],
and thus, it comes to produce a more
languid, submissive harmony, which is
therefore different in nature from that
of the first (modeJ.
Strozzi: May it not be a burden
to you to tell me what moved [Glarean]
to consider such a division in the
modes, and why the harmonic division
of the diapason and diapente pleased
the ear more than the arithmetical
division.
Bardi: As for the harmonic and
arithmetical division of the diapason,
I have never been able to find any
reason which is worthy to convince me
what causes this situation, although
many reasons are offered. I not only
do not know why the harmonic division
is more pleasing than the arithmetical
division, but I know even less why the
octave is pleasing to us and the seventh
is displeasing. I have considered carefully
that the 'samewhich happens to
the hearing with sounds likewise occurs
The reason which
led Franchino
(GaffurioJto
consider he harmonic
and arithmetic
division in
modes.
The reason why
the harmonic division
pleases
the hearing more
than the arithmetic
division.
422
to the sight with visible objects. For
example, in regarding a pyramid, the
sight [of it] delights us more when the
high part of it faces the sky and the
obtuse part faces the earth than the
opposite, perhaps on account of the
proportion and agreement which that
solid body has in that particular form
and position with our [own] perspective
in regarding it from the plane of its [74]
base. The same can result from musical
intervals with regard to pleasing and
displeasing the ear more or less, as
you will be able to comprehend easily
from the following diagram [Diagram X.).
EYou should know that] one interval of
a third appears more full of spirit to
the sense than the other. Thus, equally,
the fifth is more full of spirit above
the octave than it is below. Let us
also consider that the power of this
affection does not consist merely of
its location, that is, where the larger
part of the interval is underneath and
the smaller part is uppermost, as some
423
[DIAGRAM X]
[A MIXED DIVISION OF THE EIGHT MODES,
ACCORDING TO GAFFURIO]
Wisdo of
Fraichino,
0
Minor third----- - Sesquiquinta
Major third------------. esquiquarta
Diatessaron------------- S squitertia
d
Minor third-------------- Se uiquinta
Diapente---------
Major third------ Sesq iquarta
BisdiapasonG
plus diapente---
Diapason-----d Sesqui ertia
Diatessaron----
D
Minor third--- - i Sesquiqui ta
Major third-- Sesquiquart
i
424
believe; it consists, instead, of the
fact that what occupies that place is,
at the same time, the nearest to per.
fection, This particular fact, without
any doubt, plays a greater part
there than does the other, as you will
clearly recognize when you examine,
with the usual diligence, the minor
tenth divided by an intervening pitch
into a low fifth and a high minor
sixth, and when you also examine the
reverse i.e. a high fifth and a low
minor sixthJ. The same will be true
in hearing the major sixth separated
by an intervening pitch into a low
fourth and a high major third, although
this particular interval is
abhorred today by some persons with
delicate hearing, rather than their
being pleased by it, and it is widely
accepted by modern cithara players.
Let me tell you also that those modes
according to the modern practice which
do not have a fifth below the final note
and a fourth above, for example the fifth
and sixth modes, or to say, according to
the usage of most contemporary musicians
(although it is farther from the truth),
the seventh and the octave, their ending
is always incomplete and imperfect, and
the ear does not remain entirely appeased
as [it does] from the other available
endings. Coming, however, to deal with
the order of modes, I say this. It is
clear that in order to distinguish one
from the other, the diversity of names
which is seen in those Emodes) according
to the usage of the Greeks is necessary,
or else the numbers which are used by
the ecclesiastics. These Eecclesiastics]
wanted to call a given progression of
notes first, rather than third or some
other mode. Perhaps this was the reason.
They said that the first mode was Cexemplified]
by that composition which proceeded
to modulate among the notes of
the species of diapason which served the
Dorian mode. They were influenced by
nothing else except thefact that that
Whether the
manner of
numbering
the modes is
done at random
or in
order.
425
426
particular mode, according to the ancient
musicians, was the principal one and the
most honorable. This was the only reason
that they called such a species of dia-.
pason by the name of first.
Strozzi: What, then, motivated
Franchino EGaffurio] afterwards to consider
that particular modulation of the
first mode Eto be] divided harmonically,
rather than arithmetically?
Bardi: [He was motivated] by this.
The mese of the Greater Perfect System
separates the species of the diapason of
the EsameJ Dorian mode harmonically, and
that was divided arithmetically according
to the order of Ptolemy. In this procedure,
however, the moderns--as I have
said--went imitating Boethius. I say,
but not for this Ureason), that neither
Franchino [Gaffurioj, nor Boethius, nor
Ptolemy, nor any other more ancient
[authority] ever thought or attempted to
adapt such a foolish thing to modes, because
the unique quality Cof these tones
Why the first
mode is divided
more Eoften]
harmonically
than arithmetically,
and
likewise the
other authentics.
consisted otherwise, as they have shown,
and as I demonstrated even more extensively.
Franchino, moreover, was moved
to the consideration which you have
learned for the alleged reason. Afterwards,
since he had been obliged to consider
the quality of harmony and melody
in such a way, he did not seek to divide
it in another manner. From this division
of the diapason the modern practical
musicians took occasion to call the
larger side of the diapente by the name
of first species, and they said that
smaller side was the first [speciesJ of
the diatessaron. In this respect they
came to imitate the Greeks rather than
the Romans. Whereupon it is openly seen
that the modern musicians first sang
through h durum, rather than through
b molle. Since that species of the
diapason was so divided between D sol re
and d la sol re, and since the first
species of the diatessaron was called
re sol--as has been said--when b fa had
H durum was
used before
b molle.
427
428
been in the scale of that diapason instead
of h mi, the diatessaron in that instance
was called mi la. Otherwise, instead of
the second species, there would have been
the tritone. Their modes, moreover, had
been composed and sung originally through
h durum rather than through b molle. One
finds, according to the usage of our
practicing contemporaries, that the first
species of diapason is between D sol re
and d la sol re, that of the diapente is
between D sol re and a la mi re, and that
of the diatessaron is between a la mi re
and d la sol re. The others proceed
moving upward by conjunct steps, that is,
through h quadratum. one can argue from
this how much those have been deceived
who finally have changed, without any reason,
the order of modes and the species of
the consonances, scheming more closely to
approximate the order of modes of the
ancients. I do not believe that he
[Zarlino] has achieved this [goal) or if
80Zarlino, Istittonip
.
Zarlino8 0 in
chapters 10
and 11 of the
4th part ofhis in'stitu
429
he could (ever) accomplish other than the.
contrary effect, as much in the location
as in the distance, and equally in the
quality of the harmony.
Strozzi: I did not understand before,
and I do not understand even now, why it
is that no division other than that particular
one agreed with the quality of the
harmony and the melody of the Dorian mode?
Bardi: This was because the ecclesiastics,
having constituted their first mode
within the very same species of diapason
which served the Dorian mode, and since
that [model--as you will learn on another
occasion--was stable and quiet by nature,
without violence, and suited to induce in
the minds of the listeners grave and serious
thoughts, and customs for strong men, I say
that [the Dorian mode] did not require
being divided in any other way, since the
nature of its melody was what it was between
those pitches. It came more greatly to
manifest its particular quality when it
was applied to words which are suitable to
What agreement
the ecolesiastical
modes have
with those of
the Greeks,
it, while, on the contrary, the arithmetical
division had made it conform in many
respects to that of its plagal. This did
not agree, since it was languid, feeble,
and timorous, as one can clearly hear by
plucking together three strings which have
been disposed in such a way.
[Figure 111.)
These strings not only have such a property
when all three are plucked at the
same time, but [it is also inherent in)
the simple motion of each one of the intervals
separately when it moves away
from the mese in order to reach any of
the extremes. Thus, similarly, adding
to [the Dorian mode another part in the
high register, which proceeds with its
proper and natural movement and is consonant
with the low register, they are
manifested to the sense in the same way.
In order [to proveJthat that is true,
Arithmetical
division in
the second
mode.
430
[75)
431
first consider--as has been said--what
nature the harmony of the interval has
which is found moving away from the mese
of the Dorian in order to reach any of
the extremes of its diapason, as, for
example, descending from a la mi re to
D sol re, or ascending from this a Ia mi
re to d la sol re in the same manner.
Harmonic division
of the first mode.
[Figure 112.J
When the motion of that part, being the
low one and that which provides the melody
to the composition, is accompanied
by another in the high register which
proceeds with movement suitable to its
location, which these two cases EFigure
113.J will demonstrate, you will hear
what the harmony willbe hen each one
of these has been ung together with the
other, or at the se time
i
IT T
[Figure 113.
Strozzi: The melody of each part in
itself and the melody which issues from
both, or from all three together, when
they have been arranged in this particular
way, is indeed exactly as you have described.
Let us consider more carefully
each of these cases in the arithmetical
division, if you please.
Bardi: We will do it at the proper
place. Let us pursue Ethis present matter)
further with regard to the consideration
of modes. The ancient musicians, moreover,
divided their modes into plagals
and authentic, leaving between each pair
of these the space of a diatessaron, as
you have seen in each one of the demonstrations
according to the usage of these ancients.
Now thus, also, the ecclesiastics
intended for there to be the distance of a
diatessaron from the first mode called
authentic to the second mode called plagal,
Considerations
of the author
about modes.
VT
432
MwM N
rr--
-
433
and so this came necessarily to occupy the
place of the diapason of the Hypodorian
mode, that is, always according to the
opinion of Boethius, and not that of any
other ancient writer. The cause, then,
for their diverse divisions stemmed from
nowhere else but from wanting the harmony
which resulted from these [to be] more
consonant than was possible for the nature
of modes of the ancients from which they
had been derived. Also, as some say, [it
was] because the final pitch of the first
called authentic and Ethat] of the second
called plagal was common to both. This
rule, since it was impertinent, was not
entirely approved by the authors of ecclesiastical
songs. In this way, the first
and second modes were included within the
pitches which incorporated the same sides
as the first species of the diapason, but
with the following difference from these
[sidesJ: since the authentic has its
larger segment in the bottom part and its
smaller segment in the top part, the plagal,
on the other hand, transposing the
434
species of the diatessaron of its authentic
by an octave into the part opposite
to the high part, adds Ethis species] to
the low part of the same diapente without
moving it out of its place. Thus the
plagal comes to reach into this part a
fourth more than its authentic, and this
[authentic], on the contrary, is extended
into the high part more than that [plagal]
by a similar interval. This procedure
causes the species of the diapason to be
changed because what was first [species)
in the authentic becomes fifth [speciesJ
in the plagal. This has been considered
diversely by our practicing contemporaries.
Therefore, in that original simplicity of
canti firmi where all these conditions
had been observed, there resulted between
these canti firm a variety of harmony,
and, consequently [a variety] of affection.
That resulted mainly because of the small
number of pitches which these [canti firmi]
contained, because of their diversity, and
because of their movements. In order to
place such a truth within your grasp,
435
consider each smallest part of this particular
division. ENotice] how different
in nature it is from that first divisionsj,
not only in sound and movement of one single
part (which is what agrees with the
low segment when one wishes to move away
from D sol re, that is, its mese, Ein order
to reach anyone of the extremes of its
diapason), but also when it is accompanied
by another part placed in the high segment,
which proceeds with movement which conforms
to the low. Here is an example.
[Figure 114.]
Here is another way you really should consider
[Figure 115.]. In addition, in
hearing simultaneously the extreme pitches
of the simple diapason together with its
mese, or even the three parts together,
you will find that these--as I have said--
have a sad, flat quality. This temperament
is not far from that which is attributed
to the Hypodorian mode, nor is it at all
436
[Figure 115.J
surprising that the diversity of sound,
with regard to highness and lowness, together
with the difference of movement
and of interval produces variety of harmony
and affection. It happens that
nature ordinarily does not produce similar
things with contrary ones, nor contrary
things by means of the very same
quality, but only the opposite. With
regard to the consideration of these
matters, when the convenience of rhythm
and the conformity of conceptions had
been added, what force and effect do
we believe that that particular melody
had afterwards? It is quite certain
that it would be suited, as it already
was, to incline the spirits of the L76]
hearers in whatever direction which the
skilled musician pleased. Because some
of these things, however, have not been
understood, nor considered, nor observed
- K
437
by our practicing contemporaries in their
compositions, it happens therefore, that
the harmonic and arithmetical divisions
do not have--as has been said--any part
in them.
Strozzi: This is actually so. One
can argue from it how much our practicing
contemporary contrapuntists have misunderstood
their own very famous precept when
they have said that the parts of the composition
ought to proceed by contrary motion,
since they clearly have come to the
opposite [viewpoint), that is, the same
affection can be expressed with greater
efficacy by means of similar [motion] than
[it can] with diverse [motion). Similarly,
sadness, together with the other passions
8 1 Zarlino, The Art of Counterpoint,
p. 74. "Harmonyis madeof opposites or
contraries. This applies also to the simultaneous
movement of several parts.
Whenever possible--and this conforms to
ancient practice--when the part on which
the counterpoint is written, that is the
subject, ascends, the counterpoint should
descend, and vice versa. It is not faulty,
however, to let them move in the same direction
on occasion for the sake of
smoother voice movement."
Abuse of the
contemporary
contrapuntists.
Zarlino in
chapter 35 of
the 3rd part
of his Insti-
tutioni
.
438
can be caused in the hearer not only with
high and low sound and with fast and slow
motion, but with the different qualities
of the intervals. [This is true) even
with the same interval when it is brought
toward the low register or toward the high,
because the fifth is sad when it is ascending--as
you have said--and when it is
descending, it is joyful. On the contrary,
the fourth is joyful in ascending and sad
in descending. One observes [that] the
same [thing] happens to the semitone and
to the other intervals.
Bardi: I want to tell you even further
in this regard, that the diapente
[when it is] carried by the voice away
from the high part toward the low, or else,
on the contrary, between pitches different
from the first shown, has a dissimilar
nature from the first one, which has already
been mentioned. The same is equally
true of the diatessaron and every other interval
in the disjunct system, for when
moving away from G sol re ut with the low
Consideration
of the author
concerning the
nature of the
intervals.
439
part, and descending by a fifth or ascending
by a fourth, you will find that such
movement has [the propertiesJ of joy,
agitation, and (so to speak) virility and
naturalness. The same is true of the other
parts which are higher than that [low oneJ
which proceed with movement which conforms
to it. Here is an example.
[Figuxe 116.J
I believe, for the stated reason and no
other, since the third and the tenth are
found to be naturally major between the
parts and minor in the Cother way of proceeding),
and since they must be made this
way by means of accidentals, that the
440
implication in this way of proceeding is
that they will turn out in the manner you
have understood. This is because Nature
produces her effects more vigorously than
Art. When the parts which are above the
low [part) are made higher, the harmony
Cconcento] should become even more as we
have described it, that is, always sound
and discreetly [executed). [To prove]
that the low part is actually that which
gives the air (when singing in harmony)
to the composition, observe, in this last
example [Figure 116.], that the conjunct
movement of the notes which the contralto
part makes when ascending possesses a
high degree of virility. On the contrary,
when the very same interval is sung in the
same manner and between the same notes in
the above-mentioned [part), it should have
a sad, flat quality as the first one did,
provided that one or more parts were added
about it which performed the very same
function which they did in this last
example. The part in question should have
441
the same [sad, flat quality] when it is
transposed an octave higher.
Strozzi: What you say is undoubtedly
true, but I am surprised that these things
have not been considered before now and
put into use by our practicing contemporary
contrapuntists. Before you continue with
your explanation of the way the other modes
proceed, I would like for you to tell me
another particular besides the two already
mentioned.
Bardi: Tell me what it is.
Strozzi: Which has more part in
giving spirit to compositions, the slowness
and fastness, or the highness and
lowness of the sound? Which one of these
two methods is more effective to manifest
such compositions, exactly as they are, to
the hearing?
Bardi: Each one of these is suitable
for its own appropriate function, which
is not unlike what lines and colors do in
revealing to the sight bothj the beauty
and the ugliness of the body. Since in
doing that, the lines always have a greater
442
part than the colors, thus, similarly, in
the former [situationJ, the slowness and
fastness have lower or higher sound respectivelyj.
These lines, without the
colors, can indicate to the sight the
proportion and disproportion of the body,
since the fast and the slow movement of
the sound in a single extension Eof pitch]
can communicate to the ear the air of any
composition. There is, however, such a
close relationship between these two cases,
that one without the other cannot entirely
manifest the quality of the air, as the
lines without the colors can between those
other two. I come, returning to the invention
and usage of the ecclesiastical
modes, to tell you that the author .of
these [modesJ afterwards constituted the
third mode in the same species of diapason
which serves the Phrygian harmony. This
diapason was considered [to be] divided
in the same manner as that which serves
the first mode, since it was also one of
the authentic and was conformed, for the
reasons stated above, to such a division,
Other considerations
of the
author concerning
modes.
443
and also Lwas conformed] to the quality
of its harmony. The fourth mode, then,
since it adjoined the third, they rightfully
constituted in that species of
diapason which already served the Hypophrygian,
assigning to the fifth mode
.
the diapason of the Lydian. For the
same reason mentioned with regard to the
others, they assigned to the sixth mode
the species of the octave which served
the Hypolydian. They wanted, in addition,
to constitute the seventh [mode]
between the pitches which contain the
species of the diapason of the Mixolydian,
and the eighth and last they assigned to
the same species that they assigned to
the first [modeJ. Because of this similarity,
it was called, in imitation of
the other plagals, but the name of Hypomixolydian
(and not Hypermixolydian as
some have dreamed) since Hypo meant "below"
and Hyper "above", as you know.
82
See Gafurius, Practica music,
pp. 39-45; 52-54*
[773
Franchino8 2 in
the first of his
Practica in the
7 n in the
11th chapters.
444
Glarean indeed used these words on one
occasion. It was called the Hypermixolydian
since it was above the Mixolydian,
and it is our eighth [modej. One finds
in this description of the eight modes
(applied, however, to the modern practice
without further consideration) the same
discrepancies with regard to the distances
of highness and lowness which I mentioned
above. [This is] presumably because
their authors had not valued any more
than they did the other places, the fact
that the interval which was found between
the Lydian mode and the Mixolydian (mode]
of the ancients was the distance of a
semitone, and not a tone as one finds between
the fifth and seventh ecclesiastical
modes. They did not value it because these
[ecclesiastical modes] did not have the
power to tune together the species of the
diapason and the intervals which are found
between one and another of the ancient
modes. Thus, they have come to omit, although
very important, that which they
understood less and needed to understand
more.
445
Strozzi: What difference is there,
then, between the eighth ecclesiastical
mode and the first one, since they occupy
the same species of the diapason and the
very same pitches?
Bardi: Herein-lies the entire importance
of the affair. [It is] none
other, of course, than the fact that the
final pitch and the diversity of the
division are considered harmonically in
the first mode and arithmetically in the
eighth. Now you see, in the modern compositions
which are composed and sung
in the manner which is customary today,
that which has to do with the diversity
of the melody and the harmony between
one mode and another, the final pitch,
and the variety of the division of the
species of the diapason.
Strozzi: Since these modes of
theirs have been arranged in such a way,
it indeed reminds me of the pictures of
the singular Hermippus of Athens. This
man, in depicting males and females,
either because of the antipathy which he
The modes cannot
be more
than seven [in
number).
Hermippus, the
Athenian painter.
446
had naturally for beards and for clothes,
or through another of his particular interests,
he continually made so many
final touches that it was impossible to
tell the males from the females, if not
for the sex, as if industrious Nature had
not formed differences in a thousand other,
perceptible ways. The same thing is true
of the modes of our contemporary contrapuntists.
It is actually impossible in
hearing them sung, to tell from the highness
and lowness which is the first and
which is the second mode, which is the
third and which is the fourth, and it is
likewise impossible to tell all of these
from the other modes. It is only possible
to identify each mode from the final pitch,
although, in seeing them written, the lower
modes are quite often shown higher and the
higher modes lower.
Bardi: I want to tell you another
thing which I recall in this case, Ewhich
isJ that, while the ancient musicians, in
varying their modes, changed not only the
species of the diapason, but the entire
Another abuse
of our practicing
contem-
poraries.
447
system from low to high or, on the
contrary, from high to low, those of
our times say to vary them by singing
the same species of the diapason in
the very same intensity and slackness
of the system, and not only the neighboring
[pitchesJ but the extreme ones,
for example, the unison and the octave.
I say, also, that if each of their
modes has particular power to move in
the hearer a number of different affections
as they affirm (although contrary
to every precedent), how does it happen
that, when they have to put together a
sonnet, which has been drawn--in a
manner of speaking--from sad things,
they will sing its fourth stanza in the
second mode and its third stanza in the
ninth, or some other way? In addition,
what man is so devoid of judgement that
he does not notice that, in any of their
songs composed in not more than four
voices, if the bass sings, for example,
within the notes of the diapason which
serves the first mode, the tenor will
448
sing within those of the second, the
contralto will sing the Hypermixolydian
at the same time the bass the Hyperdorian
by means of singing an octave above it,
and the soprano will be the same as the
tenor? [What men could fail to see] that
such confusion and contrary mixing of
notes cannot induce any affection in the
one who hears it? Each part has by itself
what it does not have in any way
in this modern practice of counterpoint.
It has the specific ability to induce a
number of affections in the listener
which cannot be induced [in part songs]
due to the confusion of one part with
another, because as contraries, they
impede the natural operations. We come,
however, to our matter concerning the
addition of the four last modes, and let
us leave the foregoing matters as merely
statements of opinion. The abovementioned
Glarean, a truly scientific
man of great literary prowess, considered
that no other factor caused the eighth
ecclesiastical mode to be different from
Consideration
of Glaren
.
The addition
of four modes
to teorigi-
449
the first [modeJ except the final string
and the visualization of the harmonic
and arithmetic divisions then applied
to this new way of composing and singing
so many melodies together. He said this,
stimulated by the thirst he had to invent
such a useless novelty. It was something
which he considered most important and
necessary, which is openly manifested
by the title of his books on music; he
calls his work Dodecachordon. One reads
these incredible marvels in the [very]
first page of this where he eventually
claims to have imitated Aristoxenus. We
have already demonstrated the conformity
which they have together. I say, therefore,
that the following is what this
Glarean went around pondering:
Since the species of the diapason
which serves the first mode has
power to form the eighth Emode]
with the sole consideration that
Ethe first mode] is divided harmonically
and the [eighth mode]
is divided arithme ically, why
should the same divisions not make
a variation of harmony in the other
species which contain the other
modes?
450
On the strength of this aspect of reason, [78]
although it leaned toward weakness, he
added to the eight modes shown, four
others, which made the number of twelve.
This was the order he maintained in order
to accomplish that. He formed the ninth
mode from the same species of the diapason
which served the second mode, but transposed
an octave higher and considered
differently divided from its mese. He
formed the tenth [mode] from that of the
third assigning it and its authentic
a la mi re as the final pitch. Since the
eleventh mode did not correspond with the
fourth [mode] of the division, he formed
it from that of the sixth because he considered
that capable, causing it to end,
together with its plagal, on C sol fa ut.
This plagal, he constituted in the species
of the diapason which served the seventh
mode, observing in both of the foregoing
the very same conditions which he made in
deriving the ninth mode from the second,
omitting--as has been said--the harmonic
division of the diapason which serves the
451
fourth mode, since it is not capable of
it, just as that of the fifth Emode] is
not capable of the arithmetical division,
as one can easily comprehend in this
example.
-0
10, rd rd
xap a x
1 2 3 4 5 6
HOO
r HJcH"r
7 8 9 10 11 12
mExample 11"r Description
modes. 384
Description
of the twelve.
modes extracted
from the 1st
page of the
Dodecachordon8
of~"Glarean and
from the 7th
and 28th chaptersj
of he
2nd book.
of the twelve
8 3 See Glarean, Dodecachordon, I, 37;
I, 114-115; I, 173-175.
452
These impediments caused the number
of modes not to proceed to a greater
number. One argues from this, that
the matter of modes of the ancient
musicians has been misunderstood by
the moderns, because such foolish
things have indeed given them occasion
Eto misunderstand] and have
hindered their operation. They have
not noticed or considered at all that
the musicians of ecclesiastical songs
and, afterward, the learned Franchino
[Gaffurioj neither wanted to speculate
nor to introduce a greater number of
modes, not through ignorance, as some
believe, but in order to avoid bad
84 aGalilei., ialo0,p.78,v
Another abuse
of the modern
musicians.
The ecclesiastical
modes are
similar to those
of the ancients
in some respects.
2 3 4 r ' 7 a 9S 1v ,I 12To .e
tf-Ob"- 0' F IF
hot,
" wiw "mom
"
w.+ l..
Fowl.
momm "mom""
1
.
453
usage and to avoid the occasion of being
numbered among the deprecators of good,
true precepts. They found that each of
the fifteen pitches of the Greater Perfect
System was already occupied, and
having put into effect with their first
seven modes each species of the diapason
and quite often the other consonances
which are called today perfect as well,
they added the eighth into the [total]
number of these modestj rather in imitation
of Boethius than because they truly
knew that it was apt to introduce any
new, necessary affection. These things
were very well known by Franchino, who
had preferred to refute them because of
that, rather than form them without any
chance of innovation. Since this [eighth
mode] had already been accepted by the
general public, however, he admitted it,
thinking it was just as well. In addition
that sort of solo song, since it
was sung according to the sound of the
pitches among which it was written (as
was customary in the beginning when they
The ecclesiastics
constructed eight
modes in imitation
of Boethius.
were introduced) without any alteration,
it was not unsuitable as those of today
with many voices are, because there is
actually a difference of harmony and
melody among the pitches of their songs,
and a great deal more so when one applies
suitable words to them. However, in
singing so many melodies together according
to this new practice of figured song
canto figurato i.e. polyphonic compositionJ
(so called because of the variety
of the singable figures), two modes are
too much, let alone eight or a greater
number. The reason for this is that any
composition put into use always seeks
out the .same quantity and quality of
pitches (speaking of high and low), proceeding
in each given part with the very
same rhythm with regard to fast and slow
movement, because the contrapuntist uses
in these Ecompositions] notes of any
value, and rashly employs each interval
that suits his pleasure; he never thinks
a thing in the world of the conception of
Why measured
,
compositions
are so called.
454
455
the words. It will be proved, at the
proper place, that the diversity and
effect of the harmonies and the melodies
consist mainly of these cases, so that
the modes and the compositions of today
necessarily come to be the same in quality,
quantity, and form. Thus one comes
to have the very same color, taste, and
smell (so to speak) as the other. In
this case, it appears to me that the
example of Philoxenus, a most noble
musician, is an effective argument to
persuade us how distorted the modes of
the modern [musicians] are from those
of the ancients. let me confirm this
fact more extensively by telling you
this. The said Philoxenus, having once
upon a time begun [to play] the dithyramb
8 5 Aristotle, Politics (1342b7-12),
Richard McKeon, ed., Basic Works of Aristotle
(New York, 1941T~ TTTT ~T
'UT~ramb, for example, is acknowledged
to be Phrygian, a fact of which the
connoisseurs of music offer many proofs,
saying, among other things, that Philoxenus,
having attempted to compose his
'Mysians' as a dithyramb in the Dorian
mode, found it impossible, and fell back
by the very nature of things into the
more appropriate Phrygian."
There is only
one modern
mode.
Aristotle85 in
the 8th Ebook]
of the Politics.
456
in the Dorian mode, was forced by the
material of the subject to end it in the
adjacent Phrygian mode.
Strozzi: Would you kindly tell me
what caused this?
Bardi: In our times, one can hardly
satisfy (as far as I know) in a custom of
which there is no conception, since no
memory has remained of these things. If
you want to content yourself, however, by
knowing how greatly .Ivalue and believe it,
I am content to tell you. The dithyramb,
according to the ancient Greek poets, was
a form of song in praise of Bacchus whose
86
author was Arion. In the melody of this
dithyramb, one imitated the conceptions
and customs of excited, tipsy, joyful
worshippers of Bacchus, and others like
them. It was sung by a chorus, which
represented the deeds performed, which all
86
Arion is equally renowned for his
calming of the billows and charming of
dolphins in order to save his life. See
pp. 514-516 below.
The meaning of
dithyramb according
to the
ancient Greeks
and the identity
of its inventor.
457
the people] were accustomed to execute
(insofar as I believe) in an intense mode.
[It happens that] artists sometimes want
to have an experience with art, and thus
undertake to test it by forcing the limits
of their principles. Philoxenus, being
one of those [artists), wanted to experiment
with something new. Since it did
not prove capable of his conception,
defeated by its nature, it returned to
its normal state. Thus, when he had
commenced the dithyramb in the Dorian
mode, the harmony of which is quiet--as
has been said--and without any violence
of affection, and since he abhorred the
indecent softness of such material and
the allurements of that verse, he was
quite willing (skilled as he was) to
proceed without the aid of [that particular]
harmony in his imitation. [whereupon]
he abandoned it without respect
and went on almost violently to the
Phrygian mode which was higher than the
Dorian, and thus intense in nature and
suitable to express the conceptions which
[79]
The reason why
Philoxenus,
having begun
the dithyramb
in the Dorian
mode, was forced
to end it in the
Phrygian.
458
he had at hand. It is not at all surprising
that such a conception came to
Philoxenus, because he had the nature
and constitution to love low, quiet harmonies,
even the very lowest ones. This
argues that after all the others Chad
been found), he discovered the Hypodorian,
slowest and weakest of them all.
This is all I know to tell you in this
matter. One can also comprehend from
the example of this Philoxenus, how
greatly the composing and singing of
today is without rule, without method,
without order, at random, and according
to mere practice. The practical composers
do not know anything except that
that is a consonant interval and this
is a dissonant one, and the modern contrapuntists
(notwithstanding this
Philoxenus) sing any conception in any
mode, having abandoned entirely every
observation and law and having been
made the prey of the honest will and
power of their stratagems, without
having considered further any good
limitation or rule,
Another abuse
of our practicing
contemporary con-
trapuntists,
459
Strozzi: Did the ancient musicians,
therefore, not use the mixture of one
mode with another?
Bardi: It is not certain until Ethe
time of] Sacadas the Argive,87 a musician
and poet who was extremely famous for
three victories which he had in that type
of festival called Carneian, which the
Spartans made in honor of Apollo's birthday,
since he had been the author of proceeding,
in the same song, from the first
mode into the second, from the second into
the third, or otherwise.
Strozzi: Is this that Sacadas who
was also inventor of the elegy? Was he
also the one in Sparta who discovered that
species of song and dance called gymnopaedias,8
9 whose work is particularly remembered
by Pindar, among other writers?
87Plutarch, On Music, p. 371.
"Sacadas of Argos was aso a composer of
music and of elegiac verse set to music;
he was, furthermore, an excellent aulete
and is recorded to have won three vic
tories at the Pythian games."
8 8 Plutarch, Moralia, 15 vols. (London
and New York,:T1T93, XIV, 373.
89Ibid.
Sacadas the
Argive, inventor
of
the mixture
of modes.
The meaning
of Carneian
festivals.
Plutarch8 8 in
the 3rd of the
Apohthemata
among aheorders
and customs
of the
Laoedaemonians.
460
Bardi: This is the one.
Strozzi: May it not be a burden to
you to tell me what mixture of modes he
used, and, afterwards, what were gymno-
paedias.
Bardi: I told you all of that after
having proved to you that the Hypodorian
mode was not only discovered by Philoxenus
but [was found) last of all. Since
this is an older obligation, I should
logically satisfy it first. However,
take note! You ought to know that this
mode, according to the ancients was
called, like all the other [modes), by
diverse names, as you can understand in
part from that introduction which has
been printed both in Greek and Latin
under the name of Euclid,90 [which says
this:) "Among others, some call it
Locrian and others call it Hypodorian."
90See Strunk, Source Readin s, p. 42.
This treatise, once bue o Euclid,
has been found to be the work of Cleonides.
The H odorian
mode was) the
last o be dis-
covered.
461
91
Now Julius Pollux, praising the harmonies,
although he does not name the
Hypodorian, he says that the Locrian
was discovered by Philoxenus. Therefore,
we already know, since these two
are the very same as the writers tell
us, that Philoxenus was its discoverer.
The fact that is was the last to come
into use, besides the fact that it is,
in itself, a very likely thing since it
is the eighth mode and last in number,
92is derived clearly from Bryennius9in
the fourth chapter of his third book on
music. Having discussed the other
higher modes, telling why the Hypodorian
was called a low mode by the ancients,
since the Hypodorian was lower than it
was, he shows why this had resulted.
He says that before the Hypodorian came
into use, the Hypophrygian was by nature
the lowest of the others and therefore
481.
Julius Pollux]
in the 4th book
of his Onomasticon.
Locrian
a Hypodorian
are the same
mode.
9 13ee Bethe, I, 220.
92See Wallis, Operamathematica, III,
462
it had commonly been called [a low modeJ.
This name had remained attached to it
even though the Hypodorian, which was
lower than it in nature, had appeared.
Through this discourse, it is clearly
ascertained that the Hypodorian was the
last of all the eight modes which he
describes to be discovered. The gymnopaedias,
according to the Spartans, were
choruses of young boys who danced together
barefoot singing the praises of
the gods, and [were] in honor of those
Spartans who had fallen dead fighting
for their native land during the Therian
campaign. These, then, according to what
Plutarch tells us, are the mixtures of
modes which Sacadas used. They had in
his times knowledge of only the three
principal species of harmonies, which
were, as you know, the Dorian, the
Phrygian, and the Lydian, although
Athenaeus9 3 claims, with the testimony
of Heracleides of Pontus, that the
93
See Athenaeus, Deipnosophists, VI,
365-367.
In chapter 10
of the 4th book.
Heracleides of
Pontus.
463
Phrygian and the Lydian harmonies, perhaps
because they were foreign, were
called Aeolian and Ionian, in order to
honor the Greeks greatly. This matter
does not pass without some trouble from
those who accurately examine the names
and the order of modes according to
Socrates, Plato, and Aristoxenus. Let
is be as you wish, however. Since the
cards are not all dealt Eso to speak]
concerning these particular controversies,
mainly with regard to modes,
let us come to tell how Sacadas used
the mixture of these Emodes). He used
it mainly in the chorus in this manner.
First of all, he set forth (when it
suited the quality of the poem, however)
the Dorian mutation; after this he set
forth the Phrygian, and next the Lydian,
94Plutarch, On Music, p. 371.
"Thus, there being thee systems of tuning
in the time of Polymnestus and Sacadas,
the Dorian, the Phrygian, and the Lydian,
they say that Sacadas composed a strophe
in each, and taught the chorus to sing
the first in the Dorian, the second in the
Phrygian, and the third in the Lydian; and
that this nome was called Trimeles because
of the modulation."
Plutarch's De
Musica[deaT
NTHhJmixtures
of tones according
9
Sacadas.
'
464
This rule or law was called tripartile,
He adapted this variety of harmony to
all the songs with such industry that
his work and ability were held in greatest
esteem. He accommodated it to the
subject of the words with greatest
judgement, and expressed the affections,
which had been observed in it by the
poets, with marvellous art. The practical
musicians of today have kept no
account of this most important, most
principal concern of the art of music,
for it pleased God that their errors
ended here. There is a much worse
situation, however, because all the
rules and observations which they have
in use are in direct opposition to the
adequate expression of the affections
and conceptions of any poem. I want to
remind you of some of these Erules] at
the present Etime). These are the inviolable
laws which they have made without
alluding to [any] authority or cause.
EFirst, they say] that it is not lawful
in any way to have two perfect consonances
[80)
Other abuses
of the practical
musicians
of today,
465
of the same species one after the other.
Secondly, when one must proceed to find
these Eperfect consonances), in moving
away from imperfect Econsonances) one
goes to the nearest Eperfect consonance).
Added to this is the respect which one
should have in relation to the tritone
and the semidiapente. Now the observations
of these two precepts alone are
sufficient to preclude any affection
from ever being expressed. [There are),
in addition, other inconveniences which
I am reserving to demonstrate to you at
a more convenient place and time.
Strozzi: You will certainly require
other industry and labor, which was not
lacking recently in making me ascertain
that the lyre and the cithara of the
ancient Greeks and Latins were the same
thing, to persuade this new, rather
newest, opinion of yours to the general
public.
Bardi: I offer to convince you
(and any other rational person) of that
opinion more clearly than any other thing
466
which I have mentioned up to now, or
could possibly mention in the future.
If you are willing, however, I want to
reserve it until the proper place.