Features of Common Slavic Ablaut Alternation
Ondřej Šefčík
1. Introduction
The issue of the my paper is, first, that the system of Old Church Slavonic ablaut
could be described with the use of the set of ablaut values and features derived
from them, and second, that such ablaut features could be taken for vectors;
third, that such features form a vector and a metric space in which all differences
between ablaut grades could be exactly stated.
The analytic approach to the features which is here presented is derived
from that of Marcus (MARCUS 1967). There are some similarities to the methods
of Dependency Phonology (see ANDERSON & EWEN 1987 or DURAND 1996) and
closely to the method of phonemic description by Hubey (HUBEY 1999), who
reworked a rather intuitive approach of DP using the algebraic form. The basic
idea of Hubey’s revisited DP-method is to found and to define basic orthogonal
vectors and to describe the whole given system as a vector space. Third, our attention
is more focused on metric spaces than vector spaces, which is the shift of
focus, not a different approach at all.
The solutions of the method proposed here could be applied not only on the
OCS ablaut system, but is relevant for the reconstructed Common Slavic ablaut
system, too. The data from OCS are used purely as raw material for the presentation
of the method, although the method itself is not determined by the data.
2. Ablaut grades
The OCS ablaut is such an alternation between morphs of one identified morpheme
(most often a root-morpheme) which indicates grammatical information.
Or in other words, the ablaut is the grammatical derivation of the morpheme.
The OCS ablaut is then a non-linear alternation, i.e. such an alternation not triggered
by the phonemic context of a given morph which is different from the linear
alternation such as palatalization of velars neighboring palatal vowels and
diphthongs.
Ondřej Šefčík
(1): Ablaut grades of different verbal roots1
:
√r_k: rekD, rokъ, -rěkati, rьci, -ricati
√t_k: tečetь, tokъ, -těkati, -tačati, tьci
√g_n: ženD, goniti, -gańati, gъnati
√sl_v: slovo, slaviti, (slъšati2
), slyšati
√b_r: berD, -borъ, bьrati, -birati
√v_d: vedD, voditi, věsъ
√m_r: mrěti, moriti, -mariti, -mьrD, -mirati
√d_x: duxъ, dъxnDti, dyxati
Each of the ablaut variants of the given morpheme is termed the ablaut grade.
All ablaut grades form the set of ablaut grades.
In this phase of analysis the set of ablaut grades is an unordered set, i.e. there
is no additional information on relations between different ablaut grades.
Such an unordered set is not a system, because any system requires an organization,
or better, the structure over it.
We will try to develop such a structure using sets of values of all ablaut grades
to postulate ablaut features.
An ablaut value is then such a property which could be related to an ablaut
grade and which makes the difference between grades in at least one example.
For all ablaut grades we need to define at least a minimal set of such values.
This is equivalent to the relation between phonemes of any phonemic system
and the sets of values of its phonemes, for both systems consist of a set of
elements (phonemes, ablaut grades) and the organization itself (system of
phonemic oppositions, system of morphonemic oppositions). Other similarities
will be clear from the following lines.
3. Values of ablaut
Each ablaut grade has then its own set of such values, or in other words, any given
ablaut grade will be attached to the one and only one set of values (or better
− there is a bijective3
relation between each ablaut grade and its set of values).
__________
1 All prefixes are separated. All grades are not ordered due proportionality.
2 Probably such a form is only a variant of slyšati, but definitely of a different ablaut
value.
Features of Common Slavic Ablaut Alternation
Any value of a given ablaut grade is here symbolized V{x} in general or in
the case of a concrete value simply as {value}.
The set of values of a given ablaut grade is symbolized as V{x} (i.e. V{x} =
v1{x}∪ v2{x}…∪ vi{x}).
The set of values of the whole ablaut system is symbolized simply as V,
which is the set union of particular values of concrete ablaut grades, hence V =
V{a} ∪ V{b}…∪ V{z}i).
Any value could be classified with the help of three criteria, every one expressing
some properties of the given value in a language system. Each criterion
consists of a pair of possible incompatible properties.
The first criterion is homogeneity. Two values are homogeneous if they are
members of the same subset Vi; otherwise such values are heterogeneous
(MARCUS 1967: 46-47, KORTLANDT 1972: 57; cf. ŠEFČÍK 2008: 6, ŠEFČÍK 2009:
186-187).
(2a) The values {e-grade} and {o-grade} belong to the same subset of values,
hence they are homogeneous.
(2b) The values {reduced-grade} and {o-grade} do not belong to the same subset
of values, hence they are heterogeneous.
The second criterion is compatibility. Two values are compatible if they are
attached to the same ablaut grade {x} (members of the same set of ablaut values
V{x}). If such values are not members of the same set of ablaut values (they are
not attached to the one ablaut grade), they are incompatible values. It should be
mentioned that all compatible values are heterogeneous, but not the other way
round; thus, some heterogeneous values are not compatible (MARCUS 1967: 47-
48, KORTLANDT 1972: 57 cf. ŠEFČÍK 2008: 6, ŠEFČÍK 2009: 187).
(3a) The values {e-grade} and {reduced-grade} are attached to the same ablaut
grade; hence they are compatible.
(3b) The values {e-grade} and {o-grade} are not attached to the same ablaut
grade; hence they are not compatible.
The third criterion is contrastivity. Any values vi and vj are contrastive if there
are two ablaut grades such that V{x} – V{y} = vi and V{y} – V{x} = vj. In other
words, if replacement of one value of a given ablaut grade by another value results
in another ablaut grade, the values are contrastive. Otherwise both values
__________
3 Bijection is such a relation between sets A and B so that if every element of A is
related to exactly one element in B and if then every element in B is related to only
exactly one element of A.
Ondřej Šefčík
are incontrastive. All contrastive values are homogeneous and therefore incompatible,
but not the other way round (MARCUS 1967: 48-49, KORTLANDT 1972:
58 cf. ŠEFČÍK 2008: 6-7, ŠEFČÍK 2009: 187). Curiously, the ablaut system of
OCS is in fact structured in a way, so that any pair of homogeneous values is
necessary contrastive.
(4a) The values {e-grade} and {o-grade} are contrastive, because the absence of
the value {e-grade} necessarily means that the grade is {o-grade}.
(4b) The values {e-grade} and {reduced-grade} are not contrastive, because the
absence of one value does not necessarily mean that the second value is present.
Every value which is contrastive, homogeneous and incompatible we will term
the pertinent value (MARCUS, 1967: 46-47).
For OCS (and CSl, too) we deal with the following set of ablaut pertinent
values: {e-grade}, {o-grade}, {reduced-grade}, {non-reduced-grade}, {lengthened-grade},
{non-lengthened-grade}.
The above mentioned ablaut grades of different roots could hence be arranged in
the following table4
:
__________
4 All prefixes are removed.
Features of Common Slavic Ablaut Alternation
(5) eG oG ²G eG eG oG ²G eG
√g_b
gubiti *gъnDti
gъnuti5
gybati
√g_r žeravъ gorěti *žarъ6
garati gъrno7
√vr_t vrěmę vratiti vrьtitъ
√b_d bljusti buditi bъděti
√r_k rekD rokъ rěkati račiti8
rьci ricati
√t_k tečetь tokъ těkati tačati tьci
√g_n ženD goniti gańati gъnati
√sl_v(s)
sluxъ,
slovo
slaviti slъšati9
slyšati
√b_r berD borъ bьrati birati
√v_d vedD voditi věsъ
√m_r mrěti moriti mariti mьrD mirati
√d_x duxъ dъxnDti dyxati
Note: Marking of ablaut grades:
{reduced non-lengthened e-grade} (represented by ь, marked eG);
{reduced non-lengthened o-grade} (represented commonly by ъ, markedoG);
{non-reduced non-lengthened e-grade} (represented commonly by e, marked eG);
{non-reduced non-lengthened o-grade} (represented commonly by o, marked oG);
__________
5 Only in Russian Church Slavonic, besides gъbnuti, see DERKSEN (2008: 197)
6 The Proto-Slavic form is reconstructed on Russian žar, Czech žár, Slovak žiar, P. żar,
Serbo-Croatian, Slovene žâr, Bulgarian žar, cf. MACHEK (1971: 722), VASMER (1953:
I: 410), DERKSEN (2008: 554).
7 Only in Russian Church Slavonic, see DERKSEN (2008: 199).
8 Cf. LIV (457–8).
9 Although slъšati we consider as a variant of slyšati, formally it is {reduced nonlengthened
o-grade} = oG.
Ondřej Šefčík
{non-reduced lengthened e-grade} (represented commonly by ě, marked ²G);
{non-reduced lengthened o-grade} (represented commonly by a, marked eG);
{reduced lengthened e-grade} (represented by i, marked ²G);
{reduced lengthened o-grade} (represented by y, marked eG).
Two mutually contrastive pertinent values form an ablaut feature. Hence, any
ablaut feature could have two values, one “negative”, the second marked “positive”.
The values are attached arbitrarily (ŠEFČÍK 2008: 7, ŠEFČÍK 2009: 187).
Between both pertinent values of each given feature there is then a binary
opposition. For simplicity we will mark each “positive” pertinent value in the
features as 1, each “negative” pertinent value as 0, and hence each ablaut grade
could be expressed in a binary code.
When organizing the above enumerated pertinent values of OCS, we face
three ablaut features (AFS):
AF1: {non-reduced-grade} &{reduced-grade}
AF2: {non-lengthened-grade}& {lengthened-grade}
AF3: {e-grade} & {o-grade}
As we can see, all three AFs are mutually independent, i.e. no one is dependent
on the other AF. Such relations between AFs can be expressed in the following
graph.
(6) Systems of ablaut features of OCS:
AF2
0
AF1
AF3
In such a system there are basic vectors attached to basic ablaut grades: eG is in
the zero position, eG is the basic value on AF1 vector, ²G is basic on AF2 vector,
oG is basic on AF3 vector.
Features of Common Slavic Ablaut Alternation
4. Vector space
All three features then could be interpreted as vectors and the ablaut system as a
vector space. Because AFs are mutually independent, and their dot product is
zero, they form an orthogonal vector space.
For simplicity we will mark the AF1 vector as R, the AF2 vector as L and
the AF3 vector as O.
Any concrete ablaut grade V{x} has hence its unique set of values v1-v2-v3 of
the features of the same ordering (R–L–O). The commas will be omitted in the
following lines for simplicity. Values (and also features) for each ablaut grade
are then ordered and both sets of ablaut grades and the sets of values could then
be considered as an ordered sets due to the equivalence between both sets.
(7) AF vectors
R L O
{non-reduced non-lengthened e-
grade}
= eG 0 0 0
{reduced lengthened e-grade} = ²G 0 1 0
{reduced lengthened e-grade} = ²G 1 1 0
{non-reduced non-lengthened e-
grade}
= eG 1 0 0
{reduced non-lengthened o-grade} = oG 1 0 1
{reduced lengthened o-grade} = eG 1 1 1
{non-reduced lengthened o-grade} = eG 0 1 1
Ablautgrades
{non-reduced non-lengthened o-
grade}
= oG 0 0 1
The AF vector space is also considered as unit vector, with the size of vector
equal to 1, for the distance between two values of one feature is equal to 1 (more
on distances see below).
The unit vectors, which are orthogonal, too, are termed orthonormal vectors.
Hence our vector space of ablaut grades is an orthonormal vector space.
Any ablaut grade (i.e. V{x}) could be then described as a linear combination
of orthonormal unit vectors which could be written as (vx = value of a given ablaut
feature, i.e. 0 for the unmarked value, 1 for the marked value):
Ondřej Šefčík
V{x} = v1R +v2L + v3O
As a consequences of arbitrary ordering, symbols for vector could be omitted
and any set of values of a given ablaut grade V{x} could be expressed in a three
digit code, as above in the table 8.
Any ablaut grade V{x} could then be taken for a vertex of a unit cube. The
set of such vertices is then marked Bn
, and that set is a set of ordered n-tuples,
containing numbers 0 and 1, or in other words, as Bn
={0, 1}n
.
In that sense the elements of the set Bn
form dyadic (or Boolean) vectors
and the structure of the ablaut system of OCS is a Boolean algebra, consisting
two elements 0 and 1 (i.e. binary two-valued Boolean algebra) (see KURATOWSKI
1977: 34–35).
The distance between any pair of ablaut grades is then given by the sum of
differences between codes of given ablaut grades:
(8a) Example of minimal distance:
V{non-reduced non-lengthened e-grade} 000
V{non-reduced non-lengthened o-grade} 001
V{non-reduced non-lengthened e-grade}–
V{non-reduced non-lengthened o-grade}
001 Σ =1
(8b) Example of zero distance:
V{non-reduced non-lengthened e-grade} 000
V{non-reduced non-lengthened e-grade} 000
V{non-reduced non-lengthened e-grade}–
V{non-reduced non-lengthened e-grade}
000 Σ = 0
(8c) Example of more than minimal distance:
V{non-reduced non-lengthened e-grade} 000
V{reduced non-lengthened o-grade} 101
V{non-reduced non-lengthened e-grade}–
V{reduced non-lengthened o-grade}
101 Σ = 2
Features of Common Slavic Ablaut Alternation
Such distance between codes is known as Hamming distance10
. Hence, any difference
between set of values, considered as codes, could be expressed as a
Hamming distance.
The ablaut vector space which we now face could be considered as a metric
space.
And indeed, we can consider the difference between two values of a same
feature of two ablaut grades as a minimal distance sui generis in such a vector
space and the identity between two values of the same feature of two ablaut grades
as zero distance; then we can metricize such a vector space easily.
5. Metric space
Any metric space is generally a tuple (A, ρ) where A is a set of elements and ρ is
the distance between them given by the mapping of the set A on itself (i.e as a
Cartesian product A × A).
The metric space complies with the properties:
1) if two elements have null distance between each other, then we are dealing
with one element (axiom of identity, i.e. ρ(x, y) = 0, if x = y);
2) the distance between an element x and an element y is equal to the distance
between y and x (axiom of symmetry, i.e. ρ(x, y) = ρ(y, x));
3) the distance between x and z is equal or smaller than the sum of the distances
between elements x and y and y and z (axiom of the triangle inequality, i. e.
ρ(x, y) + ρ(y, z) ≥ ρ(x, z)). See (MARCUS 1967: 34-35, KURATOWSKI 1977:
115).
For a more obvious image of the relations inside the system of the OCS ablaut
we can draw the graph, expressing the system of OCS, in the form of 3-cube
(dyadic cube), with the length of each edge equal to 1. All grades are expressed
by their sets of values, as described in the table 7. In the following examples the
metric space of OCS ablaut is presented both with codes due to the Hamming
distance, both with enumerated ablaut grades, in the form of two mutually proportional
3-D cubes, compare with AF vectors above:
__________
10 First described by R. W. Hamming in HAMMING (1950).
Ondřej Šefčík
(9a) oG oG
eG eG
eG eG
²G ²G
(9b) 001 101
000 100
011 111
010 110
The distance between any pair of vertices is then equal to the number of edges
between them. Beside the null distance between a given ablaut grade and the
same ablaut grade (axiom of identity) we face the following examples of distances,
see example 10:
(10a) eG – oG = 000 – 100 = eG − eG = 000 – 010 = eG − ²G = 000 – 001 = 1
etc.
(10b) eG – eG = 000 – 101 = eG − oG = 000 – 110 = oG − ²G = 000 – 011 = 2
etc.
(10c) eG – eG = 000 – 111 = oG – ²G = 100 – 011 = 3 etc.
The distance between any pair of ablaut grades is the same without regard to the
order of vertices (axiom of symmetry), see example 11:
(11a) eG – oG = 000 – 100 = oG – eG = 1 etc.
(11b) eG – eG = 000 – 101 = eG – eG = 2 etc.
Such a metric space we term the fine metric space, because the distances could
be precisely stated and enumerated.
6. Conclusion
In the present paper we have demonstrated that the Old Church Slavonic (and
supposedly the Proto-Slavic) ablaut system could be expressed in the terms of
vector and/or metric space. Such spaces offer strictly formal description of the
whole system and the precise description of the relations between concrete ablaut
grades. The proposed description using vector or metric spaces is not a
Features of Common Slavic Ablaut Alternation
completely new approach to the ablaut, but rather an improvement of the standard
knowledge, using formal instruments.
The main difference between our (and Hubey’s) approach and the method of
DP is that DP “vectors” are not commutative, i.e. the “linear combination” of
such vectors differs according to the order of “vectors”. That is the reason why
we use the term “vector” for DP in the quotations marks, because vectors are
generally considered as commutative. On the contrary, our ablaut features form
commutable vectors in the full sense of the word.
It should be kept in mind that our solution offers a model not for one solitary
root, but for the complex model of the system.
The reader could see in table 5 that the distribution of reduced grades is
uneven. Some roots have only e-colored grades ({reduced non-lengthened} and
{reduced lengthened}), some roots only o-colored grades (again {reduced nonlengthened}
and {reduced lengthened}). It is in a strong contrast to distribution
of full grades that are present for both possibilities of vowel color, either ecolored
or o-colored.
Hence, any individual root has only six grades: {non-reduced nonlengthened
e-grade}, {non-reduced non-lengthened o-grade}, {non-reduced
lengthened e-grade}, {non-reduced lengthened o-grade}, {reduced nonlengthened}
and {reduced lengthened}. This model fits to any given root
(although many roots are not attested for all those grades), but has its disadvantages:
first − four grades are described using three values but two with two values,
second − reduced grades of different roots could easily differ in the vocalism
(compare roots √n_s- and √g_n- in the table 5).
The set of all roots could then be split in two disjoint (on the level of reduced
grades) subsets: e-roots and o-roots. Each subset has its own ablaut pattern
which is a sub-pattern of the general ablaut system, as described in the present
paper. This opens new possibilities for the describing of root-morphology in Old
Church Slavonic.
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Ondřej Šefčík
sefcik@phil.muni.cz
Masaryk University, Brno