We present new high-spectral-resolution observations (R = lambda/Delta lambda = 7000) of the filamentary nebula surrounding NGC 1275, the central galaxy of the Perseus cluster. These observations have been obtained with SITELLE, an imaging Fourier transform spectrometer installed on the Canada-France-Hawai Telescope with a field of view of 11 ' x11 ' , encapsulating the entire filamentary structure of ionized gas despite its large size of 80 kpc x 50 kpc. Here, we present renewed fluxes, velocities, and velocity dispersion maps that show in great detail the kinematics of the optical nebula at [S ii] lambda 6716, [S ii] lambda 6731, [N ii] lambda 6584, H alpha (6563 angstrom), and [N ii] lambda 6548. These maps reveal the existence of a bright flattened disk-shaped structure in the core extending to r similar to 10 kpc and dominated by a chaotic velocity field. This structure is located in the wake of X-ray cavities and characterized by a high mean velocity dispersion of 134 km s-1. The disk-shaped structure is surrounded by an extended array of filaments spread out to r similar to 50 kpc that are 10 times fainter in flux, remarkably quiescent, and have a uniform mean velocity dispersion of 44 km s-1. This stability is puzzling given that the cluster core exhibits several energetic phenomena. Based on these results, we argue that there are two mechanisms that form multiphase gas in clusters of galaxies: a first triggered in the wake of X-ray cavities leading to more turbulent multiphase gas and a second, distinct mechanism, that is gentle and leads to large-scale multiphase gas spreading throughout the core.