Symmetry transformations of polyphonic melody structure
From the aspect of symmetry analysis, the same symmetry transformations occur in the symmetry of counterpoint as in the symmetries of melodic structure, the only difference being that the former have an impact on the elements of one voice, transforming them into elements of a second voice. Without straying from the general subject we will examine only the symmetry analysis of two-part compositions. In this case the following analogous counterpoint isometric transformations occur: -
transposition (translation t _{x,y}), or imitation (shift) (Fig. 4.7); -
inversion (glide reflection g _{x}) (Fig. 4.8); -
retrograde inversion (central reflection O), i.e., crab imitation (Fig. 4.9); -
retrograde transposition (glide reflection g _{y}), i.e., the retrograde shift (Fig. 4.10).
With this two specific new cases of the above mentioned transformations occur: -
a) unison (parallel) imitation, i.e., translation t _{y}on the y-axis (Fig. 4.11), as a specific type of transposition; -
a) anti-parallel imitation - mirror reflection m _{x}, whose axis is parallel to the x-axis. This is a special case of counterpoint inversion where the translation vector is equal to 0 (Fig. 4.12).
The first transformation (1a) brings about the appearance of a parallel melody, and the second (2a), the appearance of an anti-parallel melody in different voices.
The symmetry analysis of polyphonic pieces may be further expanded
by a study of the counterpoint transformations of similarity,
counterpoint permutation equivalence, |