Symmetry transformations of melodic structure

When dealing with symmetry transformations of melodic structure, for the purpose of easier identification, we will be using the proposed linear system with 12 levels corresponding to half-intervals, which we will denote by 0 to 11, as well as a graphic interpretation of melodic structure - a melody graph. On the melody graph we will study the following melodic entities: motifs, phrases, periods, etc., and the symmetry transformations that they are subject to. In doing so, we always take a minimum asymmetric entity (asymmetric fundamental region) and use it for studying the action of symmetry transformations on it.

The melodic structure of a unison composition represents, from the geometry point of view, a two-dimensional modular structure. The two dimensions of this structure (the tonal pitch and the distance of the tone from the beginning of the piece, i.e., its place in the sequence of tones that make up the piece) are examined on the y and x-axis. As all tones whose pitches are equal modulo 12 are considered as equivalent, then we are speaking of a modular structure on the y-axis. In accordance with this, each tone of a melody is exactly defined with an ordered pair of coordinates (x₁,y₁). In this case x₁ is the position of a tone in the melody and y₁ is the value that corresponds to its pitch. In each melodic structure we will study its elements: its tonal pitches and their relationships - intervals. We will analyze melodic structure independently from individual tone duration, i.e., its rhythmic structure, or other properties of the studied piece. Only after registering the symmetries of its melodic structure we will proceed to examine their correlation with its rhythmic, harmonic, etc. structure symmetries.

The symmetry transformations of melodic structure can be divided into two basic types:

1. absolute, and

2. relative.

The concept äbsolute transformations" of a melodic entity implies the transformations that have the same impact on all the elements of the melodic entity and, consequently, on all the interval relations that occur among them. All absolute transformations preserve the order of the elements. From a geometry aspect, absolute melodic transformations correspond to isometric transformations (transformations of congruence, which preserve the distances between the points) and transformations of similarity (transformation of proportionality). In the case of relative (approximate) symmetry transformations of a melodic structure occurs the breaking of the structure of interval relationships among the tones.