Set Theory (music) - Symmetry

Symmetry

The number of distinct operations in a system that map a set into itself is the set's degree of symmetry (Rahn 1980, 90). Every set has at least one symmetry, as it maps onto itself under the identity operation T₀ (Rahn 1980, 91). Transpositionally symmetric sets map onto themselves for T_n where n does not equal 0. Inversionally symmetric sets map onto themselves under T_nI. For any given T_n/T_nI type all sets will have the same degree of symmetry. The number of distinct sets in a type is 24 (the total number of operations, transposition and inversion, for n = 0 through 11) divided by the degree of symmetry of T_n/T_nI type.

Transpositionally symmetrical sets either divide the octave evenly, or can be written as the union of equally-sized sets that themselves divide the octave evenly. Inversionally-symmetrical chords are invariant under reflections in pitch class space. This means that the chords can be ordered cyclically so that the series of intervals between successive notes is the same read forward or backward. For instance, in the cyclical ordering (0, 1, 2, 7), the interval between the first and second note is 1, the interval between the second and third note is 1, the interval between the third and fourth note is 5, and the interval between the fourth note and the first note is 5. One obtains the same sequence if one starts with the third element of the series and moves backward: the interval between the third element of the series and the second is 1; the interval between the second element of the series and the first is 1; the interval between the first element of the series and the fourth is 5; and the interval between the last element of the series and the third element is 5. Symmetry is therefore found between T₀ and T₂I, and there are 12 sets in the T_n/T_nI equivalence class (Rahn 1980, 148).