Symmetry In Atonal Music

Atonal music includes each of the twelve pitch-classes repeated equally within a composition. The composer then gives no preference to a particular subset of the twelve pitch-classes and avoids key-structure in the music, which is a significant part of the structure in traditional tonal music. A particular piece of atonal music is often written to favor a group of symmetric permutations of a given 12-tone row reached via the operations transposition, inversion, and retrograde. Here, we investigate symmetry in twelve-tone rows and apply these ideas to n-tone rows for microtonal systems. In terms of algebra, the goal is to count the unique groupings of permutations, or orbits, which can be reached via combinations of the three possible operations that preserve symmetry.