Tonal, Atonal and Microtonal Pitch-Class Categories

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Abstract

This paper reviews and generalizes Pitch-Class Set Theory using Group Theory (groups acting on pc-sets) and Category Theory, which provides methods for mapping the structure of a n-tone system onto another m-tone system. This paper also suggests a new implementation approach that represents pitch-class sets as bit-sequences, which are equivalent to integer values. Forte’s best normal order is shown to be equivalent to the smallest integer, among the cyclic permuted pc-sets. Further, transposition of pc-sets is shown to be equivalent to bit-shifts; and their inversion, to bit-reversal. The tonal (diatonic) pc-category is presented as a subset of the atonal (12-tone) pc-category, which, similarly, can also be contained in a microtonal pc-category. Functors between those categories present properties that preserve relationships while still using the same operations: tonal relationships are preserved even though atonal music operations, such as transposition and inversion are applied, allowing motives to be mapped into different modes, scales, or even microtonal scales. The appendix offers an implementation of this new approach to calculate and represent pc-sets with an arbitrary number of pitch-classes.