A complexity measure for musical scales
Alpar Sevgen · The Journal of the Acoustical Society of America · 2000
Equally tempered scales with N semitones and M notes and interval structures n={n1,n2,...,nM}, where nk is the number of semitones between the notes tk and tk+1, possess the following properties: Each distinct interval structure n corresponds to a multiplet of N scales. Members of a multiplet can be labeled by a set of integers {c}, modulo N, called scale labels. Each scale label is the difference between the number of sharps and flats occurring in that scale and is unique within the multiplet if N and M are relative primes. This labeling does not differentiate between different scale structures.To do this, complexity is introduced as the sum of the number of sharps and flats occurring in a scale. For N=12 and M=7, out of 462 possible scale structures, the major scale and its cyclical permutations, called modes, have the minimum complexity which allows the practical use of the key signatures in music. Complementary scales where notes and no notes are interchanged have the same complexity. The minimum and maximum complexity scales occupy the opposite ends of the energy spectrum under the force laws ± nα (α≠0), between the notes of a scale.