About the number of generators of a musical scale

TL;DR

This paper investigates the number of generators of musical scales modeled as arithmetic sequences in cyclic groups, revealing that this number is always a totient number and exploring related properties and characterizations.

Contribution

It proves that the number of generators of such scales is always a totient number and characterizes the possible cases, including the existence of scales with arbitrarily many generators.

Findings

Number of generators is always a totient number

Existence of scales with arbitrarily many generators

No scales with exactly 14 generators

Abstract

Several musical scales, like the major scale, can be described as finite arithmetic sequences modulo octave, i.e. chunks of an arithmetic sequence in a cyclic group. Hence the question of how many different arithmetic sequences in a cyclic group will give the same support set. We prove that this number is always a totient number and characterize the different possible cases. In particular, there exists scales with an arbitrarily large number of different generators, but none with 14 generators. Some connex results and extensions are also given, for instance on characterization via a Discrete Fourier Transform, and about finite or infinite arithmetic sequences in the torus R/Z.