About the number of generators of a musical scale

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.