Can a musical scale have 14 generators?

Emmanuel Amiot

pp. 349-360

in: Tom Collins, David Mérédith, Anja Volk (eds), Mathematics and computation in music, Berlin, Springer, 2015

Abstract

A finite arithmetic sequence of real numbers has exactly two generators: the sets ({a, a+f, a+2f,dots a+ (n-1)f = b}) and ({b, b-f, b-2f, dots , b-(n-1)f = a}) are identical. A different situation exists when dealing with arithmetic sequences modulo some integer c. The question arises in music theory, where a substantial part of scale theory is devoted to generated scales, i.e. arithmetic sequences modulo the octave. It is easy to construct scales with an arbitrary large number of generators. We prove in this paper that this number must be a totient number, and a complete classification is given. In other words, starting from musical scale theory, we answer the mathematical question of how many different arithmetic sequences in a cyclic group share the same support set. Extensions and generalizations to arithmetic sequences of real numbers modulo 1, with rational or irrational generators and infinite sequences (like Pythagorean scales), are also provided.