On the step-patterns of generated Scales that are not well-formed

Marco Castrillón López

Manuel Domínguez Romero

pp. 361-372

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

Abstract

It is well-known that generated scales (with irrational generator) may have two or three different steps. It is also known that the scale has exactly two steps precisely if the number of notes coincides with the denominator of a (semi-)convergent of the generator. Moreover, the step-pattern is a Christoffel word: a mechanical word with rational slope. In this article we investigate the bad case: generated scales with three different steps. We will see that their step-patterns share some properties with the Christoffel case: they are Lyndon words and their right Lyndon factorization is determined by the generator. Some conjectures on their left Lyndon factorization are also given.