Group actions, power mean orbit size, and musical scales

Abstract

We provide an application of the theory of group actions to the study of musical scales. For any group G, finite G-set S, and real number t, we define the t-power diameter to be the size of any maximal orbit of S divided by the t-power mean orbit size of the elements of S. The symmetric group acts on the set of all tonic scales, where a tonic scale is a subset of containing 0. We show that for all , among all the subgroups G of , the t-power diameter of the G-set of all heptatonic scales is the largest for the subgroup Γ, and its conjugate subgroups, generated by . The unique maximal Γ-orbit consists of the 32 thāts of Hindustani classical music popularized by Bhatkhande. This analysis provides a reason why these 32 scales, among all 462 heptatonic scales, are of mathematical interest. We also apply our analysis, to a lesser degree, to hexatonic and pentatonic scales.