Group actions, power mean orbit size, and musical scales

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.