2020/10/05 by Elliott, Jesse · 1 citation
#05E18 #26E60 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2010.01728
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 \it t-power diameter diamt(G,S) 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 S11 acts on the set of all tonic scales, where a \it tonic scale is a subset of ℤ12 containing 0. We show that, for all t ∈ [-1,1], among all the subgroups G of S11, the t-power diameter of the G-set of all heptatonic scales is largest for the subgroup Γ, and its conjugate subgroups, generated by \(1 2),(3 4),(5 6),(8 9),(10 11)\. 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.