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 ${\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}}_t(G,S)$ 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 $S_{11}$ acts on the set of all tonic scales, where a tonic scale is a subset of ${\mathbb{Z}}_{12}$ containing 0. We show that for all $t\in [-1,1]$, among all the subgroups G of $S_{11}$, 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 $\left\{\right(12),(34),(56),(89),(1011\left)\right\}$. 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.

我们提供了群体行动理论在音阶研究中的应用。对于任何群G、有限G集S和实数t，我们定义t 次幂直径 ${\mathrm{d}\mathrm{一世}\mathrm{一种}\mathrm{米}}_{吨}(G,\mathrm{小号})$为 S 的任何最大轨道的大小除以S 的元素的t次方平均轨道大小。对称群${\mathrm{小号}}_{11}$作用于所有主音阶的集合，其中主音阶是${\mathbb{Z}}_{12}$包含 0。我们证明，对于所有$吨\in [-1,1]$, 在所有子群G中${\mathrm{小号}}_{11}$，所有七子音阶的G集的t次方直径对于子群 Γ 及其共轭子群是最大的，由下式生成$\left\{\right(12),(34),(56),(89),(1011\left)\right\}$. 独特的最大 Γ 轨道由 Bhatkhande 推广的 32 首印度斯坦古典音乐组成。该分析提供了为什么这 32 个音阶（在所有 462 个七音阶中）具有数学意义的原因。我们还将我们的分析在较小程度上应用于六声音阶和五声音阶。