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.

我们提供了团体行为理论在音乐音阶研究中的应用。对于任何一组G ^，有限ģ -set小号，和实数吨，我们定义叔功率直径 ${\mathrm{d}\mathrm{一世}\mathrm{一个}\mathrm{米}}_{Ť}（G，\mathrm{小号}）$是S的任何最大轨道的大小除以S元素的t幂平均轨道大小。对称群${\mathrm{小号}}_{11}$作用于所有补品音阶的集合，其中补品音阶是${\mathbb{ž}}_{12}$ 包含0。 $Ť\in [-\mathrm{1个}，\mathrm{1个}]$，在的所有子组G中${\mathrm{小号}}_{11}$，则所有子波尺度的G集的t功率直径对于子组Γ及其共轭子组最大，由$\{（\mathrm{1个}2），（34），（56），（89），（1011）\}$。独特的最大Γ轨道由巴克汉德（Bhatkhande）推广的32遍印度斯坦尼古典音乐组成。该分析提供了为什么在全部462个七声音阶中的这32个音阶具有数学意义的原因。我们也将我们的分析（较小程度地）应用于六高音和五音阶。