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Group actions, power mean orbit size, and musical scales
Journal of Mathematics and Music ( IF 0.5 ) Pub Date : 2021-02-02
Jesse Elliott

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 d i a 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 Z 12 containing 0. We show that for all t [ 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 { ( 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.



中文翻译:

集体动作,力量平均轨道大小和音阶

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,我们定义叔功率直径 d 一世 一个 Ť G 小号 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 小号 11 作用于所有补品音阶的集合,其中补品音阶 ž 12 包含0。 Ť [ - 1个 1个 ] ,在的所有子组G 小号 11 ,则所有子波尺度的G集t功率直径对于子组Γ及其共轭子组最大,由 { 1个 2 3 4 5 6 8 9 10 11 } 。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。

更新日期:2021-02-02
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