二十面体3音乐的一般理论：音乐不变和Melakruga
General Theory of Music by Icosahedron 3: Musical invariant and Melakarta raga
摘要：
Raga是南亚，特别是印度的一个中心音乐概念，我们通过音乐二十面体来研究西方古典音乐与卡纳塔克（印度南部）古典音乐中的Raga Melakarraga之间的联系。在我们之前的研究中，我们介绍了西方音乐中连接各种音乐概念的音乐二十面体：半音阶/全音音乐二十面体，毕达哥拉斯/全音音乐二十面体和特殊音乐二十面体。本文首先介绍了音乐二十面体的种类，这些音乐二十面体通过12个音的两种排列方式：排列间和排列内连接起来，我们称之为中间音乐二十面体。接下来，我们将相邻数定义为给定音阶中相邻两个音调在给定音乐二十面体上彼此相邻的数量，我们还将音乐不变量定义为相邻数的线性组合。我们发现对于某些音乐二十面体，存在一对音乐不变量和一对音阶是常数，并分析了它们的数学结构。最后，我们定义了一个给定音阶的一个由给定音乐二十面体的相互排列所构成的扩张：排列扩张。然后，我们表明，C大调音阶的置换扩展由Melakarraga音乐二十面体，是四个中间的音乐二十面体从1型半音阶/全音音乐二十面体的类型1'毕达哥拉斯/全音音乐二十面体，是一组所有的音阶包括在Melakarraga。存在一个音乐不变量，这是恒定的所有音乐二十面体对应的音阶Melakamiraga，我们得到了一个图表表示这些尺度特征的音乐不变量。
英文摘要：
Raga is a central musical concept in South Asia, especially India, and we investigate connections between Western classical music and Melakarta raga that is a raga in Karnatak (south Indian) classical music, through musical icosahedron. In our previous study, we introduced some kinds of musical icosahedra connecting various musical concepts in Western music: chromatic/whole tone musical icosahedra, Pythagorean/whole tone musical icosahedra, and exceptional musical icosahedra. In this paper, first, we introduce kinds of musical icosahedra that connect the above musical icosahedra through two kinds of permutations of 12 tones: inter-permutations and intra-permutations, and we call them intermediate musical icosahedra. Next, we define a neighboring number as a number of pairs of neighboring two tones in a given scale that neighbor each other on a given musical icosahedron, and we also define a musical invariant as a linear combination of the neighboring numbers. We find there exists a pair of a musical invariant and scales that is constant for some musical icosahedra and analyze their mathematical structure. Last, we define an extension of a given scale by the inter-permutations of a given musical icosahedron: the permutation-extension. Then, we show that the permutation-extension of the C major scale by Melakarta raga musical icosahedra that are four of the intermediate musical icosahedra from the type 1 chromatic/whole tone musical icosahedron to the type 1' Pythagorean/whole tone musical icosahedron, is a set of all the scales included in Melakarta raga. There exists a musical invariant that is constant for all the musical icosahedra corresponding to the scales of Melakarta raga, and we obtained a diagram representation of those scales characterizing the musical invariant.
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