General Theory of Music by Icosahedron 3: Musical invariant and Melakarta raga

TL;DR

This paper explores the mathematical structure of Melakarta raga using musical icosahedra, defining invariants and permutations that connect Western and Indian classical music scales.

Contribution

It introduces intermediate musical icosahedra connecting previous models and demonstrates a musical invariant constant across Melakarta raga scales through permutation-extensions.

Findings

Existence of a constant musical invariant for Melakarta raga scales.

Introduction of intermediate musical icosahedra connecting different musical concepts.

Permutation-extension of scales reproduces all Melakarta raga scales.

Abstract

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.