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

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.

General Theory of Music by Icosahedron 2: Analysis of musical pieces by the exceptional musical icosahedra

We propose a new way of analyzing musical pieces by using the exceptional musical icosahedra where all the major/minor triads are represented by golden triangles or golden gnomons. First, we introduce a concept of the golden neighborhood that characterizes golden triangles/gnomons that neighbor a given golden triangle or gnomon. Then, we investigate a relation between the exceptional musical icosahedra and the neo-Riemannian theory, and find that the golden neighborhoods and the icosahedron symmetry relate any major/minor triad with any major/minor triad. Second, we show how the exceptional musical icosahedra are applied to analyzing harmonies constructed by four or more tones. We introduce two concepts, golden decomposition and golden singular. The golden decomposition is a decomposition of a given harmony into the minimum number of harmonies constructing the given harmony and represented by the golden figure (a golden triangle, a golden gnomon, or a golden rectangle). A harmony is golden singular if and only if the harmony does not have golden decompositions. We show results of the golden analysis (analysis by the golden decomposition) of the tertian seventh chords and the mystic chord. While the dominant seventh chord is the only tertian seventh chord that is golden singular in the type 1[star] and the type 4[star] exceptional musical icosahedron, the half-diminished seventh chord is the only tertian seventh chord that is golden singular in the type 2 [star] and the type 3[star] exceptional musical icosahedron. Last, we apply the golden analysis to the famous prelude in C major composed by Johann Sebastian Bach (BWV 846). We found 7 combinations of the golden figures on the type 2 [star] or the type 3 [star] exceptional musical icosahedron dually represent all the measures of the BWV 846.