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General Theory of Music by Icosahedron 1: A bridge between "artificial" scales and "natural" scales, Duality between chromatic scale and Pythagorean chain, and Golden Major Minor Self-Duality

Relations among various musical concepts are investigated through a new concept, musical icosahedron that is the regular icosahedron each of whose vertices has one of 12 tones. First, we found that there exist four musical icosahedra that characterize the topology of the chromatic scale and one of the whole tone scales, and have the hexagon-icosahedron symmetry (an operation of raising all the tones of a given scale by two semitones corresponds to a symmetry transformation of the regular icosahedron): chromatic/whole tone musical icosahedra. The major triads or the minor triads are set on the golden triangles of these musical icosahedra. Also, various dualities between musical concepts are shown by these musical icosahedra: the major triads/scales and the minor triads/scales, the major/minor triads and the fundamental triads for the hexatonic major/minor scales, the major/minor scales and the Gregorian modes. Second, we proposed Pythagorean/whole tone musical icosahedra that characterize the topology of the Pythagorean chain and one of the whole tone scales, and have the hexagon-icosahedron symmetry. The Pythagorean chain (chromatic scale) in the chromatic (Pythagorean)/whole tone musical icosahedron is constructed by "middle" lines of the regular icosahedron. While some golden triangles correspond to the major/minor triads in the chromatic/whole tone musical icosahedra, in the Pythagorean/whole tone musical icosahedra, some golden gnomons correspond to the minor/major triads. Third, we found four types of musical icosahedra other than the chromatic/whole tone musical icosahedra and the Pythagorean/whole tone musical icosahedra that have the hexagon-icosahedron symmetry. All the major triads and minor triads are represented by the golden triangles or the golden gnomons on each type. All of these musical icosahedra lead to generalizations of major/minor triads and scales.