  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.
