  We create a systematic theory that derives "natural" scales (Gregorian modes,
major/minor scales, hexatonic major/minor scales) from "artificial" scales
(chromatic scale and whole tone scales) by using a new concept, musical
icosahedra. We found that there exist four musical icosahedra that characterize
the topology of the chromatic scale and one of the whole tone scales. These
musical icosahedra lead to major/minor triads and the fundamental triads for
hexatonic major/minor scales with golden triangles, a duality between major
scales and minor scales and a duality between major/minor scales and the
Gregorian modes. Lastly, we propose generalizations of major/minor triads and
scales by using the symmetry of icosahedra.
