Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2103.10272

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

Yusuke Imai·Sid C. Dellby·Nobuaki Tanaka
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yusuke Imai, Sid C. Dellby, Nobuaki Tanaka. 2022-08-06. 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. https://arxiv.org/abs/2103.10272

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quelques remarques sur les vari{é}t{é}s, fonctions de Green et formule de Stokes

We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces of Sobolev spaces, the distance function, the notion of degree and a duality theorem, the variational formulation and conformal map in dimension 2, the metric on the boundary of a Lipschitz domain and polar geodesic coordinates and the Gauss-Bonnet formula and the positive mass theorem in dimension $ \geq 3 $ and in the flat and non flat case. And the Ricci flow. And fields and their relation to the equations.And obstructions in astronomy. And on strings, superstrings and D-branes. And topological solutions in the negative case, critical, supercritical and superstrings and symmetry. And geometrization. And Decision problem, SAT problem and p=np problem.

math.GM↗
arXiv2026

Counting Truchet Tile Balls

A formula is established that counts the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.

math.GM↗
arXiv2026

A Theory of Scales and Orbit Covers

This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.

math.GM↗