Wednesday, October 29, 2014

Geometric theory of harmony

For some time ago I introduced the notion of Hamiltonian cycle as a mathematical model for musical harmony and also proposed a connection with biology: motivations came from two observations (see this). The number of icosahedral vertices is 12 and corresponds to the number of notes in 12-note system and the number of triangular faces of icosahedron is 20, the number of aminoacids and the number of basic chords for the proposed notion of harmony. This led to a group theoretical model of genetic code and replacement of icosahedron with tetraicosahedron to explain also the 21st and 22nd amino-acid and solve the problem of simplest model due to the fact that the required Hamilton's cycle does not exist.

This led also to the notion of bioharmony. This article is a continuation to the mentioned article providing a proposal for a theory of harmony and detailed calculations.

  1. 3-adicity and also 2-adicity are essential concepts allowing to understand the basic facts about harmony. The notion of harmony at the level of chords is suggested to reduce to the notion of closeness in the 3-adic metric using as distance the distance between notes measures as the minimal number of quints allowing to connect them along the Hamilton's cycle. In ideal case, harmonic progressions correspond to paths connecting vertex or edge neighbors of the triangular faces of icosahedron.

  2. An extension of icosahedral harmony to tetraicosahedral harmony was proposed as an extension of harmony allowing to solve some issues of icosahedral harmony relying on quint identified as rational frequency scaling by factor 3/2.

  3. The idea that the rules of bioharmony realized on amino-acid sequences interpreted as sequences of basic 3-chords leads to highly non-trivial and testable predictions about amino-acid sequences.
If one can find various icosahedral Hamilton's cycles one can immediately deduce corresponding harmonies. This would require computer program and a considerable amount of anlysis. My luck was that the all this has been done. One can find material about icosahedral Hamilton's cycles in web, in particular the list of all 1024 Hamilton's cycles with one edge fixed (see ) (this has no relevance since only shape matters). If one identifies cycles with opposite internal orientations, there are only 512 cycles. If the cycle is identified as a representation of quint cycle giving representation of 12 note scale, one cannot make this identication since quint is mapped to fourth when orientation is reversed. The earlier article about icosahedral Hamiltonian cycles as representations of different notions of harmony is helpful.

The tables listing the 20 3-chords of associated with a given Hamilton's cycle make it possible for anyone with needed computer facilities and music generator to test whether the proposed rules produce aesthetically appealing harmonies for the icosahedral Hamiltonian cycles.

For details see the chapter Quantum model of hearing of "TGD and EEG" or the article Geometric theory of harmony.


Anonymous said...

From The Collaborative International Dictionary of English v.0.48 [gcide]:

Nome \Nome\, n. [Gr. ?, fr. ? to deal out, distribute.]
1. A province or political division, as of modern Greece or
ancient Egypt; a nomarchy.
[1913 Webster]

2. Any melody determined by inviolable rules. [Obs.]
[1913 Webster] said...

The CIDE definition of harmony based on melody alone is quite too restrictive: of course it leads to classical scales such as 7 note scale CDEFGAH and its minor variant .Pieces of this sequence appear also as sequences of basic notes of chords in some of the 11 icosahedral harmonies.

Chords are equally important for harmony: the rough rule is that one should have common notes between subsequent chords. In Icosahedral model 3-chords corresponds to faces of icosahedron (triangles) and are natural. The proposed notion of harmony relies on nearness defined by quint metric, which is basically 3-adic metric. Harmonic flow should be 3-adically smooth.

The condition that at least two vertices have minimal distance in 3-adic quint based metric implies that 1-quint chords and 2-quint chords are special where as 0-quint chords are not so harmonic and they indeed typically involve strong dissonances.

A fascinating idea is that if 20 amino acids corresponds to triangles of icosahedron and if
they are analogs of 3-chords, one would expect that evolution favors generation of harmony in this quite concrete sense at the level of proteins. Given amino acid could be followed only by 3+3 amino-acids instead of 20 as in purely random case! This is immediately checkable!

Ulla said...
:) expanded codon with 4 letters.

Jorden Thatcher said...

You wouldn't have happened to have been traveling through either Kansas City or Chicago by train in the past few years, have you? said...

No. Why are you asking?