Monday, November 17, 2014

Harmony, music, and religious myths

I have talked about the notion of harmony based on icosahedral representation of 12-note scale and its connection with biology and consciousness coming from the observation that the number of faces of icosahedron is 20 - the number of amino-acids. The link connecting music with biology would be via the geometry of Platonic solids: icosahedron and also tetrahedron.

The 12 vertices of icosahedron correspond to the notes of the 12-note scale and to some Hamilton's cycle defining a closed curved connecting nearest neighbour vertices: it is closed by octave equivalence and does not intersect itself. Quint cycle assigns to each vertex a note of the scale and thus also to every face (triangle) of the icosahedron a chord. These chords define the harmony and there are 11 cycles/harmonies allowing symmetries (6 cycles without symmetries). They fall in three types corresponding to symmetry group Z6= Z3rot× Z2refl, Z4=Z2rot× Z2refl , and Z2= Z2rot or Z2=Z2refl, which is subgroup of icosahedral isometries A5× Z2refl having 2× 60 elements. Just these groups appeared in the model of genetic code inspired by observations about the structure of the code table telling the numbers N(d) of amino-acids coded by d codons.

These symmetries define a hierarchy of symmetry breakings. This hierarchy has amazing connections with the myths, which I believe to reflect deep facts about consciousness and biology at fundamental level. The story of genesis is a good representative in this respect.

  1. The hierarchy of symmetry breakings proceeding from Z6 down to Z2refl brings strongly in mind evolution as loss of innocence. For Z6 one as 4 orbits. One orbit contains 2 triangles (chords, DNA codons assignable to ile). The other orbits correspond to six codons assignable to amino-acids ser, arg, and leu. The chords at the orbits are major chords and 7-chords, and minor chords and 6-chords for the inverse of the harmony.

    There are no dissonant chords in 0-quint sector: dissonances appear only for the remaining groups as 0-quint chords. This is musical representation of paradize. This harmony is based on 6-note scale for the basic notes of the chords and used by impressionistic composers. Amino-acids correspond to selections of preferred chord from each orbit and there are only four different chords: this sub-harmony is very simple. Life in paradize is simple!

  2. Next comes an intriguing observation. The number of amino-acids obtained as projections of the icosahedral DNA orbits is 19, not 20! One chord does not correspond to amino-acid: it is non-playable chord! Could it be impossible to have 20 amino-acids as projections of the orbits and that 19 is the maximum number? The reason for 19 is that the number of amino-acid of type Z6 is 3+1=4 rather than 5. Therefore there is one "non-playable" chord - located at the "paradize orbit" -, which does not correspond to any amino-acid. The natural identification of the non-playable chord is as one of the aug type chords (say CEG#, which is the last breath in many finnish tangos telling about unhappy love end - something between happy CM and sad Am, "raueta" is finnish word for this manner to come to an end: "expire" might be the nearest english counterpart). This chord is located at the 2-chord orbit related to the other chord of the orbit by half-octave shift (chords could be CEG# and F#BbD), the tritonus denied by church.

    One cannot avoid the associations between non-playable chord and the denied fruit hanging in the tree of good and bad knowledge in the story of Adam and Eve, and its analog in many fairy tales. The non-playable chord also brings in mind the hilarious story of Gödel-Escher-Bach about non-playable record (a truth unprovable in given axiom system).

  3. The hierarchy of symmetry breakings leading from Z6 to Z2refl encourages one to continue with the biblical analogies. Z6, Z4 and Z2rot cycles have half-octave shift as a symmetry: good and evil do not exist in paradise, but dissonances are already there for Z4 and Z2 harmonies - the evil snake! These states correspond to the consciousness of animals, children, and saints. Note that bio-harmony corresponds to the presence of one sub-harmony of type Zn, n=6,4,2.

  4. The banishing from the paradize takes place as Z2refl symmetric harmony replaces Z2rot harmony: half-octave shift is not a symmetry anymore, and one can tell between good and evil, and eventually church decides to deny tritonus as a symbol of evil! Paradise is left as icosahedral and tetrahedral code are fused to form the tetra-icosahedral code - the ordinary genetic code leading to the breaking of Z2refl symmetry.

  5. In banishment punct ("empty" amino-acid) as a counterpart of chord shared by tetrahedron and icosahedron emerges and means stopping of the music piece altogether. Death of the sinner! For unfused codes this chord is playable as Sec/Pyl and the music piece is never-ending: life is eternal in paradise! No notion of time, no sin, no death! Amusingly, impressionist music with 6-note scale is music of "now", attempt to catch this moment.

For details see the chapter Quantum model for hearing or the article Geometric theory of harmony.


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