### Thoughts on Modification of Bio-harmony

#### Abstract

I have constructed a model of bio-harmony as a fusion of 3 icosahedral harmonies and tetrahedral harmony. The icosahedral harmonies are defined by Hamiltonian cycles at icosahedron going through every vertex of the icosahedron and therefore assigning to each triangular face an allowed 3-chord of the harmony. The fascinating outcome is that the model can reproduces genetic code. The model for how one can understand how 12-note scale can represent 64 genetic codons has the basic property that each note belongs to 16 chords. The reason is that there are 3 disjoint sets of notes and given 3-chord is obtained by taking 1 note from each set. For bio-harmony obtained as union of 3 icosahedral harmonies and tetrahedral harmony note typically belongs to 15 chords. The representation in terms of frequencies however requires 16 chords per note. Consistency a modification of the model of icosahedral harmony. The necessity to introduce tetrahedron for one of the 3 fused harmonies is indeed an ugly looking feature of the model. The question is whether one of the harmonies could be replaced with some other harmony with 12 notes and 24 chords. If this would work one would have 64 chords equal to the number of genetic codons and 5+5+6 =16 chords per note. One can imagine toric variants of harmonies realized in terms of Hamiltonian cycles and one indeed obtains a toric harmony with 12 notes and 24 3-chords. Bio-harmony could correspond to the fusion of 2 icosahedral harmonies with 20 chords and toric harmony with 24 chords having therefore 64 chords. Whether the predictions for the numbers of codons coding for given amino-acids come out correctly for some choices of Hamiltonian cycles is still unclear.