Can one imagine a modification of bioharmony?
I have developed a rather detailed model of bioharmony as a fusion of 3 icosahedral harmonies and tetrahedral harmony (see this and this). 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 3chord of the harmony. The surprising outcome is that the model can reproduces genetic code.
The model for how one can understand how 12note 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 3chord is obtained by taking 1 note from each set. For bioharmony obtained as union of 3 icosahedral harmonies and tetrahedral harmony note typically belongs to 15 chords. The representation in terms of frequencies requires 16 chords per note.
If one wants consistency one must somehow modify 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. The addition of tetrahedron would not be needed.
One can imagine toric variants of icosahedral harmonies realized in terms of Hamiltonian cycles and one indeed obtains a toric harmony with 12 notes and 24 3chords. Bioharmony 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 aminoacids come out correctly for some choices of Hamiltonian cycles is still unclear. This would require an explicit construction of toric Hamiltonian cycles.
1. Previous results
Before discussing the possible role of toric harmonies some previous results will be summarized.
1.1 Icosahedral bioharmonies
The model of bioharmony starts from a model for music harmony as a Hamiltonian cycle at icosahedron having 12 vertices identified as 12 notes and 20 triangular faces defining the allowed chords of the harmony. The identification is determined by a Hamiltonian cycle going once through each vertex of icosahedron and consisting of edges of the icosahedral tesselation of sphere (analog of lattice): each edge corresponds to quint that is scaling of the frequency of the note by factor 3/2 (or by factor 2^{7/12} in welltempered scale). This identification assigns to each triangle of the icosahedron a 3chord. The 20 faces of icosahedron define therefore the allowed 3chords of the harmony. There exists quite a large number of icosahedral Hamiltonian cycles and thus harmonies.
The fact that the number of chords is 20  the number of aminoacids  leads to the question whether one might somehow understand genetic code and 64 DNA codons in this framework. By combining 3 icosahedral harmonies with different symmetry groups identified as subgroups of the icosahedral group, one obtains harmonies with 60 3chords.
The DNA codons coding for given aminoacid are identified as triangles (3chords) at the orbit of triangle representing the aminoacid under the symmetry group of the Hamiltonian cycle. The predictions for the numbers of DNAs coding given aminoacid are highly suggestive for the vertebrate genetic code.
By gluing to the icosahedron tetrahedron along common face one obtains 4 more codons and two slightly different codes are the outcome. Also the 2 aminoacids Pyl and Sec can be understood. One can also regard the tetrahedral 4 chord harmony as additional harmony so that one would have fusion of four harmonies. One can of course criticize the addition of tetrahedron as a dirty trick to get genetic code.
The explicit study of the chords of bioharmony however shows that the chords do not contain the 3chords of the standard harmonies familiar from classical music (say major and minor scale and corresponding chords). Garage band experimentation with random sequences of chords requiring conservability that two subsequent chords have at least one common note however shows that these harmonies are  at least to my opinion  aesthetically feasible although somewhat boring.
1.2 Explanation for the number 12 of notes of 12note scale
One also ends up to an argument explaining the number 12 for the notes of the 12note scale (see this). There is also second representation of genetic code provided by dark proton triplets. The dark proton triplets representing dark genetic codons are in oneone correspondence with ordinary DNA codons. Also aminoacids, RNA and tRNA have analogs as states of 3 dark protons. The number of tRNAs is predicted to be 40.
The dark codons represent entangled states of protons and one cannot decompose them into a product state. The only manner to assign to the 3chord representing the triplet ordinary DNA codon such that each letter in {A,T,C,G} corresponds to a frequency is to assume that the frequency depends on the position of the letter in the codon. One has altogether 3× 4=12 frequencies corresponding to 3 positions for given letter selected from four letters.
Without additional conditions any decomposition of 12 notes of the scale to 3 disjoint groups of 4 notes is possible and possible chords are obtained by choosing one note from each group. The most symmetric choice assigns to the 4 letters the notes {C, C #, D, D#} in the first position, {E,F, F #, G} in the second position, and {G #, A, B b, B} in the third position. The codons of type XXX would correspond to CEG# or its transpose. One can transpose this proposal and there are 4 nonquivalent transposes, which could be seen as analogs of music keys.
Remark: CEG# between Cmajor and Aminor very often finishes finnish tango: something neither sad nor glad!
One can look what kind of chords one obtains.
 Chords containing notes associated with the same position in codon are not possible.
 Given note belongs to 6 chords. In the icosahedral harmony with 20 chords given note belongs to 5 chords (there are 5 triangles containing given vertex). Therefore the harmony in question cannot be equivalent with 20chord icosahedral harmony. Neither can the bioharmony with 64 chords satisfy the condition that given note is contained by 6 3chords.
 First and second notes of the chords are separated by at least major third as also those second and third notes. The chords satisfy however octave equivalence so that the distance between the first and third notes can be smaller  even half step  and one finds that one can get the basic chords Aminor scale: Am, Dm, E7, and also G and F. Also the basic chords of Fmajor scale can be represented. Also the transposes of these scales by 2 whole steps can be represented so that one obtains A_{m}, C #_{m}, F_{m} and corresponding major scales. These harmonies could allow the harmonies of classical and popular music.
These observations encourage to ask whether a representation of the new harmonies as Hamiltonian cycles of some tesselation could exist. The tesselation should be such that 6 triangles meet at given vertex. Triangular tesselation of torus having interpretation in terms of a planar parallelogram (or perhaps more general planar region) with edges at the boundary suitable identified to obtain torus topology seems to be the natural option. Clearly this region would correspond to a planar lattice with periodic boundary conditions.
2. Is it possible to have toric harmonies?
The basic question is whether one can have a representation of the new candidate for harmonies in terms of a tesselation of torus having V= 12 vertices and F= 20 triangular faces. The reading of the article "Equivelar maps on the torus" (see this) discussing toric tesselations makes clear that this is impossible. One however have (V,F)= (12,24) (see this). A rather promising realization of the genetic code in terms of bioharmony would be as a fusion of two icosahedral harmonies and toric harmony with (V,F)= (12,24). This in principle allows also to have 24 3chords which can realize classical harmony (major/minor scale).
 The local properties of the tesselations for any topology are characterized by a pair (m,n) of positive integers. m is the number of edges meeting in given vertex (valence) and n is the number of edges and vertices for the face. Now one has (m,n)= (6,3). The dual of this tesselation is hexagonal tesselation (m,n)= (3,6) obtained by defining vertices as centers of the triangles so that faces become vertices and vice versa.
 The rule VE+F=2(1g)h, where V, E and F are the numbers of vertices, edges, and faces, relates VEF to the topology of the graph, which in the recent case is triangular tesselation. g is the genus of the surface at which the triangulation is im eded and h is the number of holes in it. In case of torus one would have E=V+F giving in the recent case E=36 for (V,F)= (12,24) (see this) whereas in the icosahedral case one has E=32.
 This kind of tesselations are obtained by applying periodic boundary conditions to triangular lattices in plane defining parallelogram. The intuitive expectation is that this lattices can be labelled by two integers (m,n) characterizing the lengths of the sides of the parallelogram plus angle between two sides: this angle defines the conformal equivalence class of torus. One can also
introduce two unit vectors e_{1} and e_{2} characterizing the conformal equivalence class of torus.
Second naive expectation is that m× n × sin(θ) represents the area of the parallelogram. sin(θ) equals to the length of the exterior product e_{1}× e_{2}=sin(θ) representing twice the area of the triangle so that there would be 2m× n triangular faces. The division of the planar lattice by group generated by pe_{1}+qe_{2} defines boundary conditions. Besides this the rotation group Z_{6} acts as analog for the symmetries of a unit cell in lattice. This naive expectation need not of course be strictly correct.
 As noticed, it is not possible to have triangular toric tesselations with (V,E,F)= (12,30,20). Torus however has a triangular tesselation with (V,E,F)=(12,36,24). An illustration of the tesselation can be found here). It allows to count visually the numbers V, E, F, and the identifications of the boundary edges and vertices. With good visual imagination one might even try to guess what Hamiltonian cycles look like.
The triangular tesselations and their hexagonal duals are characterized partially by a pair of integers (a,b) and (b,a). a and b must both even or odd. The number of faces is F= (a^{2}+3b^{2})/2. For (a,b)= (6,2) one indeed has V=12 and F=24. From the article one learns that the number of triangles satisfies F= 2V for a=b at least. If F= 2V holds true more generally one would have V= (a^{2}+3b^{2})/8, giving tight constraints on a and b.
Remark: The conventions for the labelling of torus tesselation vary. The above convention based on integers (a,b) is different from the convention based on integer pair (p,q) used in the article this). In this notation torus tesselation with (V,F)=(12,24) corresponds to (p,q)=(2,2) instead of (a,b)= (6,2). This requires (a,b)=(3p,q). In this notation one has V=p^{2}+q^{2} +pq.
3. The number of triangles in the 12vertex tesselation is 24: curse or blessing?
One could see as a problem that one has F=24>20? Or is this a problem?
 By fusing two icosahedral harmonies and one toric harmony one would obtain a harmony with 20+20+24 =64 chords, the number of DNA codons! One would replace the fusion of 3 icosahedral harmonies and tetrahedral harmony with a fusion of 2 icosahedral harmonies and toric harmony. Icosahedral symmetry with toric symmetry associated with the third harmony would be replaced with a smaller toric symmetry. Note however that the attachment of tetrahedron to a fixed icosahedral face also breaks icosahedral symmetry.
This raises questions. Could the presence of the toric harmony somehow relate to the almost exact U ↔ C and A ↔ G symmetries of the third letter of codons. This does not of course mean that one could associated the toric harmony with the third letter. Note that in the icosatetrahedral model the three harmonies are assumed to have no common chords. Same nontrivial assumption is needed also now in order to obtain 64 codons.
 What about the number of aminoacids: could it be 24 corresponding ordinary aminoacids, stopping sign plus 3 additional exotic aminoacids. The 20 icosahedral triangles can corresponds to aminoacids but not to stopping sign. Could it be that one of the additional codons in 24 corresponds to stopping sign and two exotic aminoacids Pyl and Sec appearing in biosystems explained by the icosahedral model in terms of a variant of the genetic code. There indeed exists even third exotic aminoacid! Nformylmethionine (see this) but is usually regarded as as a form of methionine rather than as a separate proteinogenic aminoacid.
 Recall that the problem related to the icosatetrahedral harmony is that it does not contains the chords of what might be called classical harmonies (the chordds assignable to major and minor scales). If 24 chords of bioharmony correspond to toric harmony, one could obtain these chords if the chords in question are chords obtainable by the proposed construction.
But is this construction consistent with the representation of 64 chords by taking to each chord one note from 3 disjoint groups of 4 notes in which each note belongs to 16 chords. The maximum number of chords that note can belong to would be 5+5+6=16 as desired. If there are no common chords between the 3 harmonies the conditions is satisfied. Using for instance 3 toric representations the number would be 6+6+6=18 and would require dropping some chords.
 The earlier model for tRNA as fusion of two icosahedral codes predicting 20+20=40 tRNA codons. Now tRNAs as fusion of two harmonies allows two basic options depending on whether both harmonies are icosahedral or whether second harmony is toric. These options would give 20+20=40 or 20+24=44 tRNAs. Wikipedia tells that maximum number is 41. Some sources however tell that there are 2040 different tRNAs in bacterial cells and as many as 50100 in plant and animal cells.
See the chapter Geometric theory of harmony, the article Geometric theory of harmony or the article New results in the model of bioharmony.
