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How to compose beautiful music of light in bio-harmony?

October 2020

DOI:10.13140/RG.2.2.22919.01444

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TGD leads to a notion of bio-harmony in terms of icosahedral and tetrahedral geometries and 3-chords made of light assigned to the triangular faces of icosahedron and tetrahedron Bio-harmonies are associated with the so-called Hamiltonian cycles , which go through every vertex of Platonic solid once. For icosahedron the number of vertices is 12, the number of notes in 12-note scale. The 64 codons of bio-harmony represented as light 3-chords formed by dark photon triplets are formed from 3 20-chord harmonies associated with icosahedron and the unique 4-chord harmony associated with tetrahedron. The surprise was that vertebrate genetic code emerged as a prediction: the numbers of DNA codons coding for a given amino acid are predicted correctly. DNA codons correspond to triangular faces and the orbit of a given triangle under the symmetries of the bio-harmony in question corresponds to DNA codons coding for the amino acid assigned with the orbit. Codon corresponds to 6 bits: this is information in the usual computational sense. Bio-harmony codes for mood: emotional information related to emotional intelligence as ability to get to the same mood allowing to receive this information. Bio-harmony would be a fundamental representation of information realized already at molecular level and speech, hearing and other expressions of information would be based on it. One topic of this article is the detailed definition of the notion of bio-harmony. A sequence of 3-chords of bio-harmony defines a music piece: what rules guarantee that this piece is beautiful? This question is interesting because the chords of bio-harmony correspond to DNA codons. One can also wonder whether the standard rather simple harmonies can be understood. Also the role of tetrahedral harmony and its relation to start and stop codons is interesting.

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Pitk¨anen, M. How to compose beautiful music of light in bio-harmony?

Article

How to compose beautiful music of light in bio-harmony?

Matti Pitk¨anen 1

Abstract

TGD leads to a notion of bio-harmony in terms of icosahedral and tetrahedral geometries and

3-chords made of light assigned to the triangular faces of icosahedron and tetrahedron Bio-harmonies

are associated with the so-called Hamiltonian cycles , which go through every vertex of Platonic solid

once. For icosahedron the number of vertices is 12, the number of notes in 12-note scale. The 64

codons of bio-harmony represented as light 3-chords formed by dark photon triplets are formed from

3 20-chord harmonies associated with icosahedron and the unique 4-chord harmony associated with

tetrahedron.

The surprise was that vertebrate genetic code emerged as a prediction: the numbers of DNA codons

coding for a given amino acid are predicted correctly. DNA codons correspond to triangular faces

and the orbit of a given triangle under the symmetries of the bio-harmony in question corresponds to

DNA codons coding for the amino acid assigned with the orbit.

Codon corresponds to 6 bits: this is information in the usual computational sense. Bio-harmony

codes for mood: emotional information related to emotional intelligence as ability to get to the same

mood allowing to receive this information. Bio-harmony would be a fundamental representation of in-

formation realized already at molecular level and speech, hearing and other expressions of information

would be based on it.

One topic of this article is the detailed deﬁnition of the notion of bio-harmony. A sequence of

3-chords of bio-harmony deﬁnes a music piece: what rules guarantee that this piece is beautiful? This

question is interesting because the chords of bio-harmony correspond to DNA codons. One can also

wonder whether the standard rather simple harmonies can be understood. Also the role of tetrahedral

harmony and its relation to start and stop codons is interesting.

1 Introduction

The topic of this article is the detailed deﬁnition of the notion of bio-harmony [3, 4, 10]. A sequence of

3-chords of bio-harmony deﬁnes a music piece: what rules guarantee that this piece is beautiful? This

question is interesting because the chords of bio-harmony correspond to DNA codons.

1.1 Bio-harmony as a realization of genetic code

TGD leads to a notion of bio-harmony in terms of icosahedral and tetrahedral geometries and 3-chords

made of light assigned to the triangular faces of icosahedron and tetrahedron [3, 4, 10]. Bio-harmonies

are associated with the so-called Hamiltonian cycles , which go through every vertex of Platonic solid

once. For icosahedron the number of vertices is 12, the number of notes in 12-note scale. The 64 codons

of bio-harmony represented as light 3-chords formed by dark photon triplets are formed from 3 20-chord

harmonies associated with icosahedron and the unique 4-chord harmony associated with tetrahedron.

The surprise was that vertebrate genetic code emerged as a prediction: the numbers of DNA codons

coding for a given amino acid are predicted correctly. DNA codons correspond to triangular faces and

the orbit of a given triangle under the symmetries of the bio-harmony in question corresponds to DNA

codons coding for the amino acid assigned with the orbit.

Codon corresponds to 6 bits: this is information in the usual computational sense. Bio-harmony codes

for mood: emotional information related to emotional intelligence as ability to get to the same mood

1Correspondence: Matti Pitk¨anen http://tgdtheory.com/. Address: Rinnekatu 2-4 A8, 03620, Karkkila, Finland. Email:

matpitka6@gamail.com.

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allowing to receive this information. Bio-harmony would be a fundamental representation of information

realized already at molecular level and speech, hearing and other expressions of information would be

based on it. For emotional expression at RNA level possibly involved with conditioning at synaptic level

see [7].

1.2 About generalizations of the notion of bio-harmony

One can consider several generalizations for the notion of bio-harmony.

1. All Platonic solids, in particualr tetrahedron, cube, octahedron and dodecahedron are possible and

one can consider the possibility that they also deﬁne harmonies in terms of Hamiltonian cycles.

Dodecahedron would have 5-chords (pentagons as faces) as basic chords and there is only single

harmony. Same mood always, very eastern and enlightened as also the fact that scale would have

20 notes.

Also octahedron gives 3-chords (triangular faces) whereas cube gives 4-chords (squares as faces).

One can of course speculate with the idea that DNA could also represent this kind of harmonies:

sometimes the 3N rule is indeed broken, for instance for introns.

2. Galois conﬁnement [11] allows the possibility to interpret dark genes as sequences of Ndark proton

triplets as higher level structures behaving like a single quantal unit. This would be true also for

the corresponding dark photon sequences consisting of 3N dark photons representing the gene in

bio-harmony as an analog of a music piece consisting of 3-chords and played by transcribing it to

mRNA.

Basic biomolecules (DNA, RNA, tRNA, amino acids) would have names represented as a sequence

of light 3-chords representing a piece of music and dark biomolecules with the same name could

recognize and communicate with each other in 3N-resonance. Dark-ordinary communications could

transform dark 3N-photon to single bio-photon so that resonance would be possible when the sum

of energies coincides with a transition energy of the ordinary biomolecule. The resonance condition

would very eﬀectively select survivors in the ﬁght for survival.

3. The picture can be viewed even more generally. Any discrete structure, deﬁning graph, in particular

cognitive representation providing a unique ﬁnite discretization of space-time surface as points

with the coordinates of the 8-D imbedding space coordinates in the extension of rationals, deﬁnes

harmonies in terms of Hamiltonian cycles. Could also these harmonies make sense? The restrictions

of the cognitive representations to 2-D partonic 2-surfaces would deﬁne something analogous to

bio-harmony as Hamiltonian cycle of 2-D graph (Platonic surfaces solids can be regarded as 2-D

graphs). The interpretation as representations of Galois groups and the notion of Galois conﬁnement

is possible although one loses the symmetries of the Platonic solids allowing to identify genetic code.

During years I have indeed considered some modiﬁcations of the original bio-harmony base on the

fusion of 3 icosahedral harmonies and tetrahedral harmony in partcular so called E8harmony and toric

harmony [5, 6] but the overall conclusion [9] is that the original model is the most plausible candidate.

1.3 The challenges of the model

The model of bio-harmony is far from complete and this article discusses a more detailed deﬁnition. Also

the question about the rules deﬁning beautiful music by posing rules on chord sequences are considered.

These aesthetic rules are also rules for the corresponding DNA and amino-acid sequences.

1. The fusion of the three harmonies having symmetry groups Zn,n= 6,4,2 has been considered but

not in the required detail. The Hamiltonian cycles of icosahedron are ﬁxed only modulo isometries

of icosahedron preserving the shape of the cycle, scalings of the cycle by a power of quint forming

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Pitk¨anen, M. How to compose beautiful music of light in bio-harmony?

group Z12 leaving the cycle of invariant but inducings transponation (change of the key), and the

change of the cycle orientation possibly related to minor-major dichotomy correlating with joyful-

sad dichotomy. For a single icosahedral cycle these transformations do not change anything but for

the fusion of 3 cycles realized at the same icosahedron the situation changes, and the number of

harmonies increases dramatically.

Are all combinations of icosahedral harmonies allowed or are there some natural restrictions on

them? I have considered this question but it seems that there is no good reason for posing any

restrictions. The spectrum of harmonies determined by dark genetic codons and therefore the

spectrum of emotions at the molecular level would be surprisingly rich.

2. Is it possible to reproduce the basic harmonies of the western music based on the 12-note system

which inspired icosahedral harmonies? In particular, can one understand the chords C, F, G of

C-major scale? By octave equivalence the nearest neighbors of the Hamiltonian cycle are related by

quint scaling scaling frequency by factor 3/2 scaling C to G. The 3-chords containing at least one

cycle edge contain quint (C→G) and quint is the basic aspect of bio-harmony. For harmonies with

opposite orientation quints become perfect fourths (C→F) and FCG corresponds to transponantion

of F by two quints.

For a single icosahedral harmony the chord-pairs analogous to C-F or C-G do not appear in any

obvious manner. If the 3 icosahedral harmonies are related by quint scalings (FCG) the analogs of

these chord pairs become natural. Could this be the solution to the problem?

3. What are the rules producing aesthetically satisfying music? I experimented with the ultraconser-

vative assumption that only chord pairs containing common quint are allowed: the result was not

ugly but it was boring. Already the transitions of CFG major scale are too radical for this option!

An attractive idea is that the sequence of 3-chords is continuous in some sense. Could the sense

be strictly geometric: could chord pairs be nearest neighbors in some sense. For Option I nearest

neighbors have a common edge (3 nearest neighbours). For Option II they have a common vertex

(10 nearest neighbors). These options do not allow all 3-chord pairs and thus not all possible DNA

pairs and all possible amino-acid pairs. A more abstract deﬁnition identiﬁes the nearest neighbors

with the orbits of nearest neighbors for Option I or II under the symmetry group Zn(n= 6,2).

Codon is replaced with the codons coding for the same amino-acid. For Option II this allows to

have all possible chord pairs and therefore DNA and amino-acid pairs.

4. Also the role of tetrahedral harmony and its relation to start and stop codons is interesting. One

wants also to understand why the genetic code at the bio-chemical level is not quite complete and

why there are several variants of it.

2 About bio-harmonies

The set of allowed 3-chords deﬁne music harmony. The 12-note scale is essential for the western view about

harmony. The TGD inspired geometric model for music harmony identiﬁes bio-harmony as a fusion of 3

icosahedral harmonies with 12-note scale represented geometrically as a Hamiltonian cycle at icosahedron

and 1 tetrahedral harmony represented as a unique Hamiltonian cycle of tetrahedron. Each icosahedral

harmony has 20 3-chords identiﬁable as triangular faces of the icosahedron whereas tetrahedral harmony

4 3-chords. This gives 20+20+20+4=64 chords - the number of genetic codons.

2.1 Symmetries of icosahedral harmonies

There are 3 types of icosahedral harmonies with symmetries characterized by a subgroup of icosahedral

isometries, which is Z6,Z4or Z2acting either as a rotation by πor as a reﬂection. The orbits of triangles

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Table 1: The number #(class) of equivalence classes of Hamiltonian cycles ad the number #(repr) of

representatives in the class for icosahedral Hamiltonian cycles. If the orientation is not taken into account

the number of representatives reduces to #(repr)/2

Symmetry #(class) #(repr)

Z61 8

Z42 12

Z2,rot 3 24

Z2,ref l 5 24

are identiﬁed as counterparts of amino-acids coded by the DNA codons assigned with the triangles of the

orbit.

1. For Z6given triangle gives rise to 3 6-orbits with 6 triangles and 1 2-orbit: Z3subgroup of icosahedral

group permutes the 3 6-orbits and acts trivially to 2-orbit.

2. For Z4there are 5 4-orbits and Z5permutes these orbits.

3. For Z2there are 10 2-orbits and Z10 permutes them. Z2can act either as reﬂections or rotations.

There are also 6 cycles without any symmetries perhaps identiﬁable as dis-harmonies. They will not

be considered in the sequel. For them the number of amino-acids coded by codon would be one.

Table 1 summarizes the numbers of equivalence classes of cycles and under icosahedral rotation group

for various symmetry groups as well as the numbers of representatives in the class. These numbers allow

to deduce the number of bio-harmonies by ﬁxing one of the icosahedral harmonies, most naturally the Z6

harmony for which one has only one class.

Remarkably, the combination of 3 icosahedral cycles with symmetries Zk,k= 6,4,2 with the tetra-

hedral Hamiltonian cycle gives 64 codons and the model correctly predicts the numbers of DNA codons

coding for a given amino acid. Could there be a connection between music and genetic code? Could one

speak of bio harmonies as correlates of emotions at the molecular level?

The natural expectation is that the symmetries Znof a given harmony leave the ratios of frequencies

of 3-chords invariant. This is true if the edge connecting nearest neighbors along Hamiltonan cycle

corresponds to a quint that is scaling of frequency by 3/2 and projection to the basic octave (octave

equivalence). Therefore the chords at the orbit of a given chord coding for the same amino-acid are

replaced by a scaling by power of 3/2 so that the scalings are mapped to unitary rotations.

The factors of 12 include indeed 6, 4, and 2 so that the 12-element group of scalings modulo octave

equivalence can be mapped to Z12 rotations. There is however a problem with rational quints due to the

fact that - as already Pythagoras found - (3/2)12 = 129.746... does not correspond exactly to 27= 128.

One reason for introducing icosahedron could be that this brings additional note allowing to get rid of

the problem. One can also construct the notes by powers of 21/12 applied to the basic frequency but now

the frequencies are not rational. Furthermore, people with absolute pitch favor rational frequency ratios,

which suggests that rational numbers and roots of unitary assignable with adelic physics as physics of

cognition are really important.

2.2 Fusion of 3 icosahedral harmonies and tetrahedral harmony to bio-harmony

There is quite a large number of icosahedral Hamiltonian cycles and therefore of bio-harmonies. Although

the isometries of icosahedron and their transponations do not matter for given icosahedral harmony, they

matter when one has 3 icosahedral harmonies. A simple example from physics helps to understand

this: although rotations are symmetries of an N-particle system the rotations of a single particle are not

symmetries anymore and represent new degrees of freedom.

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1. Bio-harmony assigns to the same icosahedron 3 Hamilton cycles with symmetries Zk,k= 6,4,2.

This means assigning to the same icosahedron 3 Hamiltonian cycles giving rise to 3 representations

of 12-note scale each giving 20 chords so that one 20+20+20 chords coding 3 classes of amino acids.

Tetrahedron gives the remaining 4 chords.

There are Ni,i= 1,2,3 cycles corresponding to Zk(i),k(i)=6,4,2: for the values of Niand

detailed 3-chord contents of icosahedral harmonies see [2]. From the table Table 1 one has for

(Z6, Z4, Z2,rot) #(class) = (#(class)1,#(class)2,#(class)3) = (1,2,3) giving 6 diﬀerent classes

and (Z6, Z4, Z2,ref l) (#(class)1,#(class)2,#(class)3) = (1,2,5) giving 8 diﬀerent classes. This

gives N= 14 diﬀerent icosahedral Hamiltonian cycles.

The numbers of reresentatives for given equivalence class are for both (Z6, Z4, Z2,rot ) (Z6, Z4, Z2,rot)

#(repr) = (2,12,24).

2. The 3 cycles go through all points of the icosahedron. This means that for each point of icosahedron

there are 3 cycles going through that point. There can be however situations in which there are

common edges. 5 edges arrive at given icosahedral vertex. There are 3 cycles entering and leaving

the vertex: this makes 6 cycle edges. There is necessarily one edge shared by two cycles. If the edge

is shared by 3 cycle edges, one edge has no cycle edge. This kind of situation - 3-edge - is achieved

by performing a suitable Z5rotation for the third cycle.

Do all bioharmonies have 3-edges? Could 3-edges have a special role concerning bio-harmony and

music experience? Could they deﬁne chords with preferred quints such as chords C, F, G in C

major scale? The bio-harmonies having chord(s) with 3-edge could give rise to simple CFG type

harmonies. Fusion of 3 icosahedral harmonies diﬀering by quint scalings gives a CFG type situation,

and one could assign all these 3 types of chords with a triangle with 3-edge. Geometrically the chord

progression would reduce to a repetition of the same triangle! Allowing also the triangle at the other

side of the 3-edge, the chord progression involving only these 2 triangles consists of 3+3=6 chords.

3. One can assume that the 3 Hamiltonian cycles start at the same almost arbitrarily chosen vertex of

the icosahedron. As a special case one can assume that it corresponds to the same basic note (C).

Since Z6allows only a single cycle, it is natural to ﬁx it: the fact this cycle has 2 orientations gives

degeneracy factor 2.

The other other cycles are determined apart from the rotation group Z5leaving the base point

invariant. Therefore the Z4and Z2harmonies give rise to an additional 52= 25-fold degeneracy of

bio-harmonies N→25N. If the cycles are required to have a common ﬁrst edge besides the base

point, one does not obtain the degeneracy factor. This argument shows that common edges are

possible and the vertices associated with them are deﬁnitely special.

Fixing the cycle types and the Z6cycle one can calculate the number of bioharmonies for a given

equivalence classes as the number #(repr(Z4)#(repr(Z2) One obtains 12 ×24 representatives for

both choices of Z2. For r Z2=Zrot the total number of bioharmonies is

N(harmony, r ot)=2×2×12 ×3×24 = 27×32

N(harmony, r efl) = 2 ×2×12 ×5×24 = 27×3×5.

The ﬁrst factor of 2 comes from the two orientations for the ﬁxed Z6cycle.

4. The transponations realized as scalings along the Hamiltonian cycle deﬁne 1-to-1 map of icosahedral

vertices which is however not an isometry but preserves the harmony. This gives a degeneracy factor

122and one has

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N(harmony, ...)→122×N(har mony, ...).

The formula for the total number of bioharmonies is

N(harmony) = N(har mony, rot) + N(harmony, ref l) = 214 ×33,

N(harmony, r ot)=211 ×34,

N(harmony, r efl)=211 ×33×5.(2.1)

(2.2)

2.3 How to understand the tetrahedral code and symmetry breaking of the

perfect code?

The precise understanding of the relationship between tetrahedral and icosahedral codes has been a long

standing challenge andI have considered several scenarios. The geometric idea has been that tetrahedron

is somehow glued to icosahedron along on faceand selects a unique codon of the icosahedron deﬁning the

basic chord. As found, another manner to ﬁx this chord as a chord to which one can assign 3 cycle edges.

There might be other faces with the same property.

One can get information about the situation by looking at the code table.

1. There are 10 unbroken icosahedral Z2doublets containing (stop,stop) plus 1 symmetry broken

doublet (stop,tyr). What could cause the symmetry breaking? The energy resonance condition

associated with the pairing of dark mRNA codons with dark tRNA codons could explain the presence

of stop codons: translation would stop when no tRNA in energy resonance is found.

Dark 3-photon representing the dark stop codons could not couple to tRNA codon in energy reso-

nance since there would not be tRNA with cyclotron resonance energy triplet resonating with that

of dark stop codon. This would be the case for the (punc,punc) doublet and also for punc member

of (puc,trp) doublet. The mimicry of dark level by biochemical level would not be complete. For

the variants of the code it would be even less complete.

2. From the table one learns that both Z6and Z4codons are realized completely for the vertebrate

code. This leaves only one conclusion: (ile,ile,ile,met) must correspond to a Z4symmetry breaking

for tetrahedral rather than icosahedral 4-plet. The AGG coding for met, which is unique in the

sense that it serves as a mark for the beginning of genes, would correspond to a tetrahedral face.

The failure of energy resonance could force the splitting of unbroken tetrahedral ile 4-plet to

(ile,ile,ile,met). Fourth codon in Z44-plet would be in energy resonance with tRNA associated

with met. Note that icosahedral code gives rise to 4+5+10=19 amino-acids and met provides the

20th amino acid. Symmetry breaking would be necessary to mark the starting and stopping points

of transcription and translation.

3-chords also depend on the icosahedral harmony and for some icosahedral harmonies energy res-

onance could fail so that the emotional state of at dark matter level would reﬂect itself at the

biochemical level. The number of icosahedral harmonies is (1,2,3,5) for (Z6, Z4, Zrot, , Z2,ref l ). For

Z4and Z2the failure of energy resonance is possible.

Remark: I must confess that many earlier texts about the problem contain a stupid error. I have

considered the proposal that (ile,ile,ile,met) could correspond to symmetry broken icosahedral 4-

plet. Vertebrate code has however 5 unbroken 4-plets corresponding to (val,pro,thr,ala,gly) as also

3 unbroken 6-plets (leu,ser,arg)! For vertebrate code the symmetry breaking can therefore occur

only for icosahedral Z2doublets and tetrahedral Z44-plet.

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2.4 Variations of the genetic code

There exists also as many as 31 genetic codes (see http://tinyurl.com/ydeeyhjl) and an interesting

question is whether this relates to the context dependence. Mitochondrial codes diﬀer from the nuclear

code and there are several of them. The codes for viruses, prokaryotes, mitochondria and chloroplasts

deviate from the standard code. As a rule, the non-standard codes break U-C or A-G symmetries for the

third code letter.

In the proposed framework the failure of energy resonance conditions could be at the level of tRNA.

The dark tRNA analog of RNA could be in energy resonance with ”wrong” amino acid.

Some examples are in order (see http://tinyurl.com/puw82x8).

1. UUU can code Leu instead of Phe (symmetry breaks for Phe doublet) and CUG can code Ser rather

than Leu (symmetry breaks for leu 6-plet). In this case it seems that the ”problem” is at the level

of tRNA. The dark RNA codon could couple with a ”wrong” amino acid.

2. In bacteria the GUG and UUG coding for Val and Leu normally can serve as Start codons. In

this case symmetry breaking for Z44-plet would be in question. The problem could be also at

tRNA level. Note however that both tetrahedral codons and icosahedral Z4codons have the same

symmetry group. Could tetrahedral codons correspond to a diﬀerent frequency scale and correspond

to Leu and Val 4-plet instead of symmetry broken ile 4-plet.

3. UGA can code to trp rather than punc: in this case the broken symmetry would be restored since

also UGG codes for trp. Both codons for (trp,trp) doublet would be in resonance: this supports

the explanation for the emergence of the third stop codon.

4. There is variation even in human mitochondrial code (see http://tinyurl.com/puw82x8). In 2016,

researchers studying the translation of malate dehydrogenase found that in about 4 per cent of the

mRNAs encoding this enzyme the UAG Stop codon is naturally used to encode the AAs trp and

arg. This phenomenon is known as Stop codon readthrough [1]. Also this phenomenon could be

understood at tRNA level.

5. There is also a variant of genetic code in which there are 21st and 22nd AAs Sec and Pyl coded by

Stop codons. UGA in (punc,trp) doublet can code for Sec and punc in the same organism. UAG can

code for (punc,punc) doublet Pyl instead of punc and UAG. This introduces additional breaking of

A-G symmetry for the third letter of codon. Energy resonance at the level of tRNA could explain

these deviations from the vertebrate code.

Peter Gariaev has introduced the notion of homonymy of genetic code meaning that the same codon

can code for several amino-acids and the coding depends on context. I have considered this phenomenon

from the TGD point of view in [8]. Resonance could explain this phenomenon.

Dark mRNA codon could be in frequency resonance with dark tRNAs coding for diﬀerent amino

acids. The fraction of particular synonymous amino-acid produced in translation would naturally depend

on how well the energy resonance condition is satisﬁed. Homonymy could also reduce to the level of

tRNA: this happens if the coupling of the tRNA analog of RNA codon has energy resonance with several

amino-acids.

3 How to produce beautiful bio-music?

Music expresses and produces emotions and harmonies in music correspond to emotions. Chemical

representation of the genetic code should be the same irrespective of the emotional state of the gene

represented at the magnetic body in terms of dark proton triplets also representing genetic codons and by

music of light represents 3-chords of light with frequency ratios determined by one of the bio-harmonies.

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Pitk¨anen, M. How to compose beautiful music of light in bio-harmony?

This is achieved naturally. The correspondence between the chords of harmony and DNA and amino-

acids does not depend on what vertex of icosahedron the base note (C for deﬁniteness in the sequel)

corresponds to. It also depends only on the shape of the Hamiltonian cycle invariant under isometries of

the icosahedron. Furthermore, transponations of the scale by power of 3/2 plus projection to the basic

octave do not aﬀect the Hamiltonian cycle and therefore leave the correspondence with DNA codons and

amino acids invariant.

The sequences of 3-chords would correspond to sequences of DNA codons mapped to sequences of

amino-acids. Genes would be like music pieces. These music pieces would also serve as kind of names

of passwords in 3N-fold resonance in communications between dark variants of basic biomolecules and

between them and ordinary basic biomolecules. They would be like theme songs of TV series catching

the attention or names essential for symbolic dynamics at the level of the basic biomolecules. The basic

biomolecules in the same emotional state - that is having the same bio-harmony - could resonate and

therefore couple.

What the rules for a beautiful bio-music could be? Could these rules select particular bioharmonies

and/or particular DNA sequences as allowed chord progressions and allow a deeper understanding of

why particular genes are selected? Note that the condition that the chords of bio-harmony deﬁne 3N-

resonances assignable to transitions of the basic biomolecules could lead to the selection of both harmony

and biomolecules. A weaker condition is that ordinary biomolecules couple only to the sum of frequencies

appearing in 3N-frequency assignable to dark codon.

3.1 Are beautiful chord sequences continuous in some sense?

The original model discussed in [2, 9] started from a very conservative idea for what harmonic change

of chord could be. The two chords should have at least a single quint. This fails for the chords with no

quints. The resulting music pieces were also boring which is not a surprise: for instance, the transitions

between basic chords C, F, G of C major scale are not possible.

This suggests that one should not start from music but from geometry. Let us consider isohedral

geometry for simplicity and the proposed picture for the bio-harmonies.

1. Continuity in some sense is a natural requirement. The natural deﬁnition of continuity is that

the sequence of 3-chords of progression should deﬁne a sequence of neighbouring triangles at the

icosahedron. But how should one deﬁne neighborhood?

2. Concerning the notion of nearest neighbor, there are 3 options to consider.

Option I: The strong form of continuity is that neighboring triangles have at least one common

edge. This allows 4 diﬀerent chord pairs. This would mean 4 possible DNA codon pairs for a

given Hamiltonian cycle. For bio-harmony the symmetry of icosahedral harmony determined by Zn

(n= 6,4,2) can change and one would have 4+4+4=12 codon pairs. This kind of correlation for

codon sequences would have been observed.

Option II: For a weaker option the neighboring triangles would have at least 1 common vertex.

A given triangle would have 4+3+2+1 =10 neighbors (”1” corresponds to the triangle itself as a

neighbor). This would give 10+10+10 =30 possible codon pairs.

Tetrahedral harmony gives further pairs but since one triangle of tetrahedron should correspond to

a ﬁxed triangle of icosahedron, this can change the situation for only a single chord. It is known

that the minimum of 32 two codons are needed to code amino acids. The optimum situation very

probably not reached for all bio-harmonies (if any), would be that the amino acid associated with

the next codon can be any aminoacid. It should be easy to demonstrate by studying a sample of

genes or more general DNA codon sequences to ﬁnd that this prediction is wrong.

Option III: For the weakest option the nearest neighbors would correspond to triangles at the

orbits of the nearest neighbors in the sense of Option II or perhaps even Option I under the

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Pitk¨anen, M. How to compose beautiful music of light in bio-harmony?

symmetry group Znof a given cycle. For instance, the transitions which would not change the

codon would be replaced with all codons coding for the same amino-acid. The notion of nearest

neighbor would reduce to the level of amino-acids: only the transitions to codons coding for the

same amino-acid would be possible.

For the generalization of Option I Z6cycle would give 4 orbits of which several must be identical

so that there are no problems. Z4cycle would give 4 orbits with 4 codons so that one amino acid

is missing. For the Z2option one obtains only 4 2-origi so that 6 amino-acids are missing.

For the generalization of Option II 10+10+10 nearest neighbours would be replaced with their

orbits. For the Z6cycle there are nearest neighbor 10 orbits and since there are only 4 orbits, there

are no problems. For the Z4cycle one there are 5 4-orbits so that the minimal degeneracy of a

given orbit is 2.

For the Z2cycle there are 10 2-orbits, and this number is obtained unless some 2-orbit occurs more

than once. The 10 nearest neighbor triangles must correspond to diﬀerent amino-acids: whether

this is possible for all bioharmonies, remains an open question. In any case, it is plausible Option

III can produce all possible codon pairs although this need not be the case for all bioharmonies.

Could preferred bioharmonies be selected by the condition that all codon pairs are possible?

3.2 What about melody?

Melody is also an important part of music. A rough rule of thumb is that a beautiful melody tends to

contain notes of the chord accompanying it. Dissonance is of course what makes music really interesting.

This can be understood as a resonant coupling of the notes of the melody with the notes appearing in

the accompanying chords.

Can one apply this picture to the music of light? Could the dark 3-photon states bound to a single

unit by Galois conﬁnement tend to decay to ordinary 3-photon states (bio-photons) and could melody

represented as a sequence of single photon states couples resonantly to these photons? Could melody

correspond to as sequence dark photons 1-plets decaying to ordinary bio-photons coupling to the the

decay products of dark photon triplets representing genetic codons?

3.3 Summary

The basic results of the article are a precise deﬁnition of bio-harmony allowing to obtain the analogs of

ordinary simple harmonies as special cases and a proposal that the 3-chord sequence deﬁnes a beautiful

music piece if it corresponds to a continuous sequence for icosahedral faces. In principle this criterion

allows bio-harmonies for which all possible codon pairings appear in chord sequences but some bio-

harmonies might be excluded.

References

[1] Hofhuis J et al. The functional readthrough extension of malate dehydrogenase reveals a modiﬁcation

of the genetic code. Open Biology, 6(11), 2016. Available at: https://www.ncbi.nlm.nih.gov/pmc/

articles/PMC5133446/.

[2] Pitk¨anen M. Geometric theory of harmony. Available at: http://tgdtheory.fi/public_html/

articles/harmonytheory.pdf., 2014.

[3] Pitk¨anen M. Music, Biology and Natural Geometry (Part I). DNA Decipher Journal, 4(2), 2014.

See also http://tgtheory.fi/public_html/articles/harmonytheory.pdf.

[4] Pitk¨anen M. Music, Biology and Natural Geometry (Part II). DNA Decipher Journal, 4(2), 2014.

See also http://tgtheory.fi/public_html/articles/harmonytheory.pdf.

ISBN: 2153-8212 DNA Decipher Journal www.DNADJ.com

Published by QuantumDream, Inc.

DNA Decipher Journal |March 2020|Vol 12.|Issue 12|pp. 1-10 10

Pitk¨anen, M. How to compose beautiful music of light in bio-harmony?

[5] Pitk¨anen M. E8symmetry, harmony, and genetic code . Available at: http://tgdtheory.fi/

public_html/articles/e8harmony.pdf., 2016.

[6] Pitk¨anen M. Can one imagine a modiﬁcation of bio-harmony? Available at: http://tgdtheory.

fi/public_html/articles/toricharmony.pdf., 2018.

[7] Pitk¨anen M. Could also RNA and protein methylation of RNA be involved with the ex-

pression of molecular emotions? Available at: http://tgdtheory.fi/public_html/articles/

synapticmoods.pdf., 2018.

[8] Pitk¨anen M. Homonymy of the genetic code from TGD point of view. Available at: http://

tgdtheory.fi/public_html/articles/homonymy.pdf., 2018.

[9] Pitk¨anen M. An overall view about models of genetic code and bio-harmony. Available at: http:

//tgdtheory.fi/public_html/articles/gcharm.pdf., 2019.

[10] Pitk¨anen M. An Overall View about Models of Genetic Code & Bio-harmony. DNA Decipher

Journal, 9(2), 2019. See also http://tgtheory.fi/public_html/articles/gcharm.pdf.

[11] Pitk¨anen M. The dynamics of SSFRs as quantum measurement cascades in the group algebra

of Galois group. Available at: http://tgdtheory.fi/public_html/articles/SSFRGalois.pdf.,

2020.

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Published by QuantumDream, Inc.

ResearchGate has not been able to resolve any citations for this publication.

The dynamics of SSFRs as quantum measurement cascades in the group algebra of Galois group

Article

Full-text available

Feb 2020

M. Pitkanen

Adelic physics, M 8 − H duality, and zero energy ontology lead (ZEO) to a proposal that the dynamics involved with "small" state function reductions (SSFRs) as counterparts of weak measurements could be basically number theoretical dynamics with SSFRs identified as reduction cascades leading to completely un-entangled state in the space of wave functions in Galois group of extension of rationals identifiable as wave functions in the space of cognitive representations. As a side product a prime factorization of the order of Galois group is obtained. The result looks even more fascinating if the cognitive dynamics is a representation for the dynamics in real degrees of freedom in finite resolution characterized by the extension of rationals. If cognitive representations represent reality approximately, this indeed looks very natural and would provide an analog for adele formula expressing the norm of a rational as the inverse of the product of is p-adic norms. The results can be appplied to the TGD inspired model of genetic code.

The functional readthrough extension of malate dehydrogenase reveals a modification of the genetic code

Article

Full-text available

Nov 2016

Translational readthrough gives rise to C-terminally extended proteins, thereby providing the cell with new protein isoforms. These may have different properties from the parental proteins if the extensions contain functional domains. While for most genes amino acid incorporation at the stop codon is far lower than 0.1%, about 4% of malate dehydrogenase (MDH1) is physiologically extended by translational readthrough and the actual ratio of MDH1x (extended protein) to ‘normal' MDH1 is dependent on the cell type. In human cells, arginine and tryptophan are co-encoded by the MDH1x UGA stop codon. Readthrough is controlled by the 7-nucleotide high-readthrough stop codon context without contribution of the subsequent 50 nucleotides encoding the extension. All vertebrate MDH1x is directed to peroxisomes via a hidden peroxisomal targeting signal (PTS) in the readthrough extension, which is more highly conserved than the extension of lactate dehydrogenase B. The hidden PTS of non-mammalian MDH1x evolved to be more efficient than the PTS of mammalian MDH1x. These results provide insight into the genetic and functional co-evolution of these dually localized dehydrogenases.

E8 symmetry, harmony, and genetic code

Article

Full-text available

Mar 2016

M. Pitkanen

Geometric Theory of Harmony

Article

Full-text available

Jan 2015

M. Pitkanen

In the earlier article 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. 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. 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 was meant to be a continuation to the mentioned article providing a proposal for a theory of harmony and detailed calculations. It however turned out that the proposed notion of bioharmony was too restricted: all icosahedral Hamilton cycles with symmetries turned out to be possible rather than only the 3 cycles forced by the assumption that the polarity characteristics of the amino-acids correlate with the properties of the Hamiltonian cycle. In particular, it turned out that the symmetries of the Hamiltonian cycles are the icosahedral symmetries needed to predict the basic numbers of the genetic code and its extension to include also 21st and 22nd aminoacids. One also ends up with a proposal for what harmony is leading to non-trivial predictions both at DNA and amino-acid level.

Biology and Natural Geometry (Part I)

Jan 2014

M Pitkänen
Music

Pitkänen M. Music, Biology and Natural Geometry (Part I). DNA Decipher Journal, 4(2), 2014. See also http://tgtheory.fi/public_html/articles/harmonytheory.pdf.

Can one imagine a modification of bio-harmony

Jan 2018

M Pitkänen

Pitkänen M. Can one imagine a modification of bio-harmony? Available at: http://tgdtheory. fi/public_html/articles/toricharmony.pdf., 2018.

Could also RNA and protein methylation of RNA be involved with the expression of molecular emotions

Jan 2018

M Pitkänen

Pitkänen M. Could also RNA and protein methylation of RNA be involved with the expression of molecular emotions? Available at: http://tgdtheory.fi/public_html/articles/ synapticmoods.pdf., 2018.

Homonymy of the genetic code from TGD point of view

Jan 2018

M Pitkänen

Pitkänen M. Homonymy of the genetic code from TGD point of view. Available at: http:// tgdtheory.fi/public_html/articles/homonymy.pdf., 2018.

An overall view about models of genetic code and bio-harmony

Jan 2019

M Pitkänen

Pitkänen M. An overall view about models of genetic code and bio-harmony. Available at: http: //tgdtheory.fi/public_html/articles/gcharm.pdf., 2019.

Preprint

Full-text available

The realization of genetic code in terms of dark nucleon and dark photon triplets

February 2022

M. Pitkanen

I have worked for more than 10 years with a proposal for two kinds of realizations of the genetic code. The first realization, bioharmony model, represents genetics as light 3-chords consisting of dark photons. The second realization is in terms of dark proton or nucleon triplets forming closed or open strings. I have considered several variants of both realizations but the details have remained ... [Show full abstract] poorly understood and I have spent a considerable time on wrong tracks. It however seems that the dust is finally settling (I am writing this in the beginning of 2022). One can see the dark nucleon model as a generalization of the quark model of nucleon and ∆ baryons obtained by replacing u and d quarks with dark nucleons. Galois confinement solves the statistics problem. The nucleons are connected by pionic flux tubes to form a closed string-like entity. The dark variants of DNA, RNA, tRNA, and amino-acids (AAs) follow as a prediction. In the sequel, the notation DDNA, DRNA, DtRNA, DAA will be used for the dark variants of the basic information molecules. One can also understand the small symmetry breaking associated with the genetic code. A concrete realization of bioharmony in terms of the dark nucleon model for codons emerges. The small symmetry breaking effects-the members of doublet that should code for the same amino acid (or act as stop codons), code for different amino acid (or amino acid and stop), are understood. Also the differences between vertebrate and bacterial codes are understood.

Preprint

Full-text available

Progress in the understanding of the icosa tetrahedral realization of the genetic code

September 2024

M. Pitkanen

TGD leads to two models of the genetic code. The first model emerges from a model of music harmony based on the combination of icosahedral and tetrahedral geometries. The second model relies on the representation of the genetic codons as entangled triplets of dark protons at the monopoles flux tubes defining the dark variant of DNA accompanying the ordinary DNA. It took quite a long time to ... [Show full abstract] understand why both icosahedra and tetrahedra are needed and how the two models are related. The solution of the puzzle came from a universal model of the genetic code based on a completely unique tessellation of 3-D hyperbolic space H 3 realized as the light-cone proper time constant hyperboloid of the Minkowski space. This icosa tetrahedral tessellation (ITT) (known also as tetrahedral-icosahedral tessellation) makes sense in all scales and I have proposed its realization at the level of DNA. The model involves several intuitive elements and the best way to proceed is to try to improve the existing understanding and to identify the possible weaknesses of the model. This article provides an answer to the question how many icosahedrons, octahedrons and tetrahedrons meet at the vertex of ITT: the answer comes by studying the vertex figure of ITT: these numbers are 12, 30, and 20. The study of the vertex figure of ITT suggests that the ITT can be constructed as a "blow-up" of the icosahedral tessellation (IT) by replacing icosahedral vertices with tetrahedra and dodecahedral vertices by pentagons and adding between icosahedral tetrahedra and dodecahedra octahedra as analogs of edges. Icosahedral and dodecahedral bioharmonies correspond to 12-note resp. assignable to Western resp. Eastern music. One can ask whether octahedral 4-codons should also be allowed. The picture provided by RID is consistent with the earlier notion of "super-icosahedron". The model of the genetic code generalizes: besides the icosahedral Hamilton cycles (HCs) and codons for the three icosahedral codes and the tetrahedral HC and corresponding codons, also a unique dodecahedral HC and associated 5-codons plus pentahedral HC and codons are in principle possible. The fundamental region deduced from RID corresponds to a sequence of 10 or 12 DNA codons as proposed already earlier on the basis "super-icosahedron model". The model allows us to understand the symmetry breaking of genetic codons. In particular, tetrahedral codons correspond to 3 stop codons and the codon coding for trp. A given codon corresponds either to I/T or D/pentahedron. The fundamental region represents a sequence of 10 or 12 DNAs so that all codons of the Hamiltonian cycle are used and the HC corresponds to a section of DNA. Fundamental region represents both DNA strands.

Preprint

Full-text available

Galois groups and genetic code

July 2021

M. Pitkanen

This article was inspired by the inverse problem of Galois theory. Galois groups are realized as number theoretic symmetry groups realized physically in TGD a symmetries of space-time surfaces. Galois confinement as an analog of color confinement is proposed in TGD inspired quantum biology. Galois groups, in particular simple Galois groups, play a fundamental role in the TGD view of cognition. ... [Show full abstract] The TGD based model of the genetic code involves in an essential manner the groups A5 (icosahedron), which is the smallest non-abelian simple group, and A4 (tetrahedron). The identification of these groups as Galois groups leads to a more precise view about genetic code. The question why the genetic code is a fusion of 3 icosahedral codes and of only a single tetrahedral code remained however poorly understood. The identification of the symmetry groups of the I, O, and T as Galois groups makes it possible to answer this question. Icosa-tetrahedral tesselation of 3-D hyperbolic space H 3 , playing centrl role in TGD, can be replaced with its 3-fold covering replacing I/O/T with the corresponding symmetry group acting as a Galois group. T has only only a single Hamiltonian cycle and its 3-fold covering behaves effectively as a single cycle. Octahedral codons can be regarded as icosahedral and tetrahedral codons so they do not contribute to the code.

Article

Full-text available

New results about dark DNA inspired by the model for remote DNA replication

February 2020

M. Pitkanen

Two objections against TGD based view about DNA and genes are discussed. TGD predicts two dark variants of genetic code realized as dark codons (DDNAs) identified either as dark proton triplets or dark photon 3-chords. The objection against dark photon 3-chords (3-photon states) is that the simultaneous emission of 3 dark photons is extremely non-probable. The proposed solution of the problem is ... [Show full abstract] that dark photons carry number theoretic color associated with Z3 subgroup of Galois group. Number theoretic color confinement would imply that only 3-chords can appear as asymptotic states analogous to baryons as 3-quark states. If also the dark protons form number-theoretic color triplet, dark codons must consists of 3 protons and therefore also ordinary codons would have 3 letters. The findings of Gariaev's group and Montagnier et al suggest the possibility or remote repli-cation of DNA. The fact that dark codons do not decompose into letters like chemical codons poses strong constraints on the replication and transcription if one assumes DDNA-DNA-pairing. These constraints strongly suggests that the nucleotides in the water environment of DNA are not actually free but form loosely bound triplets representing codons bound with DDNAs. Replication is predicted to occur in codon-wise manner: this has been observed to be possible for RNA. It might be that the loose nature of exotic DNA codons allows this to occur quite generally. Remote replication in this framework reduces to ordinary replication in TGD sense if also dark genes are present and formed by attaching flux tubes characterizing dark codons to a long flux tube associated with gene. Remote replication requires that the portion of dark gene accompanying ordinary gene is transferred from chamber A to chamber B in the experiment of Montagnier.