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Pitkänen, M., An Overall View about Models of Genetic Code & Bio-harmony

Article

An Overall View about Models of Genetic Code & Bio-harmony

Matti Pitkänen 1

Abstract

During last years kind of brain storming period has occurred in the TGD inspired models of bio-
harmony and genetic code. A lot of ideas, some of them doomed to be short lived, have emerged,
and it seems that now it its time for a thorough cleanup and integration with the general ideas of
TGD inspired quantum biology. TGD leads to 3 basic realizations of the genetic code. One can
also consider 3 realization also for bio-harmony. The question is which of them is the realistic one
or whether several options can be considered. In this article these ideas are discussed critically and
open problems are summarized. The three genetic codes correspond to a fundamental realization in
terms of dark proton sequences (dark nuclei) with 3-proton representing codon. Second realization
is the chemical realization and the third realization is in terms of dark photon 3-chords mediating
the interaction between various realizations. Frequency resonance is very natural interaction between
dark levels and energy resonance between dark level and chemical level. The possibility to modify
the value of heff for flux tube makes possible to have for given codon single resonance energy.

The homonymy of the genetic codes at various levels is discussed. At the dark level the fact that
icosahedral harmonies can have common 3-chords implies the first homonymy. The basic difficulty
of Pythagorean scale realized in terms of quint cycle realized already by Pythagoras becomes the
solution of this problem. The well-known homonymies in RNA-tRNA correspondence and even in
RNA-AA correspondence can be understood in the model in which dark photon 3-chords mediate the
interactions. Also questions related to the relationship of bio-harmony with ordinary genetic code
are considered. Why 3 copies of icosahedral harmony and only one copy of tetrahedral harmony? A
special triangle assignable to the 3 copies of icosahedron and tetrahdron is analogous to a singular
point of covering: do these 4 triangles correspond to exceptional codons breaking symmetries? How
do the dissonant 3-chords present in some icosahedral harmonies relate to stop codons? How do the
codons of bio-harmony and ordinary codons relate and is this relation consistent with what is known
about transcription and translation?

1 Introduction

During last years kind of brain storming period has occurred in the model of bio-harmony [13]. A lot
of ideas, some of them doomed to be short lived, have emerged, and it seems that now it its time for a
thorough cleanup and integration with the general ideas of TGD inspired quantum biology.

TGD leads to 3 basic realizations of genetic code: this is now relatively well established part of TGD
inspired quantum biology. One can also consider 3 realization also for bio-harmony. The question is which
of them is the realistic one or whether several options can be considered.

1.1 3 basic realizations of the genetic code

In TGD Universe there are at least 3 realizations of the genetic code.
Besides biochemical realization one has a realization in terms of dark nuclei realized as dark proton

sequences and possibly in terms of more general sequences involving effective dark neutrons. The states of
3 dark protons defining the dark codon have multiplet decomposition 64 + 64+ 40 + 20 corresponding to
dark variants of DNA, RNA, tRNA, and amino-acids (AA). I will denote these dark variants by DDNA,
DRNA, DtRNA, and DAA.

1Correspondence: Matti Pitkänen http://tgdtheory.com/. Address: Rinnekatu 2-4 A8, 03620, Karkkila, Finland. Email:
matpitka6@gamail.com.

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If one allows also dark analogs of neutrons by allowing negatively charged color bonds between protons,
the number of code letters doubles: this could relate to the recently constructed Hachimoji DNA [3] (see
http://tinyurl.com/y2mcjb4r) discussed from TGD viewpoint in [21].

Dark photon 3-chords assignable to the realization of bio-harmony with the note scale identified as
Hamilton cycle on a polytope with triangular faces gives a third realization coupling dark and ordinary
representations together. I have proposed 3 realizations in terms of icosahedral and tetrahedral [13],
icosahedral and toric [16], and icosahedral and dodecahedral [21] geometries (for the latter 5-chords
would effectively reduce to 3-chords).

If there is DDNA-DNA, DRNA-RNA, DAA-AA pairing, the negative charges of DNA, RNA, and
tRNA nucleotides finds explanation in terms of positive charge of dark proton sequence. For AAs the
situation is not clear since the charge per unit length for amino-acids varies and depends on pH. DAA-AA
pairing would require that dark analogs of neutrons are present in the dark proton sequence.

1.2 3 models of bioharmony

There are now 3 models of bioharmony [13, 16, 21] making very similar pedictions. Harmony for given
graph is defined as a Hamiltonian cycle connecting neighboring points and going through all points of the
graph without self-intersections. Scale is identified by assigning notes to the vertices and faces correspond
to the chords of the harmony obtained in this manner. Bio-harmonies are fusions of 3 or 4 sub-harmonies.

1. The original proposal - icosa-tetrahedral bio-harmony - is based on the fusion of 3 icosahedral
harmonies with symmetry groups Z6, Z4 and Z2 permuting the triangles of given orbit of Zn.
Given icosahedral harmony corresponds to an imbedding of 12-note scale as a Hamilton cycle at
icosahedron. The 12 vertices of icosahedron are identified as the notes of 12-note scale and 20
triangular faces define the 3-chords of the harmony.

The distance between nearest vertices is assumed to correspond to quint that is scaling of the
frequency by 3/2. Each cycle defines a collection of 20 3-chords defining an icosahedral harmony.
Octave equivalence is used to map the 12 frequencies obtained to single octave. There is however a
slight inconsistency since 12 quints corresponds to slightly more than 7 octaves as already Pythagoras
realized. The addition of tetrahedron to icosahedral harmony is interpreted as an addition of one
vertex adding one note which should be very near to one of the 12 notes.

Icosahedral harmonies are characterized by a symmetry group Zn, n = 6, 4, 2, 1, n = 1 corresponds
to chaotic cycles, which might serve as correlate for dis-harmony and might relate to the correlates
of emotions: at the level of genetic code is AA would be coded by single DNA codon.

Icosahedron decomposes to orbits of Zn consisting of triangles or equivalently chords. The chords
can be classified further by the frequency ratios correlating with the emotional effect. One has
the orbits 3 × 6 + 2 = 20 for Z6, 5 × 4 = 20 for Z4 and 10 × 2 for Z2. Z6 harmony is unique
but there are 3 Z4 and even more Z2 harmonies for which Z2 can correspond to rotation by π or
reflection. This can be understood as breaking of symmetry splitting the Z6 orbits to pieces. This
gives 60 = 2 + 20 + 20 3-chords. The numbers of chords at give orbit rather neatly correspond the
numbers of DNA codons coding for given AA.

4 chords and DNAs and AAs are however missing. Tetrahedral harmony would add 3 + 1 = 4
chords: Z3 would the symmetry group instead of Z4. This would be due to the symmetry breaking
due to gluing of one-tetrahedral face with icosahedral face, which is however counted as separate
face and corresponds to 1-triangle orbit under Z3 permuting its vertices. This gives 64 3-chords
corresponding to codons of genetic code.

3 + 1 decomposition would naturally correspond to (ile, ile, ile,met) 4-plet coded by codons AUX.
The numbers of codons coding given AA identified as orbit of Zn come out almost correctly. The
only exception is trp-stop doublet for which doublet decomposes to stop and singlet. One must

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understand the reason for this symmetry breaking - it might just the need to have stop codon and
this could be arranged if there is no tRNA coupling to this codon. Note that for some code variants
stop codon UAG corresponds to Pyl and UGA to Sec.

Since music generates and expresses emotions, the interpretation would be in terms of moods. Even
molecules would have moods.

2. Also icosa-dodecahedral and icosahedral-toric harmonies contain the Z6 and Z4 icosahedral har-
monies (201 and 202) so that one must only add the missing 10 doublets and 3+1 codons assigned
to tetrahedron in icosa-tetrahedral case.

The dodecahedral harmony with 6 chords arranged in doublets is unique from the uniqueness of the
Hamiltonian cycle [21]. The icosa-dodecahedral harmony would give 201 + 202 + 121 + 122 = 64. 12
decomposes into 6 Z2 doublets so that one has 12 doublets. The realization of scale for dodecahedral
harmony would in 20 powers of rational scaling x such that x20 is as near to a power of two as
possible [21]. x = 21/20 would correspond to the Eastern variant of well-tempered scale.

There are objections against icosa-dodecahedral harmony. Chords are 5-chords rather than 3-
chords. The 5-chords of dodecahedral harmony however turn out to be equivalent to 3-chords as
far as information content is considered [21]. The number of vertices for dodecahedron is 20, not
12, but one could argue that dodecahedron corresponds to Eastern harmony having micro-intervals.
Two copies of the dodecahedral harmony are needed. What could distinguish between these copies
will be discussed later. Also 3+1 is missing.

3. The icosahedral-toric harmony [16] decomposes as 201 + 202 + 24 = 64 involving torus with 24
triangles and 12 vertices. Toric harmony has Z24 as isometries and gives 12 doublets. One could
argue that the fusion of icosahedral and toric harmonies is geometrically un-natural. One must be
however cautious if the geometric realization is in extension of rationals. Also now 3+1 is missing.

The considerations in the sequel suggests that the icosa-tetrahedral option is the most realistic if not
unique.

1.3 About the geometric interpretation of icosahedral and other symmetries

The geometric interpretation of icosahedral and possible other geometries is a challenge. The 60-element
group A5 of rotations - alternating group of 5-letters - acts as orientation preserving isometries of icosa-
hedron.

1. Since Galois group is central in adelic physics, and all finite groups can appear as Galois groups,
one can ask whether icosahedral group and tetrahedral groups could act as Galois group for some
extension of rationals relevant for biology. Going to web gives an affirmative answer [1] (see http:

//tinyurl.com/y4qsea6h)! Icosahedral symmetry appears as Galois group of the general quintic
equation! The lowest order polynomial equation not allowing closed expressions for the roots.

Galois theory (see http://tinyurl.com/y6e955ke) allows to understand the situation in terms of
the discriminant defined as product D =

∏
i<j(ri − rj)2, where ri are the roots of the irreducible

polynomial considered. Sn is the symmetry group in the generic case and odd permutations of Sn

change the sign of D. If D is square of rational number in the field K considered (which can be
also extension of rationals now), Galois group reduces to alternating group A5.

Remark: For octahedron and its dual cube the group is S4 and can be realized as Galois group of
4th order polynomials. For tetrahedron the group is A4 and can be also realized as Galois group of
4th order polynomials for which discriminant is square in K.

2. Icosahedral and dodecahedral geometries having the same isometry group are common in biology,
and one can wonder whether there could be a gometric realization - perhaps at the level of magnetic

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body. This might somehow relate also to the frequent appearance of Golden mean involving
√

5 in
biology and Golden angle rated to the fifth root of unity.

3. M8 −H duality provides besides the usual formulation of TGD also a formulation in complexified
M8 identified as complexified octonions [15]. The associativity of the tangent or normal space of
space-time surface is assumed as a dynamical principle and implies quaternionicity. Quaternions
have SO(3) as automorphism group analogous to Galois group and have the finite isometry groups
of Platonic solids as finite subgroups.

Could quaternionicity give a connection with the geometric picture? In adelic physics discretizations
of space-time points as points with coordinates in the extension of rationals are in central role. Could
discretizations contain orbits of the Platonic isometries as quaternionic Galois groups? This could
also give to the geometric picture although icosahedral symmetries are not obvious in the geometry
of say DNA.

4. Is the genetic code really unique as its dark nucleus realization and the fact that the isometry groups
of Platonic solids are finite subgroups of quaternionic isomorphisms suggests? Could any Galois
group give rise to an analog of bioharmony and of genetic code? Could the recent genetic code
correspond to a first step in the process going beyond the solvable polynomial equations?

What about toric code? The group of toric isometries is Z24 and 24 is one of the magic number
of mathematics, and dimension 24 is crucial in bosonic string model. Could Z24 correspond to the
Galois group for 24:th roots of unity defining 24-D algebraic extension of rationals. We cannot
sensorily imagine higher dimensions but can do this cognitively. I have proposed that the ability
to imagine higher dimensions could be due to the possibility of higher-dimensional extensions of
rationals and p-adics.

Could one realize the icosahedron and 24-torus as imagined object in the algebraic extension of
rationals? Could the n-dimensional discrete geometric objects assignable to n-dimensional exten-
sions of rationals have quite generally this kind of representations as a generalized Platonic solid
in algebraic extension. Could they define cognitive harmonies as Hamiltonian cycles? Could one
imagine also cognitive variant of genetic code whereas as sensory/biological variant of genetic code
would be forced by dark proton physics?

1.4 Mistracks

In the attempts to understand the connection with standard realization of the genetic code I have also
considered the possibility that the frequencies of 3-chord might be mapped to their sum in the interactions.
This possibility was considered in the model of homonymy [18]. In the light of afterwisdom this proposal
looks ad hoc.

Also a proposal for how 12-note scale could quite concretely correspond DNA codons was discussed
[19]. The idea was to assign notes with individual letters of the codon such that the note depends on the
position of the letter whereas the model of harmony assignment the chord to the entire codon represented
as entangled state of 3 dark protons. It is now clear this proposal very probably cannot realize all possible
harmonies and is in conflict with the general model which as such fixes the correspondence between chords
and codons without any additional assumptions.

2 Interactions between various levels

One challenge is to understand how the various realizations of the genetic code interact with each other.
There are DX-DY interactions, DX-Y interactions and X-Y interactions and in living matter they should
occur in long length scales so that they should be mediated by dark photons.

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1. How dark photon triplets assumed to be generated by dark nucleon sequences interact with ordinary
DNA? Here one can bring in rather stable ideas of TGD inspired view about quantum biology. Dark
matter in TGD sense represents long length scale quantum coherence and bio-chemistry short scale
coherence. The interaction is therefore between long and short scales.

2. There are two manners to interact: frequency resonance and energy resonance. Frequency resonance
mediates long length scale interactions and if DX-X pairing exists, the exchange of dark photon
triplets - 3-chords - allows long range DX-DY interactions. DX-X interaction by energy resonance is
short range interaction so that X-(DX-DY)-Y interaction would give rise to long range interaction
between X-Y as interaction induced by dark level (MB).

3. DX-X interaction involves energy resonance and transformation of dark photons to ordinary photons
with the same energy. Bio-photons would be an outcome of the transition heff → h. Also the
reversal of this transition and more general transitions heff,1 → heff,2 are of course possible.

Bio-photons have a universal energy spectrum corresponding to molecular and atomic transition
energies. This is possible if they result from dark cyclotron photons if the condition heff = hgr =
GMm/v0 introduced originally by Nottale and implying that the cyclotron energy does not depend
on the mass of the charged particle producing the dark cyclotron photons.

2.1 The independence of the interaction energy on frequency

Dark matter as a hierarchy phases labelled by heff/h0 = n identifiable as a dimension of extension of
rationals implies evolutionary hierarchy: n serves as a kind of IQ. This strongly suggests that ordinary
matter is controlled by dark matter at MB and mimics its behavior.

Evolution would not proceed by change and necessity but would be a process controlled and guided
by MB. MB would be an active intentional agent guiding the evolution. Situation in biology would be
much like that in modern technological society where intentional technical progress leads to more and
more refined products. How could this be realized at the level of basic bio-molecules? One should also
understand how genetic code evolves gradually to a more refined form.

1. The selection of basic bio-molecules having energy resonance with their dark variants mediated
by dark photon 3-chords by change would be extremely in-effective process. MB should have
mechanisms of tuning the energies of dark photons to achieve energy resonance.

This is achieved if the value of heff at the flux tubes mediating the interaction can be controlled.
Since the length of flux tube is proportional to the heff by Uncertainty Principle, the variation
of heff would mean variation of the length L of the flux tube: a kind of motor action of MB.
Cyclotron frequencies are proportional to the value of monopole magnetic field B at flux tube and
by flux quantization one has B ∝ 1/S, S the area of flux tube cross section (which for monopole
flux tubes is closed 2-surface). The variation of the thickness/area of the flux tube, second motor
action of MB, would allow to vary cyclotron frequencies.

2. The ideal situation concerning the coupling to ordinary matter would be that same chemical transi-
tion with fixed energy for given molecule could couple to several frequencies. This would be achieved
if the cyclotron energy is constant.

The condition that the cyclotron energies in a coupling to a given molecule do not depend on the
frequency requires that heff,i at flux tube i compensates this dependence. MB can vary the value of
B to vary frequencies and the value of heff,i to keep energy unaffected. The areas S and length L of
flux tubes are varied so that the volume remains unaffected. B ∝ 1/S and L ∝ heff by Uncertainty
Principle. Ec ∝ ~effB = constant implies that L/S is constant. S increases like S → x2S and
L → x2L in the scaling changing fc → fc/x

2. The magnetic energy Emagn = B2SL ∝ L/S of the
flux tube is not changed. Kind of energy criticality would be in question - one would have a large

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number of flux tube configurations with the same energy and volume ideal for control purposes.
Quantum criticality is actually basic dynamical principle of quantum TGD allowing to predict the
spectrum of various coupling parameters.

3. Besides cyclotron frequences Josephson energies are central in TGD based model of nerve pulse and
EEG. Josephson energy EJ = ZeV and cyclotron frequency fc = ZeB/m do not depend on heff .
An attractive possibility is that cyclotron photons couple to Josephson junctions meaning that they
become Josephson photons and then transform to ordinary photons inducing molecular transitions.

4. In the case of bio-harmony the frequencies would be rational multiples of basic frequency and by
separating common numerator they are certain integer multiples fi = nif0 of a basic frequency
f0. The integers ni have decomposition to products of powers of certain primes: ni =

∏
pki
i and

each of pi appears as some maximal power ki,max. If one has n =
∏

i p
ki,maxn0 one can obtain

heff,i = heff/ni. In this manner one would obtain the desired independence of Ec,i on fi. For
Pythagorean scale only primes p = 2 and p = 3 would be involved.

All codons coding for given AA could have same coupling energy. Unless the values of Planck con-
stants and frequencies associated with flux tubes coupling to given codon are fixed, one could have same
transition energy for all letters but this is an unrealistic condition. Transition energies are naturally
different and can code for letters if not even codons. For this option only the correct combination of
frequencies and values of heff,i allows resonant coupling.

The 3-chords associated with different harmonies would naturally correspond to the same energy. The
physics of emotions would not be directly visible at the level of chemistry: chemist would certainly agree
with this. The values of Planck constants would characterize the frequencies: I have indeed speculated that
nucleotides could be labelled by values of heff . Number theory would be essential for the understanding
life at the level of genes: Galois groups would characterize the nucleotides. Galois groups code for
complexity at the level of dark matter so that the behavior guided by the MB of molecule would depend
on the IQ = n = heff/h0 of MB.

2.2 The independence of cyclotron energy on frequency and Nottale hypoth-
esis

Is the independence of interaction energy on frequencies consistent with hgr = GMm/v0 hypothesis [2]
[9, 10, 11]? Here one might encounter difficulties. The division by ni should change one of the parameters
appearing in the formula. The interpretation has been m corresponds to the dark proton mass at the end
of the flux tube connecting it to large mass M . If so m cannot be varied.

Could M be varied?

1. The parameter v0 ' 2−11 can be varied by powers of two, which do not affect the notes identified
by octave equivalence.

2. Could M correspond to atomic or molecular mass in good approximation equal to sum of atomic
numbers A of atoms involved? The divisors of the total atomic number Atot would define the allowed
integers ni characterizing the frequencies of Pythagorean scale in the model of bio-harmony. One
must have hgr/h > 1 with requires M > ~/Gm = 1.3× 1019mpv0. For v0 = 2−11 this corresponds
to M > ~/Gm = 6 × 1015mp. The scale of a water blob with A = 20 containing this number of
protons is about 70 µ, which is of order cell size. One can wonder how Atot could be kept as divisible
by ni characterizing the frequencies of the Pythagorean scale. The problem is that an addition of
one proton spoils the divisibility conditions completely.

3. The solution of the problem could be based on a more precise view about heff [20]. The under-
standing of the variation of Newton’s constant - too large to be due to experimental errors - led to
the realization of the meaning of the fact that space-time surfaces can be regarded simultaneously

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coverings of n2-fold M4 and n1 fold CP2 and that one has n = n1n2 in heff/h0 = n and n1 would
have interpretation as the number of flux tubes which are parallel in M4 and can be even disjoint.
This would give hgr ∝ n1 and the factors of n1 should correspond to the integers characterizing
the notes of the 12-note scale. One could perhaps say that effectively single proton is replaced with
n1 protons located at different flux tubes so that also proton mass becomes n1m. One would have
effectively a Bose-Einstein condensate like state of n1 protons (at different flux tubes).

4. In the Pythagorean representation of octave the notes correspond to powers (3/2)k, k = 0, 1, ..., 11,
if 3/2)12 ' 27 is not included. The corresponding integers are 3k211−k. Only powers of primes
p = 2 and p = 3 are involved and one just have n1 ∝ 311211. If one increases the number of octaves
involved to 14 to get a representation for chords needed to avoid the mapping of two dark codons
to same 3-chords, one must have n ∝ 323223 = 623. One can consider also simpler representations
using integers expressible in terms of powers of primes p = 2, 3, 5 but one must give up exact quint
cycle in this case. Interestingly, a good guess for the standard value h of heff is as h = 6h0 [14, 17].

5. Small p-adic primes p = 2, p = 3 and perhaps also p = 5 (Golden Mean) are expected to be of
special importance in TGD inspired biology [8]. p = 2 seems to appear everywhere and there is also
support for p = 3 in biology [5, 6] (see http://tinyurl.com/ycesc5mq): great evolutionary leaps
seem to correspond to time scales coming in powers of 3.

6. The branching of the flux tube bundle to ni sub-bundles Ni = n/ni could correspond to the
reduction heff → heff/ni. This could be seen as reduction of heff . One can also consider phase
transitions reducing n to n/ni.

3 Homonymy of the genetic code

In the following I will discuss briefly the basic facts about genetic code at Wikipedia level with emphasis
on the poorly understood aspects of the code. There are two interesting phenomena: synonymy and
homonymy. Synonymy means several names for AA or tRNA codon so that that several RNAs are
mapped to the sama AA or tRNA codon: the understanding of the genetic code is the understanding of
synonymy.

Homonymy means that the same RNA codon can correspond to several tRNAs or even AAs. A general
TGD based view about homonymy differing from that discussed in [18] based on the recent understanding
of the interaction between various representations of the genetic code is described below.

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

1. UUU can code Leu instead of Phe and CUG can code Ser rather than Leu. In bacteria the GUG
and UUG coding for Val and Leu normally can serve as Start codons.

2. UGA can code to Trp rather than Stop: in this case the broken symmetry is restored since also
UGG codes for Trp.

3. 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

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Arg. This phenomenon is known as Stop codon readthrough (see https://www.ncbi.nlm.nih.

gov/pmc/articles/PMC5133446/).

4. There is also a variant of genetic code in which there are 21st and 22nd AAs Sec and Pyl coded by
Stop codons. UGA can code for Sec and Stop in the same organism. UAG can code for Pyl instead
of Stop and introduces additional breaking of A-G symmetry for the third letter (UAA to Stop and
UAG to Pyl).

3.2 Wobble base pairing

Wobble base pairing (see http://tinyurl.com/y73se8vs) emerges from the observation that the number
of tRNAs pairing with mRNAs is smaller than 45 and considerably smaller than that of mRNAs. The
needed minimum number of tRNAs is 32. Therefore the RNA-tRNA pairing cannot be 1-1 and some
mRNA codons must correspond to several tRNA codons.

Remark: One could ask whether mRNAs code for tRNAs just like DNAs code for AAs. Homonymy
for mRNA-tRNA pairing implies that the pairing can be many-to-1 only in given context.

1. According to the standard code, the first two bases of mRNA codon corresponds to two last bases
of tRNA anti-codon and obey standard code. Wobble base pairing hypothesis applies to the pairing
of the 3rd mRNA base to the 1st base in tRNA anticodon. At the level of chemistry the hypothesis
is that the position of the first tRNA anticodon base pairing with the third mRNA base is variable
and allows it to pair with several bases appearing as 3rd base in mRNA. This homonymy would be
due to ”wobbling” of the position of the first tRNA anticodon.

2. In the original model for wobble base pairing tRNA bases contain besides standard A, C, G, U also
inosine I as a modification of G obtained by dropping NH2 from the 6-cycle of G. It has turned
out that there are actually variants of C and 5 variants of U (see http://tinyurl.com/y73se8vs).
The large amount of homonymy for tRNAs forces to ask whether chemistry alone really dictates
the genetic code.

3. The first tRNA letter is assumed to be spatially wobbling so that the association of tRNA with
RNA is not unique and mRNA-tRNA pairing involves both synonymy and homonymy as the two
tables for the pairing of the 1st 5’ anticodon base of tRNA and 3rd 3’ codon base of mRNA show.
In the second column bold letters for mRN bases allow to read the standard pairing with tRNA
codons in the first column and non-bold letters allow to deduce the non-standard behavior.

4. The first table (see http://tinyurl.com/y73se8vs) represents the original Watson-Crick proposal.

(a) The pairings of the 3rd letter of mRNA codon to the 1st letter of tRNA anti-codon are following.

• U → G.

• G → U

• {A, C or U} → I.

The 2nd and 3rd tRNA letters A and C are paired with the 1st and 2nd mRNA letters in the
canonical manner. There are only 3 tRNA letters, which implies that the number of tRNAs is
smaller than maximal.

(b) There is single 1-to-many pairing: U→ {G, I} giving rise to 2-fold homonymy.

5. Revised pairing rules (see http://tinyurl.com/y73se8vs) are more complex since the number of
tRNA bases is larger (U has 5 variants and C has 2 variants). All mRNA letters have 1-to-many
pairing. Even if one counts the variants of U as single U there is 4-fold homonymy for U and
homonymies for other codons. For A one has 9-fold homonymy.

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These variations do not induce variation in DNA → AA pairing if the AA associated with the
homonyms of tRNA are identical. This seems to be the case almost always since the variation of the
genetic code is surprisingly small. This raises the question whether there is some mechanism eliminating
to high degree the expected effects of homonymy in mRNA→ tRNA pairing.

4 TGD view about homonymies

One should understand the homonymies of the genetic code [18]. One can imagine homonymies at
the level of DDNA-3-chord and DRNA-3-chord correspondences and between RNA-AA and RNA-tRNA
correspondences.

4.1 Homonymies for DRNA-3-chord correspondence

It is possible that homonymies are present already at the dark photon level in the sense that the sub-
harmonies have common chords.

1. Are the icosahedral orbits for different symmetry groups Z6, Z4, Z2 disjoint? If they contain common
triangles, the outcome is homonymy for dark codons unless one can scale the 12-note scales with
respect to each other (different keys) to avoid common chords.

This question finds an answer from the tables of [13] representing the chords. If the two scales
considered contain 3-chords with the same frequency ratios this can happen. Z6 harmony contains
chords of same type with whole note intervals: Cx, Dx, Ex, ..., x = m, 6, 9 coding the frequency
ratios as is done in popular music. If second harmony contains several types such that they are not
separated by a multiple of whole note interval, at least one common chord is unavoidable also for
shifted harmonies.

2. From the tables 1 and 2 of Appendix one finds that for Z6 and 2 Z4 harmonies this is indeed the case
and they have 2-chords involving 2 quints in common: 6-orbit and 4-orbit containing x = 9 3-chords
have 2 common chords. One has homonymy at dark level. If entire orbits are mapped to the same
AA there would be 8 AAs in the same multiplet. Some DDNA and DRNA codons are mapped to
the same 3-chord of dark photons. This problem is shared by all 3 models of bio-harmony.

3. For the unique Z6 harmony and 3 Z2,rot (table 3 of Appendix) of harmonies common chords can
be avoided by shifting the latter harmonies by a half-note. The reason is that the chords of same
type are now separated by a multiple of whole note interval. For Z2,refl harmonics (table 4 of
Appendix) the chords of same type are separated by odd number of half-notes so that common
chords are unavoidable since 3-chords of the same type appear. There are also common chords with
Z4 harmony.

4. Z6 and Z2,rot harmonies possess no common chords by a shift by odd number of half notes. Z4

and Z2,rot and Z4 and Z6 possess at least 2 common chords. Z2,refl possesses more common chords
with Z4 and Z6.

The fusion of Z6, Z4, and Z2,rot harmonies with 2 common chords between in Z6 ∩ Z4 Z4 ∩ Zr,rot

seems seems to be best that one can achieve. This would give 1 × 2 × 3 = 6 harmonies altogether
unless one obtains new harmonies by by relative shifts of the key.

How to solve the problem?

1. The above described homonymies involving 6-plets involve either 6-plet or 2-plet as second multiplet
so that these deviations cannot be due to homonymy at the level of DRNA-3-chord correspondence.

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2. Should one take seriously the puzzle that teased Pythagoras and led him to seriously consider that
the structure of the Universe based on rationals has serious flaw in it. 12 quints give slightly more
than 7 octaves: one has (3/2)12 = 129.746337890625 rather than (3/2)12 = 128 so that one obtains
slightly more than octave under octave equivalence.

Why not represent notes as powers of algebraic number 21/12 and this is indeed done in practice
(in rational approximation of course) but very musical people notice the difference and dislike this
representation. There should be something deep in the representation of the scsale in terms of
rationals as TGD indeed predicts. Note that a strict resonance is not required, it represents only
the optimal situation.

3. Repeating the quint cycle gives slightly displaced chords: one can of course do this several times
[21]. Could these slightly displaced chords represent DDNA and RNA codons as 3-chords otherwise
mapped to the same chords? This would also mean that the corresponding DNAs and RNAs
correspond to 3-chords with at least one note differing only slightly. This kind of notes is shared by
5 chords in icosa-tetrahedral harmony. The addition of second quite cycle means that the integers
ni = 2k323−k characterize the notes of the 3-chords and 2k323−k and 2k+12311−k represent the
nearby notes.

4. The minimal modification would replace only minimum number of notes in the problematic chords
with new ones. A stronger modification would replace the problematic chords with displaced variants
with notes in the second quint cycle. One could also do the same for all chords and say that the
number of codons for non-problematic dark codons is doubled.

One could also consider the doubling of each letter of the codon so that each chord would be
replaced with 8 almost copies except in the case of homonymic AAs. A non-homonymic AA coded
by n RNAs would be coded by 8n 3-chords. If the frequency differences are small enough this is
not seen at the level of transition energies of AAs: this must be the case for non-homomymous
AAs. For homonymous RNAs the energy differences must be seen and remove the homonymy. This
DRNA-3-chord homonymy would be analogous to the RNA-tRNA homonymy.

5. One can consider the problem from a different perspective. For Hachimoji DNA [3] (see http:

//tinyurl.com/y2mcjb4r) the number of DNA letters seem to double so that codon is replaced
with 8 codons. An explanation based on the Pythagorean dilemma was discussed in [21]. In the
model it was however assumed that the doubling of dark DNA and DNA is real being due to the
possibility of having also negatively charged color bonds between dark protons so that dark proton
is effectively dark neutron (this might happen even in ordinary nuclear physics in nuclear string
model [7]). The Pythagorean double covering of 3-chords could describe the doubling of codons.
The doubling would not occur for the codons for which one has the homonymy - a prediction, which
could be perhaps tested.

4.2 The map DRNA-DtRNA by 3-chords

The map 64 → 40 for DRNA-DtRNA inducing the corresponding map for RNA − tRNA is not unique
since there are many manners to reduces 64 to 40. Could this relate to tRNA-RNA homonomy? Consider
icosa-tetrahedral code 20 + 20 + 20 + 4 = (3× 6 + 2) + (5× 4) + (10× 2) + (3 + 1) as example.

1. Suppose Z2 is the divisor group (also Z4 and Z3 ⊂ Z6 can be considered) so that the orbit can
split to two and two tRNAs are associated with given amino-acid coded by n codons. At the first
step one can take 201 + 202 + 203 + 4 → 201 + 102 + 103 + 4 = 44. Also 101 + 202 + 103 + 4 and
101 +102 + 202 +4 can be considered. Since Zn has Z2 as subgroup, the simplest manner to achieve
20k = 10k is to divide all orbits to 2 Z2 cosets. This can be carried out in 3 manners.

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2. One must get rid of 4 tRNAs. This can be achieved in several manners. In 201 = 3×6+2 one could
have 6 + 2→ 3 + 1: there are 3 alternatives. In 202 = 5× 4 one could have 5× 4→ 3× 4 + 2 + 2
(10 manners). In 203 = 10× 2 one can take two 2:s to 1 (45) manners.

3. Could all these maps be realized and could they correspond to different maps at the level of dark
codons? If the independence of resonances energies on frequencies is true with an appropriate choice
of heff,i, it would seem that in all these cases same chemical tRNA is possible.

4.3 Homonymies for RNA-AA correspondence

There are two basic types of homonymies involving bio-molecules.

1. RNA-AA correspondence can vary somewhat and there are 31 variants of genetic code. RNA-tRNA
homonymies are common and wobble phenomenon could be regarded as as such homonomy. This
homony is poorly understood.

I made the first attempt to understand homonymies in [18] but failed to realize one absolutely
essential feature. Despite RNA-tRNA homononmies there are practically no RNA-AA homonymies.
They might be completely absent for given genetic code. There must be a simple explanation for
this.

2. In TGD framework the genetic code is replaced with 3 codes. There is DRANA-DtRNA code
mapping 64 DRNA codons to 40 DtRNA codons and DtRNA − DAA code mapping 40 DtRNA
codons to 20 DAAs. The composition of these codes gives DRNA-DAA code inducing the RNA-AA
code.

The highly non-trivial fact is that one has what mathematician would call commuting triangle:
RNA-tRNA-AA = RNA-AA for given code. All the homonymies of RNA-tRNA code are possibly
completely compensated for given RNA − AA code. This must have simple explanation and once
one has made this question, one also knows its answer in TGD framework.

3. For Hamiltonian cycles the n(A) codons coding for given AA corresponds to orbit of a fixed codon
at the orbit having symmetry group Zn(A). Genetic code maps the codons at the orbit to the AA
corresponding to the orbit and replaces the symmetry group Zn with trivial group Zn/Zn = Z1.

Remark: There are 6 chaotic icosahedral Hamiltonian cycles with symmetry group Z1 so that
therefore 20 amino-acids each coded by single codon. Could one interpret the 20 amino-acids with
the chaotic representation of chaotic icosahedral Hamiltonian cycle?

For RNA-tRNA correspondence similar process is possible. Now one replaces Zn/Zk where k is
factor of n.

Consider icosa-tetrahedral code as an example. k = 2 is simplest choice since it divides n = 6, 4, 2
for icosahedral codes but not for tetrahedral code for which one has n = 3: (ile, ile, ile,met) would
naturally correspond to the 2 orbits under tetrahedral Z3. This symmetry appears only for icosa-
tetrahedral option. For other options one can explain it as an outcome of symmetry breaking
for doublets and (ile,ile) and symmetry broken (ile,met) would have ile in common. This looks
un-natural.

One can indeed construct 64→ 40 map for DRNA and DtRNA codons by replacing some orbits with
their Z2 cosets but this map is not completely unique. This is possible for all code candidates, which
all contain Z6 and Z4 symmetric icosahedral harmonies giving rise to amino-acids corresponding to
3 6-orbits and one 2-orbit for Z6 symmetry and 5 4-orbits with Z4 symmetry. The remaining orbits
are 3-orbit and 1-orbit for tetrahedral symmetry broken to Z3 and 2-plets for Z2 orbits.

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There are are however codes for which RNA-AA correspondence is non-standard. As explained above,
the simultaneous replacement UUC-Leu → UUC-Phe and UUG-Leu → UUG-Ser can take place. Also
AUG-met → CUG-met and GUG-met → GUG-met can occur.

A general explanation could be as follows. If the two homonymous amino-acids - Phe and Leu and
Leu and Ser in the first example and met and Leu and Val in the second example- have very nearly same
transition energy, and if the 3-chords correspond transition energies of AA irrespective of frequencies,
homonymy becomes possible.

This problem can be avoided if the tRNA pairing second AA with the RNA codon is not present.
Both options might be realized in the same organism. It could also happen that second AA is so far from
energy resonance that it is only rarely translated.

4.4 Homonymies for RNA-tRNA correspondence

Could the possibility of several harmonies/moods with different chords increase the number of tRNA
codons from the minimal value 40? Are these homonymies forced by necessity or do their reflect freedom
of MB to choose? Do dialects emerge already at the molecular level and do they have some practical
advantage?

1. Could the possibility of several moods demand more than the minimal number of tRNAs. Harmonies
correspond to different collections of triplets (n1, n2, n3) characterizing the chord.

It was however already noticed that the variation of the Planck constants heff → heff/ni associated
with the flux tubes can modify the cyclotron energies. This would mean that the emotions are not
directly seen at the level of molecular transitions as bio-chemist would certainly argue. If energy
resonance couples dark photons to ordinary matter it could be possible to guarantee the coupling
energy does not depend on the values of frequencies of the 3-chord at flux tubes. This would suggest
that there is no motivation to increase the number of tRNAs for the lack of required resonance
energies.

2. Could a large number of tRNAs as mediators of RNA-AA pairing be something chosen intentionally
by MB rather than being forced by chemical limitations. Could surplus of different tRNAs be a
safer option when some tRNAs are not produced. In natural languages there is large number of
dialects and new are born all the time.

No hard-wired correspondence would exist at chemical level. MB would be to some degree creative
and able to build tRNAs from the stuff that it happens to find from the lab! Biology could be
creative already at RNA-tRNA level and this flexibility could emerge from the intelligence coded
by heff = n: the larger the number of factors of n the higher the intelligence of the system would
be.

This flexibility might also explain the homonymy at RNA-AA level and different genetic codes as a
formation of dialects.

5 About the details of the genetic code based on bio-harmony

TGD suggests several realizations of music harmonies in terms of Hamiltonian cycles representing the
notes of music scale, most naturally 12-note scale represented as vertices of the graph used. The most
plausible realization of the harmony is as icosahedral harmony [13] (see http://tinyurl.com/yad4tqwl

and http://tinyurl.com/yyjpm25r).

1. Icosahedron (see http://tinyurl.com/l5sphzz) has 12 vertices and Hamiltonian cycle as a rep-
resentation of 12-note scale would go through all vertices such that two nearest vertices along the
cycle would differ by quint (frequency scaling by factor 3/2 modulo octave equivalene). Icosahedron

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allows a large number of inequivalent Hamiltonian cycles and thus harmonies characterized by the
subgroup of icosahedral group leaving the cycle invariant. This group can be Z6, Z4, or Z2 which
acts either as reflection group or corresponds to a rotation by π.

2. The fusion of 3 icosahedral harmonies with symmetry groups Z6, Z4 and Z2 gives 20+20+20=60
3-chords and 3+1 + 5 + 10 =19 orbits of these under symmetry group and almost vertebrate genetic
code when 3-chords are identified as analogs of DNA codons and their orbits as amino-acids. One
obtains counterparts of 60 DNA codons and 3+1 + 5 + 10 =19 amino-acids so that 4 DNA codons
and 1 amino-acid are missing.

3. The problem disappears if one adds tetrahedral harmony with 4 codons as faces of tetrahedron and
1 amino-acid as the orbit of the face of tetrahedron. One obtains 64 analogs of DNA codons and
20 analogs of amino-acids. I call this harmony bio-harmony. The predicted number of DNA codons
coding for given amino-acid is the number of triangles at the orbit of given triangle and the numbers
are those for genetic code.

4. How to concretely realize the fusion of harmonies? Perhaps the simplest realization that I have
found hitherto is based on union of tetrahedron of 3 icosahedrons obtained by gluing tetrahedron to
icosahedron along its face which is triangle. The precise geometric interpretation of this realization
has been however missing and I have considered several variants. I have proposed that the model
could explain the two additional amino-acids Pyl and Sec appearing in Nature.

There is also a slight breaking of symmetries: ile 4-plet breaks into ile triplet and met singlet and
trp double breaks into stop and trp also leu 4-plet can break in leu triplet and ser singlet (see
http://tinyurl.com/puw82x8). This symmetry breaking should be understood.

5.1 Why 3 icosahedral harmonies and 1 tetrahedral harmony?

The following argument suggests a more detailed solution of these problems than proposed earlier.

1. The copies of icosahedron would differ by a rotation by multiples of 2π/3 (Z3) around axis through
the common triangular face. This face unlike the other faces remains un-affected. Also tetrahedron
remains un-affected so that it is counted only once.

If the 3 copies of the icosahedral common face are counted as separate (this is important!), one
obtains 20+20+20 faces from icosahedron. If also tetrahedral shared faces is counted as separate,
tetrahedron gives 4 faces: 64 codons altogether as required. One obtains 19 orbits from the 3
icosahedra and 1 orbit from tetrahedron: 20 orbits as counterparts of amino-acids altogether.

2. But can one really counter the 4 common faces as separate? One must do so. Could these faces be
interpreted as somehow special codons? Maybe as stop codons or start codons for the vertebrate
genetic code which also corresponds to the realization of DNA, RNA ,tRNA, and amino-acids as
dark proton triplets so that DNA sequences would correspond to dark proton sequences. Could the
shared codons be assigned with various modifications of the vertebrate code involving also exotic
amino-acids Pyl and Sec.

3. Consider first the tetrahedral face. If the common face is removed from the 4-face orbit of tetrahe-
dron, the orbit has only 3 faces and correspond to an amino-acid coded by 3 DNA codons. ile is the
only such amino-acid and the interpretation could be that one ile corresponds to the 3 tetrahedral
faces and met acting as start codon to the fourth shared face.

4. Also 3 icosahedral amino-acids corresponding to orbits containing the shared face can lose 1 codon
each. To nake this more concrete, one can look for the deviations from the vertebrate code.

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(a) There are 10 doublets if the doublet UAA, UAG acting as stop codons is counted as doublet
coding for stop regarded formally as amino-acid.

(b) The second member in the doublet UGA, UGG coding for tyr in code table could correspond
to a common face and act as a stop codon.

(c) For the modifications of genetic code UAG coding for stop can code for Pyl and UGA coding
for stop can also code for Sec. UGA can also code for trp so that there would not be any
symmetry breaking in this case. Could UAG and UGA correspond to common faces for two
icosahedra?

(d) There is also third icosahedral shared face. CUG coding for leu can also code for ser. Could
this correspond to the third exceptional codon associated with the icosahedral part of the code?

5. If the answers to the questions are affirmative, all basic deviations from the vertebrate code can be
understood. The translation of the codons associated with shared face would be unstable for some
reason.

(a) 3-chord representation is more fundamental than the chemical one. This could mean that the
chords associated with the shared faces are very near to each other so that the correspondence
between 3-chord representation and chemical representation of codons becomes unstable if
based on triple resonance.

(b) The proposal has indeed been that the 13th vertex implied by tetrahedron corresponds to a
note very near to one of the notes of 12-note scale - this note is necessary since the 12-note
scale defined by quints gives 12th note slightly more than octave under octave equivalence as
discovered already by Pythagoras.

If this picture is correct, the symmetry breaking of the genetic code would be due to the presence
of the face common to icosahedron and tetrahedron and reflect the problem discovered already
by Pythagoras. The rational number based Pythagorean scale defined by quints is special:
people with absolute pitch prefer it over the well-tempered scale involving powers of irrational
number 21/12 requiring extension of rationals.

5.2 Could stop codons correspond to dissonant 3-chords?

One can approach the situation also from the point of view of harmony - or rather, dis-harmony: could
dissonance 3-chords act as stop codons. The 3-chords of icosahedral harmonies can be classified to three
groups depending on whether the triangle representing the chord contains 0, 1, or 2 sides [13]: in in
other words, whether the chord contains 0 , 1, or 2 quints. The harmonies can be labelled by the triplet
(n0, n1, n2) telling the numbers of chords with 0, 1, and 2 quints.

1. The unique Z6 harmony necessarily present in the bio-harmony has (2, 12, 6). It has two augmented
chords (transposes of Caug = CDG]) containing two major thirds and defining the 3-chord of a
harmony assignable to triangle). This beautiful chord to which finnish tangos so often end, cannot
be regarded as dissonance.

2. The 2 Z4 harmonies have (n0, n1, n2) = (0, 16, 4) and (4, 8, 8). For the latter harmony one has
genuine dissonances since the the highest and lowest note of 3-chord are separated by major or
minor third. The chords with 0 quints labelled by script ”ex1”,”ex2”,...,”ex6” (for the notation see
[13]) are dissonances in this sense. ”ex7” and ”ex8” (CDF] and CDG]) cannot be regarded as
dissonances in this sense.

3. The 3 Z2,rot harmonies have (0, 16, 4), (2, 12, 6), and (4, 8, 8). Both 2-plets and 4-plets contain 2
dissonances.

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4. There are 3 Z2,refl harmonies with (2, 12, 6) and 1 with (4, 8, 8). These harmonies have genuine
dissonances. Interestingly, (2, 12, 6) corresponds to a doublet for which only the second member
corresponds to dissonance.

5. For tetrahedral harmony single step should correspond to 1/4:th of octave (using suitable power of
3/2 as a rational approximation) so that the notes at the vertices of tetrahedron should correspond
to CE[F] defining Cdim. This does not appear in the icosahedral code table as 0-quint chord.
Although the triangles of tetrahedron and icosahedron would be shared in some sense, the chords
cannot be same. This support the idea that ile triplet and met are coded by tetrahedral faces.

The chords containing 0 quints appearing in Z4 and Z2 harmonics can be regarded as dissonant. The
minimization of dissonance would give a fusion of the unique Z6 harmony (2, 12, 6), unique Z4 harmony
(0, 16, 4) and unique Z2,rot harmony (0, 16, 4). Bio-harmony would be unique and contain no dissonances.
Recall however that the proposal is that bio-harmonies serve as correlates for moods realized even at the
level of basic bio-molecules.

For other options one would have dissonant chords. Z2,refl harmony (2, 12, 6) has only single disso-
nant chord. Since stop codons would naturally correspond to dissonances, this observation raises some
questions.

1. Could the dissonant chord of Z2,refl harmony (2, 12, 6) correspond to the triangle shared by tetra-
hedron and icosahedron? Could this correspond to (stop,trp) pair with stop coded by dissonant
chord ”ex”7 (CDF] defining part of D7 chord). This would fix the code to contain Z6 harmony
(2, 12, 6), unique Z4 harmony (0, 16, 4) and unique Z2,refl harmony (2, 12, 6). There would be single
dissonance coding for stop in stop,trp doublet.

2. The doublet coding for stop should formally code for amino-acid. One cannot realize this doublet as
a doublet of dissonances with ”ex”n, with n ∈ {1, .., 6} for single bio-harmony. The second member
of this doublet could however correspond to the shared triangle.

This tentative picture should be of course checked. There are also cycles without any symmetries.
Could these chaotic cycles be interpreted as disharmonies.

5.3 How could the representations of genetic code as dark 3-chords and nu-
cleotide triplets relate?

One of the poorly understood aspects of the model is how the various representations of the code relate.

5.3.1 Frequency coding of nucleotides is not possible

Frequency coding of nucleotides would look natural but it is easy to see that it is in conflict with bio-
harmony.

1. The representations as dark proton triplets and dark photon triplets do not involve decomposition to
ordered triplet of letters as the ordinary chemical representation does. Dark protons are entangled
and one cannot order them and there is no obvious ordering of the frequencies of dark photons.

This is not a problem for the correspondence between dark proton triplets and dark photon triplets
and one can even imagine assignment of dark cyclotron photons with 3 parallel flux tubes acting as
wave guides. This could mediate the interaction between dark variants of basic biomolecules with
same value of heff as frequency resonance.

2. The interaction between ordinary DNA/RNA/tRNA and its dark variant should involve the trans-
formation of dark photon triplet associated with flux tube triplet emanating from dark bio-molecule
to ordinary photons (possibly bio-photons) and energy resonance would be involved. Is the energy

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resonance involved with the formation of the dark-ordinary pairs or with the sustainment of these
pairings? The example of benzene suggests sustainment.

3. The assumption that energy resonance is involved with dark-ordinary pairing indeed leads to prob-
lems. The first guess would be that ordinary photon triplet somehow carries information about
the position of nucleotide in the codon. The 4 nucleotides would correspond to 4 frequencies with
frequency scale depending on the position inside the codon. There are indeed 12 frequencies in the
12-note scale so that 3 frequency scales with 4 frequencies associated with each of them would give
64 combinations of frequencies.

Frequency coding of nucleotides however leads to a problem. The first two letters of the codon
are known to determine the amino-acid coded by it to a high degree since the third letter typically
distinguishes between 1 or 2 amino-acids only, and labels codons at the orbit of DNA codon defining
amino-acid. Therefore for DNA codons coding same amino-acid the first two frequencies should be
same. This is not the case for bio-harmony for the simple reason that the frequencies of 3-chords
along the orbit defining amino-acids are different. Only the frequency ratios defining the type of
the chord are same along the orbit.

The frequency ratios determine the correspondence so that the correspondence can be only between
entire dark and ordinary codons, and cannot be reduced to correspondence between frequencies and
letters. Holism does not reduce to reductionism.

5.3.2 Does the impossibility of frequency coding of nucleotides lead to problems with the
models of replication and transription?

This becomes a potential problem in the model for DNA replication and transcription to RNA.

1. The basic picture about bio-catalysis in TGD framework is following. U-shaped magnetic flux
tubes emanate from the reactants and can reconnect to form a pair of flux tubes connecting the
reactants. The shortening of the flux tube pair by a reduction of heff brings the reactants together
and liberates the energy needed to kick the reactants over the potential wall making the reaction
rate extremely low otherwise.

The U-shaped flux tubes or flux tube triplets would be associated with dark codons of dark DNA
accompanying DNA strand, and would be formed as the flux tube pair(s) connecting the strands split
by the reversal of reconnection. The heff associated with resulting U-shaped flux tubes associated
with replicating strands would increase requiring metabolic energy. They would get longer and
could act as tentacles scanning the environment to spot similar flux tubes assignable to nucleotides
or codons by resonance.

2. In the standard picture one assumes that nucleotides defining the letters of the codons appear as
non-correlated molecules in the environment, and that each codon is built by a stepwise process
in which letters attach to it. The letters can respond only to single frequency and cannot ”know”
which position to attach to. Thefrequency coding is not consistent with the idea that dark photon
triplet assigned with the dark codon gives rise to energy resonance with the letters one by one.

Could the triple resonance occur as single step and attach all 3 nucleotides in single step? Or
could the triple resonance be a collective frequency resonance with dark codon already attached to
the ordinary codon in the environment. Ordinary-dark pairing by energy resonance would sustain
rather than generate DNA strand since otherwise the Coulomb repulsion due to the large negative
charge of DNA does not allow stability.

3. The problem is that it is nucleotides seem to appear in the environment rather than codons. Could
the nucleotides of the environment actually form loose codons connected to dark codons by long flux
tubes with large value of heff? Could the reduction of heff bringing nucleotides together induce

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the reduction of flux tube lengths giving rise to ordinary codon? If the reduction of heff for flux
tubes occurs nucleotide-by nucleotide, one would have consistency with the standard picture. The
simplest picture is following.

Dark codons are paired with the loose variants ordinary codons. The opening of DNA double strand
leads to the splitting of the flux tube pairs connecting the ordinary codons of strands to U-shaped
flux tubes, which reconnect with U-shaped flux tubes coming dark codons paired with loose ordinary
codons. The reduction of heff d pairs nucleotides of loose codons with those of ordinary codons.

4. The pairs of dark codons and loose codons would be analogous to tRNA molecules. One can
imagine even pre-tRNA molecules with loose coupling of RNA and amino-acid so that replication and
transcription would be very similar topological processes. Also RNA transcription and translation
of RNA to amino-acids would rely on similar mechanism. The only difference would be that only
the second - active - strand would form U-shaped flux tubes connecting with dark RNA codons.

5.3.3 What about remote DNA replication

This model could also explain remote replication of DNA for which Montagnier et al have reported
evidence [4]. Also remote transcription is predicted to be possible. I have already earlier considered a
model of remote replication [12] in an article written together with Peter Gariaev who has reported this
kind phenomenon already earlier. I have discussed the findings of Montagnier et al in [22].

1. The experiment involves two vessels, call them A and B. A contains genes and B only nucleotides -
at least according to the standard picture. There is irradiation using 7 Hz frequency not far from
the lowest Schumann frequency having a nominal value of 7.8 Hz. What happens is that the replicas
of genes appear in B. It is also reported that the DNA generates em radiation possibly responsible
for the information transfer.

2. The proposed model for the ordinary DNA replication generalizes easily to describe also remote
replication. The new element would be that the U-shaped flux tubes from A would extend to B -
here 7 Hz radiation could be essential - , would be parallel to each other, and have same average
length, which is natural if they have same value of heff . Also the experimental arrangement could
favor parallel flux tubes. In B the dark codons paired with loose codons formed from ordinary
nucleotides would be present, and their U-shaped flux tubes would reconnect with those coming
from A. Remote replication could take place: here it is essential that the U-shaped flux tubes are
parallel and have very nearly the same length.

The TGD interpretation would be that the Earth’s magnetic body is involved and generates quantum
coherence in the length scale at least the size of the system studied. The reported em radiation
would naturally relate to the dark photon triplets representing the codons.

5.3.4 Is ZEO needed to understand the replication?

In TGD one must give up thinking in terms of standard ontology of bio-chemistry in which the process
is a kinetic process governed by differential equations for the populations of molecules and proceeding in
step-wise manner nucleotide by nucleotide. ZEO suggests temporal holism - at least at the level of single
dark codon, which cannot be built building brick by building brick.

1. An open question is in which time scale this temporal quantum holism holds true: in the time scale
of addition of single codon or in the time scale of replication of gene or something else? In the
following the possibility that temporal holism holds in the time scale for the pairing of dark codons.

2. In ZEO one could have state function reduction in which initial state corresponds to dark codon
plus population of nucleotides and final state to dark codon paired with the ordinary codon formed

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from 3 nucleotides in energy resonance with the codon formed from nucleotides. What matters are
only the initial and final states.

3. If ”big” state function reduction (BSFR) is in question, the final state would correspond to a su-
perposition of deterministic time evolutions leading from the outcome of the reduction to geometric
past, possibly but not necessary to a state in which nucleotides do not form codon paired with the
dark codon.

4. The process would create strong correlations between the position of nucleotides of the codon and
between the positions of codon and its dark variant and therefore a generation of entanglement.
Unitary evolutions followed by ”small” state function reductions (SSFRs) would generate a state
as a superposition of the states satisfying the criteria of the desired final state and other states and
BSFR would select the desired final state. It could be followed by BSFR returning the original
arrow of time but doing nothing for the state.

6 Appendix: Tables of basic 3-chords for the icosahedral har-
monies with symmetries

The tables below give list for the three types of 3-chords for the 11 harmonies possessing symmetries.
One must remember that the reversal of the orientation for the cycle induces the transformation C ↔ C,
F]↔ F], H ↔ C], F ↔ G, D ↔ B[, E ↔ G], A↔ D] and produces a new scale with minor type chords
mapped to major type chords and vice versa. Also one must remember that all 3-chords except those
which are simple majors or minors lack the third so that their emotional tone remains uncharacterized.
For instance, C6 does could be replaced with Cm6 and G7 with Gm7. The reader can check the chords
by direct inspection of the figures. The convention used is that vertex number one corresponds to C note.

(n0,n1,n2) 0-chords 1-chords 2-chords

(2, 12, 6) (Faug,Gaug) (Cm,Dm,Em,F]m,G]m,B[m), (C9, D9, E9, F ]9, G]9, B[9).
(F6, G6, A6, B6, C]6, D]6).

Table 1: Table gives various types of 3-chords for harmonies with Z6 rotational symmetry. Note that
half-octave shift is an exat symmetry. Note that Gaug = CEG], F aug act as bridges between the groups
related by half octave shift. The chords have been arranged so that they form orbits of Z6. ”Amino-acid
chords” correspond to preferred chords at the orbits.

(n0,n1,n2) 0-chords 1-chords 2-chords

(0, 16, 4) (D7, D6, G]7, G]6), (B[9, B9, E9, F9).
(G4+, A9−, C]4+, D]9−),
(Emaj7, Gmaj7, B[maj7, C]maj7),
(C9−, A9−, F ]9−, D]9−).

(4, 8, 8) (Cex3, Eex2, F ]ex3, B[ex2). (Dmaj7, E9−, A7, A6), (B[9, F9, C9, G9).
(G]maj7, B[9−, D]7, D]6). (E9, B9, F ]9, C]9).

Table 2: Table gives various types of 3-chords for the two harmonies with Z4 = Zrot
2 × Zrefl

2 symmetry.
4-plets represent the orbits. First cycle has no harmonic loners. Second cycle gives rise to bio-harmony
(4, 8, 8) for which 0-quint chords are dissonant.

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Pitkänen, M., An Overall View about Models of Genetic Code & Bio-harmony

(n0,n1,n2) 0-chords 1-chords 2-chords

(0, 16, 4) (Em,B[m), (Cm,F]m), (D9, G]9),
(G6, C]6), (A6, D]6), (E9, B[9).
(D4+, G]4+), (B4+, F4+),
(Cmaj7, F ]maj7), (G6−, C]6−).

(2, 12, 6) (Aex4, D]ex2). (Am,D]m), (G9−, C]9−), (C9, F ]9),
(C4, F ]4), (E4+, B[4+), (A9, D]9),
(Dmaj7, G]maj7), (D9, G]9).
(Bmaj7, Fmaj7).

(4, 8, 8) (Aex2, Hex8, D]ex2, F ex8). (D7, G]7), (Amaj7, D]maj7), (G9, C]9), (A9, D]9),
(A4+, D]4+), (E7, B[7). (B9, F9), (E9, B[9).

Table 3: Table gives various types of 3-chords for harmonies with Z2 rotation symmetry acting as half-
octave shift. The doublets represent 2-chord orbits.

(n0,n1,n2) 0-chords 1-chords 2-chords

(2, 12, 6) (F]ex3, Hex4), (Am,D]), (A6, D]7), (C9, F9), (B9, F ]9),
(D7, B[6), (G6−, Fmaj7), (E9−, C]9).
(D4+, B[9−), (E9, G]4+),

(2, 12, 6) (Dex4, Hex4). (F, Fm), (C6−, B[maj7), (C9, D]9),
(D7, G]6), (Gmaj7, D]6−). (D]9, C]9),
(C]4−, A4+), (E4+, F ]6). (E9, B9).

(4, 8, 8) (Fex1, D]ex3, G]ex1, Aex2). (E7, E6), (Amaj7, B9−), (D9, B9), (C9, C]9),
(G,C]m), (D7, F ]6). (F9, G]9), (D]9, B[9).

(2, 12, 6) (Hex3, Eex7). (D7, G]6), (G,D]m), (C9, D]9),
(F, Fm), (C6−, B[maj7), (D9, C]9),
(A9−, C]4+), (E7, F ]6). (E9, B9).

(2, 12, 6) (F]ex2, F ex3). (F,B[m), (C7, G]6), (B[9, D]9),
(Amaj7, B9−), (E6, E7), (C9, C]9),
(G,C]m), (D7, B6). (D9, H9).

Table 4: Table gives various types of 3-chords for harmonies with single reflection symmetry.

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	Introduction
	3 basic realizations of the genetic code
	3 models of bioharmony
	About the geometric interpretation of icosahedral and other symmetries
	Mistracks

	Interactions between various levels
	The independence of the interaction energy on frequency
	The independence of cyclotron energy on frequency and Nottale hypothesis

	Homonymy of the genetic code
	Variations of the genetic code
	Wobble base pairing

	TGD view about homonymies
	Homonymies for DRNA-3-chord correspondence
	The map DRNA-DtRNA by 3-chords
	Homonymies for RNA-AA correspondence
	Homonymies for RNA-tRNA correspondence

	About the details of the genetic code based on bio-harmony
	Why 3 icosahedral harmonies and 1 tetrahedral harmony?
	Could stop codons correspond to dissonant 3-chords?
	How could the representations of genetic code as dark 3-chords and nucleotide triplets relate?
	Frequency coding of nucleotides is not possible
	Does the impossibility of frequency coding of nucleotides lead to problems with the models of replication and transription?
	What about remote DNA replication
	Is ZEO needed to understand the replication?


	Appendix: Tables of basic 3-chords for the icosahedral harmonies with symmetries

