MattiPitkanen_V4N3.pdf


DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 161-198 161

Pitkänen, M. Music, Biology and Natural Geometry

Article

Music, Biology, and Natural Geometry

Matti Pitkänen 1

Abstract

In an 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 observa-

tions. 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 isosahedral 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 12st 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.

1 Introduction

For some time ago 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 [13],[9, 10].
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 amino-acids and the number of basic
chords for the proposed notion of harmony. This led to a group theoretical model of genetic code and
replacement of icosahedron with tetra-icosahedron 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 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 bio-harmony
was too restricted: all isosahedral 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. This working hypothesis had to be given up. The fuel of
the minirevolution was the observation the symmetries of the Hamiltonian cycles (Z6, Z4, Z2) are nothing
but the icosahedral symmetries needed to predict the basic numbers of the genetic code and its extension
to include also 12st and 22nd amino-acids. Thus icosahedral Hamiltonian cycles predict genetic code
without further assumptions.

One also ends up with a proposal for what harmony is leading to non-trivial predictions both at DNA
and amino-acid level.

1. 3-adicity and also 2-adicity are essential concepts allowing to understand the basic facts about
harmony. The notion of harmony at the level of chords is suggested to reduce to the notion of
closeness in the 3-adic metric using as distance the distance between notes measures as the minimal

1Correspondence: Matti Pitkänen http://tgdtheory.com/. Address: Köydenpunojankatu 2 D 11, 10940, Hanko, Finland.

Email: matpitka@luukku.com.

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DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 161-198 162

Pitkänen, M. Music, Biology and Natural Geometry

number of quints allowing to connect them along the Hamilton’s cycle. In ideal case, harmonic
progressions correspond to paths connecting vertex or edge neighbors of the triangular faces of
icosahedron.

2. An extension of icosahedral harmony to tetra-icosahedral harmony was proposed as an extension of
harmony allowing to solve some issues of icosahedral harmony relying on quint identified as rational
frequency scaling by factor 3/2.

This extension is kept also now. One must however give up the idea about correlation between
polarity characteristics of proteins and properties of Hamilton cycles. One must allow all 11 icosa-
hedral harmonies with symmetries as bio-harmonies: their symmetry groups Z6, Z4, Z2 can be
identified as the symmetry groups defined the decomposition of 60 DNA codons to 20+20+20
codons in the model of the genetic code. The 4 remaining DNAs and amino-acids can be assigned
to both tetra-icosahedron and tetrahedron and icosahedron regarded as defining separate genetic
codes. This explains why stopping codons can code for the 21st and 22nd amino-acid under some
circumstances.

Tetrahedral code is second member in the hierarchy of genetic codes [?] inspired by the notion of
Combinatorial Hierarchy M(n + 1) = MM(n) = 2M(n) − 1 giving the numbers 2, 4, 7, 64, 2126, ...
as numbers of DNA codons. The fourth member would correspond to what I called “memetic
code”allowing representation of codons as sequences of 21 DNAs. It is not known whether the
Combinatorial Hierarchy of Mersenne primes continues as Hilbert conjectured.

3. The notion of bio-harmony is partially characterized by the triplet n = (n0, n1, n2), characteriz-
ing the numbers of 0-, 1-, and 2-quint chords which in turn correspond to DNA codons in con-
sistency with the observation that codons indeed correspond to triplets of nucleotides. n-quint
chord corresponds to a triangle (face of icosahedron) containing n edges of the Hamiltonian. Par-
ticular bio-harmony requires a selection of a specific Hamiltonian cycle from each class of cycles
(1 Z6 symmetric cycle having n = (2, 12, 6), 2 Z4 symmetric cycles n ∈ {(0, 16, 4), (4, 8, 8)},
3 Z2 = Zrot

2 with n ∈ {(0, 16, 4), 1(2, 12, 6), (4, 8, 8))} and 5 Z2 = Zrefl
2 symmetric cycles with

(n ∈ {(2, 12, 6), (4, 8, 8)}. Note that the are only three different triplets n.

4. The original idea was that the rules of bio-harmony could be applied to amino-acid sequences
interpreted as sequences of basic 3-chords. DNA would have represented the notes of the music.
For given choice of harmony as Hamiltonian cycle meaning selection of of 4, 5 or 10 amino-acids
coded by the 20 DNAs in question, the hypothesis had to be modified by replacing amino-acid
sequences with DNA sequences.

These DNA sequences however define also amino-acid sequences identifiable as specific triangle at
the orbit of Zn defining the DNA codons assigned to that amino-acid (there is a singular fiber space
structure). Together the three 20-plets of DNAs define an amino-acid harmony with (4+5+10 =19
chords with tetrahedral extension defining a harmony with 22 chords/amino-acids). Hence both
DNA sequences and amino-acid sequences define “bio-music”.

5. The assumption that harmonic transitions between chords (DNA codons) minimize the distance be-
tween chords defined by quint-metric leads to highly non-trivial and testable predictions about both
DNA sequences and amino-acid sequences. Negentropy Maximization Principle (NMP) [8] suggests
that evolution favors the generation of harmony which should thus increase in the proposed sense for
DNA sequences defining particular genes or other functional units of DNA during evolution. Large
quint-distances between subsequent codons/chords would tend to polished out under evolutionary
pressures.

6. Could icosahedron, tetrahedron, and tetra-icosahedron have direct physical counterparts in living
matter? For instance, water molecules form icosahedral clusters and the chlathrates associated with

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Pitkänen, M. Music, Biology and Natural Geometry

synaptic contacts have icosahedral symmetries. Tetra-icosahedron has 13 vertices with the added
vertex representing one note- say E- in C-key as note with slightly different frequency to resolve the
basic problem of rational number based 12-note scale (12 quints give slightly more that 7 octaves).
Intriguingly, microtubules consist of basic structures consisting of 13 tubulins with 2 states defining
bit: could these bit sequences define representation for the 3-chords and thus representation of
sequence of DNA codons and realization of genetic code.

7. Music is language of emotions and peptides are molecules of emotion as Candace Pert [J3] expressed
it. Could bio-harmonies serve as direct correlates for emotions? What is bio-music? A natural TGD
inspired guess is that sounds can be replaced with heff = n×h dark photons with low frequencies and
having energies in the range of bio-photons (visible and UV range maximally effective biologically)
as proposed on basis of some physical facts and theoretical ideas [9]. The frequency spectrum of dark
cyclotron photons along magnetic flux tubes would define bio-music as “music of dark light”and
bio-harmonies would correlate with emotions and moods.

If one can find various icosahedral Hamilton’s cycles one can immediately deduce corresponding har-
monies. This would require computer program and a considerable amount of analysis. My luck was
that the all this has been done. One can find material about icosahedral Hamilton’s cycles in web, in
particular the list of all 1024 Hamilton’s cycles with one edge fixed[1, 2] (this has no relevance since only
shape matters). If one identifies cycles with opposite internal orientations, there are only 512 cycles. If
the cycle is identified as a representation of quint cycle giving representation of 12 note scale, one cannot
make this identification since quint is mapped to fourth when orientation is reversed. The earlier article
about icosahedral Hamiltonian cycles as representations of different notions of harmony is helpful [13].

The tables listing the 20 3-chords of associated with a given Hamilton’s cycle make it possible for
anyone with needed computer facilities and music generator to test whether the proposed rules produce
aesthetically appealing harmonies for the icosahedral Hamiltonian cycles. Biologist with access to DNA
sequences could experiment with DNA codons to see whether their are harmonious in the sense that the
distance between subsequent chords assignable to DNA codons tend to be small in quint metric. Note
that DNA decomposes to pieces corresponding to different Hamiltonian cycles (harmonies) so that the
comparison is not quite straightforward.

2 What could be the basic principles of harmony?

It indeed seems that the idea about definition of notion of harmony in terms of Hamiltonian cycles makes
sense.

1. Chords (major and minor) are labeled by their basic tones and comes either as major or minor.
Harmony in classical sense requires that the transitions from key to another take place by a small
number of quints and that the piece does not wander too far from the major key, say C.

If quint corresponds to a step along the edge of the cycle in the direction of its orientation, the
notion of tonal closeness corresponds to the closeness in the metric of icosahedron. For instance
C,F, and G are commonly used keys in same piece and correspond to 3 subsequent points along
Hamiltonian cycle. Note that the number of ♯s of the key increases by one unit in standard direction
and the number of ♭s by one unit in opposite direction.

2. It turns out that major and minor 3-chords and are mapped to each other in the orientation reversal
for icosahedral path so that basic moods “happy”and “sad”in music have this orientation as a
geometric correlate. The effect of orientation reversal does not actually depend on the icosahedral
representation but is implied by quint cycle representation alone. C and half-octave F♯ defining
the tritonus interval are the fixed points of the orientation reversal. Orientation reversal induces
pairings (C ↔ C, F♯ ↔ F♯, G ↔ F , D ↔ B♭, A ↔ D♯, E ↔ G♯, H ↔ C♯. Quints of cycle

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Pitkänen, M. Music, Biology and Natural Geometry

correspond to the fourths of oppositely oriented cycle so that majors and minors are mapped to
each other and one can say that the moods “happy”and “sad”have geometric correlates in the sense
that majors and minors are transformed to each other in the reversal of orientation of the cycle.

The notion of harmony can be characterized in terms of numbers of basic 3-chords identified as faces
of the icosahedron and their neighborhood relationship telling when corresponding chords are near to
each other or vertex or face neighbours. The wall neighbours assignable to given edge are expected to be
in very special relationship harmonically since they possess a common quint.

The basic classification is according to the number n = 0, 1, 2 of edges of cycle contained by them and
the triplet n = (n0, n1, n2) for the numbers of faces of various kinds gives the first rough classification.
2-quint chords have common edge and thus two common notes with two 1-quint chords and are therefore
natural intermediates in transitions between them. 0-quint chords are tonal loners having no edge neigh-
bours turns out that they involve dissonances since they consists of three notes spanning length of 1 or
3/2 steps (say EFG, EF♯G or D♯EF ). Maximally symmetric harmony is an exception: 0-quint chords
correspond to augmented chords of type CEG♯ with two major thirds.

The numbers of three different kinds of face neighbor pairs for the 12 edges of the path serve as an
additional classification criterion in terms of the p = (p1,1, p1,2, p2,2) for the numbers pi,j of different kind
of edges. Note that the neighbor faces of an edge correspond to 3-chords, which possess two common
notes and are in this sense close to each other. These numbers characterize the most natural transitions
between the chords of the harmony. A further criterion is the distribution of these neighbor pairs along
the cycle.

2.1 Icosahedral harmonies

1. Chords (major and minor) are labeled by their basic tones and comes either as major or minor.
Harmony in classical sense requires that the transitions from key to another take place by a small
number of quints and that the piece does not wander too far from the major key, say C.

If quint corresponds to a step along the edge of the cycle in the direction of its orientation, the
notion of tonal closeness corresponds to the closeness in the metric of icosahedron. For instance
C,F, and G are commonly used keys in same piece and correspond to 3 subsequent points along
Hamiltonian cycle. Note that the number of ♯s of the key increases by one unit in standard direction
and the number of ♭s by one unit in opposite direction.

2. It turns out that major and minor 3-chords and are mapped to each other in the orientation reversal
for icosahedral path so that basic moods “happy”and “sad”in music have this orientation as a
geometric correlate. The effect of orientation reversal does not actually depend on the icosahedral
representation but is implied by quint cycle representation alone. C and half-octave F♯ defining
the tritonus interval are the fixed points of the orientation reversal. Orientation reversal induces
pairings (C ↔ C, F♯ ↔ F♯, G ↔ F , D ↔ B♭, A ↔ D♯, E ↔ G♯, H ↔ C♯. Quints of cycle
correspond to the fourths of oppositely oriented cycle so that majors and minors are mapped to
each other and one can say that the moods “happy”and “sad”have geometric correlates in the sense
that majors and minors are transformed to each other in the reversal of orientation of the cycle.

The notion of harmony can be characterized in terms of numbers of basic 3-chords identified as faces
of the icosahedron and their neighborhood relationship telling when corresponding chords are near to
each other or vertex or face neighbours. The wall neighbours assignable to given edge are expected to be
in very special relationship harmonically since they possess a common quint.

The basic classification is according to the number n = 0, 1, 2 of edges of cycle contained by them and
the triplet n = (n0, n1, n2) for the numbers of faces of various kinds gives the first rough classification.
2-quint chords have common edge and thus two common notes with two 1-quint chords and are therefore
natural intermediates in transitions between them. 0-quint chords are tonal loners having no edge neigh-
bours turns out that they involve dissonances since they consists of three notes spanning length of 1 or

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DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 161-198 165

Pitkänen, M. Music, Biology and Natural Geometry

3/2 steps (say EFG, EF♯G or D♯EF ). Maximally symmetric harmony is an exception: 0-quint chords
correspond to augmented chords of type CEG♯ with two major thirds.

The numbers of three different kinds of face neighbor pairs for the 12 edges of the path serve as an
additional classification criterion in terms of the p = (p1,1, p1,2, p2,2) for the numbers pi,j of different kind
of edges. Note that the neighbor faces of an edge correspond to 3-chords, which possess two common
notes and are in this sense close to each other. These numbers characterize the most natural transitions
between the chords of the harmony. A further criterion is the distribution of these neighbor pairs along
the cycle.

2.2 Why quints are near to each other harmonically?

The naive expectation would be that frequencies near to each other (using half-note as unit) are close to
each other. This is not true. Their simultaneous presence is experienced as dissonance. This probably
has a neurophysiological correlate: in ear the hair cell groups detecting notes which are near to each other
in frequency space are overlapping. This explanation does not however tell why the conscious experience
is dissonance.

The distance measure for notes could be formulated in terms of distance defined as the number of
quints connecting them. For quint the distance would be minimal. This measure applies also to chords
and allows to understand the basic rule of classical harmony stating that harmonic transitions take place
the chords related by quint shift of the basic note (adding either one ♯ or one ♭ to the scale). Also the
key changes can be understood using the same rule: consider the changes C→ G and C→F as examples.
Note that in this case the chords have common note.

One could of course question the assumption that it is possible to choose the shortest route. The
notes obtained by quint scaling are not quite same in the two directions and means that ♯ is the inverse of
♭ in well tempered scale only. Could it be that people with absolute ear are able to distinguish between
the two slightly differing scales and experience notes of quint C-G as harmonically close when 1 quint
connects them but as harmonically distant 11 quints in opposite direction connects them?

If cognition is p-adic, one can ask whether the notion of harmony can be formulated in terms of p-adic
distance concept.

1. By octave equivalence the scaling by power of two means nothing so that the scalings by 3/2 are
equivalent with scalings by 3 and the distance defined by 3-adic norm having values 3k, where k is
the number of quints makes sense. The distance defined as quints could be identified the absolute
value of k along the quint cycle in the direction in which the distance is shorter. If so, the maximal
distance is 6 units.

2. 3-adic measure of distance seems to be rather realistic. Quint corresponds to 1 unit distance. Half
step corresponds to a distance of 5 units and 6 units defines the largest distance and corresponds to
the tritonus interval which was forbidden by catholic church. Fourth (C-F) corresponds to 1- step
in opposite direction and 11 steps in standard direction.

3. There is also a problem. Second (C-D) corresponds to 3 quints but third (C-E) corresponds to 4
quints and small third to 3 quints in opposite direction. Major third would thus correspond to a
longer harmonic distance than second. This is a genuine problem, whose solution might be provided
by the extension of icosahedral scale to icosatetrahedral one bringing in one additional note which
is very near to one of the icosahedral notes and is major or minor third of icosahedral note.

4. Could one use the number of icosahedral edges as distance between notes but not as a minimal
distance along the Hamiltonian cycle but along a minimal edge path along icosahedron? The
icosahedral measure of distance would be analogous to a distance between points of object along
shortest route in space that it inhabits and depends on harmony characterized by the shape of
icosahedral cycle. C and E (and also C and F♯!) could be close to each other in some harmony

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Pitkänen, M. Music, Biology and Natural Geometry

and distant from each other in some other harmony. Icosahedral geometry would become an active
determinant of the harmony.

To sum up, music seems to have both 2-adic (octave equivalence) and 3-adic (12-note scale by quint
scalings) characters. The principle of tonal unity for classical music stating that modulations of key
should not lead too many quints away from the basic chord would have 3-adic interpretation.

2.3 What could be the rules for building a harmony?

What guarantees good harmony when one has fixed the key/harmony/representation of particular Hamil-
ton cycle?

1. One should pose conditions on the allowed transitions between chords. Are there principles would
imply harmonic smoothness in geometric sense? Could the transitions occur only between chords
with a common note? Or can one require a common pair of notes? Or can one require even a
common quint. If so, 0-quint chords would become tonal hermits and could not be used at all. In
practice their dissonant character has eliminated them in popular music and much of classical music
too.

The standard quint and fourth transitions (say C to G and C to F ) are basic examples in which
there is only one common note between chords, and it seems that one cannot require more than this
in the general case. Playing with the chords of bio-harmony however suggests that smooth bossa
nova/jazz emotionally ambivalent mood is created if common pair of notes or even quint connects
the neighboring chords. The rule is that only transitions between chords with same basic note are
allowed. Obviously this is too stringent a condition.

2. Could 2-quint chords act as bridges between two 1-quint chords? For instance, for the maximally
symmetric harmony consisting of disjoint groups of chords related by half-octave scaling the aug-
mented chords (F aug = FAC♯ and Gaug mapped to each other both by half-octave scaling and
reversal of orientation could serve as mediating bridges.

3. Could harmonic transitions take place only between neighboring faces of icosahedron (see http:

//en.wikipedia.org/wiki/Icosahedron) or should it only tend to minimize the quint distance
between subsequent chords (this distance vanishes if they have a common note)? For the 0-quint
distance harmony, the harmonic movement could be seen as a path in dodecahedron which is dual
of icosahedron. In the most general case the transition can take place to both wall and vertex
neighbors, whose total number is 3+3=6. In this geometric picture harmony and melody could be
seen as duals of each other.

Dodecahedron is dual of icosahedron and one can ask whether the harmonic motion could correspond
to a path at dodecahedron. The vertex of dodecaehdron is pentagon and has 3 neighbours (see
http://en.wikipedia.org/wiki/Dodecahedron). The above argument gives 3 + 3 > 3 neighbors
for the triangle of icosahedron. Are the wall neighbors of icosahedral triangle mapped to nearest
neighbor vertices? If so then transitions between vertex neighbor triangles should correspond to
longer steps at dodecahedron. By the duality triangles of icosahedron correspond to three pentagons
associated with the vertex of dodecahedron. The rule that comes in mind is that steps can occur
between vertices for which the 3-pentagons have one or 2 common pentagons.

Note that if the dodecahedral path is Hamiltonian cycle, it is unique apart from isometries of
dodecahedron and would define a unique chord progression. One can - and of course must - allow
self-intersecting harmonic paths. The condition that there exists a basic chord from which everything
begins and to which everything ends implies that closed but in general self-intersecting path is in
question.

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Pitkänen, M. Music, Biology and Natural Geometry

4. An interesting test for the idea would a computerized generation of random chord sequences satisfy-
ing at least one common vertex rule and finding whether they are aesthetically appealing. Incidence
matrix (see Appendix) for the icosahedral (and tetra-icosahedral) triangles wholes element tells how
many common vertices two chords have have allows computational construction of the allowed chord
sequences as random sequences.

5. For most harmonies 0-quint chords involve dissonances induced by three nearby notes (such as
CC♯D) and spanning large number of quints (maximally symmetric harmony has 2 0-quint chords,
which do not have dissonances and second harmony with 2 reflection symmetries has no 0-quint
chords). Also maj7

−
, sus4+, and 6

−
1-quint chords have half-note dissonances. Dissonances as

such are however not un-sesthetical. For instance, Bach used them to create a deeply melacholic
feeling.

2.4 More general notion of harmony

The notion of harmony discussed in previous section is rather conservative and certainly too stringent.

1. 0-quint rule is too restrictive already in chord based music. For instance, the downwards progression
Am,G,F,E appearing in Spanish music and music forms like Passacaglia would have chords with
1-quint distance. Hence one must consider also a weaker notion of harmonic chord progression
according to which this distance is minimized and below some maximum value kmax. One quint
would define the smallest non-vanishing maximal distance. One can define incidence matrices for
chords with n-quint distance. The incidence matrices with different values of kmax have disjoint
sets of non-vanishing elements and the total incidence matrix is their sum.

2. Even this is not enough. The direction of step matters for scales (major-minor difference) and it
seems to matter also for chord harmonies. The inverse E,F,G,Am of the above mentioned progres-
sion does not sound harmonic in the same Am key. The impression of achieving the goal/ending
down to something dictated by fate is lost.

Instead of EFGA one often has EF♯G♯A as a melodic progression and with E,B7, E7, Am as a
chord progression having only 0-quint steps. The rule seems to be that 1-quint steps are possible only
downwards in minor harmony, whereas upwards steps are 0-quint steps. Climbing slowly upwards
by 0-quint steps and falling down by 1-quint steps! Could this “gravitational analogy”serve as a
metaphor?

Also the number of n-quint steps between chords matters. The larger this number, the closer the
chords are. Two 0-quint steps means that chords have two common notes, 1 0-quint stet that they
have single common note. The two 1-quint steps for downwards step Am−G are between 3rd and
1st (C → G) and 5th and 3rd (E → H). For upwards 0-quint steps E − H7 1-quint steps are
between 5th and 5th (H → F♯) and 1st and 1st (E → H). For H7 → E the reversals of these steps
occur. For E7 → Am one has 3 1-quint steps: (the reversals 1-quint steps E → A and H → E steps
and 1 quint step D → A. The laste step seems to be the smallest one in a well-defined sense.

For G-F step the number of 1-quint steps is one (C → C): same is true for F-E step (A and E).

Using geometry language, for chords connected by 1-quint step(s) the mutual orientation of corre-
sponding triangles with shape defined by the intervals involved matters since the number of 1-quint
steps depends on the orientation.

The notion of chord harmony does not apply as such to polyphonic music with several simultaneous
melodies unless on can say that it involves definite chord sequence. One could try to apply the concept of
harmony for melody also in this case. The challenge is to guess what harmony for melodies could mean.

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Pitkänen, M. Music, Biology and Natural Geometry

1. A conjecture inspired by the genetic code is that the codons defining the allowed melody notes
associated with a given chord are in one-one correspondence with the triangles at the orbit of the
triangle associated with the chord under the group Z6, Z4, or Z2 characterizing the chord as a
counterpart of amino-acid. In table 4.3 the Z6 orbits are represented as groups of 6 similar chords
(2 for 1-quint chords and 1 for 2-quint chords). In table 4.3 for Z4 chords the groups consist of 4
similar chords and in the tables 4.3 and4.3 for Z2 harmony the chord groups consist of 2 similar
chords.

2. The first guess is that the union of the notes of these chords could define the chords, whose notes
are compatible with chord in the time scale shorter than the duration of the chord. Note that same
triangle can appear at orbits of several chords since the orbits of each group span entire icosahedron.

If the note lasts for a duration of several chords, the notes must be consistent with all the chords
involved. The rule would explain why fast chromatic sequences (in the scale of chord duration)
sound harmonic but slow chromatic sequences do not.

For melodies inAm key EFGA is rare and does sound harmonic being often replaced with E,F♯,G♯,A.
As far as intervals are considered, this is the inversion D♯, F,G,G♯ of AGFE shifted upwards by
5 quints. Could one regard progressions (say Am,G,F,E) breaking the strongest rule for chord
harmony as polyphonic progressions satisfying the rules for polyphonic progressions.

To conclude whether the DNA inspired notion of harmonic is realistic, one should understand how
the sub-groups Zn, n = 6, 4, 2 of the isometries of the icosahedron and defining the genetic code act on
the Hamiltonian cycles.

1. The simplest guess is that these groups are represented as subgroups of Z12 (also a subgroup of
icosahedral group) representing quint cycle. Zn generator would shift the basic note of the chord
by 12/n - that is 2, 3, 6 quints.

2. Zn maps chords of same type to chords of same type only if it is a rotational symmetry of the
harmony. For instance, the action of Z6 (see Fig. 1) on icosahedron allows doublet orbit consisting
of Xaug type chords, since Z3 maps 2 0-quint triangles in the middle of the figure to themselves and
reflection group Z2 permutes them. 6-element orbits consist of either minor or major chords. More
generally, the inspection of the cycles shows that the cyclic orbits of triangle under Zn correspond
to the orbits of corresponding subgroups of icosahedral group.

3. Z2refl maps the shape of the chord to its mirror images and so that the character of the chord
can vary along Z4 orbits. The rules are (M ↔ m),(6 ↔ 7)). For other chords the character is
unaffected.

4. Any subgroup of icosahedral isometry group A5 × Zrefl
2 having 120 elements must map chords to

chords (faces to faces). In particular any Zn) even if it is not a symmetry of a particular harmony.
The character of the chord is not preserved and the number of quints can change. Whether these
maps have interpretation in terms of music remains unclear.

These considerations forced me to finally realize that the 3 groups Z6, Z4,and Z2 that I had assigned
to 20+20+20 DNA codons in the model of the genetic code are nothing but Z6-, Z4-,and Z2-symmetric
Hamilton cycles! The numbers of amino-acids associated with various types would be 3+1=4,5, and
10 (with empty amino-acid included). Tetrahedral extension based on gluing of tetrahedron at triangle

corresponding to X6 type chord possessed by all Zrefl
2 type harmonies would give 3 additional real

amino-acids giving altogether real 22 amino-acids as required. This has implications.

1. All 11 Hamilton cycles are realized separately as DNA level harmonies. Amino-acid level harmonies
would correspond to selection of three Hamiltonian cycles, one for each Zn.

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2. To get something one must give something away. Now one must give up the idea that (4, 8, 8) is
special via the corresponding of n-quint property with polarity properties. This is a pity, since just
taking this correspondence seriously led to the extension of the icosahedral cycles to tetra-icosahedral
ones. Fortunately, the extension itself makes sense for all Hamiltonian cycles.

To understand the action of symmetries one must look how the groups Zn act on C major chord.

1. Z2 would induce half-octave shift and map C = (C,E,G) to F♯m = F♯,B♭,D♯). The assignment
of F♯ -tritonus - with C note sounds strange in the ears of harmonic conservatives.

2. Z4 would map C = (C,E,G) to A = (A,C♯,E), F♯ = (F♯,B♭, C♯) and D♯ = D♯ = (D♯,G,B♭).
These would span 8 notes since E,G,B♭, C♯, appear twice. Note that C,E,G,A are the notes
assignable to the tetrahedron in the extension of the scale and pentatonic scale corresponds to
C,D,E,G,A. Z4 orbit does not contain the notes DFG♯H but the orbit of G chord does so. The
orbit of C chord plus G7 chord alone define the notes of C major key.

3. Z6 would map C and E to the same “impressionistic”6-note scale consisting of 6 whole notes.
Together with the Z6 image of G one obtains all 12 notes of the scale.

3 Harmony and biology

3.1 Could harmonic principles be realized in biology?

The basic idea behind icosahedral harmony is the connection with biology suggested by the fact that the
number of icosahedral basic chords is 20 which is also the number of amino-acids. Actually there are
two additional amino-acids and one ends up to an extension of genetic code by attaching to icosahedron
a tetrahedron and thus adding one vertex more. The number of DNA codons increases from 60 for
icosahedral code to 64 for the real code. The triangle along which icosahedral and tetrahedral amino-
acids are attached together corresponds to punct coded by stopping codons.

Could the application of harmonic principles to biology make sense? The triangles of the icosa-
tetrahedron correspond to amino-acids or DNA codons for the amino-acids coded by 20 codons in question.

1. The strictest rule stating that there must be common edge of Hamiltonian cycle between the amino-
acids/DNAs cannot be satisfied since 0-quint amino-acids/DNA codons would be total loners and
effectively eliminated from biology.

2. The weaker “common edge or vertex” rule could however make sense. A given codon in the
group of 20 codons/amino-acid could be followed only by 3+3 different nearest neighbor similar
codons/amino-acids. If the first amino-acid is fixed there would be only 6N N-amino-acid sequences
instead of 20N sequences. This kind of symmetry would have been probably observed if exact but
one can ask whether harmonic pairs could more probable than completely random pairs.

3. A more plausible formulation is obtained by restricting the rule to the level of DNA sequences and
generalizing it so that it applies also to transitions between harmonies with different symmetries so
that a transition between corresponding amino-acids is induces.

4. An even weaker formulations states that the transitions occur with highest probabilities between
codons/amino-acids having shortest quint distance.

A natural conjecture is that evolution favors the generation of harmony even in the very concrete sense
that proteins defined by harmonious chord sequences for bio-harmony are emerge as what Darwinist would
call the fittest ones.

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3.1.1 Icosahedral water clusters made from tetrahedra

The obvious questions concern the concrete realization of the icosahedron - or more generally icosahedral
symmetries. One should also understood what the attachment of tetrahedron to icosahedron means
(note that tetra-icosahedron is not the the same thing as icosi-tetrahedron, which is Archimedean (not
Platonic) solid (http://en.wikipedia.org/wiki/Pentagonal_icositetrahedron)). What comes in
mind is attachment of an information molecule to the receptor of cell membrane.

Water molecules form icosahedral structures and - what is amazing to me - Plato regarded icosahedron
as a symbol of water (http://www.interferencetheory.com/Blog/files/4a3378c13bcad793a52213a325db7db0-30.
html)! The page “Water structure and science” of Martin Chaplin gives illustrations about the rather
complex icosahedral structures. Icosahedral structures of size 3 nm can be formed from 20 14-molecule
tetrahedral water molecule clusters containing 280 water molecules altogether. They can also consists of
cyclic pentamers and tricyclo-decamers and also from bi-cyclo-octomers. The 20 tetrahedrons correspond
to the faces of the icosahedron and tetra-icosahedron would be formed as tetrahedron is glued to the the
icosahedron along one of the faces.

The bioharmonies could manifest themselves already in the structure of water molecules. Second -
more plausible - option is that they differ only at the level of the magnetic body of the biomolecule. Bio-
harmony suggests that 3 radial magnetic flux tubes or flux tube pairs emerge from each water tetrahedron.
Hamilton’s cycle could be realized as a flux tube connecting the vertices of the icosahedron and assigning
the quint cycle to the cyclotron frequencies (magnetic field strengths).

This scenario raises several questions related to the pairings between ordinary DNA/amino-acids, their
icosahedral representations, and their representations as dark proton sequences.

Suppose that one takes seriously the idea that genetic code is represented as dark proton sequences
with the states of dark protons formed from 3 quarks representing DNA and RNA codons, amino-acids,
and even tRNA.

1. How dark proton sequences are realized? Could one regard them as icosahedral bound states of
20 dark protons? Or with a Hamiltonian cycle consisting of penta-quarks and representing dark
nuclear string? Could the icosahedral representation as dark nucleus consisting of 20 dark protons
and dodecahedral representation as dark nucleus consisting of 12 dark 5-proton states be dual
manners to interpret the state or are they different states related duality. Equivalence of the two
pictures would require that dark protons are color excited and in an entangled state.

2. Could dark proton sequences correspond to sequences of icosahedrons connected by flux tubes
connecting the dark protons assignable to the dark proton states assignable to the faces of the
icosahedrons? These dark nuclei would be definitely different from those possibly associated with
the Hamiltonian cycle.

3. What about the tetrahedral part of the genetic code in relation to dark protons sequences? What
dark proton states could tetrahedral codons and amino-acids correspond? Are they associated with
water tetrahedrons representing the faces of the water icosahedron? Note the amusing numerological
co-incidence that the vertices of tetrahedron have 3 quarks associated with them and those of
icosahedron 5 and that the quint for icosahedral edge is replaced with third for tetrahedral edge.

4. Could the chords correspond to triplets of cyclotron frequencies for quarks associated with the three
flux tubes emanating from the each face of the icosahedron? Could the breaking of the rotational
symmetry from SO(3) to SO(2) - now actually Z3 ⊂ SO(2) - assumed to occur for dark proton
states correspond to the reduction forced by the triangular geometry?

5. How DNA -amino-acid correspondence is represented at the level of dark DNA? The correspondence
should be realized in terms of magnetic flux tube triplets connecting dark DNA and dark amino-
acid and resonance condition would be essential. When the chords at the orbits of Zn are of same
type, different DNAs correspond to the same chord but with different key. When Zrefl

2 is involved,

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the two chords at the orbit are not of same type (note the analogy with left and right-handed
biomolecules). The only manner to circumvent the problem is to assume that the chord associated
with amino-acids magnetic body is that of DNA. Information is not actually lost in translation, it
is only transformed to different kind of information perhaps representing correlates of emotions.

6. Could the non-representability of one of the Z6 codons as amino-acid have an analog?

The fiber space having icosahedron as a base and 3 copies of icosahedron assigned with 3 regions of
icosahedron corresponding to Zn, n = 6, 4, 2, defines a formal geometric representation of genetic code.
Could this space represented in terms of water icosahedra?

1. Perhaps one should first try to identify the function of water icosahedrons. The first guess is that
they serve as local bridges between dark DNA/amino-acid sequences and ordinary DNA/amino-
acid sequences. This would suggest that dark proton of dark DNA forms a flux tube connection
with the face of water icosahedron dictated by the state of the dark proton : this would take place
by flux tube reconnection and cyclotron resonance. Water icosahedron in turn couples with the
DNA/amino-acid like DNA conjugate codon with codon so that kind of double helix is formed.

2. What about the pairing of ordinary DNA/amino-acids and water icosahedrons? Water icosahedron
has size of about 3 nm. The size of single DNA codon is about 1 nm. Single codon corresponds
to a twist of 3π/5=36 degrees, an angle closely related to Golden Mean. If the radius of the helix
consisting of water icosahedrons is above some minimal radius which is easy to estimate from an
equation for the helix. There are 10 DNAs per L(151) = 10 nm and they correspond to a total twist
of 3 × 2π. Therefore the twist angle is ∆Φ = π/5 = 36 degrees for single codon and corresponds
to a distance of L(151)/10 = 1 nm). From this one has equation for DNA and icosahedron helices
as z = kΦ, k = h/(6π), h = L(151) = 10 nm (radii are constant). Single codon corresponds to a

distance s =
√

dz2 +R2dφ2∆Φ along the water icosahedron helix of radius R accompanying DNA
helix. One must have s ≥ L = 3 nm defining the size of water icosahedron in order to avoid overlap.
Deltas ≥ L = 3 nm gives the condition R ≥ 10×

√
2/(3π) nm ≃ 1.5 nm.

3. If the representation of genetic code is possible, do the fiber icosahedrons correspond to subsets of
faces of the icosahedron itself? Or do they correspond to faces the of icosahedrons in some manner
associated with the amino-acid icosahedron. Direct attachment is not possible but association
could be achieved by connecting the icosahedrons by flux tubes with the tetrahedron at the ends
of flux tubes identified as representation of the same amino-acid. This kind of structure with three
icosahedra emanating from a given icosahedron could be iterated and one would obtain a fractal
structure representing a binary tree. Could the water icosahedrons organize in this manner to form
larger clusters?

What could be the physical correlates of Hamilton cycles representing harmonies?

1. Could Z6, Z4 and Z2 orbits associated with the Hamiltonian cycles be realized even in the structure
of water icosahedrons? Could they be realized as structures formed by the water tetrahedra and
correspond to three separate regions of these icosahedral structures? Could one assign to each of
the three regions of icosahedron icosahedron such that the attached icosahedron decomposes to the
orbits associated with that particular region? Could the hierarchy of the icosahedral symmetry
breakings have a direct counterpart at the level of the icosahedral structures formed by water
molecules? My intuitive feeling is that the answer to these questions is negative.

2. Could Hamiltonian cycles be realized only at the level of dark photons as quint cycles defined by
closed flux tube giving rise to dark nucleus, that is in terms of 3-chords formed by dark photons
propagating along flux tubes emanating from the icosahedron? If cyclotron frequencies of dark
quarks are in question then the magnetic fields associated with the flux tubes would define the
notes.

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3. The breaking of Zrefl
2 symmetry is of special interest since it could serve as a prebiotic analog

of chiral selection and could relate to dark variant of weak physics with effectively massless weak
bosons in nano-scales. This would require dark magnetic body. Half-octave scaling is second broken
symmetry and would have also an analog in Zrefl

2 variant of icosahedron. Note that 256 variants
of the bio-harmony are predicted and could be realized for magnetic body naturally. The presence
of electric fields at flux tubes is possible and if the electric and magnetic fields are non-orthogonal,
U(1) instanton density is non-vanishing and induces parity breaking. Is this breaking associated

with Zrefl
2 only?

3.1.2 Clathrin molecules as icosahedral structures

Clathrin (http://en.wikipedia.org/wiki/Clathrin is a structure appearing at the ends of micro-
tubules and necessary for the transmission of signals between the presynaptic and post-synaptic neurons.
Clathrin consists of triskelions - kind of triangular structures with three spiral like legs and having as
symmetries the rotational symmetry group Z3 of equilateral triangle. Clathrins can form hexagonal pla-
nar lattices and pentagonal icosahedral lattices consisting of 12 pentagonal faces - the number of vertices
of icosahedron . One can associate 3 triskelions with each pentagonal face: this makes 12 × 3 = 36
triskelions altogether. One can regard the centers of the 12 faces as vertices of icosahedron and assign to
this structure 20 faces, which are triangles formed by 3 pentagons.

If proteins and other molecules attach to the faces of clathrin, one can ask whether each icosahedral
triangle of this kind has an address formed by the three notes associated with it and serving as a password:
only those molecules, which “know”this password can attach to the face. The realization would be in
terms of three U-shaped magnetic flux tubes emerging from the 3 pentagonal faces representing the three
notes as frequencies of dark heff = n × h cyclotron photons with ELF frequencies but energies of bio-
photons (in visible and UV range). The binding of the molecule to the face triangle would be preceded
by reconnection of U-shaped flux tubes of the clathrin and molecule, by a resonant interaction by dark
cyclotron photons, and by an heff reducing phase transition bringing the molecule to the face.

3.1.3 Microtubules as music instruments?

It has become clear that microtubules have a central role in biology, neuroscience and perhaps also in
consciousness theory and the evidence that they are quantum coherent systems is accumulating. Could
music metaphor could help to understand microtubules?

1. Tetra-icosahedron has 13 vertices with the added vertex representing one note- say E- in C-key
as note with slightly different frequency to resolve the basic problem of rational number based
12-note scale (12 quints give slightly more that 7 octaves). Intriguingly, microtubules consist of
basic structures consisting of 13 tubulins with 2 states defining bit: could these bit sequences define
representation for the 3-chords and thus representation of sequence of DNA codons and realization
of genetic code.

2. The recent TGD inspired model of microtubules [12],[11] was inspired by the findings of the group
of Bandyopadhyay (https://www.youtube.com/watch?v=VQngptkPYE8) [3],[4] relies on the general
vision about bio-communications and control as being based on dark cyclotron photon radiation
travelling along magnetic flux tubes.

These dark photons have a universal energy spectrum in the range of bio-photons (visible and UV)
to which they transform as the value of heff = n × h reduces to its standard value. Frequencies
would span a wide energy range but EEG frequencies would be of special importance since they
would also couple to acoustic vibrations. The precise value of the energy scale of cyclotron photons
would be determined by the strength of the magnetic field at flux tube.

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3. Frequency modulation would be the general manner to code information in living matter: “whale’s
song”would be a good metaphor for it. This is assumed in the model for cell membrane as gener-
alized Josephson junction: the modulation would be now induced by the variations of generalized
Josephson frequency by variations of the membrane potential. Also microtubules have been pro-
posed to base their communications on frequency modulation.

4. The first possibility coming in mind is that the continually varying microtubule length codes for
the frequency [12]. The change of the frequency by say octave would however require quite fast and
large variations of microtubule length. Neither does this realization conform with the idea that the
state of single tubulin corresponds to frequency. Microtubule length could also code for the length
of the music piece represented by the microtubule serving as a music instrument or musician at
the bio-molecular level. It would also the number of microtubular units and thus the size of the
orchestra consisting of 13-units.

5. Another possibility inspired by the proposal is that magnetic flux tubes form an analog of 3-D grid
ideal for communication purposes using 12-note (or actually 13-note) system as a code equivalent
with genetic code. Also microtubules would involve three kinds of flux tubes [12] defining coordinate
grid of cylindrical coordinates: longitudinal, radial and those which rotate along the microtubule.
Radial flux tubes would be ideal for communication using 13-note system as a realization of genetic
code.

6. 13-note system as cyclotron frequency spectrum for given value of heff would be determined by
the spectrum of the magnetic field strengths going transversally through the microtubule and each
tubulin would correspond to one particular note represented as magnetic field strength. The system
would be highly analogous to the system formed by hair cells in cochlear. Note would indeed
characterize single tubulin molecule rather than entire microtubule as required if one wants to code
chords using the two tubulin conformations as a bit. Tubulin conformation would determine whether
the tubulin serves as a sending/receiving antenna or not.

7. Melody in 12-note system can be interpreted as a discretized version of frequency modulation with
frequency being piece-wise constant in time. Obviously the 13 bit sequences defined by tubulin con-
formations code for the chords of rational 12-note scale involving a representation of one particular
note (the third note of the Pythagorean scale) with two slightly different frequencies in order to
avoid problems caused by the rational number ratios of frequencies. 13th bit could also serve as a
kind of period. Also chords could be coded up to a chord with 13 notes so that microtubules would
have quite a high representative power.

The is an objection against the model.

1. One could argue that a unit consisting of 13 tubulins allows only one octave to be represented. One
can of course assume that the magnetic field strengths for subsequent units differ by octave. What
makes this interesting is that microtubules allow two variants, called A and B. B type microtubules
appear as 13-units since microtubular surface has a gap so that the helical symmetry is broken. For
variant A, which is not found in vivo or in vitro, 13-units integrate to form longer helical units.
This is assumed in Penrose-Hameroff model and the experimental absence of A type microtubules
is one of the basic objections against Penrose-Hameroff hypothesis.

2. The TGD inspired proposal is that A type microtubules corresponds to a critical state having there-
fore an enhanced symmetry and long range correlations: criticality would explain their experimental
absence. The experiments of the group of Bandyopadhyay support that the critical state is induced
by a resonant excitation at specific AC frequencies [12]. Long range correlations would mean en-
chance helical symmetry - that is fusion of several 13-units to form a longer helical structure. This
structure would allow an interpretation as a structure with frequency spectrum of several octaves

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represented coherently in terms of magnetic field strength: the 10 octave span for hearing would
mean the integration of 10 microtubule units meaning length scale of order micrometer assuming
that tubulin size is of order 10 nm.

3. If the field strength for subsequent units differ by octave, one can argue that for variant B various
octaves play their own music without knowing of each other and thus without coherence. In state
A they would play together forming something analogous to orchestra or choir.

If the octave is same for all 13-units, the phase transition would involve octave scaling of the
magnetic field strength at the flux tubes. The flux tube radius should suffer p-adic scaling by
an integer number of half-octaves, which makes sense if one accepts p-adic length scale hypothesis.
This kind of phase transition have been proposed as candidate for a basic step of energy metabolism
since they can store or liberate cyclotron energy as metabolic energy.

4. Microtubules could directly couple with both DNA and clathrin molecules if they represent 12
note system as a resonant system able to receive the radiation with corresponding frequencies. 12-
note system and the 3-chord system associated with it could define universal communication code
allowing communications between DNA, proteins, and microtubules.

To sum up, 13-note extension of 12-note system could be seen as a realization of the genetic code in
terms of frequencies. The existence of kind of realization was obvious from the beginning and I proposed
it in the model of microtubules as quantum antennas during the first years of TGD inspired theory of
consciousness [?]. Discovering the precise realization of the proposal has however required time.

3.2 Could biology help in the understanding of musical harmony?

One can also ask whether biology could provide ideas about the notion of harmony. Could icosatetrahedral
harmony possessing additional 13th note very near to the fourth of basic major chord provide a better
view about harmony?

1. The extension of the ideas about harmony to the case of isosatetrahedron is a non-trivial task. If
one assumes that the extended Hamiltonian cycle is obtained by deforming tetrahedral Hamiltonian
cycle according to the proposal made earlier, one ends up with a problem since the cycle makes a
wedge while making a side track of two steps via the new vertex. The two steps must give one quint
so that the new vertex must correspond to either minor or major third of note where it started from
(and ended to). This would add to the scale a chord of type CGD a chord of type CEG or CE♭G
(plus two other chords containing major or minor third. Depending on the orientation of the cycle
one would obtain major or minor key. The remarkable feature of icosahedral harmonies is that they
often lack a unique basic chord. Could it be that the addition of tetrahedron breaks the symmetry
and fixes the key?

2. The added third could be slightly different from the icosahedral third and this could allow to resolve
the problems due to the fact that quint cycle does not quite close ((3/2)12 = 27 does not hold true
exactly. The problems can be of course solved by introducing well-tempered scale defined in terms of
powers of 21/12: for this choices the topologically induced by these scalings is same as that induced
by real topology in frequency space. Algebraically this means introduction of an algebraic extension
of rationals. The problem is that persons with absolute ear prefer rational number based scale and
experience tempered scale as unaesthetic.

The problem with 3-adic distance of notes was already described: the distance is 4 quints for major
third (C-E) and 3 quints for minor third (C − E♭). A smaller distance is suggestive for major third.

1. The proposed extension of the scale would break symmetry by bringing a third which is indeed
nearest neighbor of the basic note plus two other notes, which are in corners of a 1-quint triangle

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in the biological realization. Thus chord CEG and and chord containing EG and third note would
be introduced.

2. Using the general results one can readily find the possible extensions of harmony if one assumes that
both major and parallel minor with same number of ♯s or ♭s are obtained. The chord chosen for
extension must be CGA, which an be seen as part of C6 or Am7. If the added vertex corresponds to
E one obtains C = CEG, Am = CEA, and the GEA which is part of C6/Am7 as also the lost chord.
In amino-acid analog CGA would become “empty” amino-acid, punct, and would be replaced with
GEA contained also in C6. One can perform this kind of realization for all 11 harmonies and/or their
mirror images. The modification induces symmetry breaking and defines a key which is otherwise
not obvious for the icosahedral harmonies. Also half-octave symmetry is broken.

3. One can perform the modification also for the inverted harmony. The transformation to reverted
harmony X → Y corresponds to X7 ↔ Y 6 and vice versa so that the presence of X7 type chords
in harmony guarantees the existence of the required type extension in the reverted harmony. One
can of course define extension also using X7 type chords. This would generate besides CEG two
dissonant chords of type GEE♭ and CEE♭.

4. In maximally symmetric harmony (2,12,6) with 6-fold rotation symmetry, there are as many as 6
manners to perform this modification so that any note of the 6-note scale spanning “impression-
istic”octave can define the key. The key is either F,G,A or Dm,E, F♯m. The harmony contains
however no X7 type chords and since the transition to the reverted harmony acts as X6 ↔ Y 7,
it does not allow a modification generating both major and parallel minor. There are also other
harmonies possessing no X6 type chords such as (2, 12, 6) and bio-harmony (4, 8, 8) with 2-fold
rotational symmetry so that the extension in the simplest form can be performed only for their
reversals.

5. For the two harmonies with 4-fold reflection symmetry there are 2 manners to perform the modifi-
cation and modified chords are related by half-octave shift. With the conventions of Table ?? the
modification introduces key which is either A (F♯m) or D♯ (Cm) for both harmonies (second one
is bio-harmony (4, 8, 8)).

3.3 About the interpretation of bioharmonies

3.3.1 How ideas about harmony evolved?

A brief summary about the evolution of the notion of bio-harmony is in order.

1. The first guess [13] was that amino-acids could be understood as chords of icosahedral bio-harmony
characterized by 3-tuples (3,10,7), where the integers tell the numbers of icosahedral triangles with
0,1, or 2 edges of the Hamiltonian cycle and identifiable as 3-chords with 0,1,or 2 quints. The
interpretation was that 3 0-quint chords correspond to 3 basic polar amino-acids, 10 1-quint chords
to the 10 non-polar amino-acids, and 7 2-quint triangles to the 7 polar and acidic polar amino-acids.
It turned out however that (3,10,7) does not appear as Hamiltonian cycle although it satisfies the
necessary conditions.

2. I introduced also a model of genetic code motivated by the properties of the code table suggesting
that 60 DNA codons are grouped into 3 groups of 20 codons. The idea that DNA codons coding for
a given amino-acid form an orbit of a subgroup of icosahedral group with order which is not smaller
than the number of these DNAs and has the aminocid at it. Three subgroups Z6, Z4, and Z2 would
predict 3 amino-acids coded by 6 codons and two amino-acids coded by 1 codon, 5 amino-acids
coded by 4 codons, and 10 amino-acids coded by 2 codons. The total number of codons would be
3× 6+ 2+4× 5+ 10× 2 = 20+ 20+ 20 = 60 rather than 64. The number of doublets is 10 instead

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of 9. Could one Z2 orbit corresponds to punct coded by two stopping codons? But what about the
codon triplet associated with Ile? Something is clearly missing.

There is also second problem: a really realistic model of genetic code should include also 21st
and 22nd amino-acids (Pyl and Sec). Pyl or pyrrolysine is modification of Lys and is basic polar
amino-acid so that the number 3 of basic polar amino-acids increases to 4. Contrary to the original
naive extrapolation Sec (selenocystein) is acidic polar rather than non-polar so that the number 2-
quint triangles increases from 7 to 8. For the properties of amino-acids see http://en.wikipedia.
org/wiki/Amino_aci{D\sharp}Physicochemical_properties_of_amino_acids. The notion of
hydrophobicity is discussed at http://en.wikipedia.org/wiki/Hydrophobicity_scales).

3. The solution of the problems came from the extension of icosahedral code with tetrahedral code
bringing 4 additional codons and 3 amino-acids assigned with the external faces of the tetrahedron
(Ile, Pyl, and some standard non-polar amino-acid), and increasing the number of stopping codons
from 2 to 3. This gives 60+3+1=64 codons but one should code also Pyl and Sec. The solution
of the problem would be that stopping codons code also these under some conditions. Are DNA
codons or their mRNA counterparts pairing with tRNAs - perhaps their magnetic body - modified
somehow?

For instance, Pyl and Sec could correspond to icosahedral codons before fusion. After fusion they
cease to be coded - most naturally because the group orbits containing punct are replaced with
those associated with tetrahedron. The 3 ordinary amino-acids represented by tetrahedron are Ile,
1-quint amino-acid and 2-quint amino-acid. As fusion is broken temporarily Pyl and Sec are coded.

4. The geometric correlate for the fusion of the codes is gluing of tetrahedron to icosahedron along
one face which corresponds to “empty” face identifiable as punct coded by stopping codons. The
icosahedral Hamiltonian cycle (4,8,8), which exists as two variants, is extended to (4,10,8) with two
new amino-acids.

5. The music analogy for the fusion of tetrahedron is symmetry breaking bringing in a definite key by
introducing the major and minor chords as 1-quint chord (but with 2-edges since tetrahedral edges
correspond to major and minor thirds).

3.3.2 Understanding the misunderstanding

This was the picture as I started to work again with the notion of bio-harmony. Just when I thought that
I understand the notion, I realized that something very essential is missing and even wrong.

1. One could argue that the assumption about the correlation of forms of amino-acid polarity with
character of Hamiltonian cycle leading to (4,4,8) identification is ad-hoc: why not allow all har-
monies? One can also wonder whether the group structure behind the genetic code leading to the
identification of sets of DNA codons coding for a given amino-acid as orbit of the corresponding
triangle can be totally dependent on the group structure emerging from the construction of the
Hamiltonian cycles.

2. The question whether the group structures associated with genetic code and with the Hamiltonian
cycles might have something to do with each other leads to the realization of the obvious: the
groups involved are the same: Z6, Z4, and Z2! The symmetries of DNA are the symmetries of
cycles. DNA code would be inherent to the Hamiltonian cycles, and the triangles of the icosahedron
representing the harmony would correspond to DNA codons! 20+20+20 icosahedral triangles to 60
genetic codons and 4 icosahedral triangles the remaining 4! The three 20-plets corresponds to 3+1
amino-acids coded by 6 (resp 2) codons, to 5 amino-acids coded by 4 codons, and to 10 amino-acids
coded by two codons.

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By direct inspection of the illustrations of the appendix one can indeed convince oneself that the
groups in question map chords to chords of same type and one obtains appropriate number of orbits.
This of course follows from group theory alone.

3. One must give up the assumption that the integers n = (n0, n1, n2) correspond to the numbers of
the basic polar, non-polar, and polar and acidic polar implying that only n = (4, 4, 8) would define

bio-harmony. All Hamiltonian cycles with symmetries define bio-harmonies and both Zrot
2 and Zrefl

2

define Z2 type bio-harmonies assignable to 10 amino-acids coded by 2 codons. This is somewhat
frustrating outcome, since just this correspondence served as guideline leading to the extension
of the icosahedral code. The extension as such is however independent of this identification and
needed in order to get the 4 missing DNA codons and to understand the coding of 21st and 22nd
amino-acids Pyl and Sec.

What do the Hamiltonian triplets n then correspond? Harmonies correlate with moods in music:
maybe the serve as mathematical correlates for emotions and moods.

4. Harmonies are not for amino-acids but for DNAs coding them. One can however identify amino-
acids as specific triangles the orbits and the chords associated with the amino-acids define much
more restricted notion of harmony involving one representative of each basic type of chord. Perhaps
the additional chords correspond to modulations of the harmony.

5. The rules of harmony generalize as such to transitions between DNA codons regarded as chords.
If chords are near to each other with respect to the distance measured as quints, the transition
between the chords respects harmony. One must think that DNA codons form a singular fiber
space such that the union of fibers for type n gives the space of 20 amino-acids. The “gauge group”
Zn acting in the fiber is different in the 3 regions of the amino-acid space and the the number of
elements in the fiber is factor of n actually equal to n for n 6= 6 and having values 6 and 2 for n = 6.
Each choice for the 3 Hamilton cycles of type Zn, n = 6, 4, 2 defines a variant of this fiber space.
The distance along the fiber isomorphic to the space of amino-acids is measured as minimal quint
distance.

Note that the DNA codons for two different variants of the fiber space need not define same kind of
chord so that also given amino-acid can correspond to several different chords. It is enough that the
notes of the chords are specified - as they indeed are. The Zn, n = 6, 4, 2 in turn can correspond to
any Hamilton cycle with symmetry Zn so that for n = 1, 4, 2 one can have 1, 2, 3 + 5 = 8 different
fiber spaces. The hierarchy of Fibonacci numbers is involved. A hierarchy of symmetry breakings
is highly suggestive and leads to increasingly richer harmonies.

Z6 has maximal symmetry but Z4 is not a subgroup of Z6 so that only the symmetry breakings
Z4 → Zrot

2 and Z4 → Zrefl
2 can be said to occur. Note that transition between different realizations

of the covering space has interpretation as a phase transition and that it could occur at RNA
rather than DNA level. These phase transitions need not relate to the biochemistry but to serve
as correlates for emotions and moods. Also the degeneracy due to the existence of several DNAs
coding given amino-acid could have similar interpretation.

One can of course play with more stringent scenarios for the transitions between DNAs or RNAs).
For instance, the assumption that transitions can occur between chords of same type, leads to
contradiction since the Xaug chords of Z6 harmony do not appear in any other harmony.

In any case, the quint-rule in its various forms is readily testable for DNA sequences.

6. An open question concerns the change of the key. The convention of the illustrations is that 1-2
edge corresponds to C-G quint. Should one allow the DNAs at various sheets of covering space to be
in different keys? Change of the key could be identified as a rotation by some number of quints. It
would change the graph representing icosahedron and change the chords. Z12 would allow to realize

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all keys. Z12 is not however a subgroup of the icosahedral isometries (whereas Z6 = Z3×Zrot
2 is) so

that the transformation should be interpreted as a translation in quint space acting as coordinate
transformation.

The active transformations induced by isometries of icosahedron do not change the graph and would
map chords to new ones. The action of Z6 is well-defined also for other harmonies than Z6 symmetric
ones. Could the modulations of the basic key correspond to Z6 transformations. If so, one would
have 6 keys. Unfortunately, the most common modulation by quint (G → G) would be missing.

The change of key could correspond also the change of the chords defined by the extension to tetra-
icosahedral harmony. One can choose the chord for extension in several manners for Zrot

2 and Zrefl
2

and these choices could define the allowed modulations of the key.

7. What would be the correlates of different keys the level of DNA? An attractive assumption is that
notes are realized in terms of dark photons, which could also transform to ordinary sound since
living matter is piezo-electric system. The general hypothesis is that dark photons have universal
energy spectrum, which is that of bio-photons. Change of key corresponds to a change of frequency
scale and would correspond the change of either Planck constant or of magnetic field strength
the flux tubes of the magnetic body associated with DNA codon (or amino-acid perhaps). This
would mean that 12-note scale would correspond to 12-note scale for the magnetic fields strength
to which cyclotron frequency is proportional or equivalently for the thickness of the flux tube since
magnetic flux is quantized if monopole fluxes are in question. 12-note scale could mean in biology
a standardization of frequencies used.

One must modify the extension of the icosahedral Hamiltonian cycles to tetra-icosahedral ones appro-
priately.

1. The Z6 symmetric 20-plet contains 3 6-plets and 1 doublet and the Z2 symmetric code contains 10
doublets so that here is one 11 DNA doublets in the icosahedral code. “Ordinary” amino-acids have
only 9 doublets. The interpretation is that the Z6 doublet corresponds to ile and the additional ile
is coded by tetrahedral codon. The second surplus doublet can be identified as 2 codons coding for
punct , “punct”. This gives 4+5+ 10 =19 amino-acid if “punct”is counted.

2. What is lacking is one ile, met, trp, plus Pyl and Sec. Also 4 DNA codons are needed. One
of them must code ile, one met, one for punct, and one for trp. The tetrahedral codons would
thus correspond to orbits of Z1. This is actually the only possible subgroup since for the choices
Zn = 2, 3, 4 the numbers of codons and amino-acids are not correct. This exhausts all DNA codons.

3. The only manner to proceed is to assume that icosahedral and tetrahedral codes can appear also as
unfused versions. This would naturally occur for Zref

2 for which all cycles contain X6 type chord
but can occur also for Zrot

2 if the completion is done for the inverse harmony and then mapped
to the harmony back. The icosahedral code would be as already described. The “free”tetrahedral
codes would correspond to Z1 and the faces coding punct in the two codes would code for Pyl and
Sec. The fusion of the tetrahedral and icosahedral codes codes gives just the ordinary genetic code
so that the proposal is consistent with the proposal that dark proton sequences realize genetic code
[7].

4. Note that geometrically this extension means only that the amino-acid sheet of the fiber space is
extended by tetrahedral sheet.

The challenge is to construct the covering space of the icosahedron representing amino-acids.

1. The has as a local fiber the orbit under Zn associated with the amino-acid defining base point. The
space of amino-acids decomposes to disjoint regions corresponding to the 20+20-20 DNA codons.

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Zn is the analog of gauge group and by symmetry breaking is different from three different regions
of amino-acid space. There are 1 × 2 × 8 = 16 variants of this space due to existence of several
harmonies for given symmetries. There are actually only three different options for n given by
n = (0, 16, 4), (2, 12, 6, and (4, 8, 8).

2. The Zn orbits of the three disjoint amino-acid regions (containing 3+1=4, 5, resp. 10 amino-acids)
intersect each other. The challenge is to choose the representative amino-acids from the orbits of
Zn in such a manner that the chosen amino-acids belong to the three disjoint regions. It remains
to be proven that this is possible. One must also understand how uniquely this can be done.

3. One could think of choosing a set P2 of 10 representatives from the 10 orbits of Z2 related by
6-quint scaling along Hamiltonian cycle. The 3+1+5=9 amino-acids associated with Z6 and Z4

would belong to the mirror images P (S) of this 10-element set. P (S) decomposes into set P6 of
3+1 triangles and set P4 of 5 triangles and there are 2-element, 4-element and 6-element orbits
connecting the elements of the sets P2, P4, and P6.

The following observations lead to a rather detailed and surprisingly simple picture.

1. The key observation is that the construction of the covering space - that is identifications of amino-
acids at the orbits of the groups involved - depends only on whether the choice of Z2 as Zrot

2 or

Zrefl
2 ! Thus the two codes (ordinary one and code with Pyl and Sec coded by stop codons) are

distinguished by different DNA-amino-acid covering spaces. The details of the Hamiltonian cycle
do not matter. Only the structures and mutual relationships of the groups Z6 = Z3 × Zrefl

2 ,

Z4 = Zrot
2 ×Zrefl

z and Zrot
2 and Zrefl

2 matter. Furthermore, the actions of the groups Zrot
2 , Z3 and

Zrefl
2 determine also the actions of Z6 and Z4. Only Zrot

2 and Z3 are non-commuting actions.

2. One can decompose amino-acids to 10 pairs of Zref
2 orbits and visualize the 20 codons involved as

two layers on top of each other such that two on top of each other correspond to the same 2-orbit -
2 boxes on top of each other. The choice of the two layers is not unique since one can permute the
members of any vertical box pair.

3. By a suitable choice of the members of vertical box pairs one can arrange that Z3 and Zrot
2 act along

the two layers horizontally. Zrot
2 orbits divide each layer to 5 pairs of horizontal boxes. One can

also permute the vertical pairs horizontally in such a manner that the 5+5 Zrot
2 orbits correspond to

neighboring horizontal boxes along upper and lower layer giving 2+2+2+2+2 decomposition. This
still leaves the possibility to permute these 5+5 horizontal pairs defining 4-orbits of Z4 horizontally
with each other.

Simply by drawing one find that Z3 orbits divide each layer to 3 triplets and 1 singlet and by a
suitable choice Z3 singlets correspond to the 10th box on the right for both layer. The Z3 orbits
and Zrot

2 orbits overlap in such a manner that the middle Z3 orbit contains entire Zrot
2 orbit.

4. It is clear how to choose amino-acids from the orbits.

(a) Consider first the Z2 = Zrefl
2 case. The lower layer corresponds to the 10 Zrefl

2 amino-acids
(punct included) coded by 2 codons. One must choose from each Z4 orbit consisting of a
square of 4 boxes one upper box to represent Z4 amino-acid (ala,val, gly, pro, thr). Each 4-
unit contains one free upper box to which one can assign 1 Z6 amino-acid. One cannot however
put two amino-acids on 3-orbit. There are 3+1 Z6 amino-acids and 5 boxes so that one box
remains unused. This must be the case. The used box must belong to either second or third
horizontal Zrot

2 2-box: if it were filled, the middle Z3 3-orbit would contain 2 Z6 amino-acids
and the fiber space-structure would fail.

Contrary to the original intuition, the unfilled box is not at the 2-orbit of Z6 containing as Ile
but at the middle upper 3-orbit, which would contain 2 amino-acids if filled. It is associated

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with one of the 10 amino-acids coded by two codons and is same for both Zrot
2 and Zrefl

2 . One
expects that this amino-acid is somehow special: maybe it is punct. Also the corresponding
6-amino-acid (Ser, Arg, or Leu) might be somehow special.

(b) Z2 = Zrot
2 can be treated similarly. The upper row of boxes is filled in the same manner as in

the previous case. The horizontal box pairs in the lower row contain one Z2rot box and one
Z4 box. The difference to the previous case is that Z2 boxes are now shared by the both rows:
in the previous case they belonged to the lower row.

5. The assignment of amino-acids to the orbits is not unique: for n similar orbits there are n! different
assignments. Inside orbit there is also some non-uniqueness.

The following table represent the two situations graphically.

4 6 4 6 4 4 6 4 6(2)
2 2 2 2 2 2 2 2 2 2

2 6 2 6 2 2 6 2 6(2)
4 2 4 2 4 2 4 2 4 2

Table1: The representations of the associations of amino-acids to the orbits of of Zn, n = 6, 4, 2 for
Z2 = Zrefl

2 (upper two rows) and Z2 = Zrot
2 (lower two rows). The integer n in box tells that the

amino-acid associated with that box corresponds to Zn type amino-acid. “(2)” tells that the Z6 orbit in
question consists of 2 codons.

3.3.3 Music and physical correlates of emotions

Peptides are regarded as molecules of emotion and also information and positive/negative coloring of
emotions would naturally correlate with the increase/reduction of negentropic resources of the system
as negentropy is transferred to or from it away or increases as a whole. Music induces and expresses
emotions. Therefore the idea that music in generalized form - say represented by dark photons with ELF
frequencies and having energy spectrum in visible and UV energy range of bio-photons- could be the
fundamental correlate of emotions and whether tetra-icosahedral music could be in special role (note that
one can associated Hamilton’s cycles and “music”with any graph).

There are 11 candidates for the icosahedral harmony and its extensions. The candidates have either
Z6 (Fig. 1, Z4 reflection symmetry (Figs. 2, 3), or Z2 rotation symmetry (Figs. 4, 5,6), and Z2 reflection
symmetry (Figs. 7, 8, 9, 10, 11). For the first case Z2 reflection symmetry and for the second case
Z2 rotation symmetry are represented as as half-octave shift. Second reflection symmetry corresponds
geometrically to reflection in horizontal direction. The extension assigns to them definite key and adds to
1-quint chords minor and major chords absent for the icosahedral bio-harmonies. The question is whether
one of these harmonies is selected in biology or whether all three can appear and are perhaps realized at
the level of magnetic bodies of amino-acids.

The reversal of the harmony differs from the original one and major-minor transformation takes place.
Could it be that both “moods” are realized at the level of magnetic body and even serve as the physical
correlates of moods and emotions? Could emotions be realized at the level of amino-acid magnetic bodies
as phase transitions affecting parts of organism or even entire organisms and in this manner changing
the mood. Peptides are regarded as molecules of emotion: could these phase transitions occur only for
peptides and other information molecules involving proteins? Could peptides also serve as seeds of these
phase transitions? Could even the Hamiltonian cycle be changed for the magnetic body of the entire
organism and correspond to some importance two-valued characteristic of emotional profile?

Could orientation reversal relate to time reversal, which in Zero Energy Ontology (ZEO) corresponds to
state function at opposite boundary of causal diamond (CD)? This reversal would occur in volitional acts:

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Pitkänen, M. Music, Biology and Natural Geometry

the subsequent reduction would not affect the quantum state in positive energy but in TGD framework
they affect the state at opposite boundary CD and in this manner give rise to the experience flow of time.

The simplest extension of the harmony in the proposed form requires that harmony possesses X6

chord. It does not exist for for the candidate with Zrot
2 symmetry but for its reversal 4 of them are

present as images of D7, E7 and G♯7, B♭7 which are chords of type X6. One can however map the
harmony to its reversal, perform the completion for it, and perform the reversal back to the original
harmony. The reversal depends on what note remains invariant in the reversal. One can require that it is
the basic note of the chord to itself: with this condition one would obtain Dm,Em,G♯m,B♭m and major
keys C♯, F,A,H. 4 different harmonies would result. Without the restriction the number of harmonies is
different and each has different emotional characteristics.

3.3.4 Religious myths, music, and biology

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

1. The hierarchy of symmetry breakings proceeding from Z6 down to Zrefl
2 brings strongly in mind

evolution as loss of innocence. For Z6 one as 4 orbits. One orbit contains 2 triangles (chords,
DNA codons assignable to ile). The other orbits correspond to six codons assignable to amino-acids
ser, arg, and leu. The chords at the orbits are major chords and 7-chords, and minor chords and
6-chords for the inverse of the harmony.

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

2. Next comes an intriguing observation. The number of amino-acids obtained as projections of the
icosahedral DNA orbits is 19, not 20. Could it be impossible to have 20 amino-acids as projections
of the orbits and that 19 is the maximum number? The reason for 19 is that the number of amino-
acid of type Z6 is 3 + 1 = 4 rather than 5. Therefore there is one ”non-playable” chord -perhaps
located at some ”paradize orbit” -, which does not correspond to any amino-acid.

The first guess for the non-playable chord is as one of the aug type chords (say CEG♯, which is the
last breath in many finnish tangos telling about unhappy love end - it is something between happy
CM and sad Am, ”raueta” is finnish word for this manner to come to an end: ”expire” might be
the nearest english counterpart). This chord is located at the 2-chord orbit related to the other
chord of the orbit by half-octave shift (chords could be CEG♯ and F♯B♭D), the tritonus denied by
church.

Unfortunately, this identification is not consistent with the argument identifying the amino-acid
chords at Zn orbits (see table 3.3.2) the non-playable chord must belong to an intersection of 6-
orbit and 4-orbit and is not completely unique without further assumptions. It belongs to a 2-orbit
of Zrefl

2 : if it is somehow special, it could belong to the 2-orbit assignable to punct. If the chords
at the 2-orbit have basic notes differing by tritonus, the inspection of the Table 4.3 shows that it is
possible to find a unique chord pair having this property for all 5 Zrefl

2 cycles.

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

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3. The hierarchy of symmetry breakings leading from Z6 to Zrefl
2 encourages one to continue with

the biblical analogies. Z6, Z4 and Zrot
2 cycles have half-octave shift as a symmetry: good and

evil do not exist in paradise, but dissonances are already there for Z4 and Z2 harmonies - the evil
snake! These states correspond to the consciousness of animals, children, and saints. Note that
bio-harmony corresponds to the presence of one sub-harmony of type Zn, n = 6, 4, 2.

4. The banishing from the paradize takes place as Zrefl
2 symmetric harmony replaces Zrot

2 harmony:
half-octave shift is not a symmetry anymore, and one can tell between good and evil, and eventually
church decides to deny tritonus as a symbol of evil! Paradise is left as icosahedral and tetrahedral
code are fused to form the tetra-icosahedral code - the ordinary genetic code leading to the breaking
of Zrefl

2 symmetry.

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

6. Also the holy trinity finds an analog as Z6−Z4−Z2 trinity of the bio-harmony. Holy Spirit, Father,
Son: perhaps in this order. Even more, Zrot

2 can be associated with Son in Heaven and Z2refl with
Son at Earth as ordinary mortal!

3.3.5 What do DNAs/amino-acids sound like?

If DNA/amino-acid sequences correspond to chord sequences of tetra-icosahedral harmony, one can ask
what they sound like. The best manner to study this question is to build concrete simulations of the
DNA/amino-acid sequences.

1. This requires specification of harmony by selecting one Hamiltonian cycle from the cycles belonging
to the groups of cycles with Zn, n = 6, 4, 2 symmetry and decomposing amino-acids to 3 groups
correspondingly (those coded by 6, 4, and 2 codons). One must include tetrahedral codons and
amino-acids.

2. The basic rule of harmony would be the minimization of quint distance between initial and final
chords of the transition. One can consider probabilistic versions of this rule or pose strict form
of the rules stating in the most stringent form that only transitions with vanishing quint distance
(between neighboring triangles) are possible.

3. The transitions between different amino-acid regions would be governed by this rule. Aso the
transitions between different variants of the DNA-amino-acid space defined by different choices of
the Hamilton cycles would be governed by the same rule

4. The most plausible looking model considers only transitions between DNA codons since DNA se-
quences induce amino-acid sequences.

Appendix represents an example about randomly generated chord sequence assignable to bio-harmony
defined as a composite of 3 harmonies - one from each symmetry type and Z2 = Zrefl

2 involving tetra-
icosahedral extension. Anyone having garage band skills in guitar playing can check what these chord
sequences sound like and maybe try to build a melody on the background. One could also test the
proposal that codons at the orbit of amino-acid define the melody by finding a concrete representation
for the orbits and building random melodies defined by DNA sequences coding for the chord sequence.

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4 Icosahedral harmonies

In the following the icosahedral harmonies are discussed in detail. This includes overall summary and
tables giving the 20 3-chords of the harmonies and illustrations of the Hamiltonian cycles.

4.1 About symmetries of the icosahedral harmonies

Some words about the symmetries associated with the icosahedral harmonies and genetic code are in
order.

There are 3 different kind of bio-harmonies characterized partially by the symmetry group which can
be Z6, Z4 or Z2 which acts either as rotations or reflections.

1. The first variant as Zrot
3 × Zrefl

2 subgroup of icosahedral group as symmetries and its orbits corre-
spond to 3 6-plets and 1 2-plets for which Z3 leaves the triangle invariant. The counterparts for the
orbits are 3 DNA 6-plets and one 2-plet.

2. The second variant has Z4 symmetry generated by two commuting reflection as symmetries as is
obvious from figures 3,??: the reflections act on vertical and horizontal coordinates. The orbits are
five 4-plets of chords. Vertical reflection induces half-octave shift and horizontal one permutes the
note sequences B♭CDG♯F♯E and D♯C♯HFGA.

3. Zrot
2 or Zrefl

2 acts as symmetries of the remaining 3+5 cycles. The covering space of 10 amino-acids
involved defined by 20 DNA codons decomposes to 10 2-plets.

The 2-fold rotation symmetry of the Hamiltonian cycles is obvious from the illustration ??: it
corresponds to 6-quint rotation and the chord sets must be invariant under this rotation. This
rotation corresponds to the 1/2 octave shift realized as rotation. These symmetries are realized as
“coordinate transformations”for the cycle - a curve in the “imbedding space”defined by icosahedron
but induced from the “imbedding space symmetries”acting as isometries of icosahedron.

DNA codons have also almost exact Z2 symmetry discussed in [?, 5, 6].

1. For the last codon the reflection A-T, C-G is an almost symmetry broken only for special cases.
This approximate symmetry could be understood as following from the fact that the number of
DNAs coding given amino-adic is even in most cases. The exceptions are ile, met, trp coded by
odd number of DNA codons. By mapping DNAs to binary sequences one can order the situation
so that the 6:th binary digit is the almost-symmetry digit.

2. What is trivial is that RNA has chosen the third bi-digit to be the almost symmetry digit with the
ordering UCAG of the nucleotides so that a genuine physical symmetry is in question. An interesting
question is how this symmetry relates to the model of genetic code based on tetra-icosahedral orbits.

The restriction of DNAs to 60 icosahedral DNAs demonstrates that this symmetry originates from
the icosahedral Z2. The tetrahedral extension of the code breaks this symmetry by extending ile
and punct multiples by one codon and introducing also 4 singlets met, trp, Pyl, and Sec.

The detailed correspondence between chords of the harmony and DNA codons is also a problem to be
solved.

1. The correspondence matters in the proposed scenario since the chords at at the orbits are different
and the gluing of tetrahedron breaks the symmetry in Z2 sectors so that quint rule determining
harmonic DNA sequences is different.

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2. The common face of tetrahedron and icosahedron corresponds to punct so that the quint rule for
different representations says something about the pairs of form codon-stop codon that is about
the codon preceding the last codon of gene! This codon could allow to recognize what Hamiltonian
cycle is in question. If C-major is one of the added chords, stop codons correspond to what was
C6 = CGA chord and its Z2 image, which is X7 type chord. By the strongest form of the quint
rule only the chords having common notes with these chords would correspond to DNA codons of
Z6 and Z4 cycles which can precede stopping codon.

3. There are some restrictions on the correspondence. Zrefl
2 symmetry would correspond to the flipping

of the 6th bit for the bit representation defined by nucleotides representing 2-bits in the case of
Z3 = Z3 × Zrefl

2 . Z4 = Zrot
2 × Zrefl

2 . For Z2 = Zrot
2 the role of Zrefl

2 must be taken by Zrot
2 .

One can of course ask whether Zrot
2 cycles are realized at all. For Z4 cycles Zrot

2 would correspond
to symmetry permuting the AT, CG doublets for the first nucleotide. For Z6 subgroup Z3 would
cyclically permute the 3 doublets with respect to third nucleotide. These constraints do not fix the
correspondence completely.

To sum up, there is a connection between genetic code and the groups acting along the Hamiltonian
cycle. The simplest option fixes the orbits of the triangles and therefore also the representation of genetic
code.

4.2 Summary of the basic results

One can find the list of Hamiltonian cycles at http://cs.smith.edu/~orourke/MathOverflow/hpaths.
html. The edge {1, 2} is fixed and cycles are oriented so that there are 1024 of them. All of them are
relevant from the point of music interpretation and the change of orientation corresponds to major-minor
duality, albeit not in the simplest sense. Note that this duality does not affect the characteristics listed
above.

The general following general results hold true as one can learn at http://mathoverflow.net/

questions/37788/why-are-there-1024-hamiltonian-cycles-on-an-icosahedron. One can classify
the cycles using their symmetries which can correspond to isometries of icosahedron leaving them fixed
or to a reflection taking the vertex n at the cycle to vertex 12− n. This symmetry is not same as change
of orientation which is purely internal operation and cannot change the cycle.

One can even find images of the cycles possessing symmetries at https://www.flickr.com/photos/
edwynn/sets/72157625709580605/ and deduce the triplets n and p characterizing them by visual inspec-
tion. Also one can write explicitly the 3-chords defined by the three kinds of faces. I have deduced the
triplets n and the 3-chords defining the harmony by the inspection of the images. “Bio-harmony” (4,8,8)
forced by the model of extended genetic code involving also the 21st and 22nd amino-acids is of spe-
cial interest.The classes of cycles with symmetries 6-fold rotational symmetry and two distinct reflection
symmetries realize it.

Before continuing some terminology and notation is in order. Take C as the major key. Submediant
or relative minor corresponds to Am, subdominant (sharp or flat) to F major (F ) or Fminor (Fm),
dominant to G. The notation for chords is such that quints correspond to subsequent notes in the chord.
For 1-quint chords this means that first two notes define the quint. The following table summarizes
notation inspired by the popular music notation. The basic different is that the third is in most cases
excluded so that the emotional character of the chord is not fixed.

CEG ≡ C , CD♯G ≡ Cm , CD♯F♯ ≡ Co , CEG♯ ≡ Caug ,
CFG ≡ C4 , CF♯G ≡ C4+ , CGG♯ ≡ C6

−
, CGA ≡ C6 ,

CGB♭ ≡ C7 , CGB ≡ Cmaj7 , CGC♯ ≡ C9
−

, CGD ≡ C9 .
(4.1)

Besides these notions it is convenient to introduce additional notations for various dissonant chords
appearing as 0-quint chords.

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CC♯D ≡ Cex1 , CC♯D♯ ≡ Cex2 , CDD♯ ≡ Cex3 , CDE ≡ Cex4 ,
CD♯E ≡ Cex5 , CC♯E ≡ Cex6 , CDF♯ ≡ Cex7 , CDG♯ ≡ Cex8 .

(4.2)

Clearly, the sets {ex1}, {ex2,ex3}, {ex4,ex5,ex6}, {ex7}, {ex8}, corresponds to the span of 2,3,4,6,8 half
notes for the chord. The following summarizes the results. Note that Cex7 can be seen as part of D7
chord.

1. There are 6 collections of cycles without any symmetries containing 48 cycles each: these 48 cycle
are mutually isometric so that one can say that there 6 different harmonies.

2. There is a collection with 6-fold rotational symmetry, 48/6=8 examples. n = (2, 12, 6). The chords
of this scale define 6-note scale involving only total steps. CDF and and its 6 translates by integer
number of steps define 6 1-quint chords. CE♭G (Cm) and its 6 translates (they obviously correspond
to the 6-fold rotational symmetry) define also 6 1-quint chords. The reflection transforms these series
to those defined by GB♭G and its translate and byFAC (F major) and its translates. Impressionists
like Debussy used 6-note scale of this kind. Half-octave shift is an exact symmetry. 1-chords lack
the third so that one cannot assign to 3-chords any emotional quality. The extension to 4-chord
can however bring either “happy”or “sad”quality. Clearly, these harmonies have “jazzy”character.

0-quint chords are Faug ≡ FAC♯ and Gaug ≡ GHD♯ are transformed to each other by both
half-octave shift and inversion.

3. There are 2 collections with 2 distinct reflectional symmetries with 12=48/4 representatives in each.
Half-octave scaling is a symmetry of both these scales as one might guess.

The first cycle (see Fig. 2) has n = (0, 16, 4) so that there are no 0-quint chords which in general are
dissonant. Second cycle (see Fig. 3) realizes n = (4, 8, 8) bio-harmony and deserves some comments.
It will be discussed in detail later.

(a) The 8 2-quint chords consist of B♭FG ≡ B♭9, C9, F9, G9 and their half-octave scalings.
Clearly, the simple four-note scale appears here.

(b) Using the popular notion introduced earlier 1-quint chords consist of two 4-plets Dmaj7, E9
−
,

A7, A6 and G♯maj7, B♭9
−
, D♯7, D♯6 related by half-octave shift. The harmony contains

no “simple”major or minor chord and only the extension to tetrahedral harmony can provide
them. The same is true for the second bio-harmony.

(c) The 4 0-quint chords are Cex3 ≡ CDD♯ and Eex2 ≡ EFG and their half-octave scalings
F♯ex3 ≡ F♯G♯A and B♭ex2 ≡ B♭BC♯G.

4. There are 3 collections with Z2 rotational symmetry with 48/2 = 24 representatives in each. The
triplets n are (0, 16, 4) (see Fig. 4), (2, 12, 6) (see Fig. 5), and (4, 8, 8) (see Fig. 6). All these
harmonies are symmetric with respect to half-octave shift (tritonus), which obviously corresponds
to the Z2 rotation. Tritonus would not have been tolerated by catholic church! This symmetry
characterizes all 3 harmonies. Basic 3-chords do not contain pure minor and major chords. The
reflection of the scale does not leave the collection of chords invariant but it is not clear whether
this corresponds only to a change of scale, probably not.

Consider the (4,8,8) case (see Fig. 6).

(a) The 8 2-quint chords appear as four-plet H9, C♯9, D♯9, F9 and its half octave shift (tritonus
interval) acting as a symmetry of the harmony. 2-quint chords are always of type X9 (note
that the third is missing) but also 1-quint chord can be of form X9 as explicit construction of
chords demonstrates: I have denoted these 1-quint chords by symbol X4 (CDG is obviously
equivalent with CDG).

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(b) Using the popular music notation introduced earlier, the 8 1-quint chords areD7, Amaj7, A4+, E7
and their half-octave shifts G♯7, D♯7, D♯4+, B♭7.

No major and minor chords are included and only the extension to tetra-icosahedral harmony can
provide them and also break the symmetry giving rise to well-defined key.

5. The four 0-quint chords appear in two types. D♯ex2 ≡ D♯EF♯ and its half-octave shift Aex2 ≡
AB♭C plus Hex3 ≡ HC♯G and its half-octave shift Fex3 ≡ FGC♯. According to usual thinking
these chords involve dissonances. This dissonance character is a rather general phenomenon for the
harmonic loners and classical views about harmony would exclude them as asocial cases! In the
case of maximally symmetric harmony the loners are diminished chords and thus not so dissonant.
In some cases there are no 0-quint chords.

There are 5 collections with Z2 reflection symmetry having 24 representatives in each (see Figs. 7,
8,9,10,11). The integer triplets n are (2, 12, 6), (2, 12, 6), (4, 10, 6), (2, 12, 6), (2, 12, 6). Bio-harmony has
representative also in this class (see Fig. 9). The half-octave scaling symmetry is broken for these
harmonies. I have not found simple characterization for the symmetry which corresponds to reflection in
the direction of x-axis since it changes the interval structure of the chords.

Some comments (4, 8, 8) case are in order (see Fig. 9).

1. 2-quint chords appear as reflection related multiplets C9, D9, H♯9, D♯9 and C♯9, H9, F9, B♭9.

2. 1-quint chords appear as symmetry related mutiplets G,D7, Amaj7, E7 and C♯m,F♯6, H6
−
, E6.

Key G major and C♯ minor would be natural looking keys even without tetrahedral extension. For
the mirror image B♭ minor and E major would be the natural looking keys. For extension E major
would be the key.

To sum up, half octave shift is a symmetry of all harmonies expected those having only Z2 reflection
symmetry, and fails thus also for the corresponding bio-harmonies.

4.3 Tables of basic 3-chords for the icosahedral harmonies 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 2. The 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).

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Table 3. The 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. Both cycles have Z2 rotation symmetry acting as a
vertical reflection symmetry in figures and realized also as half-octave shift so that 4-plets contains chords
and their half-octave shifts. The genuine reflection symmetry acts as a horizontal reflection symmetry in
figures. The cycles correspond to figures 2, 3.

(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 4. The table gives various types of 3-chords for harmonies with Z2 rotation symmetry acting as
half-octave shift. The doublets represent 2-chord orbits. The cycles correspond to figures 4, 5, and 6.

(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 5. The table gives various types of 3-chords for harmonies with single reflection symmetry. The
cycles correspond to figures 7, 8, 9,10, 11.

References

Mathematics

Icosahedral graph. Wolfram MathWorld. http://mathworld.wolfram.com/IcosahedralGraph.html.

Why are there 1024 Hamiltonian cycles on an icosahedron? http://mathoverflow.net/questions/

37788/why-are-there-1024-hamiltonian-cycles-on-an-icosahedron.

Neuroscience

A. Bandyopadhyay. Experimental Studies on a Single Microtubule (Google Workshop on Quantum
Biology), 2011.

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DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 161-198 188

Pitkänen, M. Music, Biology and Natural Geometry

A. Bandyopadhyay G. Ghosh, S. Sahu. Evidence of massive global synchronization and the consciousness:
Comment on ”Consciousness in the universe: A review of the ’Orch OR’ theory” by Hameroff and Penrose.
Biosens Bioelectron, 2013.

C. B. Pert. Molecules of Emotion. Simon & Schuster Inc., 1997.

Books related to TGD

M. Pitkänen. DNA as Topological Quantum Computer. In Genes and Memes. Onlinebook. http:
//tgdtheory.fi/public_html/genememe/genememe.html#dnatqc, 2006.

M. Pitkänen. Evolution in Many-Sheeted Space-Time. In Genes and Memes. Onlinebook. http://
tgdtheory.fi/public_html/genememe/genememe.html#prebio, 2006.

M. Pitkänen. Homeopathy in Many-Sheeted Space-Time. In Bio-Systems as Conscious Holograms.
Onlinebook. http://tgdtheory.fi/public_html/hologram/hologram.html#homeoc, 2006.

M. Pitkänen. Negentropy Maximization Principle. In TGD Inspired Theory of Consciousness. Online-
book. http://tgdtheory.fi/public_html/tgdconsc/tgdconsc.html#nmpc, 2006.

M. Pitkänen. Quantum Model for Hearing. In TGD and EEG. Onlinebook. http://tgdtheory.fi/
public_html//tgdeeg/tgdeeg/tgdeeg.html#hearing, 2006.

M. Pitkänen. Three new physics realizations of the genetic code and the role of dark matter in bio-
systems. In Genes and Memes. Onlinebook. http://tgdtheory.fi/public_html/genememe/genememe.
html#dnatqccodes, 2006.

M. Pitkänen. Quantum Mind, Magnetic Body, and Biological Body. In TGD based view about liv-

ing matter and remote mental interactions. Onlinebook. http://tgdtheory.fi/public_html/pdfpool/
lianPB.pdf, 2012.

Articles related to TGD

M. Pitkänen. New results about microtubules as quantum systems. http://tgdtheory.fi/public_

html/articles/microtubule.pdf, 2014.

M. Pitkänen. Pythagoras, music, sacred geometry, and genetic code. http://tgdtheory.fi/public_

html/articles/pythagoras.pdf, 2014.

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