TGD diary: What could be the physical origin of Pythagorean scale?

I was contacted for a couple years ago by Hans Geesink and we had long discussions about consciousness and quantum biology. The discussion stimulated new ideas and this inspired me to write a chapter and article comparing our approaches. Now Hans sent me two prepublications by him and D. K. F. Meijer.

The first preprint "Bio-Soliton Model that predicts Non-Thermal Electromagnetic Radiation Frequency Bands, that either Stabilize or Destabilize Life Conditions" is in arXiv ). The abstract reads as:

Solitons, as self-reinforcing solitary waves, interact with complex biological phenomena such as cellular self-organisation. Soliton models are able to describe a spectrum of electromagnetism modalities that can be applied to understand the physical principles of biological effects in living cells, as caused by electromagnetic radiation. A bio-soliton model is proposed, that enables to predict which eigen-frequencies of non-thermal electromagnetic waves are life-sustaining and which are, in contrast, detrimental for living cells. The particular effects are exerted by a range of electromagnetic wave frequencies of one-tenth of a Hertz till Peta Hertz, that show a pattern of twelve bands, if positioned on an acoustic frequency scale. The model was substantiated by a meta-analysis of 240 published papers of biological radiation experiments, in which a spectrum of non-thermal electromagnetic waves were exposed to living cells and intact organisms. These data support the concept of coherent quantized electromagnetic states in living organisms and the theories of Davydov, Fröhlich and Pang. A spin-off strategy from our study is discussed in order to design bio-compatibility promoting semi-conducting materials and to counteract potential detrimental effects due to specific types of electromagnetic radiation produced by man-made electromagnetic technologies.

Second preprint "Phonon Guided Biology: Architecture of Life and Conscious Perception are mediated by Toroidal Coupling of Phonon, Photon and Electron Information Fluxes at Eigen-frequencies" is in Research Gate. The abstract is following.

Recently a novel biological principle, revealing discrete life sustaining electromagnetic (EM) frequencies, was presented and shown to match with a range of frequencies emitted by clay-minerals as a candidate to catalyze RNA synthesis. The spectrum of frequency bands indicate that nature employs discrete eigen-frequencies that match with an acoustic reference scale, with frequency ratios of 1:2, and closely approximated by 2:3, 3:4, 3:5, 4:5 and higher partials. The present study shows that these patterns strikingly resemble eigen-frequencies of sound induced geometric patterns of the membrane vibration experiments of E. Chladni (1787), and matches with the mathematical calculations of W. Ritz (1909). We postulate that the spectrum of EM frequencies detected, exert a phonon guided ordering effect on life cells, on the basis of induction of geometric wave patterns. In our brain a toroidal integration of phonon, photon and electron fluxes may guide information messengers such as Ca²⁺-ions to induce coherent oscillations in cellular macromolecules. The integration of such multiple informational processes is proposed to be organized in a fractal 4-D toroidal geometry, that is proposed to be instrumental in conscious perception. Our finding of an "acoustic life principle" may reflect an aspect of the implicate order, as postulated by David Bohm.

A very concice and very partial summary about the articles would be following.

12-note scale seems to be realized in good approximation as frequency bands (rather than single frequencies) for a membrane like system with the geometry of square obeying four-order partical differential equation studied numerically by Ritz. Since the boundary conditions are periodic this system has effective torus topology. This is rather remarkable experimental fact and extremely interesting from TGD point of view.
The papers also argue that also the octave hierarchy is realized. p-Adic length scale hierarchy indeed predicts that subset of powers of 2, and more generally of 2^1/2 defines a hierarchy of fundamental p-adic scales with p-adic prime p near to power of two.

In the following I will discuss first the condensed matter realization of 12-note scale and after that considder the significance and realization of 12-note scale from TGD point of view.

Condensed matter realization of 12-note scale in terms of oscillations of square plate

The article discusses a condensed matter physics based realization of 12-note. Acoustic waves are seen as fundamental. Certainly the sound waves are important since they couple to electromagnetic waves. My feeling is however that they provide a secondary realization.

The realization of 12-note system as 12 bands discussed in the articles is as eigen frequencies of deformations of square plate. Periodic boundary conditions imply that one can regard the system also as a torus. One has bands, not eigenfrequencies. I do not know whether one can pick up from bands frequencies, whose ratio to the fundamental would be rational and same as for Pythagorean scale. Since the system can be treated only numerically, it is difficult to answer this question.
So called Chladni patterns (see An Amazing Resonance Experiment) are associated with vibrating thin square plate and correspond to the node lines of the deformation of the plate in direction orthogonal to the plate. As one adds very small particles at the plate and if the vibrational acceleration is smaller than the gravitational acceleration the particles get to the node lines and form Chladni pattern. Hence the presence of gravitation seems to be essential for the Chladni patters to occur. These patterns make visible the structure of standing wave eigenmodes of the plate. It is also possible to have patterns assignable to the antinodes at which the deformation is maximum but vibrational acceleration vanishes as in the harmonic oscillator at the maximum value of the amplitude.
The vibrations of square plate obey fourth order partial diff equation for the Chladni pattern having the general form

∂_t² u= K (∇²)² u .

Here u is the small deformation in direction orthonormal to the plate. The equation can be deduced from the theory of elasticity about which I do not know much. For standing wave solutions the time dependence is separable to trigonometric factor sin(ω t) or cos(ω t), and one obtains eigenvalue equation

K(∇²)² u =-ω² u .
The natural basis for the modes is as products of 1-D modes u_m(x) for string satisfying ∂_x² u_m=0 at the ends of the string (x={-1,1}): this in both x and y directions. This must express the fact that energy and momentum do not flow out at boundaries.

The modes satisfy

d⁴u_m/dx⁴= k_n⁴ u_m .

Boundary conditions allow modes with both even and odd parity:

u_m= [cos(k_m)cosh(k_mx) + cosh(k_m) cos(k_nx)] /[cosh²k_m+cos²(k_m)] ,

tan(k_m)+tanh(k_m)=0 , m even .

u_m= [sin(k_m)sinh(k_mx) + sinh(k_m)sin(k_nx) )]/[sinh²k_m+sin²(k_m)]

tan(k_m)-tanh(k_m)=0 , m odd .
The 2-D modes are not products of 1-D modes but sums of products

w^ε_mn= u_m(x)u_n(y) + ε u_m(y)u_n(x) , ε=+/- 1 .

Modern physicist would notice classical entanglement between x and y degrees of freedom. The first ε=1 mode is analogous symmetric two-boson state and second ε=-1 mode to antisymmetric two-fermion state.
The variational ansatz of Ritz was superposition of these modes (this variational method was actually discovered by Ritz). Ritz minimized the expectation value of the Hermitian operator (∇²)² in the ground state and obtained an approximation for the frequencies which holds true with 1 per cent accuracy.

Unfortunately, 4-D geometry does not give rise to this kind of equations: time and space are not in democratic roles. TGD inspired vision would be different. The magnetic flux tubes and even strings could be the fundamental objects concerning biology and consciousness. The acoustic realization of the 12-note scale would be secondary one. Even genetic code would have fundamental realization at the level of dark nuclear physics and chemical realization of genetic code would be secondary realization.

2. Why 12-note scale?

Why I am convinced that 12-note scale should be so important?

The mysterious fact about music experience is that frequencies whose ratios come as rationals are somehow special concerning music experience. People with absolute pitch prefer the Pythagorean scale with this property as aesthetically pleasing. Pythagorean scale is obtained by forming the 3^k multiples of fundamental and by dividing by a suitable power 2^m of 2 to get a frequency in the basic octave. This scale appears in TGD inspired model for music harmonies, which as a byproduct led to a model of genetic code predicting correctly the numbers of DNA codons coding for given amino-acid. The appearance of powers of 2 and 3 suggest 3-adicity and 2-adicity. Furthermore, rationals correspond to the lowest evolutionary level defined by the hierarchy of algebraic extensions of rationals.

This gives excellent reasons to ask whether 12-note scale could be realized as some physical system. One might hope that this system could be somehow universal. Geometric realization in terms of wave equation would be the best that one could have.
The model of harmony is realized in terms of Hamilton cycles assignable to icosahedron and tetrahedron. Hamilton cycles at icosahedron are closed paths going through all 12 points of icosahedron and thus can define a geometric representation of the Pythagorean scale. The rule is that curve connects only nearest points of icosahedron and corresponds to scaling of frequency by 3/2 plus reduction to basic octave by dividing by a suitable power of 2. The triangles of the icosahedron define allowed 20 chords for given harmony and one obtains 256 basic harmonies characterized by the symmetries of the cycle: symmetry group can be cyclic group Z₆, Z₄ or Z₂ or reflection group Z₂ acting on icosahedron.

Bioharmonies are obtained by combining Z₆, Z₄ and Z₂ of either type. One obtains 20+20+20 =60 3-chords defining the bio-harmony. One must add tetrahedral harmony with 4 chords in order to obtain 64 chords. It turns out that it corresponds to genetic code under rather mild assumptions. DNA codons with 3 letters could correspond 3-chords with letter triplets mapped to 3-chords. Amino-acids would correspond to orbits of given codon at icosahedron under one of the symmetry groups involved.

How to realize 12-note scale at fundamental level universally?

How could one realize 12-note scale at the fundamental level - that is in terms of 4-D geometry? The realization should be also universal and its existence should not depend on special properties of physical system. Vibrating strings provide the simplest manner to realize 12-note scale. Harmonics do not however allow its realization. They are in higher octaves and define only the color of the note. There are actually two realizations.

The simplest realization relies on the analogy with piano.

The string of piano corresponds to a magnetic flux tube/associated fermionic string and the frequency of the note would be determined by the length of the flux tube. The quantization for the length as certain rational multiples of p-adic length scale gives rise to the 12-note scale. Tensor network would be like piano with the flux tubes of the network with quantized lengths defining the strings of piano.
Why the length of the flux tube defining the fundamental frequency would correspond to a frequency of Pythagorean scale? Could this be due to the preferred extremal property realizing SH and posing very strong conditions on allowed space-time surface and 3-surfaces at their ends at boundaries of causal diamonds? If so, 12-note scale would be part of fundamental physics!

The rational multiples f(m,n)= (m/n)f₀, m=0,1,..n-1, of the fundamental f₀ with m/n≤ 2 (single octave) are in a preferred position mathematically since the superpositions of waves with these frequencies can be represented as superpositions of the suitable harmonics of the scaled down fundamental f₁=f₀/n. For Pythagorean scale m/n= 3^k/2^l the new fundamental is some "inverted octave f₁= f₀/2^k_max of the fundamental and the allowed harmonics are of form m=2^r3^l.

Second realization would be dynamical and based on the analogy with string instruments.

String instruments allow to realize 12-note scale by varying the length of the vibrating string. The note of scale corresponds to the fundamental frequency for the portion of the shortened string, which is picked. Why the lengths of shortened strings should correspond to inverses of frequencies of 12-note scale? One should have powers of 3 divided by powers of 2 to get a frequency in fundamental octave. Could p-adic length scale hypothesis, which generalizes and length scales coming as powers of square roots of small primes help?
Strings bring in mind magnetic flux tubes connecting partonic 2-surfaces. They behaving in good approximation like strings and are actually accompanied by genuine fermionic strings and corresponding string world sheets. Flux tubes play a fundamental role in living matter in TGD Universe. Flux tubes carrying dark matter identified as large h_eff=n× h phases would serve as space-time correlates for negentropic entanglement and gives rise to tensor nets with partonic 2-surfaces as nodes and flux tubes connecting them (see this). Could magnetic flux tubes or associated fermionic strings provide the instruments using Pythagorean scale?

Partonic 2-surfaces and string world sheets dictate space-time surface by strong form of holography (SH) implied by strong form of general coordinate invariance. It is quite possible that not all configurations of partonic 2-surfaces and string world sheets allow SH that is realization as space-time surface: perhaps only the flux tubes with length corresponding to Pythagorean scale allow it. For p-adic counterparts of space-time surfaces the possibility of p-adic pseudo-constants (failure of strict determinism of field equations) makes this possible: the interpretation is as imagined p-adic space-time surface which cannot be realized as real space-time surface.

How these flux tubes could behave like strings of guitar? When my finger touching the guitar string it dividing it to two pieces. The analogy for this is the appearance of additional partonic 2-surface between the two existing ones so that one has two flux tubes connecting the original partonic two-surface to the new one. A change of the topology of 3-space would be involved with this stringy music!

More precisely, the flux tubes would be closed if they carry monopole magnetic flux: they would begin from "upper" wormhole throat of wormhole contact A (partonic 2-surface), go along "upper" space-time sheet to the throat of wormhole contact B go the "lower"space-time sheet through it, return to the "lower" throat of wormhole contact A and back to the "upper" throat. Shortening of the string would correspond to a formation of wormhole contact at some point of this flux tube structure splitting the flux tube to two pieces.
Another realization could be in terms of the quantization of the distance between partonic 2-surfaces connected by flux tubes and associated strings in given p-adic length scale, which by p-adic length scale hypothesis would correspond to power of square root of 2 so that also octaves and possibly also half octaves would be obtained (note that half octave corresponds to tritonus, which was regarded by church as an invention of devil!). Also now the justification in terms of SH.