Against Arturo Reghini, the Golden Scale and for the Blessed Dance of the Muses: Number Theory, Music Theory and Orthodox Pythagorean Philosophy

So then I had my math-music research published on Amazon as Do-Ohm Up the Yin-Yang!

Around the same time last year a new Ph.D. thesis was published as pdf on the Freemason Pythagorean Traditionalist Arturo Reghini

Only Arturo Reghini was not a "real" Pythagorean despite the claims. Reghini was just promoting Western science materialism from Archytas - and Plato - not the real Pythagorean Orthodox philosophy.

We can see this from the Western alchemy website that has translated his work into English -

As I noted in my math-music research there is a "bait and switch" going on between geometry and arithmetic when it comes to music. Math professor Luigi Borzacchini also discovered this cover-up of the music origins of Western math - a cover-up he called "really astonishing" and "shocking."

So people think they research the "occult" and music when in fact they are just promoting Platonic philosophy from Archytas - for example I asked the religion studies music author Peter Bebergal to consider my research but he didn't read any of it - he just clicked on the website and left the site real quick. haha. Bebergal is working on a new book called "strange frequencies" but his first music occult book was published by Mitch Horowitz who just promotes Freemasonry like it's a great thing.

Instead of getting bogged down in an ideological debate on morality - we can just stick to the empirical facts about music theory and math or number theory. So as I detailed before - and it is recorded free online in the conspiracy forum - abovetopsecret - here people tried to debunk me. And so this forced me to clarify my thinking. Finally I discovered the top mathematician Alain Connes corroborating my view - that the empirical truth of music theory is really a "universal scaling system" that is "noncommutative" (unlike the symmetric logarithmic math promoted by Plato and Archytas from the wrong music theory).

I quoted Connes research in my previous blog posts here. But this issue is quite subtle, yet also radical and profound. So as Borzacchini states -

the ancient arithmetic perception of numerical magnitude...the basic Pythagorean 'perception' of linear number

broke down in a "crisis" of cognitive bias that was supposedly solved using materialistic geometry. And it is this materialistic geometry that is promoted by Freemasonry - and so then Pythagorean philosophy is also promoted as being this same materialistic geometry - when it is clearly not!

So it is Borzacchini who notes that even in current times mathematicians assume that the discrete/continuous dichotomy is naturally embedded in the geometric continuum concept, when in fact it was an ideological construct secretly derived from music theory. It is then math professor Alain Connes who points out that the real natural numbers as music theory are not only empirically true but also noncommutative and therefore a quantum universal scaling system - that he calls "music of the quantum spheres."

This whole noncommutative issue is completely ignored by 99.99% of supposed Pythagorean or Occult New Age music healing theorists and teachers, etc. But in fact traditional Taoist philosophy is also noncommutative and based on the same music theory as Orthodox Pythagorean philosophy, just as Samkhya, the oldest philosophy of India. And more so, all human cultures use the 1-4-5 music intervals of natural numbers that are in fact noncommutative and the focus of the spiritual trance training from the original human culture in Africa - the Bushmen.

So in terms of number theory analysis - in current times - we find this issue of noncommutative resonance passed over - i.e. in Cartwright J.H.E., González D.L., Piro O. (2009) Nonlinear Dynamics, the Missing Fundamental, and Harmony. In: Klouche T., Noll T. (eds) Mathematics and Computation in Music. MCM 2007. Communications in Computer and Information Science, vol 37. Springer, Berlin, Heidelberg pdf link

So let's first consider Reghini - and his error in reasoning - which again is another "bait and switch" example that I have already analyzed similarly in that abovetopsecret thread discussion. First of all in Orthodox Pythagorean number theory only the Octave-Fourth-Fifth is allowed, not the number 8, as I have referenced in the academic analysis. So again even the claims of writing down the extended ratios of 9/8 by Archytas and Philolaus are not Orthodox Pythagorean philosophy. Secondly in Orthodox Pythagorean philosophy, several years of silence meditation was required - 5 years.

In the special case where both a = 2c the Babylonian proportion is:

and if a = 1

This quaternion contains the numbers that are the respective measures of the length of four as well as the tetrachord of Philolaus.

 So Arturo Reghini makes the above claim as if it is Orthodox Pythagorean philosophy.

But we can now reveal what his "bait and switch" trick is. He admits that he is "averaging" and that Archytas did not "average" but as other mathematicians have argued - indeed Archytas and Plato promoted irrational magnitude based on this averaging concept.

In fact, if in a long segment h one takes its midpoint, the segment is divided into two segments each of length 1: of 2 h.

 So here we have his conversion from number to geometry. Reghini assumes that geometry came first when in fact Pythagoras taught "all is number and harmony" and it's the Freemasons who teach "all is geometry" really from Plato.

Let’s start from the tetrachord of Philolaus C, F, G, C (do, fa, sol, do) whose chords are respectively long 1, 3 : 4, 2 : 3, 1 : 2 such that

and where the second term is the arithmetic median of the extremes and the third is the harmonic median of the extremes, while the fourth is half of the first.

The last two terms can be considered as the first two terms of a new proportion in which the fourth term is, as in the case of the previous ratio, half of the first that’s to say 1: 3 and the third term x should be accordingly calculated. Therefore it is

That is

G    C     X    G ( sol    do   x    sol)

and the new tetrachord  G    C    D     G  ( sol    do     re      sol)

So here Reghini is neglecting to point out that actually the harmonic mean was created by Archytas in order to create the geometric mean, whereas for Philolaus, the harmonic mean was still the "subcontrary mean." Only unless someone is already an expert on the topic would they note this cover-up. And so as I have referenced, for the harmonic mean, only a value greater than 1 is allowed to match the equation "harmonic mean x arithmetic mean = geometric mean squared." This condition of the equation covered up the noncommutative empirical truth of the music theory. So in point of fact - 2/3 is F not G as Reghini claims and then 2/3 as F had to be converted from a Perfect Fifth to 4/3 as F not 3/4 as F as Reghini claims.

It's quite possible that Christian Giudice, the author of the Ph.D. thesis on Reghini is not even aware of these logical fallacies in Reghini's argument - again since this "bait and switch" cover-up of the noncommutative truth is pervasive in the Western math-music tradition, even those claiming to be Pythagorean.

So now we go on to the "current" number theory music analysis - Nonlinear Dynamics, the Missing Fundamental, and Harmony.

Since this is a "hard science" analysis we get a more honest approach but the same paradoxes of mathematical logic arise - paradoxes that only Luigi Borzacchini is not afraid to discuss openly in terms of music theory.

So in this analysis of pitch perception at first the spectral versus Fourier model is presented, but still the assumption is that the math is symmetric, whereas we know that time-frequency uncertainty is from noncommutative resonance.

Then it is assumed in their analysis that the Fibonacci number series starts with zero or 0 and therefore can be the reciprocal of the golden mean, based on geometric convergence. But this concept of zero is only after the symmetric geometric mean number system is introduced by Archytas and Plato - the "negative infinity" that Aristotle was against!

Third the bias against rational numbers is revealed by stating that the synchronized resonance is too easily destroyed by "external forces." Apparently this is the limitation of their listening process! For we now know that internal listening as the limit of time-frequency uncertainty, is bypassed by up to ten times a greater level than the noncommutative limit in geometric math. The highest frequency to be heard, sounds like a click, in the trade off of time as shorter wavelength, yet the perceived pitch only shifts to the limit of the noncommutative Perfect Fifth. I cited this research in an earlier blogpost.

At that point sound transcends the limitation of an "external force" and the listening interacts with the internal resonance of the mind-body as ultrasound is resonated from the highest pitch heard externally, while ELF subharmonics are "side-band" resonances of the ultrasound activating the vagus nerve and finally the quantum consciousness of the microtubules is activated.

So in this "nonlinear" analysis then states how the "convergent to the golden number" has 2/3 after the octave but this "breaks the symmetry of the scale." That is their "honest" confession of the loss of the commutative property. So instead they switch back to the original logarithmic symmetric approach - the "bait and switch" that covered up the noncommutative truth.

Symmetry is meant in the Greek sense....If we take logarithms of all quantities the symmetry becomes the usual sort and the geometric mean can be viewed as a mirror.

And so this more honest approach, unlike Reghini, starts with pure number, not assuming a geometric line. So then assuming you have the music notes as 1, 1/2 and then a mystery "x" - suddenly the symmetry is lost again when the music notes are maintained as ratios.  1/2x = 3/4 as "a" while "b" = 2/3 and this is called the "hidden solution" or golden mean solution that breaks the symmetry.

If we equate the two ratios, a=b, this gives for x the geometric mean of 1/2...For the geometric mean the two ratios are equal; for the rational 2/3 there is one interval greater than the other and the permutation corresponds to the exchange of these. If we include this hidden solution, 3/4, we reestablish the symmetry as if a mirror were placed at the golden mean.

So we now have the "bait and switch" out in the open - because at first the golden mean convergence is the "hidden solution" that breaks the symmetry and now it's claimed that 3/4 as the geometric mean convergence is the new "hidden solution" that re-establishes the symmetry! In other words literally two wrongs makes a right. This is a quite fascinating argument and similarly I have made the connection between the Golden Ratio and the geometric mean via this secret reversing of the order of infinity - to the negative infinity quadratic solution. Kepler was against this reversal to the negative infinity - but Newton was not only for it - he also realized this secret music connection!!

So they state the golden mean convergent of 2/3 is then divided by the geometric mean convergent that then corresponds to the golden mean convergent. This "breaks the symmetry" but the symmetry can be re-established again by another geometric mean convergent inversion. This is called the Golden Scale.

I actually revealed this secret in similar form when I discovered the equation of converting the continued proportion ratios to the cube root of two. As the writers state, this method they use of exchanging the Golden Mean convergents and geometric mean convergents then meets at the square root of two irrational number as the tritone music interval.

This is curious because we are forced to choose a note that is essentially different to the others, having an irrational interval.

No they are not "forced" - but on the contrary it was their bait and switch method to maintain symmetry that assumed a geometric irrational foundation to the logic from the beginning, as I have revealed.

...the symmetry axes are geometric means between neighboring convergents of the golden section.

So their math is fascinating and does reveal a deeper layer to the "bait and switch" symmetry secret - and it corroborates the equation that I discovered as well! But in the end by ignoring or covering up noncommutative time-frequency resonance, the real truth of the Pythagorean Orthodox philosophy, the writers limit the internal listening resonance as real alchemy training based on quantum biology or Eco Echo.