So the above is the Philolaus system...
But it had to deal with the "geometric mean" as irrational or alogon Number...
Hence Boethius translating Philolaus - by Boethius MISunderstanding Philolaus:
So we can NOW see that those waves are the overlapping 2/3 and 3/2 wavelengths to the octave...
As geometric ratios but as soon as the root tonic as a NUMBER is considered then you get the noncommtuative number paradox....
Cartwright, J.H.E., González, D.L., Piro, O. (2021). Dynamical Systems, Celestial Mechanics, and Music: Pythagoras
Revisited. Math Intelligencer 43, 25–39.
https://arxiv.org/abs/2104.00998
Gioseffo Zarlino reintroduced the Pythagorean paradigm into Renaissance musical theory. In a similar fashion, Nicolaus Copernicus, Galileo Galilei, Johannes Kepler, and Isaac Newton reinvigorated Pythagorean ideas in celestial mechanics; Kepler and Newton explicitly invoked musical principles. Today, the theory of dynamical systems allows us to describe very different applications of physics, from the orbits of asteroids in the Solar System to the pitch of complex sounds. Our aim in this text is to review the overarching aims of our research in this field over the past quarter of a century. We demonstrate with a combination of dynamical systems theory and music theory the thread running from Pythagoras to Zarlino that allowed the latter to construct musical scales using the ideas of proportion known to the former, and we discuss how the modern theory of dynamical systems, with the study of resonances in nonlinear systems, returns to Pythagorean ideas of a Musica Universalis.
So as ratios they can change the direction in disregard to the root tonic value...
So then Kepler took the math from Archytas!
So how is the noncommutative phase currently covered up in the argument of Cartwright?
If you read this quote - you discover the "bait and switch" - it is claimed that the "geometric mean was NOT used for the calculation of harmonic intervals.
YET the Octave was indeed the "MOST" Harmonic Interval as the Geometric Mean Squared!!
Oops. So indeed the Geometric Mean WAS used to cover up the "double octave" necessary to hide the noncommutative phase to the Root Tonic of the "one" as a listening meditation.
And thus the "bait and Switch" scam continues!!
So Cartwright, et. al. quote Rameau that the fundamental is ALWAYS the lowest - but the "Phantom Tonic" means that a Perfect Fourth interval above the "fundmental" - due to it NOT being a natural overtone of the root tonic, thereby creates a NEW root tonic as an octave below - or the "Phantom Tonic" as the Perfect Fifth, noncommutative to the original root tonic....
And that is the secret of the Taiji or "voidisyinyang" that Cartwright neglects to mention, by quoting Rameau:
Yet as the same time Cartwright establishes that if there is a removal of the fundamental frequency and nearest overtones - and then a phase shift of the higher overtones all by the same amount - there is also a phase shift of the fundamental frequency.
What this proves is that the overtone series is the true listening based on the ABSOLUTE pitch of the root tonic - and not the Harmonic series already assuming a "symmetry" of invertible ratios with no relationship to the fundamental frequency or "residue" or "trace."
So Alain Connes is emphasizing we do not hear the scale as just a symmetric ratio that adds and subtracts the "harmonics" - rather we hear the ratio as a multiplication or power factor and thus to reverse the order is the noncommutative reversal of the exponentiation.
thanks for reading my master's thesis. Yes the error of the math in that work was revealed to me when a physics major who published music books offered to publish my master's thesis. His name is "Charles Madden" and he admitted he did not understand my argument and rather it was his wife who wanted to publish my master's thesis. So then I read Charles Madden book on music and fractals - so I could try to determine what it was we were misunderstanding. There I found him explaining that the Taiji symbol is not a fractal since the math is not symmetric. In my master's thesis I had claimed that the Taiji was symmetric as a paper I read claimed the Taiji was explained by a logistic equation.
So that was in 2001 and I knew the symmetric math was wrong. Then when I discovered Alain Connes book in 2001, his 'Triangle of Thoughts" - I realized it was the noncommutative logic that I had been called "complementary opposites" up till then. Connes was not explicit in the music theory so I was not sure if he was being metaphorical.
So in 2007 then I published what my music theory noncommutative claim was - that again I had previously been called "complementary opposites."
Then in 2011 Connes published his music theory claim and I discovered his lecture that was posted on youtube - maybe around 2015....
So that's when I realized that indeed the music theory claim that I was making was the same that he is making.
Yes the 5th dimensional math of the noncommutative matrices are difficult to understand.
Since I had learned the "Dirac Dance" as nonlocality of quantum entanglement - intuitively I knew this was connected to the same music theory claim I had realized from Pythagoras with the Perfect Fifth as 2/3 ( C to F) not being allowed in the Western equal-tempered music scale.
So then when I learned that Eddie Oshins argued the Dirac Dance was the same noncommutative logic as the Neigong or qigong training - then that corroborated the noncommutative music theory claim as well.
Now Alain Connes has checked out my research. The thing to realize again is that the "inner product" or inner automorphism of the matrices are noncommutative due to the imaginary number as a phase shift that harmonizes the time and frequency BEFORE any classical amplitude measurement. So technically it's considered a "nonobservable" since it's based on the spectrum of light measurements being noncommutative.
So then the thing to realize is that the one unit bloch sphere is the phase and so it has to be a 720 degree geometric rotation as the "Dirac Dance" - the noncommutativity is not limited to the microquantum scale. So for example as Eddie Oshins points out the outside of the hands are yang and inside are yin and the upper body is yang and lower body is yin. So then the Dirac Dance puts yang against yin and vice versa as a noncommutative rotation.
Eddie Oshins even made a video of this - that Louis Kauffman, math professor, then continued with his math students. You can find it on youtube. It's called the Quaternion Handshake that Kauffman and Oshins devised together.
So then in terms of music theory the frequency is the modulation of the phase in the opposite coordinate. So then when you have the square root of negative one as the imaginary number that is based on the noncommutative phase included with zero and 1. This is what I quote Connes on but the quote is not really clear.
So that is the mathematical structure that is equivalent to then what is squared as the square root of two for the equal-tempered tuning. In other words Connes is revealing that the discrete "double quotient" of the rational numbers are a more dense noncommutative spectrum of time and frequency. This can then be squared due to the phase shift of the imaginary number since the negative frequency axis then "cancels out" due to the squaring as the number -1 number. So you get the square root of number 1 as the inverse frequency of the Dirac Operator along with the zero and the one.
The "chords" then are Odd or even - just as with the 2 and 3 of the Perfect Fifth being noncommutative - such that the overlapping of the chords determines whether they are odd or even. These rational discrete ratios as chords then overlap and twist as the 5th dimension of time that can not be seen - INSIDE the 3 dimensional torus. So there is a higher dimensional "rotational" due to the time-frequency phase shift inside the 3 dimensional space as an irrational symmetric continuum.
So therefore space itself along with time are both originating from the noncommutative time-frequency phase shift of the 5th dimension that is inherently unobservable. Based on my own quantum biology analysis from my own music training experience - therefore this 5th dimension can be listened to as the secret of the paranormal spiritual training.
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