Monday, February 14, 2022

The Residue Pitch: The phase shift of difference tones is proven to be linear and absolute of the Fundamental Frequency as a residue: Cartwright, Gonzalez, Piro

 It is the Pythagorean one but there is coherence due to noncommutativity as Alain Connes explains.

I guess we disagree about what Connes thinks about music theory. haha. I have a new article about to be published that gives the details from Connes. The quotes from Connes and Michio Durdevich quoting Connes confirms my claim - with further details of course.

As Connes [8] emphasizes
“It is precisely this lack of commutativity between the line
element and the coordinates on a space [between ds and the
a ∈ A] that will provide the measurement of distance.”

And as it relates to music theory, Connes again:
"It is precisely the irrationality of log(3)/ log(2) which is responsible for the noncommutative nature of the quotient corresponding to the three places {2, 3,∞}. "
And Connes elaborates in his lecture posted on youtube:

"If you want the dimension of the shape you are looking at, it is by the growth of these eigenvariables. When talking about a string it's a straight line. When looking at a two dimensional object you can tell that because the eigenspectrum is a parabola.... They are isospectral, even though they are geometrically different ....when you take the square root of these numbers, they are the same [frequency] spectrum but they don't have the same chords. There are three types of notes which are different....What do I mean by possible chords? I mean now that you have eigenfunctions, coming from the drawing of the disk or square .... If you look at a point and you look at the eigenfunction, you can look at the value of the eigenfunction at this point.... The point [zero in space] makes a chord between two notes. When the value of the two eigenfunctions will be non-zero. ...The corresponding eigenfunctions only leave you one of the two pieces; so if there is is one in the piece, it is zero on the other piece and if it is non-zero in the piece it is zero there...You understand the finite invariant which is behind the scenes which is allowing you to recover the geometry from the spectrum....Our notion of point will emerge, a correlation of different frequencies...The space will be given by the scale. The music of the space will be done by the various chords. It's not enough to give the scale. You also have to give which chords are possible."

In Connes’ later published book version, there is crucial clarification:
“The fact that the ratio log 3/log 2 is only approximated by the rational number 19/12 is responsible for the difference between the ‘circulating temperament’ of Baroque music (e.g. the Well Tempered Clavier) and the ‘equal temperament’ of XIX century music” (Connes & Marcolli, 2007, p. 388).

And Connes again:
"the ear is only sensitive to the ratio, not to the additivity...multiplication by 2 of the frequency and transposition, normally the simplest way is multiplication by 3...2 to the power of 19 is almost 3 to the power of 12....time [spacetime] emerges from noncommutativity.... What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers qn for the real number q=2 to the 1/12th∼3 to the 1/19th. Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number. (Connes, 2012)
And Connes again:
"It explains the spectrum of the guitar because when you raise the number 2 to the power of 19, we get practically the number 3 raised to the power 12. It can’t be a tie because when we raise 2 to the power 19, we get an even number. When we raise 3 to the power 12, we get an odd number. So it can’t be an equality….Because if we calculate its size using what I told you before, we obtain that it is an object of dimension 0, an object of dimension 0 in the sense that its dimension is smaller than any number, not zero but positive. (Connes & Prochiantz, 2018)"
And Connes again:
"But the inverse space of spinors is finite dimensional. Their spectrum is SO DENSE that it appears continuous but it is not continuous.... It is only because one drops commutativity that variables with a continuous range can coexist with variables with a countable range."
And Connes again:
"multiplication by 2 of the frequency and transposition, normally the simplest way is multiplication by 3...2 to the power of 19 [524288] is almost 3 to the power of 12 [531441]....You see what we are after....it should be a shape, it's spectrum looks like that...We can draw this spectrum...what do you get? It doesn't look at all like a parabola! It doesn't look at all like a parabola! It doesn't look at all like a straight line. It goes up exponentially fast...What is the dimension of this space?...It's much much smaller. It's zero...It's smaller than any positive.... Musical shape has geometric dimension zero... You think you are in bad shape because all the shapes we know ...but this is ignoring the noncommutative work. This is ignoring quantum groups. There is a beautiful answer to that, which is the quantum sphere..:"
So Connes calls this the "double quotient" of rational integers.
The noncommutative secret - if you study it is the same as what this academic points out:
Dr. Oscar Abdounor explains:
"Anachronistically speaking, this means that, supposing both these both cycles meet, there would be m and n integers such that (2:3)n = (1:2)m, that is, 3n = 2m+n, which is impossible, since the left term is odd and the right is even. (Abdounur, 2015)"
and this academic:
"This musical property is the counterpart of the principle mathematical characteristic of the Pythagorean diatonic, very Pythagorean indeed, constituted by the fact that each interval of the scale is expressed by the ratios of type 2 to the m divided by 3 to the n or 3 to the m divided by 2 to the n. (Bellissima, 2013)"
So just study the math again as Connes is inverting the ratios to prove they are noncommutative.
So that the exponential uses 3 to the 12th and 2 to the 19th while the inverse logarithm uses 2 to the 1/12th and 3 to the 1/19th.
THAT's what makes them noncommutative as the irrational number.
If one studies quantum physics more then it's understood he's making the same point about the "inner automorphisms" of the Dirac Operator whereby the discrete diagonals of the matrices are noncommutative.
 Here is a good intro vid on it:
https://www.youtube.com/watch?v=PnQRfZ05_Fg
We just have a different understanding of Connes. I'll leave you with this quote from math professor Micho Durdevich who cites Connes:
"On the other hand, the ancient Pythagorean musical scales naturally lead to a simple quantum circle…. will be quantum (noncommutativity of the algebra V) ....Let us outline a hybrid re-formulation, applicable to every discrete group G freely acting (say, on the right) on a compact topological space X . The Pythagorean
example is given by G = Z and X = © [the circle], with the described irrational twist defining
the action,...
The whole non-commutativity of the algebra comes from non-commuting U and
U ∗. If they commute, then we are back in classical geometry, at the ‘one-particle’
level (and in particular U ∗ = U −1 ). However, even in this case there is a highly
non-commutative world of higher-order collectivity algebras Bn . This can be used
to capture the geometry of rotations, like those appearing in classical Pythagorean
octave versus perfect fifth considerations."
and
"In the irrational case, as the one associated to the Pythagorean musical
scale, the torus space exhibits a total wholeness, no divisibility at all (the C*-algebra
is simple). However these projectors can be used to construct a surjective map from
the quantum torus space, over the extremely disconnected Cantor triadic set."
You can read the whole article if you use "sci-hub" - and enter in the doi:
https://sci-hub.se/10.1007/978-3-319-47337-6_11

So what Durdevich is explaining is that the rational numbers in the noncommutative torus create a nonlocal overlap while the Pythagorean noncommutative Perfect Fifth and Octave (as I quoted Connes in the previous response to you) provides a noncommutative limit or "irrational twist" as Durdevich calls it. This noncommutative limit is due to the "double quotient" of the discrete Pythagorean numbers.
So that is how the quantum noncommutativity is "converted" back to the symmetric commutative math. But it originates from the rational numbers due to the 2/3 undertone as the Perfect Fifth (or C to F) - not being allowed by the ET spectrum scale.
"With his theories of “Music before Medicine” and “Music is also Medicine”, he has revealed the lost remedies of ancient Chinese music therapy and became the first person to introduce Five Tones Therapeutic Music to the modern world. Over the last decade, Professor Wu spent a great deal of time studying ancient medical texts such as the “I-Ching—Five Tones and Eight Sounds”, “The Yellow Emperor’s Classic of Internal Medicine – Five Tones and Five Major Organs” and “Twelve Scales and Twelve Meridians”. He has combined music melody with physiology to create a systematic and organized subject with a series of musical therapeutic methods. It contains a strong root in ancient philosophy and is differentiated from the Western method of seven scales. The Chinese "lǜ" tuning is closest to the ancient Greek tuning of Pythagoras. "

https://www.swjoyoflife.com/master-shen-wu-and-the-song-of-five-tones/


 I know of the Connes lecture you're referring to as there are two or three versions of it posted on youtube and linked on his website also. I first discovered Connes in 2001 in his book "Triangles of Thought." He is discussing orchestration and how a conductor reading a score is then transposing different clefs as different spacetimes but they are all phase harmonized. So that is his view of equal-temperament having a higher dimension of time that is a cubic noncommutative time. I trust you have done orchestrating training as per your Ph.D. in music (that you refuse to document). haha. 

So you state your math professor got irate when explaining to you how the approximation of rationals is not the same as the irrational number limit. And that Connes is explaining how the real word is noncommutative at its core on a quantum level but on the classical level it is commutative. Yes I understand that argument but that is not the claim in quantum biology and quantum psychology nor in traditional Pythagorean or Daoist music training or other nonwestern meditation music training.

So in the nonwestern tradition the noncommutative rational "double quotients" then keep resonating up to the macrolevel because noncommutativity is not limited to the quantum level. For example quantum physicist Basil J. Hiley who collaborated with David Bohm also emphasizes this point - that there is no need for a "collapse" of the quantum noncommutative math back into a commutative classical math.
Also Professor Hiley collaborated with Roger Penrose who argues that the dark energy is the gravitational entropy or gravitation potential of the photon. So the photon has a hidden "relativistic mass" or reverse time and negative frequency energy from the future. This was first realized by de Broglie in his "Law of Phase Harmony" and it is also noncommutative math.

Essentially the "one" as the fundamental frequency is not a vibrating string or pipe or even an air molecule as Western science assumes but rather we LISTEN in music to the source of the one as a quantum coherence that is noncommutative. So the source of the one is the light that has mass from quantum frequency directly proportional to momentum. 

So it resonates directly back to the noncommutative math so that there is resonance with a negative frequency such that 2/3 as C to F undertone is allowed as the "hidden" or spectral invariant - going on behind the scenes. The music melody is thus BEFORE the "scale" that creates a zero point in spacetime.

It sounds like your math professor friend who got irate truly did not listen close enough to Connes music lecture. I myself had to study his claim from several different sources to figure it out.
I have give you the quotes and details - it's just a matter of you studying it and realizing what he is stating about how the DISCRETE numbers are MORE DENSE than the continuous line. So Connes is "redefining" what "distance" as "length" means.

And this argument goes right back to what an irrational number means - as Charles Sayward and Philip Hugley detail in their 1999 Philosophy article, "Did the Greeks Discover the Irrational?" They argue NO because geometric length is not the same as arithmetic distance. For there to be a "positive" proof of the square root of two as the irrational number there first has to be a geometric "unit" that is inherently tied to the number as distance.

This is Connes point also that the noncommutative spectral frequency with time is BEFORE any coordinate system of a unit because the 2/3 and 3/2 overlaps as a nonlocal "double quotient" ratio of the future and past overlapping.

So this is Basil J. Hiley's point also - that it is ONLY in quantum ALGEBRA as a PROCESS of time that the noncommutative truth is revealed - and there is no need to rely on a visual geometry that attempts to contain infinity with a symmetry.

So this also explains that as Connes points out all of science has been based on commutative algebraic geometry thus far - and therefore considers noncommutative math to be a "nuisance" but also as Penrose explains that commutative gravity is thus a "collapse" of the quantum nonlocal superposition - and therefore the increase in the entropy of matter on Earth as the destruction of life is due to this "collapse" of nonlocal noncommutativity into a lower frequency consciousness that considers gravity to be an objective measurement. So gravitational entropy as the gravitational potential is the noncommutative source of reverse time negentropy that powers the Sun as Schroedinger points out - through quantum entanglement creating fusion - but also powers life on Earth!!

OK so there's absolutely no record of your Ph.D. thesis or you getting any music Ph.D. on the interwebs - and it would be recorded somewhere. You repeatedly keep misspelling noncommutativity and noncommutative and also commutative. So "communitative" is not the correct spelling. Sorry. Clearly you're not versed with the topic. Thirdly your claiming "secret" sources as second hand evidence. That is not valid in an academic argument or any kind of valid evidential argument. Fourth your claiming that because Alain Connes gave his music lecture privately therefore I could not know the subject. I disagree. Connes published his music lecture and he's repeated the SAME lecture precisely at more than two different locations at more than two different times. Finally you keep claiming Alain Connes makes claims but you are not quoting Alain Connes at all while I actually provided the quotes of Alain Connes.
So for all those reasons your argument simply does not have any validity. I tried to give you as much attention as possible. Clearly you're just biased towards a certain perspective against the evidence stating otherwise.
I appreciate you sharing your views as this only inspired me to research Alain Connes more and clarify my own views. haha.
Have a nice life.
 
So in my previous research I pointed out that the Pythagorean Comma has a slightly different value as 3/2 to the 12 approximating 2 to the 7th as Sir James Jeans defines it in his book on Music. Whereas the usual definition of the Pythagorean Comma assumes the logarithmic conversion of 3/2 against 2 into 3 to the 12th approximating 2 to the 19th. So because it is based on multiplicative as listening that is an exponential growth factor then the conversion of "halving" of the ratios back into the octave has to invert the exponentials as being noncommutative! This is why is it then 3 to the 1/19th approximating to the 2 to the 1/12th. And this is why the discrete Pythagorean noncommutative ratios are MORE DENSE than the irrational number continuum. The noncommutative ratios are a "double quotient" as the noncommutative cross multiplication or "inner automorphism" of the matrix math.


I'm clarify for you again about the Pythagorean Comma and Connes discrete noncommutative math. Here is a quote from Connes:
"So if we take the [quantum] 2-sphere, if we take the round sphere, its spectrum this time is very very simple. It is also formed by integers, exactly as in the case of a string. But these integers appear this time with a certain multiplicity, that is to say it's not exactly integers. It is more exactly the root of J(J + 1). ...The shapes on the sphere are different, the sound we hear is the same. [Isospectral but not isomorphic]. And that is what we call spectral multiplicity, that is to say that in the spectrum, what will happen is that we will have the same value, but it will happen multiple times. I will come back to this for the musical shape, that, we will see that later....
"when you make music, in fact, it is not at all integers 1, 2, 3, 4, 5,
etc., as frequencies which are used ? Absolutely not, these are the powers of the same
number, the powers of the same number, that is to say we have a number q. And we
look at the numbers qn, that is what counts, because it is the relationships between
frequencies that count. And the wonder that makes piano music exist, called The
harpsichord well temperated, etc., it is the arithmetic fact that exists, which means
that if we take the number 2 to the power of a twelfth, if you take the twelfth root
of 2, that’s very, very close to the nineteenth root of 3.
See, I gave those numbers. You see that the twelfth root of 2 is 1.059..., etc. The
nineteenth root of 3 is 1.059... Where does 12 come from ?"
"The 12 comes from the fact that there are 12 notes when you make the chromatic
range. And the 19 comes from the fact that 19 is 12 + 7 and that the seventh note
in the chromatic scale, this is the scale that allows you to transpose. So what does
it mean ? It means that going to the range above is multiplying by 2 and the ear is
very sensitive to that. And transpose is multiplication by 3, except that it returns to
the range before, i.e. so it is to multiply by 3 / 2, that agrees.
Well, that’s the music, well known now, to which the ear is sensitive, etc. Okay.
But... there is an obvious question ! It is "is there a geometrical object which range
gives us the range we use in music ?". This is an absolutely obvious question."
"If you look at what is going on, like these are the powers of q, you notice that the
dimension of the space in question is necessarily equal to 0. Why ? Because earlier,
I had shown you its limits. ...So I had shown you earlier that the objects had a range that
looked like a parabola when they were of dimension 2.
When an object is larger, it will be a little more complicated than a parabola.
For example, if it is in dimension 3, it will be y = x to the 1/3, okay, but here, it’s not at
all a thing that is round like a parabola like that...This is something that pffuiittt !
that gets up in the air like that. And what it tells you is that the object in question
must be of dimension 0. So you say to yourself, "an object of dimension 0, What does
it mean ? etc. Well..."
"What I hope one day is that we will find the noncommutative sphere in Nature and
one will be able to use it as a musical instrument and it will be a wonderful instrument because it will never detune." (Connes, 2011)
Connes, A. (2011). Transcript of a conference given by Alain Connes, Duality Between Shapes and spectra, given at the Collège de France, on October 13, 2011.
http://denisevellachemla.eu/mai8-en.pdf

So you see Connes is explaining that BECAUSE it is an inverse discrete frequency of the exponential as 3 to the 1/19 and 2 to the 1/12th against 3 to the 12th and 2 to the 19th THEREFORE it is a quantum sphere noncommutative shape that is a zero dimension in geometry and more "dense" than the commutative irrational numbers.

You probably just need to study Connes more on the math.
Good luck.

Just so you know that chaos theory is based still on logistic symmetric algebraic geometry - not the noncommutativity quantum algebra math. The key point that Connes is making is that because of the exponential growth of the inverse frequency music as chords - therefore it is a geometric dimension of zero yet a positive volume! haha. You are projecting your need for a closed system as a "cognitive bias" as Borzacchini points out on the music origins of Western science. It could be that you lack the early age music training required to increase your corpus callosum that integrates the left and right brains. You ask for what I have added to Connes and Connes himself says that if such a 2-sphere quantum noncommutativity existed in nature then it would be a perfect natural tuning of the Universe. I am adding that it has been proven that humans can hear up to ten times faster than time-frequency uncertainty and this is indicative of the symmetry breaking as noncommutativity from the Josephson Effect via listening to the highest pitch we can hear externally (that then resonates the brain internally as ultrasound). In music theory this is called the "Hypersonic Effect" from natural tuning of nonwestern music.

Thanks for your feedback and take care!
haha.
 
 You need to take more time to examine Conne's mathematics of music theory. His noncommutative claim is due to the discrete rational ratios as noncommutative exponentiation as a 2-sphere or quantum sphere. So the Pythagorean ratios are 2 to the 19th approximating 3 to the 12th with the "double quotient" noncommutative "inverse frequency" as 2 to the (1/12th) as 3 to the (1/19th). The reason they have to be inverted is, as Connes points out, due to the original Pythagorean ratio of (3/2) to the 12th approximating 2 to the 7th. The reason it is 2 to the 7th as Connes points out is due to the 7th interval of the chromatic scale being the Perfect Fifth that creates the noncommutative transposition.

So you started out by stating that the Western scale is based on overtones. Yes that is true but it is also true that the Perfect Fifth as a 2/3 undertone or "C to F" is also true. This is called the "Phantom Tonic" in music theory as I point out. So the Pythagorean Scale does increase the tuning error over the "scale" as you point out but the Perfect Fourth is not created from the Perfect Fifth exponential multiplication of 3/2. Rather the Perfect Fourth is created from 2/3 as the Perfect Fifth ("Phantom Tonic"). The reason it is a Phantom Tonic is because if you are just relying on LISTENING to music then the Perfect Fourth as 4/3 is NEVER an overtone of the "one" or root tonic - despite the Harmonic Series just claiming that 4/3 is a Perfect Fourth from reversing the direction of the octave C to G. 

The reason the direction of the Harmonic Series can be assumed to be reversed is because it already assumes the symmetric ratio from the irrational magnitude.

This is the "cognitive bias" or "cognitive error" that Math Professor Luigi Borzacchini refers to. All human cultures use the Octave, Perfect Fifth and Perfect Fourth but, as I reference, for example in traditional Indian tuning it is well known that if there is a drone of two notes with the Perfect Fourth as the higher frequency it is actually heard as the root tonic due to the Perfect Fourth never being a natural overtone.

So again Connes is pointing out that the discrete Pythagorean Ratios are a geometric zero dimension but that are still a positive volume of frequency energy due to the noncommutative time of the imaginary number as the 5th dimension based on the fact that the exponentiation is an inverse ratio as 3 to the (1/19th) and 2 to the (1/12th).

Thanks for your interest in Connes research and please just keep studying - You'll figure it out. haha.

"Pythagorean comma, 531441/524288, 3 to the 12/2 to the 19th,"
That's the well-established definition. So not sure what "alternative facts" you are relying on. haha.
You keep making claims with no evidence. While I provide quotes.
https://en.wikipedia.org/wiki/Pythagorean_interval
https://en.wikipedia.org/wiki/Pythagorean_comma

Lore and Science in Ancient Pythagoreanism
https://books.google.com/books?isbn=0674539184
Walter Burkert - 1972 - ‎Literary Criticism
to this, the apotome would be 2187: 2048, and the komma 531441 : 524288 — pure frivolity.41 Philolaus' treatment is different:42 He establishes as the basis of tone the number which first makes the cube of the first odd number and was highly honored among the Pythagoreans [i.e., 27] ... a number which is separated by a ...
The Science of Harmonics in Classical Greece
https://books.google.com/books?isbn=1139468626
Andrew Barker - 2007 - ‎Philosophy
It is again obvious that Philolaus is not thinking in terms of ratios alone. The ratio of the komma can be computed;it is 531441:524288, but this–in Burkert's phrase– is 'pure frivolity'.21 Boethius has already told us, in fact (Inst. mus. 3.5), that Philolaus identified the komma [531441 : 524288 aka the Ditonic Comma] with the unit, 1, as being the difference between a diesis ...
From the Beginning to Plato - Page 269 - Google Books Result
https://books.google.com/books?isbn=0415062721
Christopher Charles Whiston Taylor - 1997 - ‎Reference
The 'diesis' should be 256:243 and the 'comma' 531441:524288. Neither of these intervals can be divided in half in the sense of the Sectio Canonis. Since Philolaus seems clearly to recognize that the tone cannot be divided in half, it is rather surprising that he apparently takes for granted—what is false in terms of the ..
"The 4th is the same interval as the 5th except that the Root is then the Octave above. The Root being a term only for reference."
https://nmbx.newmusicusa.org/IV-The-Phantom-Tonic/
Start your learning! haha

Yes you can't get past the word "approximation" to the square root of two.
I already explained that the square root of two is for the SCALE as the Spectrum with a definition of a "zero point" in space as a geometric length.
The Pythagorean ratio is from time-frequency as the fifth dimension that is a ZERO geometric dimension so it is NOT a geometric scale as a line but rather it is a "mathematical structure" that is "hidden" behind the scenes all the time.
Because it is noncommutative therefore the undertone as 2/3 or C to F is NOT allowed in the logarithmic scale.
This is proven due to Philolaus.
You can read Richard McKirahan for details.
https://brill.com/view/journals/bapj/27/1/article-p211_8.xml

Colloquium 7: Philolaus on Number
In: Proceedings of the Boston Area Colloquium in Ancient Philosophy
So what Philolaus did is CHANGE the root tonic since 2/3 was not allowed to create the FIRST logarithmic equation in western science! haha.
So Philolaus flipped his Lyre around and then used 6/8 as the wavelength for an 8/6 frequency or 4/3.
So to get the 4/3 frequency the root tonic was changed to 8 instead of the previous root tonic of 12 for 2/3 as 8/12.
So then the logarithmic equation was then 12 to 8 as 3/2 PLUS 8 to 6 as 4/3 = 2 as the octave.

https://en.wikipedia.org/wiki/Pythagorean_tuning
for more details.
 
 Well the problem is that I have references to back up my claim.
In fact, traditional Indian music tuning, like traditional Chinese tuning, is based on the "three gunas" yoga philosophy that also recognizes the Phantom Tonic truth of music as noncommutativity.
"Our whole universe is trigunatmaka, as it is pervaded by the three gunas....Therefore one gets a wonderful visualization of cosmogony, while doing the elementary Nada-sadhana, with the help of a tanpura, which is one of the oldest musical instruments…. As a result the first musical vibration (Spandan) that has emanated out of the trigunatmaka (relating to the three gunas), equipoise of Cosmic Nature - that alone is the AUM or OMkara....
The most common tuning for the four-string tanpura sets the first string at "Pa," the fifth degree of the scale, while the second and third strings are tuned to the tonic above, and the last string is tuned to the tonic below." (Chandra Dey, 1990, p. 82)
Jairazbhoy notes the Phantom Tonic concept as key to Indian music meditation:
The thesis can be expressed in the following way:
"If two drones either a fourth or fifth apart are sounded, one of these will 'naturally' sound like the primary drone. It is not always the lower of the two which will sound primary, but the one which initiates the overtone series to which the other note (or one of its octaves) belongs. By amplifying a prominent overtone the secondary drone lends support to the primary and intensifies its 'primary' character.
Ma [Perfect Fourth as 4/3], although consonant to Sa (root tonic), is alien to the overtone series and is not evoked in the sound of Sa. On the other hand, Sa is evoked in the sound of Ma, since Sa is a fifth above Ma and is its second overtone."
Jairazbhoy N. A. (1995). The Rags of North Indian Music: Their Structure and Evolution, Popular Prakashan.
Chandra D. S., The Quest for Divine Music (Ashish Publishing House, Delhi, India, 1990)
"While the fundamental is usually heard most prominently, overtones are actually present in any pitch except a true sine wave. The relative volume or amplitude of various overtone partials is one of the key identifying features of timbre, or the individual characteristic of a sound."
Sorry I'm gonna have to disagree with your claim about needing an "instrument." haha.
Music has been made for tens of thousands of years from just singing and hand clapping - no need for external measurements or instruments.

Yes the Chinese scale does not use the octave equivalence but rather the "infinite spiral of fifths" is based on the Perfect fourth and Perfect Fifth as yin and yang.

Most music theorists would dismiss the infinite spiral of fifths as just modal math and thus no different than an "infinite spiral of thirds." The key logical secret missing was noncommutativity. As I quote the Pythagorean music theorists and Connes also explains, the 12 note scale is the commutative line element but the music listening as frequency is multiplicative. 

So the “halving” back into the octave therefore requires an inverse of the ratio of the Perfect Fifth since it is noncommutative to the octave. This is why it is 2 to the 19th and 3 to the 12th because the (3/2) is multiplied for the 12 notes of the octave as the exponential but the (3/2) is also the 7th note in the commutative scale of the 12 notes as the modal transposition against the octave doubling.
So the 2 is divided out and added as 2 to the 19th. 
 
So again as I quote and document the basic Pythagorean ratio is noncommutative due to the multiplying of 3/2 with its inverse within the same octave. It does not have to limit to 19 or 12 as some Pythagorean scales go up to 53 notes, as Alain Danielou constructed. This is why it's called the "Infinite Spiral of Fifths" and it's a Perfect Fourth in the opposite direction assuming the octave equivalence as pitch of doubling. Connes is pointing out that we do not hear pitch as an additive math but as a multiplication of the ratios that is noncommutative and that is the Pythgorean secret. This is why the Perfect Fourth is actually called the "Phantom Tonic" since it's not a natural overtone of the root tonic and therefore requires the noncommutative Perfect Fifth as the undertone to be doubled based on the different root tonic, thereby revealing the secret infinite spiral of fifths truth. In Daoism this is called the "Single Perfect Yang" so that the Perfect Fourth or "yin" music interval is only due to the attempt for an octave equivalence in pitch as doubling. Whereas the empirical truth is the infinite spiral of fifths or what Connes calls, (2, 3, ∞).
Dr. Oscar Abdounor explains:
"Anachronistically speaking, this means that, supposing both these both cycles meet, there would be m and n integers such that (2:3)n = (1:2)m, that is, 3n = 2m+n, which is impossible, since the left term is odd and the right is even.... "It is confirmed by the fact that the notes F and C are the last one...produced by the cycle of fifths if one considers the approximation mentioned before of 7 octaves with 12 fifths" (Abdounur, 2015).

Connes is emphasizing, think again:
"It explains the spectrum of the guitar because when you raise the number 2 to the power of 19, we get practically the number 3 raised to the power 12. It can't be a tie because when we raise 2 to the power 19, we get an even number. When we raise 3 to the power 12, we get an odd number. So it can't be an equality.... Because if we calculate its size using what I told you before, we obtain that it is an object of dimension 0, an object of dimension 0 in the sense that its dimension is smaller than any number, not zero but positive." (Connes & Prochiantz, 2018)

"This musical property is the counterpart of the principle mathematical characteristic of the Pythagorean diatonic, very Pythagorean indeed, constituted by the fact that each interval of the scale is expressed by the ratios of type 2 to the m divided by 3 to the n or 3 to the m divided by 2 to the n. " (Bellissima, 2013)

 There's something called "noncommutative calculus" using reverse time and negative frequency. That's what I thought of when I was taught the Pythagorean Theorem. I didn't know the math but I knew the Pythagorean Theorem was from music theory as an approximation of 9/8 cubed = the Tritone as square root of two (the Devil's Interval). So I was always secretly against the Pythagorean Theorem and then I was taught "Dirac's Dance" my first year of college when I took quantum mechanics from Professor Herbert J. Bernstein who is now having his quantum teleportation satellite communication system tested by NASA.

So essentially at each zero point in space there already is a noncommutative phase of the Perfect Fifth as 2/3 and 3/2. Why? Because due to the octave in the music scale the undertone as C to F or 2/3 is not allowed yet 4/3 is also not allowed as the Perfect Fourth because the overtone needs the root tonic in the denominator (or an equivalent octave pitch). 
 
This is called the "Phantom Tonic" in music theory but one electrical engineer called it the "Quantum Undertone." He just passed on last year. Anyway he said that the ancient Greeks understood this "quantum undertone" as reverse time, negative frequency if you really study their projective geometry - that also inspired the Dedekind Cut to justify the irrational number continuum.
 
It's a proven fact now that the very first logarithm was created from music theory that covered up this truth of the Perfect Fourth being created from a Perfect Fifth undertone of the double octave. This is why the "Fundamental Frequency" in physics has a 2/1 wavelength while the "first harmonic" has an octave or 1/2 wavelength as the "string" within the wavelength. The 2/1 wavelength of the "fundamental frequency" is assuming a "harmonic mode" as an infinite wave in Fourier.
 
But Fourier also assumes a linear operator of time-frequency as "time-frequency uncertainty." Yet it's now proven in music that we can listen up to ten times faster than Fourier Uncertainty and this also is due to the noncommutative quantum coherence as nonlocality at the foundation of reality. A good talk on this is by Fields Medal Alain Connes math professor - on youtube. He has several versions of the talk on youtube. He first gave the talk in 2011 on the "shape of spectras." He was studying the spectra of a drum - the classic math study, "Can you hear the shape of a drum?" The answer is no because it is noncommutative. So he realized that sound is also inherently noncommutative as an acoustic system and therefore it has a geometric dimension of zero but it does have positive volume due to the eternal frequency-time nonlocal resonance.
 
It technically has to be called a new quantum "force" since it's before the squaring of the imaginary number to get the real number magnitude. So the negative frequency and reverse time is in the imaginary frequency axis but it has a virtual photon phase that can be listened to and resonated.

 OK yes the convergence is due to the noncommutativity. This is Connes point. I am not plagiarizing as you are accusing. It's quote the opposite. I have provided you with the quotes from Connes and you have chosen to ignore them. haha.
Also I have provided you with the actual historical proof that the first commutative magnitude was created from music theory by Philolaus flipping his lyre around! This is what noncommutative means - the change in direction then changes the value.

So it's quite simple if you think about it. The logarithm means that "subtracting" the Perfect Fourth from the Perfect Fifth = 9/8. Yet from the Pythagorean 3/2 it is multiplied as listening to the frequency as Connes emphasizes and therefore it is NOT a logarithmic adding of the intervals - again as Connes clearly states. So therefore the 9/8 was actually derived from 8/6 as the music interval of 4/3 Perfect Fourth from the root tonic of 8 as the wavelength. 8 also being the Perfect Fifth as the undertone of 12 for the frequency of 8/12 and wavelength of 12/8.

Now you are claiming they do not converge but this is also proven wrong.
"Aesthetics, dynamics, and musical scales: a golden connection"
JHE Cartwright, DL González, O Piro, D Stanzial
Journal of New Music Research 31 (1), 51-58
https://www.academia.edu/1795960/Aesthetics_Dynamics_and_Musical_Scales_A_Golden_Connection
So you can see he is arguing the Perfect Fourth and Perfect Fifth converge to the square root of two.
But he ignoring the noncommutative truth of the Perfect Fifth that Alain Connes recognizes.
Alain Connes grew up playing music from a young age and his mom was a music teacher - so Connes understands the truth of music. haha. He still practices Chopin at home, etc.

"The octave interval, defined by the notes of frequency 1=2, the tonic, and 1=1, its superior octave, is divided by its geometric mean p1=2 as shown. The interval is defined by the first two convergents of the golden number, 1=1 and 1=2, to which we have added the next convergent, 2=3. However, this breaks the symmetry of the scale. There exists another solution which consists of the permutation of the short and long intervals defined by 2=3, i.e. 3=4. This solution can be viewed as that symmetric to 2=3 through the symmetry axis p1=2. Symmetry is meant here in the Greek sense, that is, as an equality of ratios, i.e. (2=3)=p1=2 = p1=2=(3=4). If we take logarithms of all quantities the symmetry becomes the usual sort and the geometric mean, p1=2, can be viewed as a mirror."

 

So in other words the 9/8 as major 2nd is from 9/4 that is halved back into the octave but in fact it originates from 6/8 wavelength (such that the denominator is not of the same root tonic as the fundamental frequency) or 8/12 frequency (from a 12/8 wavelength) now as 8/6 or 4/3 frequency with the 8 as the new root tonic because Philolaus flipped his lyre around.


Nonlinear Dynamics, the Missing Fundamental, and Harmony

 


Fascinating!!



The residue pitch


Where does the "quadrivium" come from? I do not know, but I am very
interested in 'divining' about Pythagoreanism.... but the best is, in my
opinion, the history of Athens' organization, as performed by Clistenes,
Pericles, etc. and 'sublimated' in Plato's "Laws", where base 12 appears
throughout when there is a whole to divide and base 10 when there is a
group to build. (The easiest and trivial rationale could be the base-12
in astronomy in the passage from the lunar to the solar calendar, and
the base-10 in hands-counting.)

 

You can find this online as "The Rotten Root."
https://www.nonduality.com/whatis8.htm
So the phrase "rotten root" is a pun on the square root of two being rotten.
You can find the details in my "Actual Matrix Plan" expose wherein I expose the "music logarithmic spiral" as evil.
https://www.bibliotecapleyades.net/ciencia/ciencia_matrix43.htm


Bellissima, F. (2013). Epimoric Ratios and Greek Musical Theory, in Language, Quantum, Music edited by Maria Luisa Dalla Chiara, Roberto Giuntini, Federico Laudisa, Springer Science & Business Media, Apr 17.

Notice how it's in a book about quantum physics and music? haha.
 
the "measurement problem" of quantum physics.

Yes this is what Connes is pointing out - the "line element" is the "system" or "scale" of the coordinates but it is NOT the "chord" or "melody" of the coordinates.

So Connes is arguing that frequency and time enable a location in space to be found without any SYSTEM of coordinates but purely based on noncommutative time-frequency of a zero dimension of geometry.

So the ONLY way the "zero" point in space is created is because the Perfect Fifth as 2/3 is not allowed since it is noncommutative to the octave even though empirically as the undertone it exists as the inverse frequency.

So let me corroborate the claim about the importance of the "root tonic" for listening to music.
 
Yes music is found in all human cultures and if you look at our original human culture they did not rely on musical instruments - it was singing and hand clapping. So this is my point - there is no external measurement needed for a musical instrument.

So I did grow up playing a Steinway grand piano - so I am aware of the practicalities of equal-tempered tuning.

As I mentioned I did study orchestration so I know how to "transpose" different "spacetimes" into a zero point of harmonic phase.

But Connes is stating that before space is created as a zero point there already is a 5th dimension that is noncommutative time-frequency and also nonlocal.

 
 Actually I have precedence over Connes because I published in 2007 my "Secrets of Psychic Music" article that argues for the noncommutative Perfect Fifth of C to G and G to C and I reiterated that claim in 2008.

So then I point out that listening to music - as Connes emphasizes - is not just noncommutative but it is proven to be up to ten times faster than Fourier Uncertainty. The Heisenberg Uncertainty that you refer to originates from Fourier Uncertainty because as Louis de Broglie discovered the quantum momentum is directly proportional to the frequency.


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