"Because as you know an octave is the combination of 2/3 and 3/4. Not 2/3 and 2/3. That is after all rather simple mathematics."
Yes you are doing the "bait and switch" - 2/3 is C to G and 3/4 is C to F but as you know 3 can not be the denominator of the overtone series. So the 3/4 is constructed from 2/3. I've detailed that from Professor Richard McKirahan translating Philolaus. Did you not read that? The Harmonic Series assumes a logarithmic ratio but is not the same as the overtone series. As I've detailed - Dr. Josh McDermott, MIT Professor, has documented that people hear consonance based on overtone frequencies - NOT Western logarithmic ratios. As I've detailed math professor Alain Connes emphasizes that music theory is from noncommutative phase due to the above inverse relation to the octave pitch. As I've detailed nonwestern music also emphasizes this truth.
I'm sorry xxx - I didn't read your quote closely enough to notice your error.
"Because as you know an octave is the combination of 2/3 and 3/4. Not 2/3 and 2/3. That is after all rather simple mathematics."
As I quote Archtyas - the Western pitch is made from 3/2 as C to G and 3/4 as C to F. So you are correct it's NOT made from "2/3 and 2/3." It's made from 3/2 and 2/3. You'll just have to actually study my research in more detail. thanks.
"I didn't say I hadnt read his [Connes] papers, only the detail. I haven't read the detail because it is not relevant to Music Theory or to Pythagorean Intervals."
This is not true at all. Read Connes work. I'll give you some quotes of Connes.
“standard music suggests dealing with new shapes which are quantum such as the quantum 2-spheres....On the other hand the stretching of geometric thinking imposed by passing to noncommutative spaces forces one to rethink about most of our familiar notions. ...And it could be formalized by music….
"I think we might succeed in this way to educate the human mind to deal with polyphonic situations in which several voices coexist, in which several states coexist, whereas our ordinary logical allows room for only one. Finally, we come back to the problem of adaptation, which has to be resolved in order for us to understand quantum correlation and interrelation which we discussed earlier, and which are fundamentally schizoid in nature. It is clear that logic will evolve in parallel with the development of quantum computers, just as it evolved with computer science. That will no doubt enable us to cross new borders and to better integrate the mathematical formalism of the quantum world into our metaphysical system....
"When Riemann wrote his essay on the foundations of geometry, he was incredibly careful. He said his ideas might not apply in the very small. Why? He said that the notion of a solid body of a ray of light doesn't make sense in the very small. So he was incredibly smart. His idea, I have never been able to understand his intuition...But however he wrote down explicitly that the geometry of space, of spacetime, should be encapsulated, should be given by the forces which hold the space together. Now it turns out this is exactly what we give here...One day I understood the following: That we are born in quantum mechanics. We can not deny that... Quantum mechanics has been verified. The superposition principle has been verified. The spin system is really a sphere. This has been verified. This has been checked so many times. That we can not say that Nature is classical. No. Nature is quantum. Nature is very quantum. From this quantum stuff, we have to understand our vision, our very classical, because of natural selection way of seeing things can emerge. It's very very difficult of course. ...
"Why should Nature require some noncommutativity for the algebra? This is very strange. For most people noncommutativity is a nuisance. You see because all of algebraic geometry is done with commutative variables. Let me try to convince you again, that this is a misgiving. OK?....Our view of the spacetime is only an approximation, not the finite points, it's not good for inflation. 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....What is a parameter? The parameter is time...If you stay in the classical world, you can not have a good set up for variables. Because variables with a continuous range can not coexist with variables of discrete range. When you think more, you find out there is a perfect answer. And this answer is coming from quantum mechanics....The real variability in the world is exactly is where are you in the spectrum [frequency] of this variable or operator. And what is quite amazing is that in this work that I did at the very beginning of my mathematical studies, the amazing fact is that exactly time is emerging from the noncommutivity. You think that these variables do not commute, first of all it is that they don't commute so you can have the discrete variable that coexists with the continuous variable.
"What you find out after awhile is that the origin of time is probably quantum mechanical and its coming from the fact that thanks to noncommutativity ONLY that one can write the time evolution of a system, in temperature, in heat bath, the time evolution is really coming from the noncommutativity of the variables....You really are in a different world, then the world of geometry, which we all like because we all like to draw pictures and think in a geometric manner. So what I am going to explain is a very strange way to think about geometry, from this point of view, which is quite different from drawing on the blackboard...I will start by asking an extremely simple question, which of course has a geometrical origin. I don't think there can be a simpler question. Where are we?....
"The mathematical question, what we want, to say where we are and this has two parts: What is our universe? What is the geometric space in which we are? And in which point in this universe we are. We can not answer the 2nd question without answering the first question, of course....You have to be able to tell the geometric space in an invariant manner....These invariants are refinements of the idea of the diameter. The inverse of the diameter of the space is related to the first Eigenoperator, capturing the vibrations of the space; the way you can hear the music of shapes...which would be its scale in the musical sense; this shape will have a certain number of notes, these notes will be given by the frequency and form the basic scale, at which the geometric object is vibrating....The scale of a geometric shape is actually not enough....
"However what emerges, if you know not only the various frequencies but also the chords, and the point will correspond to the chords. Then you know the complete thing....It's a rather delicate thing....There is a very strange mathematical fact...If you take manifolds of the same dimension, which are extremely different...the inverse space of the spinor doesn't distinguish between two manifolds. The Dirac Operator itself has a scale, so it's a spectrum [frequency]. And the only thing you need to know...is the relative position of the algebra...the Eigenfunctions of the Dirac Operator....a "universal scaling system," manifests itself in acoustic systems....There is something even simpler which is what happens with a single string. If we take the most elementary shape, which is the interval, what will happen when we make it vibrate, of course with the end points fixed, it will vibrate in a very extremely simple manner. Each of these will produce a sound...
"When you look at the eigenfunctions of the disk, at first you don't see a shape but when you look at very higher frequencies you see a parabola. 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 [frequency with the same area], even though they are geometrically different [not isomorphic]....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 [triangle, etc.]. 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 [2, 3, infinity] 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....The only thing that matters when you have these sequences are the ratios, 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 [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... .There is a quantum sphere with a geometric dimension of zero...I have made a keyboard [from the quantum sphere]....
"This would be a musical instrument that would never get out of tune....It's purely spectral....The spectrum of the Dirac Operator...space is not simply a manifold but multiplied by a noncommutative finite space......It is precisely the irrationality of log(3)/ log(2) which is responsible for the noncommutative [complementary opposites as yin/yang] nature of the quotient corresponding to the three places {2, 3,∞}. The formula is in sub-space....Geometry would no longer be dependent on coordinates, it would be spectral...The thing which is very unpleasant in this formula is the square root...especially for space with a meter....So there is a solution to this problem of the square root, which was found by Paul Dirac....It's not really Paul Dirac, it is Hamilton who found it first...the quaternions is the Dirac Operator....
"Replace the geometric space, by the algebra and the line element...for physicists this thing has a meaning, a propagator for the Dirac Operator. So it's the inverse of the Dirac Operator.... You don't lose anything. You can recover the distance from two points, in a different manner....but by sending a wave from point A to point B with a constraint on the vibration of the wave, can not vibrate faster than 1; because what I ask is the commutator of the Dirac Operator is less than 1...It no longer requires that the space is connected, it works for discrete space. It no longer requires that the space is commutative, because it works for noncommutative space....the algebra of coordinates depends very little on the actual structure and the line element is very important. What's really important is there interaction [the noncommutative chord]. When you let them interact in the same space then everything happens....You should never think of this finite space as being a commutative space. You have matrices which are given by a noncommutative space...
"To have a geometry you need to have an inverse space and a Dirac Operator...The inverse space of the finite space is 5 dimensional....What emerges is finite space... a point of the geometric space "X" can be thought of as a correlation ...which encodes the scalar product at the point between the eigenfunctions of the Dirac operator associated to various frequencies, i.e. eigenvalues of the Dirac operator....
"It's related to mathematics and related to the fact that there is behind the scene, when I talk about the Dirac Operator, there is a square root, and this square root, when you take a square root there is an ambiguity. And the ambiguity that is there is coming from the spin structure.... We get this formula by counting the number of the variables of the line element that are bigger than the Planck Length. We just count and get an integer.... There is a fine structure in spacetime, exactly as there is a fine structure in spectrals [frequencies]....Geometry is born in quantum space; it is invariant because it is observer dependent....
"Our brain is an incredible ...perceives things in momentum space of the photons we receive and manufactures a mental picture. Which is geometric. But what I am telling you is that I think ...that the fundamental thing is spectral [frequency]....And somehow in order to think we have to do this enormous Fourier Transform...not for functions but a Fourier Transform on geometry.
"By talking about the "music of shapes" is really a fourier transform of shape and the fact that we have to do it in reverse. This is a function that the brain does amazingly well, because we think geometrically....The quantum observables do no commute; the phase space of a microscopic system is actually a noncommutative space and that is what is behind the scenes all the time. They way I understand it is that some physical laws are so robust, is that if I understand it correctly, there is a marvelous mathematical structure that is underneath the law, not a value of a number, but a mathematical structure....
"A fascinating aspect of music...is that it allows one to develop further one's perception of the passing of time. This needs to be understood much better. Why is time passing? Or better: Why do we have the impression that time is passes? Because we are immersed in the heat bath of the 3K radiation from the Big Bang?...time 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 12th∼3 to the 19th. [the 19th root of 3 = 1.05953 and 12th root of 2=1.05946] 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....This means it is a zero dimensional object! But it has a positive volume!...
"As explained in the talk, there is a beautiful space which has the correct spectrum: the quantum sphere of Poddles, Dabrowski, Sitarz, Brain, Landi et all. ... We experiment in the talk with this spectrum and show how well suited it is for playing music. The new geometry which encodes such new spaces, is then introduced in its spectral form, it is noncommutative geometry, which is then confronted with physics....Algebra and Music...music is linked to time exactly as algebra is....So for me, there is an incredible collusion between music, perceived in this way, and algebra....
"I believe that this variability is more fundamental than the passing of time. And that it's behind the scene, meaning that the passing of time is a corollary of this....it's exactly the variability of which value you get in the spectrum of the operator....It's associated with a specific spin structure; You have the chirality and the charge conjugation operator....
"Chirality is put in in your algebraic formalism. You PUT the chirality. It's an operator and two spinors, so you put it there. ..This is quite tricky. If you want, what happens...when you write it because,...it's a two by two matrix....it's a map to the [quantum] two-sphere....This is a totally anti-symmetric tensor.... "
Fields Medal math professor Alain Connes (compilation of quotes)
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