The octave harmonic is an identity function that includes both 1 and 0. The square matrix is (1, 0) top row and (0, 1) bottom so the determinant is 1. The fundamental is (0, 1) which is a unit and direction orthogonal to the frequency line. The octave is the point 1,1, equivalent to (0, 1).
That makes the space orthonormal. The inner product is always zero and the distance is always 1. That's an eigen function, isn't it?
The octave is the metric of the space and any subset of the octave must have the same metric.
The point here is that the next overtone does not have the octave metric because the fundamental is 1/3.
https://www.researchgate.net/post/How_can_a_string_have_more_than_one_mode_of_vibration_at_the_same_time
The question of defining my terms does not solve your problem of explaining how the string can have more than one mode of vibration. Obviously, it can't. It is the followers of Mersenne that are confused. Think about it: the string is not twice as long as it looks! The frequency we hear is not twice the wavelength. Any string detained by two points can only have one mode of vibration. The amplitude and the frequency are independent operators (that is, inner product is zero).
I just sent him a personal message on facebook through my relative's account. I told him to watch Alain Connes video on music theory.
And earlier Terence B. Allen states:
I think that the series 1/n is correct because its inverse induces a series of whole numbers that are the simple multiples of the fundamental 1, 2, 3, ...
In classic wave theory they say the series is 2/n but the reciprocal of this induces a series of rational numbers. Then you count the second overtone as 1.5 times the wavelength of the fundamental. But wait only simple multiples are allowed! The operators we have are multiplication, same as adding, wavelengths. Dividing wavelength or frequency is not defined! Only intervals are defined, and the collection of all intervals is a Borel set. That's your eigen function.
The eigen function comes because the string is an orthogonal projection, an arrow with a 1. The string is an orthogonal matrix where the map between the 12-tone pitch and position sets S X S maps on to the pitch set. You don't seem to have the standing wave equation correctly stated. It is a matrix with determinant 1 (like the octave identity matrix which is a Boolean atom.
People TRY to dismiss him but those people don't understand noncommutative phase logic!!
What seems so incredible to me is that this theory of string vibration is over 300 years old and nobody seems to realize that its obviously not correct.
For instance, I was taught that plucking the string and then detaining the string at the 12th fret shows that the first octave harmonic is inside the string and merely revealed by dampening all the other modes of vibration. That implies the string has 2 states of system at the same time.
Exactly! I learned this secret from the piano! And PRESTO - suddenly I had my Pythagorean insight that Pythagoras was correct!!
Classic string theory then implies that the string can have many different fundamentals at the same time so the state of system of the string is not a constant but an infinite series.
But frequency must be a 1 or a 0 in measure theory. 1 is a homomorphism which is not a function of time.
So Terence B. Allen has not discovered that what he is trying to solve is called noncommutative phase logic!
It seems to be overlooked that music is a theory of strings that are at once open to polyphonic union and closed by the octave. Strings are open and nonreducible, so we cannot separate the pitch from the string state of system. This is not the modern string theory where strings are either open or closed, but not both at once. The problem of how the string is a musical set is resolved in projective space where lines are always circles too. The natural overtones are not a closed system, and to make 12-tones you have to go to many higher partials.
so that link EXPLICITLY goes into noncommutative phase - only he just doesn't know to call it that.
The point here is that the next overtone does not have the octave metric because the fundamental is 1/3.
Terence gets closer!
https://www.researchgate.net/post/Is_the_sound_envelope_produced_by_a_plucked_musical_string_a_natural_experimental_that_shows_how_many_modes_of_vibration_exist
Real analysis of frequency is compelling and successful theory but does not apply when objects in the frequency domain are defined by a discrete topology, because the discrete values like pitch values in music are vectors and not scalar values. In harmonic systems all defined values have a common point of origin which is the fundamental. Any two vectors are perpendicular, if they are not the same.
The implications of the real and discrete topology are quite different, but it does not seem to be well-known.
Using the language of category theory, if every defined value in the harmonic system is a simple multiple of the fundamental frequency, and if the fundamental is thought of as an arrow with a 1, or just a 1, then the fundamental F is itself a vector in a field and every frequency defined by a simple multiple of the fundamental must also be a vector. (please see attached diagram) My question is why isn't that true?
The important difference between the Stewart and Ray answer is how the operation of multiplication is defined on the fundamental F. Stewart's answer assumes the harmonic operation of multiplication is the same thing as the multiplication of real numbers. The topology of real numbers makes the assumption the frequency is a real valued, continuous function the same as the real number line. But the frequency is a discrete value, not contiuous. But the octave space is continuous because the octave interval is precise to a point.and so Terence emphasizes the same secret of Alain Connes - only Terence has NOT discovered Alain Connes yet!
You asked a big question with a very important answer that somehow is beyond the grasp of musicians and mathematicians. This is anomalous to the idea that everything is available Google. Try asking whether the binary path of tonal movement is 2-fold or 3-fold. Simple question.
So just in case Terence discovers this blog post - I will repost my excerpts from Alain Connes here:
“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....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 is almost 3 to the power of 12....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. 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. 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.
Fields Medal math professor Alain Connes,
OK I Emailed Terence as he posted an email from 2015. So I hope it is still active.
Hi Terence B. Allen: The reason no one can understand you is that
hardly anyone is aware of Fields Medal math professor Alain Connes'
noncommutative phase music analysis. You have precisely recreated it
without realizing it as well. Below are quotes from Alain Connes on
music theory: Most of the quotes are from his music lecture on youtube. https://www.youtube.com/watch?
take care, drew hempel http://elixirfield.blogspot.My abstract "Music Automata Theory: Machines and Languages" was accepted for the AAAS Pacific Division Meeting (Computer Science and Information Section) at Ashland, OR June 18-21.Oops - he's regressed since no one could understand his earlier analysis from 2016!!! Oh well.
https://www.researchgate.net/post/What_is_the_axiomatic_basis_for_music_theory
Precisely!! Terence realizes that the spiral of fifths using integers is precisely empirically true in contrast to the real number continuum....
We can agree that we have at concert pitch the set A1 = 55 Hz, A2 = 110, A3 = 220, A4 = 440.We don't say that A4 is exp 440 or log 440. We have p = f but it is also clear that each higher note doubles the frequency of the note before, and each lower note halves the frequency. So we have a tower of octaves with powers of 2 going up and in effect square roots of 2 going down. Log 2 it turns is counted in 1 and 0, like a binary code.Now it is not true that any log can replace log 2 because in fact there is only 1 point in log 2 f that is actually true to p = f. The log curve and the line have only 1 point in common and that point is the octave where they intersect. This is true because log 0 = 1 and log 1 = 2, which we treat like 0 and 1. That is, we have only two values: 2 to power 0 is 1 and 2 to power 1 is 2. Those are the only point that are defined and the only points we need.Now suppose we have a chromatic or harmonic circle (5ths). Then there is a spiral above the circle that is formed if we allow the pitch to continue rising. The circle is the closed form of the spiral.The way that this happens is described in mathematics like a spiral staircase, where you are walking up the steps and there is always another person above you at the same octave position. This makes the spiral continuous and the octave is the winding number that counts how many time the spiral has made one full revolution.The octave is precise to a point and that makes the pitch values and the pitch value intervals continuous.The point is very important because in mathematics frequency is not continuous. This means that if we say " A1 = 55 Hz, A2 = 110, A3 = 220, A4 = 440" is the definition of pitch we have a trivial function and a connected space where arithmetic is just real number arithmetic which includes fractions and integers.It sounds like "trivial" and "simply connected' are good things but they lead to paradox, while in mathematics "nontrivial" and "disconnected" are good things.
I think that you've made an extra leap and applied the (correct) idea of halving (as the inverse operation) to the power of the two and not just to the two itself.Yes but that's not the point! What Terence is pointing out is that the "doubling" got turned into SQUARING via the octave! That is what I call the "bait and switch" - by ignoring the noncommutative phase origin that is from REVERSING the direction of the octave.
So you square it (and thereby coverup the noncommutative phase) and then you HALF it.
The values are DIFFERENT as I have proven!
Hi
Terence B Allen - in this discussion/debate with fellow music theorists
- you are CORRECT but there is a secret math solution I solved that
proves you are correct.
I have several blog posts on this - I'll find the relevant ones. I've been researching this topic since the 80s!
Notes on the noncommutative phase origin that was covered up:
Sir James Jeans, the
quantum physicist, author of Science and Music he describes the Pythagorean Comma as the difference between 3 to the 12th and 2 to the 7th.
“To put the same thing in another way, we have just identified the frequency ratio 1.5 with the interval of a [perfect] fifth, although our table gave the value as 1.4983. The difference is only small – 1.13 parts in a thousand – but by the time we have taken the twelve steps needed to pass completely around the clock-face, it has been multiplied twelvefold into the difference of 13.6 parts in a thousand, which represents the aforesaid difference in pitch of almost a quarter of a semitone. When this is allowed for, the true clock-face is that shown in fig. 55; it extends to infinity in both directions and all simplicity has disappeared.” Sir James Jeans book Science and Music, (Dover Publications, 1968), p. 166- so that above definition of the Pythagorean Comma is not how it is defined based on the logarithmic math as per the "Pythagorean Comma" (as in the Wikipedia page).
1.05946309436
So that is the value of the 12th root of 2 for equal-tempered tuning.
So then you take that to the 7th as 7 half steps from the tonic and you get = 1.49830707688
And so that appears to be close enough to 1.5 for 3/2 as the Pythagorean harmonics.
But as Sir James Jeans points out - this "error" multiples.
So 2 to the 7th is 128 and 3 to the 12th is 531441
And
so the assumption is that the logarithms can be added so you get 2 to
the 19th=524288
And so the Pythagorean Comma is 531441/524288=1.01364326477 This is an
ERROR that covers up the noncommutative phase truth of the "Phantom
Tonic."
Whereas if you take 3/2 to the 12th instead of assuming that the octave can be divided out of the 3 though "halving" -
then you get 129.746337891/128= 1.013671875
So you have 8 to the 6th power and 9 to the 6th power (531441) based on the rule of doubling/halving for octaves but the ONLY way that the Perfect Fifth - the Perfect Fourth = 9/8 and the Perfect Fifth + the Perfect Fourth = the Octave is to define the Perfect Fifth and Perfect Fourth not as the fractions 3/2 or 2/3 but as the logarithms of 3/2 squared being 9/4 with the Octave 2 defined as the Geometric Mean Squared. This is the subtle difference between multiplying as doubling and squaring as geometric mean.
So - from Professor Richard McKirahan we learn how Philolaus had to flip his Lyre around so that 3/4 as the Perfect Fourth is the wavelength of 0 to 8 root tonic as 1 while 4/3 is the geometric magnitude of 8/6 as the root tonic of 12 to 0 in the opposite direction as 1. So the question is - is 1 a square inherently as geometry? or is 1 a number that can not be seen but rather listened to.
For the Harmonic Series to derive the standard Pythagorean Comma we:
(start counting at zero):Oh yeah? REALLY? Now the Harmonic Series in math starts at 1 and diverges but still defines frequency as square root of wavelength divided by PI (assuming commutative phase time at zero). This covers up noncommutative time-frequency energy.
So for Hertz the frequency is actually defined as the square root of
the wavelength divided by PI - it is already assumed to have the
symmetric logarithmic math.
note that all is geometric magnitude so that 12:8 (3/2) plus 8:6 (4/3) = 2/1 as geometric magnitude from the double octave. This has to use 0 to 8 as one "root tonic" for 6/8 wavelength as 4/3 frequency and then 0 to 12 as the other root tonic for 12/8 frequency as 3/2 with 8/12 wavelength. This is called the "phantom tonic" in music theory since the Perfect Fourth can not be created from the harmonic series as 3 does not go into 2 from 1 as the root tonic denominator. In other words the root tonic changes due to complementary opposites.
Professor Richard McKirahan reveals the secret:
Or as math Professor Luigi Borzacchini states:
The word translated epogdoic is not a musical term but a mathematical one. An epogdoic ratio is the ratio of 9 to 8. The occurrence of a mathematical term here [in Philolaus]is unexpected. It has been treated as an unimportant anomaly but in fact it is the key to the entire fragment....The word magnitude normally refers to physical size, but here it is given a new application, extending the notion of magnitude to include musical intervals.
These remarks raise the question of the difference between the ancient Pythagorean ‘musical’ perception as displayed in the Pythagorean idea of ‘linear number’ in Boethius [Philolaus] or in Nicomachus, and the modern ‘geometrical’ perception of the linear numerical magnitudes.
note that all is geometric magnitude so that 12:8 (3/2) plus 8:6 (4/3) = 2/1 as geometric magnitude from the double octave. This has to use 0 to 8 as one "root tonic" for 6/8 wavelength as 4/3 frequency and then 0 to 12 as the other root tonic for 12/8 frequency as 3/2 with 8/12 wavelength. This is called the "phantom tonic" in music theory since the Perfect Fourth can not be created from the harmonic series as 3 does not go into 2 from 1 as the root tonic denominator. In other words the root tonic changes due to complementary opposites.
And so the use of "zero" is to create the geometric mean as logarithm by covering up the noncommutative phase. So at "zero" energy there is STILL a noncommutative phase quantum energy. Professor Richard McKirahan:
So you have 8 to the 6th power and 9 to the 6th power (531441) so that the 9 is equated to 3/2 squared as 9/4 halved to 9/8 with the 9 being 3 squared and so 9 to the 6th = 3 to the 12th.
So instead of taking 12:9, which is 3/4 of 12, we take 8:6, which is 3/4 of 8. And so by adding the length 12 to 8 [as geometric magnitude not wavelength!!] with the length 8 to 6, [as geometric magnitude, not wavelength!!] we get the length 12 to 6, which corresponds to the ratio 2:1.
We are told that 6:8:9:12 is just multiplying (1:4/3:3/2:2) times 6 but in actually 9/8 is assuming that 3/2 is also the geometric mean squared.
Yes
Philolaus taught that a "double octave" is necessary to create music
theory with 0 to 6 as first octave and 6 to 12 as second octave and then
the wavelength is reversed. So 3/2 is the first harmonic as C to G
overtone, Perfect Fifth but then 8/6 is the 2nd harmonic as 4/3
frequency from 6/8 or 3/4 wavelength that is really the 2nd octave (as 0 to 8 turned into geometric magnitude)!
See the bait and switch! This then enabled Archytas to claim that
Perfect Fifth plus Perfect Fourth = the Octave since the arithmetic mean
x harmonic mean = geometric mean squared or 3/2 x 4/3 = 2. Notice that
2/3 is also the Perfect Fifth as C to F subharmonic - but that can no
longer be used based on the lie of ARchytas. So everyone learns the
wrong music theory based on symmetric math when the truth is
noncommutative phase or G=3=F at the same time as complementary
opposites of overtones and subharmonic undertones.
https://www.researchgate.net/profile/Terence_Allen2/publication/335841044_Regular_Guitar_Music_Learning_Algorithms_and_Unified_Atomic_Field_Theory_of_Music/links/5d7fabfe458515fca16cc91a/Regular-Guitar-Music-Learning-Algorithms-and-Unified-Atomic-Field-Theory-of-Music.pdf?origin=publication_detail
Terry responses:
Thank you for your interest in my work.I do not agree that music is non-commutative. The octave is precise to a point. Music on one instrument can be transformed to any instrument or notation system. That is commutation, is it not?Perhaps you mean that the string modes are non-commutative? There I agree with you.But the 12-tone scale, which has the octave metric, is not defined by any overtone above the octave.Alan Connes is a great mathematician but I do not see any formal treatment of music in the list of publications on his site.There are so many mathematic theories about music but there can only be one that is correct. Apparently no one knows what it is. Of course we all understand music in a trivial mathematic way, but science cannot explain consonance. There is, after all, nothing pleasing about a Pythagorean comma. If music is not commutative then it makes no sense. Its not natural math, its subjective.
Hi Terry: thanks for your nice reply. I really appreciate you
willing to discuss this topic. The Chinese music tuning does not go to
the octave - so there are scales but the octave is not "doubled" in
terms of the tuning math. Patrick Edwin Moran, Chinese Professor, has
the details - and he also has wikipedia posts on quantum physics! So he
first studied physics at Stanford before he finished a degree in Chinese
philosophy. So I'm sure you accept that all human cultures use the
octave, Perfect Fifth and Perfect Fourth. But the Chinese tuning does
not "double" the ratios back into the octave (i.e. turning the 2/3 as C
to F subharmonic into 4/3 as C to F within the same octave of 6:8:9:12
as Philolaus first did. Philolaus, as professor Richard McKirahan notes,
was the first to introduce the mathematical term for geometric
magnitude into music tuning. So the "Orthodox" Pythagorean tuning is
like the Chinese - it JUST uses the the Octave, Perfect Fifth and
Perfect Fourth. As Sir James Jeans, a quantum physicist who wrote a
"formal publication" on music theory noted - if the empirical truth of
the infinite spiral harmonics is used then "all simplicity disappears."
So
Terry - there IS something "pleasing" about the Pythagorean Comma -
because Philolaus, Plato and Archytas LIED about what the real
Pythagorean Comma is!! Philolaus had to literally flip his Lyre around
to create two different root tonic values of "one" so that 0 to 8
created the Perfect Fourth from 6/8 so that 8/6 could be the new
geometric magnitude tuning for commutative math. Math professor Luigi
Borzacchini did publish a "formal" article on music tuning, and he
states elsewhere this error - a cover-up that he calls "really shocking"
and "astonishing" is also a "deep pre-established disharmony."
So
the octave value DOES change if you look at the infinite spiral of
fifths and YES you are correct that the octave is inherently a geometric
mean squared value in the Western tuning model. So you think a Fields
Medal math professor is wrong? A Fields Medal is harder to get than a
Nobel Prize. Indeed there is lots of discussion of this acoustic music
tuning in the math "formal" publications. Sorry but a medical degree is
much easier to get than a Ph.D. research degree from what I have read. I
can give you some examples - but I'd rather just stick to Alain Connes.
One of his recent published articles is an actual music composition
that you can listen to! haha.
Take care,
Hi Terry: Here's the "formal" sources you need on Alain Connes and
music tuning (I recommend you give some time to this topic as you seem
to have devoted your life to it. Please don't just pass off a Fields
Medal math professor!) So in other words READ the links. thanks.
Duality between shapes and spectra The music of shapes
A. Connes. • How to specify our location in ... shapes and CKM invariant. • Towards a musical shape. ... Eigenfrequencies of sphere. 20. 40. 60. 80. 100. 120. 2. 4.
arXiv:math/0404128v1 [math.NT] 6 Apr 2004
by A Connes - 2004 - Cited by 14 - Related articles
discrete scaling manifests itself in acoustic systems, as is well known in western classical music, where the two scalings correspond, respectively, to passing to the octave (frequency ratio of 2) and transposition (the perfect fifth is the frequency ratio 3/2), with the approximate value log(3)/ log(2) ∼ 19/12 responsible for.
Terry: I will test you: which of those two music math documents
published by Fields Medal (hard to get than a Nobel Prize) math
professor Alain Connes contains this quote?
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,∞}.
thanks,
drew
Hi Terry: Here is Professor Moran:
Using this system would never permit derivation of the true octave. But the method used here continually divides frequencies by two) to make a gamut all the notes of which fall within one octave, so it is clear that the early Chinese had a definite idea of how to define and use the octave."
And maybe next you need a quote from Sir James Jeans, in his book "Science and Music"? haha. good luck!
drew
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