@MRPancakes - Well so humans can listen faster than Fourier Uncertainty or time-frequency uncertainty. So rather than defining an octave as a symmetrical ratio of 1/2 wavelength and 2/1 frequency instead the time as period is not something that can be seen.
So Louis de Broglie discovered this as the Law of Phase Harmony and it's based on phonons as acoustic energy. So if we listen to the shortest time period as the highest frequency externally this beats any technological linear operator of changing time into frequency as a symmetric spatial measurement. Humans are faster than Fourier Uncertainty but up to ten times.
And so in fact as Math professor Alain Connes states, he sums up the music theory as (2, 3, infinity) with a geometric dimension of zero due to the noncommutative phase of the music theory. So this means - to explain it simply as ratios -
"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."
Professor Fabio Bellissima,"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, 2013
Or as this:
Alain Connes:
"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 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 q (to the n) 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. This means it is a zero dimensional object! But it has a positive volume!"
So that is Alain Connes on music theory. So the octave is actually the number 2 as a resonance of an infinite noncommutative phase. So for example if the root tonic is 1 then 2/3 is C to F as the subharmonic and 3/2 is C to F as the overtone harmonic but notice that BOTH ratios are the Perfect Fifth is PITCH. So the EAR as listening does not distinguish the PITCH as Perfect Fifth even though the frequency and time are noncommutative phase at the same time. So that G=3=F at the same time as noncommutative phase which is also quantum nonlocality.
Connes:
"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...."
Or this math professor who cites
Connes:
"On the other hand, the ancient Pythagorean musical scales naturally lead to a simple quantum circle....There is something profoundly quantum in all music. A discrete space–the skeleton hosting any musical score, morphs into a true musical form, only after being symbiotically enveloped by a geometry of sound. And this geometry is inherently quantum, as it connects the points of the discrete underlying structure, invalidating the difference between now, then, here and there; thus creating an irreducible continuum for a piece of music: continuous discreteness and discrete continuity....
By taking the inverses L to 1/1 + L and 1/infinity=0, we can identify M = {0, 1, 1/2, 1/3,...}. The geometrical picture is that we have a circular object, unifying infinitely circular 'oscillating modes.' The limiting oscillating mode is the classical mode....All other modes are purely quantum 'virtual modes,' so we can not distinguish separate fibers over the classical points labeling these modes. The entire structure is a unified and irreducible quantum circle.... The oscillating modes base space M...will be quantum (noncommutativity of the algebra V)."
Durdevich, Micho Institute of Mathematics, UNAM (Mexico City) “Music of Quantum Circles” 10/2015
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