More questions and answers on Daoist Harmonics as alchemy meditation

Ok, I've been spending a while thinking about this. I hope I'm getting there by now...

In Summary?:

1. Adding a Perfect fifth (3:2) to a perfect fifth (3:2) does not ever create an overtone octave, no matter how many perfect fifths one adds, because a perfect fifth's frequency is an irrational number.
2. And likewise taking away a perfect fifth (2:3) from an octave does not create a perfect fifth of any lower octave. However, dividing the octave frequency by the perfect fifth frequency of its octave results in the perfect fourth frequency (4/3x) of the lower octave.
3. Perfect fifth (3:2) x Perfect fourth (4:3) equals an octave (1.5x x 1.333x = 2x, whereas 1.5x + 1.333x = 2.8333x).
4. One uses multiplication, not addition, because working with music is working with logarithms. 
5. When a perfect fifth is played with the tonic, both the undertone perfect fifth (2:3) and overtone perfect fifth (3:2) frequencies are produced. The undertone perfect fifth is called the phantom tone. This you call 'the secret of the undivided Single Perfect Yang'.
6. C x G does not equal G x C, as each octave is logarithmic.

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I have a few questions for clarification if you don't mind.

1. You say that you can multiply frequencies (like the frequencies that are a perfect fourth from the tonic and a perfect fifth from the tonic). Can you do the same with other frequencies too to arrive at a meaningful frequency, like the:
   1. Other perfect intervals (like the perfect fourth with the octave, perfect fourth with the tonic, perfect fifth with the octave, perfect fifth with the tonic, tonic with the octave, or any of these intervals with itself).
   2. Frequencies which aren't resonant at all. 
   3. The major intervals. You said previously that the second interval (9:8) was not used by the pythagoreans, so what significance does the thirds, sevenths or sixths have?
2. What significance does two different frequency geometries, overtone and phantom undertone, when played with the tonic, have exactly? 
3.   Could you please explain what you mean by this? Suppose I have a piano, and I play a key at 440hz, which is my fundamental. Then I play it with the perfect fifth of that frequency (which is an irrational number, which also produces a 'phantom undertone'), how does that change the frequency of 440hz? I don’t know personally, but I’m quite sure one can still hear the 440hz fundamental frequency. 
4. Why exactly is yin 4:3, and yang 3:2? What properties do these ratios have, which causes them to be so? Why couldn't I call 4:3 yang, and 3:2 yin? I know odd numbers are yang, and even numbers are yin, but maybe there’s more to it than that.
5. How does 3:2 yang change into 4:3 yin? In your previous post, you said that 12 perfect fifths (3:2) transduces (into yin (4:3) I assume), but I thought it just produces a pythagorean comma at the 7th octave, not a perfect fourth (4:3).
6. Where do the three gunas come into all of this. You say that the three gunas equal 2, 3 and infinity, but I don’t see what they have got to do with the perfect fourth or perfect fifth. Apologies for my ignorance.

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a perfect fifth's frequency is an irrational number.

No, a Perfect Fifth is 3/2. The concept of frequency assumes already a logarithmic definition of number as geometry which is irrational. 3 is a prime number and so does not go into 2.

One uses multiplication, not addition, because working with music is working with logarithms.

No - the whole point of logarithms is you can add the numbers as geometric magnitude while you multiple the fractions.
And then to subtract the geometric magnitude you divide the fractions - or multiple by the inverse. And so the logarithms are ratios that are commutative since you can reverse the direction as multiplication and get the same result. This is NOT true for the empirical reality of natural harmonics as yin and yang which are noncommutative.

C x G does not equal G x C, as each octave is logarithmic.

Yeah octave is a doubling - doubling is NOT the same as a logarithm. A logarithm assumes an irrational number as a geometric mean "containment" of infinity into a materialistic dimension. So the reason C x G does not equal G x C is because 2 does not go into 3 - due to the odd and even value of the number or their inherent complementary opposites as the T'ai Chi that is infinite natural resonance. It can never be "contained" as a boundary condition of visual geometry.

Can you do the same with other frequencies too to arrive at a meaningful frequency,

Again the concept of frequency assumes this logarithmic containment but the other ratios or numbers - yes you can do this but the point is that the first overtones are noncommutative. So you can create ALL the "frequencies" through infinity just from 2 and 3 - just as the Tao Te Ching teaches - the secret is that because 2 does not go into 3 and the change of direction of the number changes the geometric value - then all the other "frequencies" are derived - the 10,000 things, etc. from 2 and 3.

The major intervals. You said previously that the second interval (9:8) was not used by the pythagoreans, so what significance does the thirds, sevenths or sixths have?

The reason 9/8 is not used is because it was derived by a logarithmic geometric mean equation - used by Philolaus. So you "squared" 3/2 as 9/4 and then you divide by 2 for the "average" as 9/8. And so 9/8 becomes the geometric mean interval as the major 2nd and so 9/8 cubed is then the tritone - which is also the origin of the Power Set Axiom or the Pythagorean Theorem. So this assumes again a logarithmic definition so that you can "subtract" the frequencies - and get 9/8. But again we are talking about three factors here - Pitch is the Perfect Fifth as both 2/3 and 3/2. The only way 9/8 works is if you pretend that 2/3 is NOT a Perfect Fifth and ignore it - in relation to 4/3.

What significance does two different frequency geometries, overtone and phantom undertone, when played with the tonic, have exactly?

O.K. so in Western music tuning - the claim is that 4/3 is the harmonic mean for the equation "arithmetic mean (3/2) x harmonic mean (4/3) = geometric mean squared (2)." So the problem with this equation is that if 4 is the octave as the geometric value of C then 3 is G as the Perfect Fifth or 3/2 and so 4/3 is C to G. But if the octave is a value of 2 as frequency then 4/3 and 3/2 and not BOTH but C to G - because it is greater than the octave when added together. So 4/3 is constructed from the "phantom tonic" by doubling 2/3 which is C to F as the subharmonic.

So that is actually the T'ai Chi. And so 2/3 then if doubled as 4/3 means now that the "one" as the root tonic is now 3 and not 1. So then you have eternal resonance with no root tonic - or as the Pythagoreans stated "one is not a number." It is light as consciousness - which perceives the future and the past at the same time - as nonlocal awareness (the hidden momentum of light)....

So in Indian music tuning - this is well recognized as the three gunas - because the first overtone of the Perfect Fourth as F is C which then reinforces C as the root tonic while the first overtone of the Perfect Fifth as G is D which does not reinforce C. So there is an inherent pushing and pulling of the Perfect Fifth and Perfect Fourth based on their relation to the "one" which is always changing.

So this is also the case in Blues music - which is why the harmonics are "bent" or "crushed" to be closer to the natural overtones. So that instead of the logarithmic third - then 5/4 is used - or instead of the 7th - then the flat 7th is used - since that is closer to the natural harmonics. And then the whole music emphasis is on this hidden tension of the 1-4-5 music intervals or 1:2:3:4 as yin-yang-Emptiness. So the idea of Blues music - as one of the original blues musicians stated - is to empty out the mind - to clear out the thoughts, as healing energy.

So Westerners will typically think blues is boring, simplistic, repetitive - because they don't understand the secret of the Mantra as nonlinear harmonics - the same intervals are repeated by the resonance means it is ever changing and ever new and ever fresh.

So the one is light as shen but light does not experience space or time. But light also has relativistic mass -  which is the noncommutative phase. This means that when the one doubles and then goes to three as harmonics - the G=3=F as yang at the same time. So the undivided yin-yang is the one since the 3 changes what the one was - as an Emptiness transformation.

So the realize this alchemically means to transform the yin of 3:4 which is C to F back into 2/3 as the subharmonic Perfect Fifth yang. And beyond that - the harmonics are infinite resonance because 2 does not go into 3 - and so they can never line up geometrically.

So Alain Connes gives his lecture on music theory and the unified field theory - and I quote him in the thread. So the Western standard math that is symmetric is contained within the noncommutative phase that is a 5th dimension that is just pure time-frequency resonance as eternal listening or logical inference. So at each "zero" point in space the future and the past are interwoven as noncommutative phase - the G=3=F at the same time because the 2 is the same geometry as the 1, it is just a change in the frequency. So you have a zero point in space but you have a "chord" of 2 frequencies - you have the Perfect Fifth and the Octave as the two frequencies - but it is a "triple spectral" since the Perfect Fifth is noncommutative phase with the root tonic and the octave. So again the root tonic is zero but it is not zero - since it is eternal motion as time-frequency. Standard science calls this "time-frequency uncertainty" but actually it is noncommutative phase as nonlocal entanglement or formless awareness.

I keep reading this, and I know that 2 does not go into 3, and 3 does not go into 2, because they are both prime numbers, but what relevance does this have? And if neither line up, then that means they aren’t harmonic/resonant with each other, no? Only multiples of something are resonant.

So every human culture uses the octave because the brain neurons register the octave as the same "pitch" and then the Perfect Fifth has the strongest neuron amplification next to the octave. I posted the image for this in the thread. So the question is why is this the case? Western science at first tried to dismiss this - because of Western science relying on logarithms from the wrong music theory. So the Pythagorean natural number ratios were called a "coincidence of consonance." But again the neurons are now proven to prefer the small number ratios. So one music analyst describes it as this:

the ratio of overtone 3 of the 2nd note to overtone 3 of the 1st note is pure

 Meaning pure as in the same pitch. The third overtone of the root tonic is the same as the Perfect Fifth and the 3rd overtone of the Perfect Fifth is the root tonic.

During the Middle Ages (c476-1453) a Pythagorean tuning, a tuning derived from a series of six 3:2 fifths, was an accepted "standard" for tuning the seven tones of a diatonic scale.

So this is based on "horizontal" harmony - meaning one note after another note. But as "vertical" harmony developed - the focus is no longer on the natural harmonics but rather on trying to standardize the tuning - so that when the notes of the scale are developed then they can have more intervals but also still line up with the octave.

So this is when the tuning had to be "fudged" because if you try to fit the notes to line up at the same time, then you end up getting the Perfect Fifths being greater than the octave. In other words if you "build" the scale based on the most harmonious note - it does not come back to a circle. And so at first in Western science they used "fudge factors" - staring with extending the Tetrakytus - from 1:2:3:4 into 5:4 as the major third - which is also the cube root of two approximation. So this ushered in the Renaissance since with the cube root of two then you get perspective drawing.

Then by the late 1500s - Simon Stevin said - who cares about the consonance low integers - it is just a coincidence - so we can have as much vertical harmony as we want if we just evenly space the notes as the 12th root of two. So - this is just a continuation of the same logic from Philolaus - using the geometric mean but more precisely. So instead of using 9/8 as the major 2nd interval to build the scale from 3/2 "squared" - the divide and average math is more precise. And actually Philolaus did this math - down to the evenly spacing of the half step, for the chromatic 12 notes of the scale. It is just that Simon Stevin used more precise math.

So now you can "line up" the octaves - from the notes that build up the scale - because now the octave is defined as "geometric mean" - just as Archytas and Philolaus did originally. the number 2 now refers to a geometric mean as a wavelength. But this discounts that the original consonance ratios are Noncommutative phase.

So when Helmholtz developed the harmonics of the natural numbers - the definition is this:

Helmholtz proposed that the degree of dissonance produced by an interval is related to the degree of roughness produced by partials of the two fundamental tones that make up the interval. The term roughness was a quality Helmholtz associated with the number of semitone and whole-tone conflicts among the partials of the two fundamental tones.

O.K. and so above I noted the "purity" of the overtones - but that was for 2/3 NOT for 3/2. Remember the difference? So for C is the Perfect Fifth is G then the first overtone difference is D. And this is how Helmholtz defines it. Whereas if for C, the Perfect Fifth is F as the subharmonic 2/3 then it is pure. 

it rather reflects the circumstance that the third partial of the lower tone coincides with the second of the upper 

And so finally Helmholtz relied on "beats" produced - so in the case of C and G - the beats dissonance in the overtones between them is for C and E then "beating" with D.  But the "beating" actually determines the time to perceive the harmonics! In other words there is a trade off with the difference in the "frequency" and the time it takes to perceive the difference. So if the "frequency" difference is 3 hertz then it takes 1/3 of a second to perceive the difference. I’ll repeat from above, Physicist Iori Fujita:

“But even Δt is 1.000 sec, the bandwidth remains about 2 Hz.”

If the "frequency" difference is an octave - the beats are so fast that the brain perceives the note as the same "pitch."

The most frequently used intervals—the octave,

perfect 5th and 4th—correspond to those considered the most consonant by culturally diverse listeners (Bowling & Purves, 2015; Burns, 1999), an observation that is all the more impressive given that the frequency resolution of the human auditory system is sufficient to distinguish hundreds of unique intervals (Parncutt, 1989; Roederer, 2008).

  The Nature and Nurture of Musical Consonance (PDF Download Available).

So what that article states is that because there is no mathematical solution to harmonics - due to time-frequency uncertainty - then actually we have to turn to biological evolution to find an answer. Science defines "time" as based on radians of "cycles" and so the uncertainty is due to the conversion of the cycles into a frequency. So again because 2 does not line up with 3 - in terms of a VISUAL measurement of geometry - there is inherently an uncertainty, in terms of a visual measurement. This is converted to pure geometry as "transcendental" or "irrational" numbers - but it means there is an inherent uncertainty to the visual measurement that converts time to frequency.  Δf.Δt > ~ 1     but this is "averaged" as a "geometric mean" - based on 2pi as radians of the visual cycle.  So a sphere is 4pi and so as an uncertainty the average is based on 2pi. But this is NOT how hearing works as perception.

When Δν is 0.1Hz, Δt should be more than 1 /(4π *0.1) sec which is about 0.80 sec. Middle C of the equal temperament is calculated approximately as 261.6256 Hz. If you want to determine C with this precision 0.0001Hz, you need Δt of 800 sec, or 13 min 20 sec. If you want to get the exact Middle C, you need an infinite time and a continuous wave of C. In other words, theoretically we cannot get the perfect fifth tone with a frequency f ( that is ν ) from a root tone with a frequency f₀ by calculating f = f₀ × 3/2.” 

Uncertainty Principle for Temperament by Iori Fujita, http://www.geocities.jp/imyfujita/wtcuncertain.html  

So based on science it is impossible to HEAR the Perfect Fifth music interval of 3/2. But we know this is not true. In fact quite the contrary:

Let us draw up a fundamental vibration with its first and second harmonic overtones [1:2 and 2:3]….if we then shift the phase relationship, we get a totally different course of the combined curve; that is, of the pressure course [amplitude and frequency] which is apparently effective in the final analysis. When looking at both curves, one might suppose that in the two cases we should hear two sounds that are just as different. But in fact our ear does not notice any difference.

biophysicist Vitus B. Dröscher states in his book, Magic of the Senses, p. 168

So here is how Alain Connes describes this amazing ability of the brain:

space is not simply a manifold but multiplied by a noncommutative finite space. There is behind the scene, there is a square root and when you take a square root there is an ambiguity and the ambiguity that is there is from the spin structure....Finite space which is there is essentially the simplest finite space which has dimension zero, as far as the [frequency] spectrum is concerned...  a "universal scaling system," manifests itself in acoustic systems....They are the same [frequency] spectrum but they are not the same chord. There are three types of notes....What do I mean by possible chords? I mean now that you have eigenfunctions.... 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. ... Musical shape has geometric dimension zero....There is a fine structure in spacetime, exactly as there is a fine structure in spectrals [frequencies]...Our brain is an incredible .... receives moments of 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 an enormous 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....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,∞}.  

Alain Connes, 2012

https://arxiv.org/html/1202.4212v1/#sec_2_1_2

This conjecture on the brain creating virtual pitch seems to hold even for non-humans, as pointed out in "This is Your Brain on Music" by Daniel J. Levitin [Levitin2006, p. 41] (emphasis in the original):

When I was in graduate school, my advisor, Mike Posner, told me about the work of a graduate student in biology, Petr Janata.... Peter [sic] placed electrodes in the inferior colliculus of the barn owl, part of its auditory system. Then, he played the owls a version of Strauss's "The Blue Danube Waltz" made up of tones [by "tones" here he means what we are calling "notes": each note is an entire series of overtones] from which the fundamental frequency [what we are calling the fundamental tone of the overtone series] had been removed. Petr hypothesized that if the missing fundamental is restored at the early levels of auditory processing, neurons in the owl's inferior colliculus should fire at the rate of the missing fundamental. This was exactly what he found. And because the electrodes put out a small electrical signal with each firing -- and because the firing rate is the same as a frequency of firing -- Petr sent the output of these electrodes to a small amplifier, and played back the sound of the owl's neurons through a loudspeaker. What he heard was astonishing; the melody of "The Blue Danube Waltz" sang clearly from the loudspeakers: ba da da da da, deet deet, deet deet. We were hearing the firing rates of the neurons and they were identical to the frequency of the missing fundamental. The harmonic series has an instantiation not just in the early levels of auditory processing, but in a completely different species.

Michael O'Donnell pointed out to me that there is an ambiguity here [O'Donnell, 14 February 2009]:

[The above story] doesn't allow one to distinguish whether the Owl, or the human listener, is experiencing the virtual pitch.

I passed this on to Daniel J. Levitin; his response [Levitin, 24 May 2010]:

You're absolutely right that these two possibilities need to be distinguished. The electrodes that were placed in the brain of the owl (in the inferior colliculus) were analyzed using specotrograms[sic] and fourier[sic] analysis. It was clear that the signal itself coming from the owl's brain had replaced the missing fudnamental[sic]. It was only after this analysis that Petr thought to hook it all up to play the signal over loudspeakers (so that humans could hear the output) as a cool demonstration.

Female Mosquitoes only mate when rate of the wing-beats of the male harmonize at a Perfect Fifth above the rate of her wing-beats (we start introducing musical terminology such as the Perfect Fifth in Section 3.1 "The Major Triad"). From "Mosquitoes make sweet love music" [Mosquito-harmony]:

The familiar buzz of a flying female mosquito may be irritating to humans, but for her male counterpart, it is an irresistible mating signal. Males and females each have their own characteristic flight tone - which they create by beating their wings.
But when scientists from Cornell University listened in on a male Aedes aegypti pursuing his mate, they were surprised to hear a new kind of "music" playing....
The amorous couple began to beat their wings together at a matching frequency - 1,200 hertz. This love song is a "harmonic", or multiple, of their individual frequencies - 400 Hz for the female and 600 Hz for the male....
"So we're trying to discover what makes a male more attractive. It's a mystery. It could be his odour[sic], or his bright black and white markings.
"But we think females are assessing the fitness of males based on how well they can sing."