I posted the link in the sidebar of what can be considered the "top" academic overview of the latest - "and greatest" - Pythagorean philosophy research.
And so that full pdf - it is quite fascinating - specifically the chapter by Richard McKirahan:
Philolaus on Number. The chapter on Philolaus emphasizes that he could not consider the music ratios to be fractions (i.e. the number 2.5 as 2 1/2)- and so the paradox of how to "subtract" the Perfect Fourth from the Perfect Fifth to get 9/8 could only be solved by expanding the "monochord" to the "greater perfect system" or double octave. So McKirahan states for Philolaus the first octave is 0 to 6 on a string length of 0 to 12 and the 2nd octave is 12 to 6. So then the ratio 3:2 is the string length 2/3 and the ratio 4:3 is actually 8/6 as 3/4 of the double octave 12 to 8 string length. So the ratios only work as fractions of the double octave, which is the secret of the "bait and switch" math. I go into this in detail in my "Alchemy of Rainbow Heart Music" book.
This also enabled this conversion of ratios to fractions to be commutative math.
But my understanding is that it was not allowed to write out the actual teachings of Pythagorean philosophy and so - although Pythagoras himself is considered to have possibly written a couple books - for Philolaus to have done so, indicates he was not a high level student. The main view is Pythagoras himself was against writing his teachings down. The understanding is that in fact he was not a high level initiate or most definitely a "later Pythagorean."
I have previously cited academic analysis that the "Orthodox Pythagoreans" did not use the ratio 9:8 even though Philolaus had to use 9:8 and so the only way he could do so was by creating the double octave analysis.
And, as I argued before, this enabled the "bait and switch" whereby half of the double octave could then be switched to a square root - (square root of 4 is 2, same as half).
Because Westerners learn symmetric math since an early age and most are unaware of noncommutative math - then it is not realized what the actual truth of Pythagorean philosophy is!
And so Peter Kingsley had disagreed with Carl Huffman's rendering of Philolaus - the subject of a whole book by Huffman.To quote
Kingsley's review see : In Search of Pythagoreanism (2013 googlebook)
Kingsley definitely views Philolaus as a later attempt at "scientific" rendering of Pythagorean mysticism.
So then Carl Huffman states that Kingsley is just "appealing to the authority" of the older Babylonian mystic teachings.
The whole discussion is about the "central fire" (Hades or Hell) and so alchemically, as Kingsley points out that Empedocles teaches to focus the mind on the lower body stomach - that the "central fire" is in fact the light that comes out of the darkness of the lower emotional energy itself.
Even though Kingsley does not elucidate the noncommutative math, the philosophy he focuses on is clearly nonwestern or "pre-Socratic" in the true sense of the term.
So on the other hand Philolaus states that the Number One is the central fire - and as we learn "one is not a number" meaning the I-thought is actually the Yuan Shen as the spiritual ego, the "positive" or yang fire.
The cosmos arises from the breathing of the One from the Unlimited as the void. And so the One as the spiritual ego fire or shen then arises from the breathing of the Yuan Qi, the formless awareness of the Wu Chi. The void refers to the physical world and so as the number 2 - the jing is created as the shen is "under it" - the fire from the steam or qi energy, arising from the water, number 2.
Harmony is what binds together things and is provided by reason - the Logos - and so it is not that all is number. It was rather only with Plato and Archytas that the moniker "all is number" became definitive via geometry (Freemasonry).
As math professor Luigi Borzacchini has pointed out then music theory was actually a "vanishing mediator" (to use Zizek's term) - that was concealed - and replaced by algebraic geometry.
And so we are told that the Unlimited or Apeiron as formless awareness is "even" and so a reflection of the female number 2 as matter and then limited by the number 1 as the fire, or shen, positive fire, spiritual ego. So how can this be? We know that in fact the formless awareness Yuan Qi is actually, alchemically, the process of putting shen "underneath" the jing - and so the formless awareness is "even" as shen-jing.
Or to put this in terms of physics - Shen is frequency as light to time as the Emptiness (the void) as momentum (superluminal yuan qi) is to wavelength (jing). So then superluminal phase is shen x jing and the superluminal phase equals the momentum (yuan qi) x time (void).
So the only way to properly understand Philolaus is to "reverse engineer" his philosophy by using noncommutative phase analysis of music theory.
So we are told that the Pythagoreans believed that the eternal soul is in constant motion - and so also it is the void that gives substance to the numbers - this is a way of stating noncommutative time-frequency is the truth of reality, the relationship of light and spacetime.
Whereas the analysis of Philolaus assumes music is constructed based on a string or physical matter - the real issue is that the early Pythagoreans required five years of silence as music-meditation. In other words no instrument is needed to listen to the source of harmonics. And because no instrument is needed then noncommutative phase can be understood, philosophically. So for example Carl Huffman considered the Philolaus definition of Pythagorean Unlimited to be geometric magnitude.
So to just accept the view of Philolaus is to then already misunderstand the secret sophistication of real or early Orthodox Pythagorean philosophy based on the tetraktys or tetractus.
The number 4 was the number of justice - just as my master's thesis was titled, "Epicenters of Justice" - the secret is from noncommutative phase. So that if the Perfect Fourth, C to F, has to be 4/3 this actually changes the value of the root tonic since 3 does not resonate as the harmonic of 1 from the octave 2 for the harmonic series. This is called the "ghost tonic" in music theory.
Again I have gone over this math-music analysis in more detail elsewhere.
The answer then is that noncomutative phase arises from the discrete numbers but is not a symmetric continuum as irrational magnitude. Instead it is a non-local time-frequency noncommutative phase that is in eternal motion as light to spacetime, creating matter.
So then Robert McKirahan states he is going to argue that Philolaus was correct to realize that "ratios" are the truth of reality and not "number" as the early Pythagoreans believed. But what McKirahan is not noticing is that the truth of "number" is actually noncommutative phase! Whereas the truth of the "ratios" that Philolaus constructed is symmetric math as geometry - not real harmonia as the emptiness.
McKirahan details how Philolaus was definitely referring to the tuning of a lyre music instrument when he constructed his "Pythagorean" scale that Plato then used for Timaeus and Archytas further developed.
But the key point again is that we listen to music - no instrument is necessary and by limiting the numbers to geometric materialism than the true meaning of Pythagorean philosophy is lost!
So then a self-proclaimed James Randi follower writes a book on PreSocratic Magic! Hilarious.
