The actual link that the article refers to is a taken down website. hahaha. The below article demonstrates my attempt to explain "noncommutativity" without using the term - and so no one could understand me. hahahaha.
https://indigus.wordpress.com/2007/12/24/how-the-west-lost-alchemy/
Against Archytas: How the West Lost Alchemy or Paranormal Complimentary Opposite Harmonics
by drew hempel, MA
(anti-copyright, free distribution)
The early Greek mathematics used the 60-based number system of Babylon from which Archytas, a collaborator with Plato, received the harmonic tetrachord or the continued proportion 6:8::9:12. This tetrachord creates a geometric mean between the octave, perfect fourth and perfect fifth music intervals, or 1:2:3:4, through “divide and average” logarithmic-based mathematics. So 6:8 and 9:12 are in the continued proportion 3:4, the perfect fourth music interval, while 6:9 and 8:12 are 2:3, the perfect fifth music interval, and 6:12 is 1:2, the octave. The geometric mean is A:B::B:C or B squared = AC or the square root of AC = B. What Archytas added to this Babylon “divide and average” harmonic mathematics was the concept of the Greek “incommensurable” – the algebraic axiomatic proof of “alogon” or a precise irrational number, the square root of two. This process ushered in what’s called “The Greek Miracle” that continues to be the structure of science: symmetry-based mathematics.
Instead
of the above system, the alchemical Pythagorean Tetrad relies on
complimentary opposite harmonics so that an equilateral triangle of
geometric points equals the continued proportion 1:2:3:4 as the octave,
perfect fifth and perfect fourth music intervals. In “orthodox”
Pythagorean harmonics this was known as the “subcontrary mean” whereby
the complimentary opposites of the Tetrad were maintained in violation
of “divide and average” mathematics. So for the Tetrad A:B is 2:3 and
B:A is 3:4 against the commutative property, A x B = B x A. In music
theory this complimentary opposite inversion of the perfect fifth and
perfect fourth is taught as 2:3 is C to G while 3:4 is G to C. This
process of complimentary opposites is listened to, as the perfect fifth,
perfect fourth harmonics, which create all the notes. Most importantly
the complimentary opposite harmonics transduces sound throughout the
whole energy spectrum, as I’ve described in previous articles…
Philolaus, one of the early Pythagorean writers, detailed that this “subcontrary mean” or complimentary opposite harmonic caused any attempt at subdividing the scale into symmetry as a failure. In contrast Archytas changed the “subcontrary” complimentary opposite mean into the “harmonic mean” using “divide and average” mathematics. The outcome has precisely opposite the meaning of “harmony” which for Pythagoreans referred to the paranormal source of sound as the Goddess Harmonia or what I call female formless awareness. For Philolaus the perfect fifth as 2:3 could be inverted to 3:2 and then extended another fifth to 9:4 and then divided back into the octave, below 2, for the major second interval of 9:8 or C to D. Yet 9:8 cubed or three major second music intervals equaled the 3:2 perfect fifth music interval, plus a tiny ratio called “the comma of Pythagoras.”
This “comma of Pythagoras” is the difference between the “divide and average” octave system adopted by Archytas and the complimentary opposite fifths inverting into fourths, used by the orthodox Pythagoreans. The “comma of Pythagoras” is the key to harmonic alchemy whereby 2:3, the perfect fifth, is yang in Taoism and 3:4, the perfect fourth, is yin. As Gurdjieff desribes the “shock” of the diatonic scale, whereby the “inverse ratios” do not line up with the octaves, is intensified as the octaves expand. In contrast Archytas argued that 9/8 cubed or three major second intervals equals the square root of two as the Greek Miracle, the axiomatic algebra of the precise incommensurable irrational number. What Archtyas essentially did, as I’ll describe, is equate the perfect 5th or 2:3 with the perfect 4th or 3:4 as equally-divided or symmetric ratios through a “divide and average” mathematics.
Archytas took the Babylonian geometric mean of 6:8::9:12 used for harmonics and then applied the Pythagorean Tetrad 1:2:3:4 so that the 2:3 ratio of complimetary opposite frequency was converted to 3:2 as a materialistic vibrating string length. This became known as the Law of Pythagoras even though it goes against the true meaning of the complimentary opposites when this “inverse ratio” is combined with the “divide and average” commutative property. Gurdjieff, for example, still relies on the “inverse ratio” of density or string length versus frequency or consciousness. But Gurdjieff does not use Archytas’ “divide and average” symmetric-based mathematics, instead Gurdjieff relies on the Law of Three aka the Tetrad, or octave-fifth-fourth, to resonate through the comma of Pythagoras as the “shocks” of alchemy. In Taoism this system of alchemical shocks is taught as the 12 harmonic nodes along the outside of the body, enabling healing and paranormal energy, an exercise called “the small universe.”
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