Dear Dr. Oscar Abdounur: Thank you for your clearly written and fascinating article on music and math

Dear Dr. Oscar Abdounur: Thank you for your clearly written and fascinating article on music and math. I have corresponded since 2000 with math professor Luigi Borzacchini on this topic. I would ask you to consider Professor Richard McKirahan on Philolaus and Number, detailing the use of geometric irrational magnitude by Philolaus. In other words, as musicologists acknowledge, the "orthodox" or early Pythagoreans only used the Tetraktys, while Philolaus, not being a real initiate, tried to convert the noncommutative phase secret into geometric magnitude. Are you familiar with math professor Alain Connes music theory analysis? He points out that music theory is a "Universal scaling system" with a geometric dimension of zero, because it is noncommutative phase, as a triple spectral. So at zero spacetime there is still (2, 3, infinity) as noncommutative frequency-time phase. Meaning that 2/3 is the Perfect Fifth as C to F subharmonic while 3/2 is the Perfect Fifth as C to G overtone harmonic. In other words the reason 4/3 was used as frequency was because of the need to create geometric magnitude within the same octave from the Greater Perfect System of the double octave. So that 0 to 8 is the root tonic to create 6/8 as 4/3 frequency with 0 to 12 to create 8/12 as 3/2 frequency so that 12 to 8 is geometric magnitude as Perfect Fifth PLUS 8 to 6 as geometric magnitude of 4/3. This way for Archytas the equation from Babylon is Arithmetic Mean (3/2) x Harmonic Mean (4/3) = geometric mean squared (the octave). So then 9/8 cubed approximated the tritone. Boethius went against this tuning of course - instead insisting on the Orthodox Tetraktys but by then the noncommutative phase secret had been lost.

The emergence of the idea of irrationality in Renaissance theoretical music contexts pdf

Thank you again, and I have argued that the ancients knew this noncommutative phase secret as the basis for meditation philosophy training, also found in the Chinese tuning math system. Notice how the Chinese used a pitchpipe instead of a monochord - and the pitchpipe was also used in West Asia. Hence the transfer of Hermes to Apollo from Pitchpipe to Lyre with Dionysis using the Pitchpipe and Apollo using the Lyre and Orpheus overseeing this transfer. Then the Greeks, using Babylonian tuning from IndoEuropean culture, then canceled out the Mother Goddess "incubation" meditation - the need to go into the counter-Earth as the Void that is the source of the 1. So alchemically the Perfect Fifths are noncommutative with Fire going into Earth as 2/3 frequency and then Water turning into Air as 3/2 frequency. Air being Aion as the kundalini ionization of neurohormones in meditation. 
I have research details on this - http://elixirfield.blogspot.com
thanks and please let me know any comments, questions or criticisms.
Happy Equinox (full moon)!
drew hempel