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.
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