Thanks to Alina Abraham, music theory teacher, for challenging and engaging w/ noncommutative music

alina abraham
..once a G is 3 (3rd overtone in a harmonic series in which C=1), the same G3 cannot be undertone of 1! but undertone of 4 in my opinion. Thank you..
ra sixa
Hi, I think so too. In oldest music notation the undertone 4 was as drone note? needs to be researched in detail yet, imho.
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drew hempel
Jairazbhoy notes the Phantom Tonic concept as key to Indian music meditation: page 72
" Ma [Perfect Fourth as 4/3], although consonant to Sa (root tonic), is alien to the overtone series and is not evoked in the sound of Sa. On the other hand, Sa is evoked in the sound of Ma, since Sa is a fifth above Ma and is its second overtone.
Now for the historical origin that covered up this change in the root tonic as per the introduction of the term "magnitude" in Western mathematics from the irrational number music theory we rely on Professor Richard McKirahan's analysis of Philolaus. "The word translated epogdoic is not a musical term but a mathematical one. An epogdoic ratio is the ratio of 9 to 8. The occurrence of a mathematical term here is unexpected." (McKirahan, 2013)
"Can we do something similar with Philolaus's other claim that 3:2 + 4:3 = 2:1? We might think that this would amount to the claim that 12:8 plus 12:9 = 12:6, but this is not an obvious result, since it is immediately not clear how to perform the addition. ... So instead of taking 12:9, which is 3/4 of 12, we take 8:6, which is 3/4 of 8. And so by adding the length 12 to 8 we get the length 12 to 6, which corresponds to the ratio 2:1." (McKirahan, 2013)
So that proves indeed Philolaus flipped his lyre around - as McKirahan explains - thereby changing the root tonic in order to transform the Perfect Fourth into an irrational geometric magnitude ratio! "In other words we start at 12 and take 2/3 of its length. Then we add a length corresponding to the ratio 4:3, but this time we are starting not at 12 but at 8 and we want to take 3/4 of that. (McKirahan, 2013) ...This is true because Philolaus is dealing no longer with the musical intervals between the notes made by a particular pair of strings, but with the magnitude of those intervals ... (1, 4) = (7, 5)." (McKirahan, 2013) see McKirahan, R. (2013). Philolaus on Number From the book On Pythagoreanism Colloquium 7: Philolaus on Number. (2012). Proceedings of the Boston Area Colloquium of Ancient Philosophy, 27(1), 211–239. doi:10.1163/22134417-90000137 
But if we rely on the truth of reality and not this "trick" of changing the root tonic then we confront noncommutativity in music theory as Fields Medal math professor Alain Connes explains (and ONLY Fields Medal math professor Connes - the rest of the teaching in the West is mass mind control of commutative geometry!). Any musician knows from internal listening that 1 to 4 is C to F while 7 to 5 is C to G.
"When you permute A and B, and you make the A pass on the other side, you have to make it evolve with time. And the time in which it has to evolve is in fact the purely imaginary number. This is what is behind the scenes." (Jackson & Connes, 2021)
has the details thanks