Music theory never made any sense to me at all because the western scale of 12 notes seems completely arbitrary. Then someone finally explained to me the only reason for the 12 note scale is the fact that 2^(7/12) is very close to 3/2 and most people's ears aren't good enough to tell the difference. Mind was forever blown. It turns out the 12 note equal temperament scale is just the smallest one that happens to come close enough to hitting a perfect fifth to fool our ears.
the problem with what you were told is that the explanation wrongly assumes that the octave power can be reduced as a logarithm to the fifth as 3. So Sir James Jeans book "Science and Music" uses the correct analysis of 3/2 in comparison to the octave but he is a quantum physicist (or was). Most Westerners think that the Pythagorean Comma is based on already assuming a "squaring" or logarithmic value of the octave. The math turns out to be different. I go into this on my blog. The basic truth was rediscovered by Fields Medal math professor Alain Connes who emphasizes that music theory provides the logical framework for the unified field physics (relativistic quantum reality). So Connes has a good lecture on this - on youtube - just search music and Alain Connes (a Fields Medal is much harder to get than a Nobel Prize). So when people learn music theory (as Neely did), then this foundational fact - this empirical truth - of the "noncommutative phase" is overlooked. Essentially the Perfect Fifth as subharmonic to the "one" is C to F as 2/3 while the overtone Perfect Fifth to the "one" is C to G as 3/2. So that inherently CAN NOT be "compromised" into the logarithmic equal-tempered tuning that you refer to. you say our "ear is fooled." Well actually Blues Music is based on the natural tuning that is NOT fooled (hence the bending of the notes that are closer to the natural harmonics). So also Daoist philosophy is based on the truth of the complementary opposites (yang is the Perfect Fifth and yin is the Perfect Fourth). So the C to F subharmonic is NOT just "divided by 2" back into the octave. Instead the octaves do not line up!! Why? Because the Daoists were being true to empirical reality - and so this reveals what Eddie Oshins at Stanford Linear Accelerator Center realized was the secret of Neigong or Neidan (Daoist alchemy training) - the noncommutative phase of relativistic quantum physics is the SAME as the truth of the empirical resonance that Connes calls, (2, 3, infinity). I'll find the blog post I did that gives the details on that math. But essentially the West wrongly assumes that you can just ADD the multiplication as you did - so you get 2 to the 19th compared to 3 to the 12th with the difference being very close. Again the 7/12 of the octave as an exponential can NOT just be "added" as a logarithmic equation because the Fifth is NON-commutative to the octave. ONLY Alain Connes has realized the crucial importance of this empirical truth and he's made it the foundation of his noncommutative geometry unified field science. The Ancients also realized this truth as alchemy.
http://elixirfield.blogspot.com/2018/06/why-hertz-hurtz-as-ditonic-comma-lie.html OK this shud have it - just scroll way down to get to the blog post. I'll quote some tid bits.
1.05946309436 So that is the value of the 12th root of 2 for equal-tempered tuning. So then you take that to the 7th as 7 half steps from the tonic and you get = 1.49830707688 And so that appears to be close enough to 1.5 for 3/2 as the Pythagorean-Daoist harmonics. But as Sir James Jeans points out - this "error" multiples. So 2 to the 7th is 128 and 3 to the 12th is 531441 And so the assumption is that the logarithms can be added so you get 2 to the 19th=524288 And so the Pythagorean Comma is 531441/524288=1.01364326477 Whereas if you take 3/2 to the 12th instead of assuming that the octave can be divided out of the 3 though "halving" - then you get 129.746337891/128=1.013671875
What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers qn for the real number q=2 (to the 1/12)∼3 (to the 1/19). Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number. As explained in the talk, there is a beautiful space which has the correct spectrum: the quantum sphere http://noncommutativegeometry.blogspot.com/2012/10/the-music-of-spheres.html http://elixirfield.blogspot.com/2018/08/revisiting-noncommutative-music-of.html
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