So to revisit this pdf - as Adam Neely does my blog post gives the details
So this is the book Adam is referring to, using the term "The Cosmic Joke" for infinite pitch drift - The Arithmetic of Listening by Kyle Gann
So in his book, Kyle Gann is incorrectly claiming that the 5/4 as Major Third is from the 5th overtone harmonic. This is not true at all - it's from Archytas use of geometric mean.
"However, he [Archytas] noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5."A Truman State University review on Scriba, Christoph J. “Mathematics and music.” (Danish) Normat 38 (1990), no. 1, 3–17, 52.
So that's the ORIGIN of Adam incorrectly using 5/4 (and the 16th C. WEstern music theorists) as an extension of 3/2. It's not a "circle of fifths" but in fact it's an infinite spiral of fifths as noncommutative phase (2/3 is C to F while 3/2 is C to G). 6/5 (harmonic mean) x 5/4 (arithmetic mean) = Perfect Fifth as Geometric Mean Squared. 3/2 is NOT Geometric Mean Squared as Adam Neely is claiming.
"Archytas will have assigned this interval the ratio 5:4 (the nearest epimoric smaller than the ditone: (9:8)squared = 81:64 and 5:4 = 80:64)."The Monochord in Ancient Greek Harmonic Science
https://books.google.com/books?isbn=0521843243
David Creese - 2010 - History
So in his book, Kyle Gann is incorrectly claiming that the 5/4 as Major Third is from the 5th overtone harmonic. This is not true at all - it's from Archytas use of geometric mean.
And so as I've pointed out Philolaus only got 9/8 by using a double octave that CHANGES the root tonic pitch of the overtone series! You can't do that - it goes against the overtones and undertones! So in fact 2/3 is C to F as the subharmonic and this is called the Phantom Tonic in music theory. Yep
Kyle Gann does not even mention the "phantom tonic" in his book! Oops. So Philolaus used the root tonic of 0 to 8 so that 3/4 as 6/8 then enabled a new use of geometric magnitude for logarithmic math. So then 12 to 6 as the octave 2 is 12/8 or 3/2 plus 4/3 as 8/6. But the 4/3 was NOT from 0 to 12 - rather it was from 0 to 8! Professor Richard McKirahan gives the details in his essay, "On Philolaus."
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