https://arxiv.org/ftp/arxiv/papers/1312/1312.5020.pdf
So this is a fascinating paper by Math Professor Robert Schneider.
Submitted 17 December, 2013; originally announced December 2013.
Comments: Proceedings of Bridges: Mathematics, Music, Art, Architecture, Culture Conference (June, 2012)
And there are is a follow up paper as commentary!
Difference Tones in "Non-Pythagorean" Scales Based on Logarithms
In order to explore tonality outside of the `Pythagorean' paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones in this way must consist solely of integer ratios. With this in mind, we present some methods for producing logarithmic scales which contain many, but not all, of the difference tones they generate.
OK this is precisely what I was just thinking!!
I haven't read that paper yet but here's the deal.
"the Pythagorean music school was rapidly able to calculate the tone fa-sol as the difference between the fifth do - sol and the fourth do - fa, and consequently as the ratio 3/2 : 4/3 = 9/8 [logarithm]....The Pythagorean tradition denied that it was possible to divide the tone into two equal parts (semitones [based on rational ratios])....Dividing the Pythagorean tone into two parts would mean admitting the existence of the proportional mean between 9 and 8, that is to say, 9 : [a] = [a] : 8, where 9:[a] and [a]:8 are the proportions of the required semitone....Clearly [a]= [square root of 9 x 8] and therefore [a] = (3 x 2) x [square root of 2]!"
Math Professor Tito M. Tonietti, University of Pisa, Italy
So the "difference" here is subtraction as a logarithm thereby requiring division as inverse or reciprocal multiplication. This means in terms of music theory the G to C as 3/2 is multiplied by 3/4 or G to C' (the C as the octave). This inherent reciprocal inverse math then converts the noncommtuative phase of 4/3 as F to C derived from 2/3 as the noncommtuative undertone or Phantom Tonic....
And then we have this quote:
"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)
Here the "dividing" actually means ADDING as a logarithm so that the product is just straight multiplication based on the "arithmetic mean." And so 3/2 was originally already the Arithmetic Mean based on the noncommutative inverse of the Harmonic Mean of 4/3 - from SUBTRACTING as the difference between the Perfect Fifth and Perfect Fourth. Once the noncommutative phase was covered up - THEN the "division" could be extended as a continued "arithmetic mean" ADDING such that the 3/2 as the arithmetic mean originally can now "produce" the logarithmic addition of 5/4 (the cube root of two) and 8/5 (the cube root of four).
But the ONLY way this can work is due to the original 3/2 as arithmetic mean and 4/3 as harmonic mean assuming that the original octave as the 2 was not a "doubling" but rather already the Geometric Mean SQUARED of the "one." And therefore the continued arithmetic mean derivations assume an octave equivalence from the original 2 as a geometric mean squared value.
This was my original math-music equation that I had sent to Math Professor Luigi Borzacchini but since it was not explicated by Archytas then we can only logically infer that it existed. Clearly the mathematical logic is already there - only the precise equation was not written down. I already wrote this equation down in 2001 and mailed it off to Professor Borzacchini and he stated my math was "good" and then Math Professor Joe Mazur asked me to submit this question to the most read math journal for publication and that I had done "very impressive" research.
OK so now let's go back to this second follow up paper.
The key point I made previously https://elixirfield.blogspot.com/2021/07/grant-sanderson-stanford-trained.html
was that assuming Archytas was already using the logarithmic logic - as is undisputed - therefore the "rate" of growth of the mathematics means that the Square root of two converges much FASTER than the Golden Ratio of the 8/5 as originating from the continued fraction of 1 plus 1 divided by 1.
And yet as I point out in my 2012 book - the "closed" form solution of the Golden Ratio also had to reverse the order of infinity.
OK Now I'll read these two follow up papers and comment on them.
First I need to explain what inspired me to discover the work of Schneider.
I'm reading Dr. Ruth E. Kastner's first book on relativistic quantum physics and she writes in a very clear lucid form - to make the concepts extremely accessible for anyone to understand. So she uses a 3-4-5 Pythagorean triangle to explain imaginary numbers.
In other words we know from Abraham Seidenberg and Plutarch that the Pythagorean Triangle was based on the height or 4 as Osiris and so Osiris was also considered the number 17 as the irrational secret proof between doubling 9 and 8 as 18 and 16. This was tied to the Lunar calendar with 17 as 3 days after the Full Moon and therefore the lunar energy going down.
So now we can realize that the octave doubling of 4/3 as the Harmonic Mean is dependent on the noncommutative reversal of 3/2 due to the logarithmic subtraction as inverse multiplication.
And so the 4 as the octave of the 1 and 2 is in reality hiding the imaginary number of the 1 as the 4 as the same pitch of the root tonic as the C. Therefore the 1 as the All Seeing Eye or I-thought is originally noncommutative as the imaginary time iteration, as explained by math professor Louis Kauffman.
So in Egyptian alchemy then the hypotenuse as 5 is actually an eternal rational harmonization between Set and Horus (satan and jesus). And so this means the "closed" or Power Axiom Set of the irrational square root of two is based on reversing the order of infinity of the Golden Ratio 1 series - so that 5/8 is now 8/5 just as 4/3 is now 3/4.
So again by "difference tones" he means subtracting logarithms as inverse multiplication. Since they are noncommutative then they HAVE to be rational ratios as noncommutative phase!
This is not explicated of course but that is the logical truth.
so...
so....
There you go.
So then....
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So finally:
Hence the number one is inherently a reciprocal ratio of itself containing noncommutative time as its secret imaginary dimension - just as Professor Kauffman explains!!
So to convert this back into the Fibonacci Series:
The key point here is that Fn is based on reciprocal subtraction and hence "closes in" back on 1 by assuming a "final cause" of time that can be reversed in order.
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