So this is precisely the same argument I made - well I didn't rely on Euclid but modern analysis of Euclid - for a math paper that math professor Joe Mazur asked me to submit for publication. It's original form is online somewhere - I found it a few years ago on some forum. My Archytas analysis - when I had a dream of an equation and I sent it to Borzacchini, another music-math professor who has recently retired. I wonder if Bellissima references Borzacchini! Also I wonder if Bellissima is willing to acknowledge the noncommutative phase truth origin of music theory math logic!!
Let's take a look at a few of his papers on this topic.
https://sci-hub.tw/10.1080/17498430.2015.1044697
This first one is the Euclid analysis above. I can't wait to dig in!
Propositions VIII.4–5 of Euclid's Elements and the compounding of ratios on the monochord
in BSHM Bulletin Journal of the British Society for the History of Mathematics 30(3):1-17 · August 2015So what I like about his analysis so far is he is getting into the nitty-gritty of the cognitive style involved in the mathematical logic. Most math papers just whizz through the logic as if they're a computer doing calculations. Bellissima is willing to slow down to analyze just what type of thinking is going on!! As Borzacchini points out - this "willful ignorance" about the music origins of math is due to a "cognitive bias."
So next Bellissima emphasizes that in Music the "interval" or pitch (he does not use that term yet) is PERCEIVED (that implies pitch) to be the same if the RATIO is the same (not the "difference"). So here he is explicating the geometric connection to music.
And then we get to the zinger:
Ok - fascinating stuff if you want to truly dissect Euclid, which he does.
So I had this precognitive thought yesterday - I was thinking how the concept of what I've been researching and conveying - the REASON that Westerners can not delve into this easily - is because we automatically associate music theory with sound, as a materialistic tuning device. Rather Pythagorean theory is the origin of sound from harmonic numbers - not sound itself. It is rather listening to the origin of sound. Listening does not require sound but it is STILL based on harmonics. And this is what becomes meditation. Or as Bellissima cites this concept:
Soundless Harmony.
Ok so Bellissima is emphasizing that the fact that Euclid did not associate compounding ratios with multiplication and that the same is true in music theory, therefore corroborates the music origins of the compounding ratios as theorems.
So then Bellissima goes into a detailed discussion of Zarlino's use of Euclid....
And wraps up with this:
So he is then HINTING at noncommutative phase logic - through his analysis of Zarlino. This is quite fasincating.
So by concatenation he also means "the Interval Question" (which is based on the order of the numbers)....in which case you can get the same result from multiplication as with compounding but the multiplication implies a reversal of order that is commutative - something that MISSES the noncommutative process or "mediation" - the means being different - as Zarlino had emphasized.
OK Now onto Bellissima's next paper!
It's on "mode" - I'm not sure I'm going to include this as it does not seem focused on the issue at hand... What's after this?
Logarithmic anamorphosis of Pythagorean intervals
in Bollettino di Storia delle Scienze Matematiche 31(2) · December 2011right - an earlier version. We'll see what we can find. No link to read it.
same with the other one.... So it's just an early summary of Zarlino.
Dear Professor Bellissima: Thank you for your wonderful analysis on the music origins of Western math. I did not notice you citing math professor Luigi Borzacchini?
(see below). It's readable also online. Also Professor Richard McKirahan:
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