Exploring the Connection Between Mathematics and Music | The Chinese Remainder Theorem & Polymetres

In this video, I delve into the intricate relationship between mathematics and music, showcasing how the Chinese Remainder Theorem can be applied to calculate polymetres-a key concept in progressive music. As a mathematician and musician, I explore the beauty of these two disciplines and I am (recreationally) interested in how they intersect in fascinating ways.
About Me:
I'm Will Troiani, a mathematician, musician, and creator of Obdurate, a performance art piece celebrating progressive music. My research spans algebraic geometry, mathematical logic, and theoretical computer science.
Connect with Me:
Instagram: @Obduratefleet34
Bandcamp: Obdurate - obduratefleet....
Personal Website: williamtroiani.github.io
Mathematics Research: williamtroiani.github.io/Mathematics.html
Music Project: williamtroiani.github.io/Obdurate.html
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[@mikey-rc7fo]

Visually position, velocity, and acceleration are foreground, middle ground, and background. They are in order less obvious to the eye. They are related by the derivative and the derivative of the derivative. I have tried to develop something similar for the ears. melody being the foreground, then harmony (the distance between the melodies) and then ...Oh I don't know...The distance between the harmonies. The math of the harmonies of the melodies (or voices SATB) would be division of the frequencies (or subtraction of the exponents) and the "subtraction of the subtraction" would be akin to acceleration for the eyes. It is interesting to note that if one takes the reciprocal function 1/x (the one where the string length gets shorter, the frequency gets higher) and then take its derivates, the intersection points of the foreground 1/x and the middle -1/x^2 and the background 2/x ^3 yield credence to the use of Equal temperament and basic ratios in music. The use of contrary motion in Western Harmony is extended by more divisions to the octave. For example the major third in contrary motion can go the the minor third. and the true tritone (true as exemplified by the derivative intersection points of 1/x) resolves to a perfect fourth or fifth in the microtonal realm. Much like n! is a subset of the continuous gamma function, these discrete resolutions in contrary motion are subsets of some yet undiscovered (or invented) continuous curve. Or maybe it is already known. I have worked in solitude and insanity for years. These continuous curves would be in contrary motion and continuous yet somehow tonal in the Western tradition. It would be part of a music theory that would encompass all possible equal divisions of the octave. The actual contrary motion curves that I am seeking would be in infinite divisions to the octave.

[@tashwinters3355]

Based on my limited understanding: We hear different notes because of different frequencies of sound, right? So a note that sounds higher has a higher frequency than a note that sounds lower. The "frequency" of a note is literally just that - it's how many times per second the note "repeats". Every sound is just a ripple in the air, and our ears interpret each ripple based off of how close the waves are together. Which means ultimately, all sounds are polymetres, just on incredibly small scales. A G1 has a frequency of 49 Hz for example, which means that it is repeating in our ears 49 times per second. A G2 has a frequency of 98 Hz, exactly double that, so they "line up" together perfectly (a G2 happens exactly twice in the time a G1 happens once). Discordant sounds are just sounds which line up less. So yeah, all music is math, and polymetres do control everything. Our brains just "summarize" the information to make it easier for us to parse, and then we play around with it because it we like it when the noises line up.

[@LucianoVeryNiceHi]

mmm mathematics is our way of explaining the world to ourselves. one of the ways, anyways. as goedel and russel showed us - every system is inherently incomplete in itself and it's nature... therefore we invent new systems to understand more because falacies of one system might be advantges in different discipline - chemistry, biology, physics, ... music theory. all those are just systems we use to explain the world to ourselves. that being said, music is not a system, music theory is. music precedes any theory about it. is it legitimate to connect music, or should we be connecting music theory to mathematics instead as comparable referent systems? is there a difference? i want to say there is, regardless of the factual truths such as correlation between series /intervals and music. mandelbrot fractal is just one beautiful shape that was derived from theory, but music is reverse -- we first create beautiful things and then we try to quantize them so we could understand them better. mathematics are beautiful and so is music theory; i am s big afficionado of both but i still want to think that ... first of all, music theory is a full subset of mathematics, secondly music itself is beautiful and while we do use systems to understand it there will never be mathematical formula for beauty... it is true that we live in the age where music is written according to a formula but thats just worthless pop music, not "real" music that moves you ... there's a formula to understand giant steps but there will never be a formula to write something as epic and monumental... AI can write like bach but it will never be bach because, well, bach existed and ai music is modeled after his works. how can you model something that is yet to be invented?? don't get me wrong, music and math are really important in my life and i think i would like the idea that we might be able to formulaicly predict great music but that will never happen. how do you invent new anticipation, culmination and resolution - there is a sea of functions but only one will ever produce mandelbrot set... there is more musical combinations and possibilities then we would ever be able to calculate, let alone to inspect them or find beautiful ones...

[@jakeaustria5445]

Thank you

[@joycebyersfanacc]

very sick 🤩🤩 feel like i’m in one of your tutes

[@mcpr5971]

What matters is the psychological element that establishes what sounds like a half step. That's completely arbitrary.

[@erik2602]

It's not arbitrary. It's based on the natural physical overtones from which you can deduce the major scale.

[@mcpr5971]

@@erik2602 What I meant is there is nothing natural about A440, or that it's divisible by 10, which only seems natural because it's how many fingers we have. Nature doesn't care how many Hz it takes to go up or down a half step. That part is arbitrary based on our hearing, psychology, and eons of history.

[@erik2602]

@@mcpr5971 That's not true. Choosing A = 440Hz is arbitrary, yes. But octaves and overtones are naturally occuring resonances. From these 'sympathy vibrations' you can deduce the half steps. It's only one point as a base that's arbitrarily chosen, but everything around that does in fact appear in nature.

[@LucianoVeryNiceHi]

mathemstics as we know it is completely arbitrary; remove axioms on which our math sits and it's all gone with the wind. doesn't make it less valuable for our understanding of the world around us. music theory is arbitrary and so is western equal temper scale in it's core but that doesn't mean they don't affect real life and perceptions