Mike Battaglia @battaglia01@qoto.org
Joined Nov 2022
@mrlyonsmaths Another interesting thing: the ear tends to like very musical intervals whose frequencies are in simple rational relationships (just intonation), which often sound "consonant." Since we can compose two intervals by stacking them on top of one another to form another one, the resulting structure is isomorphic to the multiplicative group of strictly positive rationals.
But what if we want the most "dissonant" intervals as well? Then in some sense we want to add "maximally irrational" numbers. Which numbers are these?
Well, there are a few ways to do it, but one reasonable candidate for this is to look at those numbers whose continued fraction ends in a tail of all 1's (noble numbers). These have a few number-theoretic properties that give them a pretty solid claim to being the "most irrational" numbers. Also, in a musical setting, these tend to be pretty close to the perceptual "most dissonant" interval point in between two neighboring consonant intervals (given a few extra criteria I'm handwaving here), at least approximately as measured by models such as harmonic entropy. So, we may just say they're good enough as a basic idealized model to get started.
The resulting structure we get is the set of positive elements from the quadratic number field Q[√5]. It makes perfect sense, but I still think it's somewhat magical and bizarre. You want the most consonant intervals and you get Q, and if you want to add in the most dissonant as well you get Q[√5].
@mrlyonsmaths My favorite fact is probably the relationship between the Riemann Zeta function and musical tuning theory. The zeta function has this interpretation of telling you how well the various equal temperaments approximate the harmonic series, which I wrote a little bit about here: https://qoto.org/@battaglia01/109338636561305902
I've been reading the past few days about the history of Mastodon - the Gab invasion and backstory regarding which instances defederated them and etc, and where qoto stands on this.
For all of this drama it doesn't look like people on qoto can see much on Gab anyway. For instance, if you look up Andrew Torba (Gab CEO) his account does appear in search from here, but only toots from him prior to like 2020 appear. The rest seem to be censored from somewhere. The same is true for other Gab accounts; it looks like most of them either show nothing or only old toots.
I am super not interested in participating in Gab but am kind of curious about the tech here. Has qoto blocked newer toots from Gab users? Or has Gab blocked the rest of the fediverse? Or what? @freemo
@vihart I listened to this at 4:30 AM and now I wish I had enchiladas. Or at least a tortilla of some kind. But at least the song provided an immersive enchilada experience which I much appreciated, even in the absence of a true enchilada. 5 stars
@robfielding good stuff! Looks like a Dedicated SoundCloud basically?
Some of these Mastodon instances are pretty neat - do there exist any instances for audio or digital signal processing stuff? Maybe @Sevish would be tuned in
Are there any other famous music/math folks on here besides Vi Hart?
@trinsec makes sense, thank you for explaining! That sounds very reasonable. Maybe I will just have a few accounts on here on different instances. BTW is there any way to see which instances block others? Would be nice to get e.g. a large graph
@awwbees Neat stuff! Does it have any microtonal tuning support?
Sheesh, looks like this "QOTO" server I joined is blocked/defederated from a bunch of other instances so people can't see me. I just picked this one randomly because of the 65535 char max and LaTeX support. Is there a better Instance I should join?
@Sevish yeah for real man! I was looking at PeerTube as well, thinking of mirroring my videos on it. Are you on there?
@Sevish man I'm trying! What instance do you think I should join? (Is there any difference between instances?)
While we're talking about mathematical music theory it is very nice to see @vihart on here of whom I have been a huge fan for many years...
Any interest in microtonality? Was just talking to @keenanpepper about how the Riemann zeta function can be thought of as measuring the harmonicity of an equal temperament. Maybe of interest to you? https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning
@freemo just trying the same LaTeX here: \[ G = \left\langle a,b,c \mid a^2, b^2, c^2 \right\rangle \]
And some other LaTeX
\[ x+y+\frac{z}{w} \]
\(\log x^2\)
@freemo Hi, new to QOTO but enjoying it so far! I noticed a bug with LaTeX rendering on Android as it relates to federating w/ Mathstodon... Things that render there don't seem to render here sometimes. For instance, look at this tweet on Mathstodon from earlier today:
https://mathstodon.xyz/@glakeland/109338986335014383
For whatever reason the LaTeX rendering is partly broken. Loading it three times gave three different weird results: either a "Math Processing Error," not parsing it at all or a very strange rendering with huge angle brackets. The fourth pic (in a different font) is Mathstodon's rendering which seems correct. This is on Android using both Chrome and Tusky.
Anyway though, great server, not sure where to post bug reports but hopefully this gets seen! Cheers
Hi everyone, new on here. My focus is on digital signal processing, some ML/CS stuff, and microtonal music. I'm looking for what instance to join and who to follow. Is QOTO good for this kind of thing?
@johncarlosbaez @keenanpepper before reading the derivation I would first just look at a simple graph of the zeta function on the critical line!
https://www.wolframalpha.com/input?i=plot+abs%28zeta%280.5%2Bi*t*%282*pi%29%2Fln%282%29%29%29+from+t%3D0..40
You can see there are some pretty prominent peaks at some familiar looking numbers, such as 12, 19, 22, 31, 41, 53, 72 divisions of the octave, all of which have turned up pretty frequently in tuning theory (and sometimes historically, throughout the world) for having excellent representations of various just intonation ratios (relative to their size).
The derivation is not that difficult if one looks at Re[s] > 1 where the Dirichlet series is convergent, although probably best to just look at the results first! It is very easy to see manually that, for instance, 31 equal has better approximations of simple harmonic ratios than its neighbors 30 and 32 (as the zeta function correctly predicts).
@keenanpepper we should really clean this up and publish it properly sometime!
@johncarlosbaez OK, this is long enough, and I didn't expect it'd be that long, but hopefully people who are interested in microtonality will think this is interesting.
The TL;DR:
1. As far as small equal temperaments go, 12-equal has about as good of an approximation to the perfect fifth and fourth as you can get.
2. The "problem" with it is the other intervals, not the fifths and fourths.
3. The Pythagorean comma isn't even the comma most people were focused on, historically, at least in the setting of Western polyphonic music.
4. The real comma of interest in that regard is the syntonic comma, and tempering it out gives you meantone temperament.
5. 31-equal is about as good as meantone gets; chords sound much crunchier.
If you want to hear what any of this sounds like, here are some musical examples in 31 equal:
Sweet Lorraine: https://www.youtube.com/watch?v=RGZ0JlMwZpY
Speak Like a Child:
https://www.youtube.com/watch?v=gKT3W2aF4LA&t=54s
Infant Eyes: https://www.youtube.com/watch?v=uIYg8b2p8JY
Anyway, this was fun though longer than I expected! End rant. [6/6]
Joined Nov 2022
