@johncarlosbaez A very interesting post with a long thread of comments
https://mathstodon.xyz/@johncarlosbaez/115774416563837297

@johncarlosbaez Holy shit. This information is EXACTLY what I needed right NOW! Thank you John!

@johncarlosbaez

Hi! A couple of thoughts from someone who is fascinated with musical tuning but in a fairly superficial way.

First, in looking at the hexagonal grid pages about 2/3 of the way through I wondered: You touch on scales with more than 12 tones, but only in the guise of equal temperament (29-TET). Would it be possible instead to develop a family of "just" intonations with 29 tones? Would it be described by a similar kind of grid, just with more elements, or would the shape be different?

Second, I went looking in OEIS for the list of "best approximation to 3/2" and found https://oeis.org/search?q=5%2c7%2c12%2c29%20id:A060528 -- but it goes 1,2,3,5,7,12,29,... and I can't figure out what difference between the two statements of the problem leads to 3 being/not being on the frontier of best approximations.

Finally, just an observation: It's unusual to see a discussion of tuning without an appeal to the overtone series somewhere. I assume this was a conscious decision.

@johncarlosbaez i do NOT understand math or music, but still enjoyed reading all of this. Thanks :D

@johncarlosbaez Hey that's great. I don't know the first thing about music theory. But one of the kids is learning an instrument and I've got a math background... This is very welcome, thanks! And happy holidays.

@johncarlosbaez For people curious about the subject, I want to mention the lovely 2023 ICFP FARM talk by Gloria Cheng https://youtu.be/k_htPyDDIwk?si=TpvGiQL1Nq-dE4Kl

@johncarlosbaez Would love to see the video recording of the talk, if any.

@johncarlosbaez That's really good! Thanks!

@johncarlosbaez
Wow! Thanks for this, it's fascinating!

@johncarlosbaez

I'm sure you're well aware of all those other notes there are, that appear in, for example, Ottoman scales, Indian music, and so on...

https://www.doctor-dark.co.uk/rebetiko/hydra2004fri.html

@johncarlosbaez i like this web page https://aatishb.com/dissonance/ and this video https://youtu.be/tCsl6ZcY9ag which explain how harmony and dissonance are related to the overtones of notes

especially because that helps to explain how different tuning systems arise in other musical traditions: western instruments make sounds wih overtones on the harmonic series, but that isn’t true for bells or bars

@johncarlosbaez (Typo in first post...missing the 'h' in 'http' for the slides link.)

@johncarlosbaez Recently got curious about temperaments - long story related to Rob Reiner and "This Is Spinal Tap". Found some fascinating information including a standard for defining tunings called Scala - see https://huygens-fokker.org/scala/ if curious.

@johncarlosbaez

But not Tmas?

The world is, after all, Minkowskian. That 4th time-like tree is essential, because otherwise how will you know when it's the holidays?

@johncarlosbaez I was expecting to see the 12th root of 2 somewhere in there.

@johncarlosbaez Math, music and time travel - nice!

@johncarlosbaez One if my favorite topics. The impossibility of a "perfect" tuning has always fascinated me. Kind of like Gödel for musicians :)

If you look at a piano keyboard you'll see groups of 2 black notes alternating with groups of 3. So the pattern repeats after 5 black notes, but if you count you'll see there are also 7 white notes in this repetitive pattern. So: the pattern repeats each 12 notes.

Some people who never play the piano claim it would be easier if had all white keys, or simply white alternating with black. But in fact the pattern makes it easier to keep track of where you are - and it's not arbitrary, it's musically significant.

(2/n)