Talk:Just intonation point

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Would anyone who understands the first part of this page be able to revise it a bit for clarity and/or extend it until it actually explains what JIP is? I'm completely lost. Here's my experience reading this text. It tells me that J is a point in a space. I am now in eager suspense about which point that is. Instead of answering that, a new object M is introduced. Whatever it is, it might be a monzo, it might not. This begs the question: what else could it be? And if it's not a monzo, it's not evaluated in octaves? Who cares if it is or isn't -- what does that have to do with JIP? We then do a "shift" which is weird because we weren't in any acknowledged state or action to shift away from yet, and we take my favorite arbitrary monzo I was implicitly asked to imagine as M (maybe!), which we haven't even used yet, and throw it away, now "representing" it as something using a variable "e" which is certainly not Euler's constant but isn't explained at all, and the grand conclusion is that J "becomes" something. But I don't want to know what it becomes; I want to know what it is in the first place. And the thing that it is shown to become doesn't make any sense from the typical interpretation of val: what kind of val could map all the primes to a single step? If you're going to use a bizarre val like that, certainly it has to be explained. --Cmloegcmluin (talk) 15:58, 6 May 2021 (UTC)

The original text dates back to Gene Ward Smith, who was a mathematician, and - as far as I know - rarely in the mood to make his ideas understandable to non-mathematicians, unfortunately! --Xenwolf (talk) 16:17, 6 May 2021 (UTC)
I know Gene prioritized going deep on the ideas over making them broadly accessible, and I respect that. But now we can take the gifts he did give us and just massage them a bit so their value opens up to more people (and that's what I've been trying to do full-time for almost a month now). The page is becoming much clearer to me now, thanks to the recent help from Flora and Inthar. But I still don't have any idea what's going with the middle paragraph, the monzo part. Can we just remove it? What value does it have? It's meaning or purpose are not clear to me. Mostly, probably, because mechanically I still have no idea what's happening. Tenney-weighting's purpose isn't explained and it's not linked anywhere, and I can't find a straightforward explanation of what it is on the wiki. I can see some similar patterns. But I still don't know what these e numbers are (and what their relationship to the m numbers is). And I don't understand what it means for J to become something other than what it is, or how that could be val of all 1's. Can anyone help explain those things, or if the answers to them are beyond the scope of this page, add some links to give the reader who lands on this page navigate closer to the bedrock where they might eventually be able to decipher what they're reading? --Cmloegcmluin (talk) 18:14, 7 May 2021 (UTC)
I also didn't understand much of the article, but let's see if someone is willing to make it even more accessible for non-mathematicians. So I wouldn't suggest to remove anything by now. Thanks so far for Flora, Inthar and you. --Xenwolf (talk) 11:41, 8 May 2021 (UTC)
It's just the tuning map which maps monzos to cent values; I added a high-level blurb. Mike Battaglia (talk) 23:12, 9 May 2021 (UTC)

The logarithm and generators

Math rant: The p-limit is a submodule of Q (the rational numbers) under multiplication. The only reason to take a logarithm is because addition is easier for us (and allows us to use the tools developed in linear algebra). So really, the logarithm is an isomorphism between p-limit JI and the free Z-modules under addition we use in RTT. The generators of JI are just the primes, so the JIP is nothing more than the generators under this isomorphism. I know this sounds pretty abstract but maybe this helps: you can think of the p-limit JI as a temperament with the identity matrix as its map, and the JIP is its generator (list). As such there should probably be a mention of generators, and I propose renaming JIP to something like "just generators".

EDIT: I have written up an explanation here.

- Sintel (talk) 22:20, 18 December 2021 (UTC)

The JIP isn't just a list of generators; it's a function that maps from monzos to their size in cents (or whatever unit you want). In non-weighted coordinates, the coefficients are the sizes of the basis monzos in your subgroup, in cents; in weighted coordinates it'll typically be <1 1 1 1 1 ...| if you're in some subgroup generated by primes. Mike Battaglia (talk) 10:58, 19 December 2021 (UTC)
Yes, this fact follows directly from the definition. Sintel (talk) 14:35, 19 December 2021 (UTC)
It doesn't if we go with saying the JIP is just a list of generators. Mike Battaglia (talk) 17:00, 19 December 2021 (UTC)