Talk:Just intonation point: Difference between revisions

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:: 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? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:14, 7 May 2021 (UTC)

::: I also ~~don~~'t understand the article ~~right now~~, 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 and ~~Inthar~~. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 11:37, 8 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. --[[User:Xenwolf|Xenwolf]] ([[User talk: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. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk: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 [[User:Sintel/The_mathematics_of_temperaments|here]].

- [[User:Sintel|Sintel]] ([[User talk: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. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 10:58, 19 December 2021 (UTC)

::: Yes, this fact follows directly from the definition. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 14:35, 19 December 2021 (UTC)

:::: It doesn't if we go with saying the JIP is just a list of generators. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 17:00, 19 December 2021 (UTC)

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 :: 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? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:14, 7 May 2021 (UTC)
-::: I also don't understand the article right now, 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 and Inthar. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 11:37, 8 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. --[[User:Xenwolf|Xenwolf]] ([[User talk: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. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk: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 [[User:Sintel/The_mathematics_of_temperaments|here]].
+- [[User:Sintel|Sintel]] ([[User talk: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. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 10:58, 19 December 2021 (UTC)
+::: Yes, this fact follows directly from the definition. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 14:35, 19 December 2021 (UTC)
+:::: It doesn't if we go with saying the JIP is just a list of generators. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 17:00, 19 December 2021 (UTC)