Talk:Just intonation point: Difference between revisions

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::: 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 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)