Talk:Just intonation point: Difference between revisions

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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 [[~~https://en.xen.wiki/w/~~User:Sintel/The_mathematics_of_temperaments|here]].

: 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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 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 [[https://en.xen.wiki/w/User:Sintel/The_mathematics_of_temperaments|here]].
+: 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)