Talk:Direct approximation: Difference between revisions

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 ::::: Nice.  Now that I see what's being talked about on that page, it makes sense.  Regardless, I think we should wait for Sintel to weigh in, just to make sure I at least have all my facts straight. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 14:55, 22 December 2021 (UTC)
+:::::: Just to give a sort of overview here, there are two ways you could go about constructing a mapping for 12edo.
+::::::
+:::::: The first is the construct a regular temperament, with mapping M = [12 19 28]. To calculate the output you just take the prime factorization and multiply, as we're used to:
+::::::* M(5/4) = 4
+::::::* M(81/80) = 0
+::::::* M((81/80)^3) = 0
+::::::
+:::::: Now because M is linear, it satisfies the conditions:
+::::::* M(a) + M(b) = M(a * b)   (for two JI intervals a, b)
+::::::* n*M(a) = M (a^n)  (for some integer n)
+::::::
+:::::: So "linear map" really means: a map that conserves interval arithmetic. I think this is what the word "regular" refers to. ("linear temperament" already means something else sadly.)
+::::::
+:::::: (Factoring through the map, addition becomes multiplication because of the way interval arithmetic works on <math>\mathbb{Q}(\times)</math>)
+::::::
+:::::: The second way to define the 12edo temperament is to just round. The map is then:
+:::::: F(x) = round(12 * log2(x))
+::::::* F(5/4) = 4
+::::::* F(81/80) = 0
+::::::* F((81/80)^3) = 1 (!)
+::::::
+:::::: Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations,  I think most people prefer conserving interval interval arithmetic, which is why we care about RTT in the first place.
+::::::
+:::::: Note that all of this is unrelated to the fact that we found the regular 12edo map by rounding. It's just a sort of shortcut to find linear maps that have reasonable errors. Note also that in the definition of the linear map above, you never have to take any logarithms, they're also just convenience.
+::::::
+:::::: "Direct approximation" is a good name imo. Regarding "simple map" and "integer uniform map", I have some more thoughts on that but I'll talk about those there.
+::::::
+:::::: - [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 18:26, 22 December 2021 (UTC)
 : I second that this page had better be moved to ''direct approximation''. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:39, 22 December 2021 (UTC)