Talk:The Riemann zeta function and tuning: Difference between revisions
add extra explanation for why i believe the adjustment is correct |
→Criticisms of and possible improvements to new list: offer alternate list and optimal edo sequences |
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::: I've been trying to figure out how to make it more clear why I believe that the alteration is correct. So it seems like it must be correct to think of cosine as being a function that gives a reward or punishment in the -1 to 1 range so that confirms it is in some sense directly related to the step size. The question is rather whether multiplying by ''x'' is for sure the right adjustment to make it absolute. The reason I believe it is correct is because of how we expect the relative and absolute scoring function to behave w.r.t. contorted systems. Consider 53edo which is very strong in the 5-limit, so that its 5-limit mapping is preserved for quite a few multiples. The cosine will consider 53''x''-edo as having underlying relative errors as ''x'' times as off, and in the absolute sense this corresponds to the maximum offness being 1/''x'' the size. Therefore, if we want to adjust for the fact that we are judging with a -1 to 1 range for harmonics that can only be up to 1/''x'' as off, isn't the correct adjustment necessarily to multiply by ''x''? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 15:42, 17 April 2025 (UTC) | ::: I've been trying to figure out how to make it more clear why I believe that the alteration is correct. So it seems like it must be correct to think of cosine as being a function that gives a reward or punishment in the -1 to 1 range so that confirms it is in some sense directly related to the step size. The question is rather whether multiplying by ''x'' is for sure the right adjustment to make it absolute. The reason I believe it is correct is because of how we expect the relative and absolute scoring function to behave w.r.t. contorted systems. Consider 53edo which is very strong in the 5-limit, so that its 5-limit mapping is preserved for quite a few multiples. The cosine will consider 53''x''-edo as having underlying relative errors as ''x'' times as off, and in the absolute sense this corresponds to the maximum offness being 1/''x'' the size. Therefore, if we want to adjust for the fact that we are judging with a -1 to 1 range for harmonics that can only be up to 1/''x'' as off, isn't the correct adjustment necessarily to multiply by ''x''? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 15:42, 17 April 2025 (UTC) | ||
=== Alternate list based on unmodified zeta function === | |||
Here's a list based purely on unmodified zeta, in case someone proves the alterationis wrong (though I'd still be interested in an "absolute" version of the list that includes an equal temperament if it's better than the 3rd-best-scorer so far). Because there's no accounting for the size of the equal temperament, I'm giving equal temperaments a lot of chances to appear to try to account for this bias, so that an equal temperament appears if it's better than the 9th-best-scorer so far. The other reason I give so many chances is that the resulting list is very similar and also surprisingly high-quality. | |||
Take a look at [[User:Godtone/zeta]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences. | |||
A rather strange recurring theme is 60edo is liked by zeta a surprising amount, but looking at its low- and high-limit tuning profile it doesn't seem that remarkable to me. (A strange coincidence is some time ago I had a dream that this was a good edo. That doesn't happen often at all (dreaming about edos, let alone a specific one being good). Also happens to be significant as the simplest way to represent fourth-order ambiguities in my theory of functional harmony which I derived from first principles starting from [[Ringer scale]]s (especially Perfect Ringer scales), so that (other than the 12edo intervals) it represents the most xenmelodically nontrivial categories available (which correspond to areas of nontrivial harmony).) | |||
--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 18:39, 17 April 2025 (UTC) | |||