Talk:The Riemann zeta function and tuning: Difference between revisions

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 : Secondly, the reasoning you wrote doesn't make sense to me. You go from relative error to absolute error by ''dividing'' by the edo, not multiplying. Zeta does kind of look like a relative error measurent, but it isn't really, so it doesn't work out.
 : – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 20:00, 16 April 2025 (UTC)
+:: Take a close look at the expression at the end of Gene's derivation:
+:: <math>\displaystyle F_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math>
+:: This has the behaviour that it "rewards" for being in-tune and "punishes" for being out-of-tune, and it looks only at harmonics that are prime powers, with a simplicity weighting.
+:: It should hopefully be obvious from this expression that what it is measuring is in terms of steps of ''x'' equal temperament (because "(near-)perfectly in tune" means the cosine in the numerator evaluates to (nearly) 1 and "(near-)perfectly out of tune" means it evaluates to (nearly) -1), which we can confirm by noticing that the unaltered zeta graph has its record peaks grow very slowly/subtly, which should intuitively make sense: if you are measuring by cosine, for every interval that's in there's gonna be an interval that's out; you can't beat this, you can only try to prioritise the intervals that are lower complexity, and even that only works to some extent.
+:: If we divide by ''n'', then we would get a very strange graph that as far as I can tell doesn't make any sense, because the "valleys in error" (which here, because of the sign flip, are peaks) would become smaller and smaller, so how would you even tell what a record is? (Because the result would cross the zero line many times, suggesting those tunings are "perfectly in tune". So if you apply this reasoning correctly by using something based on 1 - cos(2pi x), then you get something fairly different.)
+:: In other words, what zeta gives is a kind of "score" to an EDO based on contributions of prime power harmonics as being in- and out-of-tune, and this score is in terms of pure relative error, therefore it seems to me that the correct alteration must be to multiply by the EDO, because that corresponds exactly to how much of an advantage (in terms of maximum cent error) a larger EDO necessarily has over a smaller one.
+:: Also, zeta contains a lot of interesting tuning information which is basically completely absent in the main lists because of being purely in terms of relative error and only looking at strict records.
+:: Also, "5, 7, 12, 19, 31, 41, 53" is guaranteed in practically list derived from zeta.
+:: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 21:19, 16 April 2025 (UTC)