Talk:The Riemann zeta function and tuning: Difference between revisions
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== Zeta troughs? == | |||
Would it be possible to use this function to calculate increasingly deep troughs as well as peaks, so as to produce a sequence of EDOs that have increasingly high levels of relative error for their size & inconsistency between various limits? That could be an interesting addition to the lists for people who want to produce intentionally perverse & dissonant music. --[[User:Yourmusic Productions|Yourmusic Productions]] ([[User talk:Yourmusic Productions|talk]]) 17:42, 6 April 2021 (UTC) | |||
: Yes, although since we are looking at the absolute value of the zeta function can be no greater than 0. So, any EDO which has a zeta value of 0 is "maximally inharmonic," in a sense. According to the Riemann hypothesis, anyway, the function can only get to 0 if we're on the critical line, which means we're weighting rationals as <math>1/(nd)^{0.5}</math>. But even on other lines there will always be local minima. Strangely, the touch tone "DTMF" frequencies were chosen, they say, to be maximally inharmonic, because they didn't want to confuse harmonics of the DTMF tones with other tones. You may think that this would mean they went for a zeta zero, but if you look you will see that all of the touch tone frequencies (which form a subset of an equal temperament) magically happen to synchronize with a local *maximum* of the zeta function - though a relatively small one - to within rounding error in Hz. In other words, the DTMF folks went with a relatively inharmonic region of EDO space (14-EDO ish), but then they maximized the harmonicity within that region. Very strange... [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 11:13, 9 April 2021 (UTC) | |||
== Review needed == | == Review needed == | ||
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:: 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 a slight extent (as is evident from the flatness of the graph). | :: 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 a slight extent (as is evident from the flatness of the graph). | ||
:: 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 (regardless of the sign flip), so how would you even tell what a record is? ( | :: 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 (regardless of the sign flip), so how would you even tell what a record is? (If you apply this reasoning correctly to something based on 1 - cos(2pi x) (like the previous step in Gene's derivation), then you get something fairly different, which ''would'' be a relative error score, because it'd always be strictly positive (except for at 0 equal temperament).) | ||
:: 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 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 relative to its step size, 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, 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. | ||
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:: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 21:19, 16 April 2025 (UTC) | :: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 21:19, 16 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 alteration is 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 10th-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#Top 10]] and compare edos you're unsure about to [[User:Godtone/optimal_edo_sequences]] by looking for number of occurrences | |||
(please share any findings/concerns here or on {{nowrap| [[User talk:Godtone/zeta|the talk page for the zeta page I made]]). }} | |||
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; the dream said its 11-limit was good; maybe that's true in the sense that the high errors of 5, 7 and 11 can easily cancel each-other out in ratios or composites, since zeta doesn't obey a val). 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) | |||
== Top 20 (and top 10) zeta edos == | |||
Since I got increasingly curious how high-quality the lists can be using the "top n" strategy applied to zeta peaks, I've detailed them and shared the lightly-modified code here: [[User:Godtone/zeta]] (same page I linked before). | |||
--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 17:06, 24 April 2025 (UTC) | |||