Talk:The Riemann zeta function and tuning: Difference between revisions
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* The construction itself remains largely unmotivated. Why the specific error functions? There is a large amount of handwaving and heuristics. We can use different cyclic error functions which will not lead to zeta, but they seem equally valid. If there is no specific reason to use cosine functions with these specific weights, then this should be clearly mentioned. Currently it seems like the page is actively trying to obscure these facts to make it seem like the connection is more "natural" than it actually is. | * The construction itself remains largely unmotivated. Why the specific error functions? There is a large amount of handwaving and heuristics. We can use different cyclic error functions which will not lead to zeta, but they seem equally valid. If there is no specific reason to use cosine functions with these specific weights, then this should be clearly mentioned. Currently it seems like the page is actively trying to obscure these facts to make it seem like the connection is more "natural" than it actually is. | ||
* Why focus specifically on the critical strip? The page currently states | * Why focus specifically on the critical strip? The page currently states | ||
:: As s approaches the value s = 1/2 of the critical line, the information content, so to speak, of the zeta function concerning higher primes increases [...] | :: As s approaches the value s = 1/2 of the critical line, the information content, so to speak, of the zeta function concerning higher primes increases [...] | ||
: This is very vague and should be clearly explained. | : This is very vague and should be clearly explained. | ||
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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, 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.) | :: 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? (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, 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 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. | :: 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. | ||