The Riemann zeta function and tuning: Difference between revisions

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==== Valley edos ====

Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs|1, 8, 18, 39, 55, 64, 79, 5941, 8294}}... These tunings tend to deviate from ''p''-limit JI as much as possible, and can serve as "more xenharmonic" tunings. Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that |Z(x)| grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval [0, ''c'' log ''x''], the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos.

Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs|1, 8, 18, 39, 55, 64, 79, 5941, 8294}}... These tunings tend to deviate from ''p''-limit JI as much as possible, and can serve as "more xenharmonic" tunings. Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that |Z(x)| grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval [0, ''c'' log ''x''], the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.

Note that ''tempered-octave'' zeta valley edos make no sense, since any zero of Z(x) would qualify for a |Z(x)| minimum.

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 ==== Valley edos ====
-Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs|1, 8, 18, 39, 55, 64, 79, 5941, 8294}}... These tunings tend to deviate from ''p''-limit JI as much as possible, and can serve as "more xenharmonic" tunings. Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that |Z(x)| grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval [0, ''c'' log ''x''], the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos.
+Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs|1, 8, 18, 39, 55, 64, 79, 5941, 8294}}... These tunings tend to deviate from ''p''-limit JI as much as possible, and can serve as "more xenharmonic" tunings. Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that |Z(x)| grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval [0, ''c'' log ''x''], the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.
 Note that ''tempered-octave'' zeta valley edos make no sense, since any zero of Z(x) would qualify for a |Z(x)| minimum.