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''. 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.

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

@@ Line 295: / Line 295: @@
 ===== Valley edos =====
-Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima''. 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.
 Note that ''tempered-octave'' zeta valley edos make no sense, since any zero of Z(x) would qualify for a |Z(x)| minimum.