The Riemann zeta function and tuning: Difference between revisions

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Alternatively (as [[groundfault]] has found), if we allow no octave stretching and thus only look at the record |Z(x)| zeta scores corresponding to exact edos with pure octaves, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973}} ... of ''zeta peak integer edos'', Edos in this list not included in the previous are {{EDOs|87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} ... and edos not included in this list but included in the previous are {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} ... with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on the peak of 53. This definition may be better for measuring how accurate the edo itself is without stretched octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos." Similarly, we can look at pure-tritave EDTs, etc.

~~===== 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.

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

==== Integral of zeta edos ====

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==== Strict zeta edos ====

We may define the ''strict zeta edos'' to be the edos that are in all four of the above lists. The list of strict zeta edos begins {{EDOs|2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578, 8539, 14348, 58973}}... .

==== 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.

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

=== ''k''-ary peak edos{{idiosyncratic}} ===

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 Alternatively (as [[groundfault]] has found), if we allow no octave stretching and thus only look at the record |Z(x)| zeta scores corresponding to exact edos with pure octaves, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973}} ... of ''zeta peak integer edos'', Edos in this list not included in the previous are {{EDOs|87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} ... and edos not included in this list but included in the previous are {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} ... with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on the peak of 53. This definition may be better for measuring how accurate the edo itself is without stretched octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos." Similarly, we can look at pure-tritave EDTs, etc.
-===== 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.
-Note that ''tempered-octave'' zeta valley edos make no sense, since any zero of Z(x) would qualify for a |Z(x)| minimum.
 ==== Integral of zeta edos ====
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 ==== Strict zeta edos ====
 We may define the ''strict zeta edos'' to be the edos that are in all four of the above lists. The list of strict zeta edos begins {{EDOs|2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578, 8539, 14348, 58973}}... .
+==== 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.
+Note that ''tempered-octave'' zeta valley edos make no sense, since any zero of Z(x) would qualify for a |Z(x)| minimum.
 === ''k''-ary peak edos{{idiosyncratic}} ===