The Riemann zeta function and tuning: Difference between revisions

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{{EDOs|1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691,}} ... of ''zeta peak edos''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}.

Alternatively, if we demand no octave stretching and thus only look at the record peaks corresponding to exact EDOs, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494,}} ... of ''zeta peak integer EDOs''. EDOs in this list not included in the previous are {{EDOs|87, 311, 472,}} ... and EDOs not included in this list but included in the previous are {{EDOs|27, 72, 99, 152, 217, 342, 422, 441,}} ... 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.

Alternatively, if we demand no octave stretching and thus only look at the record peaks corresponding to exact EDOs, 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}} ... 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. As [[User:Ks26|ks26]] has found, the previous definition is better for measuring how near an edo is to especially accurate non-integer edos, rather than measuring how accurate the edo itself is. Hence the previous list may better be named ''zeta rounded peak edos''.

Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the ''zeta integral edos'', goes {{EDOs|2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973,}} ... This is listed in the OEIS as {{OEIS|A117538}}. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.

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 {{EDOs|1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691,}} ... of ''zeta peak edos''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}.
-Alternatively, if we demand no octave stretching and thus only look at the record peaks corresponding to exact EDOs, we get {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494,}} ... of ''zeta peak integer EDOs''. EDOs in this list not included in the previous are {{EDOs|87, 311, 472,}} ... and EDOs not included in this list but included in the previous are {{EDOs|27, 72, 99, 152, 217, 342, 422, 441,}} ... 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.
+Alternatively, if we demand no octave stretching and thus only look at the record peaks corresponding to exact EDOs, 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}} ... 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. As [[User:Ks26|ks26]] has found, the previous definition is better for measuring how near an edo is to especially accurate non-integer edos, rather than measuring how accurate the edo itself is. Hence the previous list may better be named ''zeta rounded peak edos''.
 Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the ''zeta integral edos'', goes {{EDOs|2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973,}} ... This is listed in the OEIS as {{OEIS|A117538}}. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.