The Riemann zeta function and tuning: Difference between revisions
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More can be found at the page on [[Harmonic entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|harmonic entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>. | More can be found at the page on [[Harmonic entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|harmonic entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>. | ||
== Zeta | == Zeta edo lists == | ||
=== Record edos === | |||
The prime-approximating strength of an edo can be determined by the magnitude of Z(x). Since a higher |Z(x)| correlates to a stronger tuning, we would like to find a sequence with succesively larger |Z(x)| values satisfying some property. | |||
==== Peak edos ==== | |||
If we examine the increasingly larger peak values of |Z(x)|, we find they occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[edo]]s | |||
{{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, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} ... of ''zeta peak edos''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks typically do not occur at exact integer values, but are close to integer values; this can be interpreted as the zeta function suggesting a "stretched octave" tuning for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos." | |||
=== Integral of zeta | 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. | ||
==== Integral of zeta 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. | 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. | ||
=== Zeta gap | ==== Zeta gap edos ==== | ||
Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of ''zeta gap edos''. These are {{EDOs|2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, 8539, 14348, 58973, 95524,}} ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as {{OEIS|A117537}}. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line. | Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of ''zeta gap edos''. These are {{EDOs|2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, 8539, 14348, 58973, 95524,}} ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as {{OEIS|A117537}}. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line. | ||
=== Strict zeta | ==== 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}}... . | 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}}... . | ||
The following list of edos are not determined by successively large measured values, they are edos that purely satisfies some property relating to zeta peaks instead. | === Local property edos === | ||
The following list of edos are not determined by successively large measured values, they are edos that purely satisfies some local property relating to zeta peaks instead. | |||
=== Local zeta edos === | ==== Local zeta edos ==== | ||
Edos with a higher zeta peak than the edos on either side of them: | Edos with a higher zeta peak than the edos on either side of them: | ||
{{EDOs|5, 7, 10, 12, 15, 17, 19, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 79, 72, 75, 77, 80, 82, 84, 87, 89, 91, 94, 96, 99}}… | {{EDOs|5, 7, 10, 12, 15, 17, 19, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 79, 72, 75, 77, 80, 82, 84, 87, 89, 91, 94, 96, 99}}… | ||
=== Anti-zeta edos === | ==== Anti-zeta edos ==== | ||
Edos with a lower zeta peak than the edos on either side of them: | Edos with a lower zeta peak than the edos on either side of them: | ||
{{EDOs|6, 8, 11, 13, 16, 18, 20, 23, 25, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 86, 88, 90, 92, 95, 97}}… | {{EDOs|6, 8, 11, 13, 16, 18, 20, 23, 25, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 86, 88, 90, 92, 95, 97}}… | ||
=== Sloped-zeta edos === | ==== Sloped-zeta edos ==== | ||
Edos which are neither local zeta edos, nor anti-zeta edos: | Edos which are neither local zeta edos, nor anti-zeta edos: | ||
{{EDOs|1, 2, 3, 4, 9, 14, 21, 26, 32, 39, 45, 51, 54, 62, 67, 70, 74, 70, 85, 93, 98}}… | {{EDOs|1, 2, 3, 4, 9, 14, 21, 26, 32, 39, 45, 51, 54, 62, 67, 70, 74, 70, 85, 93, 98}}… | ||