The Riemann zeta function and tuning: Difference between revisions
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==== Zeta peak edos ==== | ==== Zeta peak edos ==== | ||
If we examine the increasingly larger peak values of {{nowrap|{{!}}Z(''x''){{!}}}}, we find they occur with values of ''x'' such that {{nowrap|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,}} | If we examine the increasingly larger peak values of {{nowrap|{{!}}Z(''x''){{!}}}}, we find they occur with values of ''x'' such that {{nowrap|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 occur close to integer values, but are never exactly located at an integer; this can be interpreted as the zeta function suggesting [[stretched and compressed tuning|detuned (stretched or compressed) octaves]] 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." | ||
==== Zeta peak integer edos ==== | ==== Zeta peak integer edos ==== | ||
Alternatively (as [[groundfault]] has found), if we | Alternatively (as [[groundfault]] has found), if we instead only look at the record {{nowrap|{{!}}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,}}{{ellipsis}} of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}}{{ellipsis}} and edos present in the previous list but not present here include {{EDOs|4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}}{{ellipsis}}. 72's removal is perhaps being the most surprising, showing how strong 53 is in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate EDOs are without stretched or compressed 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. | ||
==== Zeta integral edos ==== | ==== Zeta integral edos ==== | ||
Similarly, if we take the integral of {{nowrap|{{!}}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,}} | Similarly, if we take the integral of {{nowrap|{{!}}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,}}{{ellipsis}} 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 edos ==== | ==== 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, }} | 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,}}{{ellipsis}} 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 edos ==== | ==== 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,}}{{ellipsis}}. | ||
It's debatable whether the constraint on pure octaves is a good idea, as it could be interpreted as allowing for accounting for tuning tendency sharpward or flatward, in which case the two smallest additional edos that are strict zeta edos are [[72edo]] and [[954edo]]. | It's debatable whether the constraint on pure octaves is a good idea, as it could be interpreted as allowing for accounting for tuning tendency sharpward or flatward, in which case the two smallest additional edos that are strict zeta edos are [[72edo]] and [[954edo]]. | ||
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=== Anti-record edos === | === Anti-record edos === | ||
==== Zeta valley edos ==== | ==== Zeta valley edos ==== | ||
In addition to looking at {{nowrap|{{!}}Z(x){{!}}}} maxima, we can also look at {{nowrap|{{!}}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, }} | In addition to looking at {{nowrap|{{!}}Z(x){{!}}}} maxima, we can also look at {{nowrap|{{!}}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,}}{{ellipsis}} These tunings tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all. | ||
Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that {{nowrap|{{!}}Z(x){{!}}}} grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[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. | Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that {{nowrap|{{!}}Z(x){{!}}}} grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[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. | ||
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==== Parker edos ==== | ==== Parker edos ==== | ||
Named after the Parker square in mathematics, ''Parker edos'' may be defined as non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo. A helpful list for finding an alternative to any given zeta peak edo of similar size and still-okay accuracy, but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19). | |||
{{EDOs|6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87, 111, 140, 183, 243, 301, 311, 460, 472, 525, 571, 581, 814, 836, 882, 1205,}} | {{EDOs|6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87, 111, 140, 183, 243, 301, 311, 460, 472, 525, 571, 581, 814, 836, 882, 1205,}}{{ellipsis}} | ||
We can then remove those secondary peaks again to get '''tertiary-peak edos'''. | We can then remove those secondary peaks again to get '''tertiary-peak edos'''. | ||
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==== Local zeta edos ==== | ==== Local zeta edos ==== | ||
We may define ''local zeta'' edos as a generalization of the ''zeta peak'' and ''zeta peak integer'' EDOs as those that do not necessarily have successively higher zeta peaks but simply have a higher zeta peak than the edos on either side of them. This is a helpful list for finding EDOs that approximate primes well | We may define ''local zeta'' edos as a generalization of the ''zeta peak'' and ''zeta peak integer'' EDOs as those that do not necessarily have successively higher zeta peaks but simply have a higher zeta peak than the edos on either side of them. This is a helpful list for finding EDOs that approximate primes well (but are not necessarily the best at doing so) for their size, or for finding EDOs in size ranges that lack any record-holding zeta edos (e.g. between 60 and 70 tones). | ||
{{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, 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, 72, 75, 77, 80, 82, 84, 87, 89, 91, 94, 96, 99,}}{{ellipsis}} | ||
==== Local anti-zeta edos ==== | ==== Local anti-zeta edos ==== | ||
We may define ''anti-zeta'' edos as the opposite of zeta peak and local zeta edos (i.e. those with a ''lower'' zeta peak than the edos on either side of them). This is helpful for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge to inspire creativity. | We may define ''anti-zeta'' edos as the opposite of zeta peak and local zeta edos (i.e. those with a ''lower'' zeta peak than the edos on either side of them). This is helpful for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge to inspire creativity. | ||
{{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,}}{{ellipsis}} | ||
==== Indecisive edos ==== | ==== Indecisive edos ==== | ||
Finally, ''indecisive'' edos can be defined as edos which are neither local zeta, nor anti-zeta. These tunings are more restrictive than local zeta edos, but not as far off the deep end as anti-zeta edos. They might narrow down the range of compositional choices available so as to be not so many to promote indecision, but not so few as to promote frustration. | Finally, ''indecisive'' edos can be defined as edos which are neither local zeta, nor anti-zeta. These tunings are more restrictive than local zeta edos, but not as far off the deep end as anti-zeta edos. They might narrow down the range of compositional choices available so as to be not so many to promote indecision, but not so few as to promote frustration. | ||
{{EDOs| 9, 14, 21, 26, 32, 39, 45, 51, 55, 62, 67, 70, 74, 79, 85, 93, 98,}} | {{EDOs|9, 14, 21, 26, 32, 39, 45, 51, 55, 62, 67, 70, 74, 79, 85, 93, 98,}}{{ellipsis}} | ||
== Optimal octave stretch == | == Optimal octave stretch == | ||