The Riemann zeta function and tuning: Difference between revisions

One way to frame this in the usual group-theoretic paradigm is to consider the group in which each strictly positive rational number is given its own linearly independent basis element. In other words, look at the [https://en.wikipedia.org/wiki/Free_group free group] over the strictly positive rationals, which we'll call "'''meta-J''i'." The zeta function can then be thought of as yielding an error for all meta-JI [[Patent_val|generalized patent vals]]. Whether this can be extended to all meta-JI vals, or modified to yield something nice like a "norm" on the group of meta-JI vals, is an open question. Regardless, this may be a useful conceptual bridge to understand how to relate the zeta function to "ordinary" regular temperament theory.

The prime-approximating strength of an edo can be determined by the magnitude of Z(x). Since a higher {{nowrap|{{!}}Z(x){{!}}}} correlates to a stronger tuning, we would like to find a sequence with succesively larger {{nowrap|{{!}}Z(x){{!}}}}-associated values satisfying some property.

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 a stretched or compressed octave 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."

Alternatively (as [[groundfault]] has found), if we do not allow octave stretching or compression and only look at the record {{nowrap|{{pipe}}Z(x){{pipe}}}} 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 not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … 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,}} … 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.

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,}} … 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.

Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that {{nowrap|{{pipe}}Z(x){{pipe}}}} 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.

{{Idiosyncratic terms|the term "k-ary-peak edos" itself, as well as the names for the different types of k-ary-peak edos. Proposed by [[User:Akselai]] and [[Budjarn Lambeth]].}}

where the product is over all primes ''p''. The product converges for values of ''s'' with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(''s'') by the corresponding factors {{nowrap|(1 − ''p''^−''s'')}} for each prime ''p'' we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for ''s'' with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 − 2^−s)ζ(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

Along the critical line, {{nowrap|{{!}}1 − ''p''^{−{{frac|1|2}} − it}{{!}}}} may be written

Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before.

Removing 2 leads to increasing adjusted peak values corresponding to the division of 3 (the "tritave") into {{EDTs|4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316...}} parts. A striking feature of this list is the appearance not only of [[13edt|13edt]], the [[Bohlen-Pierce|Bohlen-Pierce]] division of the tritave, but the multiples 26, 39 and 52 also.

From this we may deduce that {{nowrap|θ(''t'')/π ≈ r ln(''r'') − ''r'' − 1/8}}, where {{nowrap|''r'' {{=}} ''t'' / (2π) {{=}} ''x'' / ln(2)}}; hence while x is the number of equal steps to an octave, ''r'' is the number of equal steps to an "e-tave", meaning the interval of ''e'' {{nowrap|1200 / ln(2) {{=}} 1731.234}} cents.

Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{!}}ζ{{!}} {{=}} {{!}}Z{{!}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} x/ln(2)}}, and if {{nowrap|''n'' {{=}} ⌊''r'' ln(''r'') − ''r'' + 3/8⌋}} is the nearest integer to {{nowrap|θ(2π''r'') / π}}, then we may set {{nowrap|''r''⁺ {{=}} (''r'' + ''n'' + 1/8) / ln(r)}}. This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.

For an example, consider {{nowrap|''x'' {{=}} 12}}, so that {{nowrap|''r'' {{=}} 12/ln(2) {{=}} 17.312}}. Then {{nowrap|''r'' ln(''r'') − ''r'' − 1/8 {{=}} 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|n {{=}} 32}}. Then {{nowrap|(''r'' + ''n'' + ''1/8'') / ln(r) {{=}} 17.338}}, corresponding to {{nowrap|x {{=}} 12.0176}}, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents.

The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for {{nowrap|θ(2π''r'') / π}}, which was 31.927. Then {{nowrap|32 − 31.927 {{=}} 0.0726}}, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo ''x'' by computing {{nowrap|⌊''r'' ln(''r'') − ''r'' + 3/8⌋ − ''r'' ln(''r'') + ''r'' + 1/8}}, where {{nowrap|r {{=}} ''x'' / ln(2)}}. This works more often than not on the clearcut cases, but when x is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.

The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|z {{=}} 1}} which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of {{nowrap|(z − 1)}}, but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of {{nowrap|(1 − 2^(1 − z))}}, leading to the eta function:

The Dirichlet series for the zeta function is absolutely convergent when {{nowrap|s > 1}}, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points {{nowrap|1 + 2πix / ln(2)}} corresponding to pure octave divisions along the line {{nowrap|s {{=}} 1}}, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [[Wikipedia:Euler summation|Euler summation]].