The Riemann zeta function and tuning: Difference between revisions
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=== Into the critical strip === | === Into the critical strip === | ||
So long as {{nowrap|''s'' ≥ 1}}, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when {{nowrap|''s'' < 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 < ''s'' < 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{frac|2}}}} of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [https://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{frac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{frac|1|2}}}}, which is of the most interest. | So long as {{nowrap|''s'' ≥ 1}}, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when {{nowrap|''s'' < 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 < ''s'' < 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{frac|1|2}}}} of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [https://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{frac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{frac|1|2}}}}, which is of the most interest. | ||
As {{nowrap|''s'' > 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' >> 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>−''s''</sup>}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +∞}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' >> 1}} the derivative is approximately {{nowrap|−ln(2) / 2<sup>''s''</sup>}}, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as ''s'' decreases. The zeta function approaches 1 uniformly as ''s'' increases to infinity, so as ''s'' decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where {{nowrap|''s'' {{=}} {{frac|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|2}} + i''g'')}} are real are called "Gram points", after [[Wikipedia:Jørgen Pedersen Gram|Jørgen Pedersen Gram]]. We thus have associated pure-octave edos, where ''x'' is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos. | As {{nowrap|''s'' > 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' >> 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>−''s''</sup>}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +∞}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' >> 1}} the derivative is approximately {{nowrap|−ln(2) / 2<sup>''s''</sup>}}, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as ''s'' decreases. The zeta function approaches 1 uniformly as ''s'' increases to infinity, so as ''s'' decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where {{nowrap|''s'' {{=}} {{frac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + i''g'')}} are real are called "Gram points", after [[Wikipedia:Jørgen Pedersen Gram|Jørgen Pedersen Gram]]. We thus have associated pure-octave edos, where ''x'' is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos. | ||
Because the value of zeta increased continuously as it made its way from +∞ to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if −ζ'(''z'') is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [[Wikipedia:Bernhard Riemann|Bernhard Riemann]] which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the [[Wikipedia:Riemann-Siegel formula|Riemann-Siegel formula]] since [[Wikipedia:Carl Ludwig Siegel|Carl Ludwig Siegel]] went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of {{nowrap|ζ({{frac|2}} + i''g'')}} at the corresponding Gram point should be especially large. | Because the value of zeta increased continuously as it made its way from +∞ to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if −ζ'(''z'') is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [[Wikipedia:Bernhard Riemann|Bernhard Riemann]] which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the [[Wikipedia:Riemann-Siegel formula|Riemann-Siegel formula]] since [[Wikipedia:Carl Ludwig Siegel|Carl Ludwig Siegel]] went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of {{nowrap|ζ({{frac|1|2}} + i''g'')}} at the corresponding Gram point should be especially large. | ||
=== The Z function === | === The Z function === | ||
The absolute value of {{nowrap|ζ({{frac|2}} + i''g'')}} at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of {{nowrap|ζ({{frac|2}} + i''t'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the [[Wikipedia:Riemann hypothesis|Riemann hypothesis]] is equivalent to the claim that all zeros of {{nowrap|ζ'(''s'' + i''t'')}} occur when {{nowrap|''s'' > {{frac|1|2}}}}, which is where all known zeros lie. These do not have values of ''t'' corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the [[Wikipedia:Z function|Z function]]. | The absolute value of {{nowrap|ζ({{frac|1|2}} + i''g'')}} at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of {{nowrap|ζ({{frac|1|2}} + i''t'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the [[Wikipedia:Riemann hypothesis|Riemann hypothesis]] is equivalent to the claim that all zeros of {{nowrap|ζ'(''s'' + i''t'')}} occur when {{nowrap|''s'' > {{frac|1|2}}}}, which is where all known zeros lie. These do not have values of ''t'' corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the [[Wikipedia:Z function|Z function]]. | ||
In order to define the Z function, we need first to define the [[Wikipedia:Riemann-Siegel theta function|Riemann-Siegel theta function]], and in order to do that, we first need to define the [http://mathworld.wolfram.com/LogGammaFunction.html Log Gamma function]. This is not defined as the natural log of the Gamma function since that has a more complicated branch cut structure; instead, the principal branch of the Log Gamma function is defined as having a branch cut along the negative real axis, and is given by the series | In order to define the Z function, we need first to define the [[Wikipedia:Riemann-Siegel theta function|Riemann-Siegel theta function]], and in order to do that, we first need to define the [http://mathworld.wolfram.com/LogGammaFunction.html Log Gamma function]. This is not defined as the natural log of the Gamma function since that has a more complicated branch cut structure; instead, the principal branch of the Log Gamma function is defined as having a branch cut along the negative real axis, and is given by the series | ||
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There are three major differences between our "cosine error" functions, and the way we're incorporating them into the result, and what TE is doing: | There are three major differences between our "cosine error" functions, and the way we're incorporating them into the result, and what TE is doing: | ||
# First, the function here is flipped upside down - that is, we're measuring "accuracy" rather than error - as well as shifted vertically down along the y-axis. Since it is trivial to convert between the two, and since we only care about the relative rankings of EDOs, it is clear that we're measuring essentially the same thing. | |||
# Instead of weighting each interval by <math>1/\log(nd)</math>, we weight it by <math>1/(nd)^\sigma</math>. | |||
# Instead of only looking at the primes, as we do in TE, we are now looking at 'all' intervals, and in particular looking at the best mapping for each interval. | |||
The last one is nontrivial, and we will go into detail below. | The last one is nontrivial, and we will go into detail below. | ||
There are also a few notes we will only write in passing, for now, perhaps to build on later: | There are also a few notes we will only write in passing, for now, perhaps to build on later: | ||
# If we do want <math>1/\log(nd)</math> weighting, we can derive this kind of weighting from an antiderivative of the zeta function. | |||
# If we only want the primes, rather than all intervals, we can use something called the "Prime Zeta Function" to get those kinds of summations. | |||
# If we do want the true TE squared error rather than our cosine error, then we would end up getting something called "parabolic waves" rather than cosine waves for each interval. A parabolic wave is the antiderivative of a sawtooth wave, and as it is a periodic signal, it has a Fourier series and can be expressed as a sum of sinusoids. We can use this to get a derivation of the squared error as an infinite sum of zeta functions. | |||
For now, though, we will focus only on the basic zeta result that we have. | For now, though, we will focus only on the basic zeta result that we have. | ||
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== Zeta EDO lists == | == Zeta EDO lists == | ||
=== Record edos === | === Record edos === | ||
The prime-approximating strength of an edo can be determined by the magnitude of Z(x). Since a higher |Z(x) | 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. | ||
==== Zeta peak edos ==== | ==== Zeta peak edos ==== | ||
If we examine the increasingly larger peak values of |Z(x) | 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 typically do not occur at exact integer values, but are close to integer values; this can be interpreted as the zeta function suggesting a detuned 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." | ||
==== Zeta peak integer edos ==== | ==== Zeta peak integer edos ==== | ||
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==== Zeta integral edos ==== | ==== Zeta integral edos ==== | ||
Similarly, if we take the integral of |Z(x) | 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. | ||
==== Zeta gap edos ==== | ==== Zeta gap edos ==== | ||
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{| class="wikitable" style="text-align: center; margin: auto auto auto auto;" | {| class="wikitable" style="text-align: center; margin: auto auto auto auto;" | ||
|+ style="font-size: 105%;" | Zeta record EDOs | |+ style="font-size: 105%;" | Zeta record EDOs up to 1000 | ||
|- | |- | ||
! colspan="2" | EDO | ! colspan="2" | EDO | ||
{{EDOs|{{!}} 1 {{ | {{EDOs|{{!}} 1 | ||
{{!}} 2 | |||
{{!}} 3 | |||
{{!}} 4 | |||
{{!}} 5 | |||
{{!}} 7 | |||
{{!}} 10 | |||
{{!}} 12 | |||
{{!}} 19 | |||
{{!}} 22 | |||
{{!}} 27 | |||
{{!}} 31 | |||
{{!}} 41 | |||
{{!}} 46 | |||
{{!}} 53 | |||
{{!}} 72 | |||
{{!}} 87 | |||
{{!}} 99 | |||
{{!}} 118 | |||
{{!}} 130 | |||
{{!}} 152 | |||
{{!}} 171 | |||
{{!}} 217 | |||
{{!}} 224 | |||
{{!}} 270 | |||
{{!}} 311 | |||
{{!}} 342 | |||
{{!}} 422 | |||
{{!}} 441 | |||
{{!}} 472 | |||
{{!}} 494 | |||
{{!}} 742 | |||
{{!}} 764 | |||
{{!}} 935 | |||
{{!}} 954}} | |||
|- | |- | ||
! rowspan="2" | Zeta | ! rowspan="2" | Zeta peak !! Detuned<br />octaves | ||
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! colspan="2" | Zeta integral | ! colspan="2" | Zeta integral | ||
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! colspan="2" | Zeta gap | ! colspan="2" | Zeta gap | ||
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