The Riemann zeta function and tuning: Difference between revisions

Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' = 80}}, ''x'' reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen-Pierce|Bohlen-Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' = 8.202}}.

This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ_''p'' for these minima is the square of the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

Another consequence of the above definition which might be objected to is that it results in a function with a [[Wikipedia:Continuous_function#Relation_to_differentiability_and_integrability|discontinuous derivative]], whereas a smooth function be preferred. The function ⌊x⌉² is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is {{nowrap|1 − cos(2π''x'')}}, which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any {{nowrap|''s'' > 1}}:

For any fixed ''s'' > 1 this gives a real [[Wikipedia:analytic function|analytic function]] defined for all ''x'', and hence with all the smoothness properties we could desire.

So long as {{nowrap|''s'' &geq; 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 &minus; ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' = {{frac|2}}}} and {{nowrap|''s'' = 1}}, and especially the zeta function along the critical line {{nowrap|''s'' = {{frac|2}}}}, which is of the most interest.

As ''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^&minus;''z''}}, 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^&minus;''s''}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' = +&infin;}} 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 &minus;ln(2)/2^''s'', 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|&zeta;({{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.

Because the value of zeta increased continuously as it made its way from +&infin; 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 &minus;&zeta;'(''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|&zeta;({{frac|2}} + i''g'')}} at the corresponding Gram point should be especially large.

The absolute value of {{nowrap|&zeta;({{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|&zeta;({{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|&zeta;'(''s'' + i''t'')}} occur when {{nowrap|''s'' > 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]].

Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor {{nowrap|''t'' = 2π''x''/ln(2)}}, for ''x'' a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here].

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' = 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.

Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other ''s'' values, such as {{nowrap|''s'' = 1}} or {{nowrap|''s'' = 3/4}}, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.

Now let's do two things: we're going to expand {{nowrap|s = σ + it}}, and we're going to multiply &zeta;(s) by its conjugate &zeta;(s)', noting that {{nowrap|&zeta;(s)' = &zeta;(s')}} and {{nowrap|&zeta;(s)·&zeta;(s)' = |&zeta;(s)|²}}. We get:

where the last equality makes use of the fact that {{nowrap|cos(&minus;x) = cos(x)}} and {{nowrap|sin(&minus;x) = &minus;sin(x)}}.

Now, let's decompose the sum into three parts: {{nowrap|n = d}}, {{nowrap|n &gt; d}}, and {{nowrap|n &lt; d}}. Here's what we get:

First, in the leftmost summation, we can see that n=d implies {{nowrap|ln(n/d) = 0}}. Since {{nowrap|sin(0) = 0}}, the sin term in the numerator cancels out, yielding:

Now, noting that {{nowrap|ln(p / q) = &minus;ln(q / p)}} and that sin is an odd function, we can see that the sin terms cancel out, leaving

Finally, by making the mysterious substitution {{nowrap|t = 2π / ln(2) · x}}, the musical implications of the above will start to reveal themselves:

For every strictly positive rational n/d, there is a cosine with period 2π log₂(n/d). This cosine peaks at {{nowrap|x = N/log₂(n/d)}} for all integer N, or in other words, the Nth-equal division of the rational number n/d, and hits troughs midway between.

Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable t, which was the imaginary part of the zeta argument s, can be thought of as the number of divisions of the interval {{nowrap|''e''^2π ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "natural interval.")

As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function <math>(1 - cos(x))/2</math> - which is "close enough" for small values of x. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have <math>-0.5 < x < 0.5</math> we have a decent enough approximation.

Going back to the infinite summation above, we note that these cosine error (or really "cosine accuracy") functions are being weighted by 1/(nd)^σ. Note that σ, which is the real part of the zeta argument s, serves as sort of a complexity weighting - it determines how quickly complex rational numbers become "irrelevant." Framed another way, we can think of it as the degree of "'''rolloff'''" formed by the resultant (musical, not mathematical) harmonic series formed by those rationals with {{nowrap|d = 1}}. Note that this rolloff is much stronger than the usual 1/log(nd) rolloff exhibited by TE error, which is one reason that zeta converges to something coherent for all rational numbers, whereas TE fails to converge as the limit increases. We will use the term "rolloff" to identify the variable σ below.

So, for instance, in 16-EDO, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps {{nowrap|(9 * 2 - 16 = 2)}}. However, there is a better mapping for 9/8 at 3 steps - one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16-EDO's "direct mapping" for 9 is useful when playing the chord 4:5:7:9, and the "indirect" or "prime-based" mapping for 9 is useful when playing the "major 9" chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good "extra" approximations of rationals which can be used in this way. And although 16-EDO is pretty high error, similar phenomena can be found for any EDO which becomes [[consistency|inconsistent]] for some chord of interest.

Now, the common factor c/c cancels out inside the log in the numerator. However, in the denominator, we get an extra factor of c ² to contend with. This yields