The Riemann zeta function and tuning: Difference between revisions

As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is ever-present in the background of tuning theory — the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function," which is closely related and can also be derived as an simple expression of the zeta function.

Suppose ''x'' is a variable representing some equal division of the octave. For example, if ''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 ''x'' = 8.202.

Now suppose that ||''x''|| denotes the difference between ''x'' and the integer nearest to ''x''. For example, ||8.202|| would be 0.202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||''x''|| denotes the function |''x'' - ⌊x+1/2⌋|.

<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{||x \log_2 q||}{\log_2 q}\right)^2</math>

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."

<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{||x \log_2 q||^2}{q^s}</math>

<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{||x \log_2 n||^2}{n^s}</math>

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 1 - cos(2π''x''), which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any ''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 ''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 ''s'' < 1, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when 0 < ''s'' < 1. As s approaches the value ''s'' = 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 1-''s'' and ''s'' having the same information content. Hence it is the zeta function between ''s'' = 1/2 and ''s'' = 1, and especially the zeta function along the critical line ''s'' = 1/2, 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 1 + 2^-''z'', which approaches 1 as ''s'' = Re(''z'') becomes larger. When ''s'' >> 1 and ''x'' is an integer, the real part of zeta is approximately 1 + 2^-''s'', and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from ''s'' = +∞ with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when ''s'' >> 1 the derivative is approximately -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 ''s'' = 1/2, it produces a real value of zeta on the critical line. Points on the critical line where ζ(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 ζ(1/2 + i''g'') at the corresponding Gram point should be especially large.

The absolute value ζ(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 ζ(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 ζ'(''s'' + i''t'') occur when ''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]].

<math>\displaystyle \Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln(1 + \frac{z}{k})</math>

Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that θ is holomorphic in the strip with imaginary part between -1/2 and 1/2. It may be described for real arguments as an odd real analytic function of ''x'', increasing when |''x''| > 6.29. Plots of it may be studied by use of the Wolfram [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta online function plotter].

Since θ is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.

Now let's do two things: we're going to expand s = σ+it, and we're going to multiply ζ(s) by its conjugate ζ(s)', noting that ζ(s)' = ζ(s') and ζ(s)·ζ(s)' = |ζ(s)|<span style="font-size: 90%; vertical-align: super;">2</span>. We get:

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 (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, (1-2^(-s))ζ(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

When [[Gene_Ward_Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]

From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where 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, 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 |ζ| = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(r ln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set 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.

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 θ(2πr)/π, which was 31.927. Then 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 floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where 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.

* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of ζ(1/2 + it)] by Tadej Kotnik

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