The Riemann zeta function and tuning: Difference between revisions

The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann Hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an incredible musical interpretation as measuring the "harmonicity" of an equal temperament. Put simply, the zeta function shows, in a certain sense, how well a given equal temperament approximates the harmonic series, and indeed *all* rational numbers, even up to "infinite-limit JI."

Much of the below is thanks to the insights of [[Gene Ward Smith]]. Below is the original derivation as he presented it, followed by a different derivation from [[Mike Battaglia]] below which extends some of the results.

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.

so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.

So long as s is greater than or equal to one, 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 is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. 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 [http://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.

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]].

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 s=1 or 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.

=== Peak EDOs ===

If we examine the increasingly larger peak values of |Z(x)|, we find they occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[EDO|edo]]s

We may define the ''strict zeta edos'' to be the edos that are in all four of the zeta edo lists. The list of strict zeta edos begins {{EDOs|2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578}}... . [[16808edo]] is also known to be strict zeta.

Another use for the Riemann zeta function is to determine the optimal tuning for an EDO, meaning the optimal octave stretch. This is because the zeta peaks are typically not integers. The fractional part can give us the degree to which the generator diverges from what you would need to have the octave be a perfect 1200 cents. Here is a list of successively higher zeta peaks, taken to five decimal places:

11664.01488

14347.99444

16807.99325

28742.01019

34691.00191

The [http://mathworld.wolfram.com/EulerProduct.html Euler product] for the Riemann zeta function is

Removing 2 leads to increasing adjusted peak values corresponding to the division of 3 (the "tritave") into 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.

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]]

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.

There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [[Wikipedia:Dirichlet eta function|Dirichlet eta function]] which was introduced to mathematics by [[Wikipedia:Johann Peter Gustav Lejeune Dirichlet|Johann Peter Gustav Lejeune Dirichlet]], who despite his name was a German and the brother-in-law of [[Wikipedia:Felix Mendelssohn|Felix Mendelssohn]].

The Dirichlet series for the zeta function is absolutely convergent when 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 1 + 2πix/ln(2) corresponding to pure octave divisions along the line 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]].

Above, Gene proves that the zeta function measures the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]], sometimes called "Tenney-Euclidean Simple Badness," of any EDO, taken over all 'prime powers'. The relative error is simply equal to the tuning error times the size of the EDO, so we can easily get the raw "non-relative" tuning error from this as well by simply dividing by the size of the EDO.

Now, one nitpick to notice above is that this expression technically involves all 'unreduced' rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for reduced rationals:

Let's go back to this expression here:

Now, since we're fixing σ and letting t vary, the left zeta term is constant for all EDOs. This demonstrates that the zeta function also measures cosine error over all the reduced rationals, up to a constant factor. QED.

So far we have shown the following:

Note that, although the last four expressions were all monotonic transformations of one another, this one is not - this is the 'real part' of the zeta function, whereas the others were all some simple monotonic function of the 'absolute value' of the zeta function. The results, however, are  very similar - in particular, the peaks are approximately to one another, shifted by only a small amount (at least for reasonably-sized EDOs up to a few hundred).

The expression

More can be found at the page on [[Harmonic_Entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|Harmonic Entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.

* [http://front.math.ucdavis.edu/0309.5433 X-Ray of Riemann zeta-function] by Juan Arias-de-Reyna

* [http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/ Selberg's limit theorem] by Terence Tao [http://www.webcitation.org/5xrvgjW6T Permalink]