The Riemann zeta function and tuning

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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."

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 - Harmonic Entropy 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.

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.

Gene Smith's Original Derivation

Preliminaries

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 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 .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function |x - floor(x+1/2)|.

For any value of x, we can construct a p-limit generalized patent val. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. Now consider the function

[math]\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^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 tunings of the octaves of the associated vals, while ξ for these minima is the square of the Tenney-Euclidean relative error of the val - equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:

[math]\sum_2^\infty \frac{||x \log_2 q||^2}{q^s}[/math]

If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the Von Mangoldt function, an arithmetic function on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, as Λ(n), and in terms of it we can include prime powers in our error function as

[math]\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{||x \log_2 n||^2}{n^s}[/math]

where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.

Another consequence of the above definition which might be objected to is that it results in a function with a 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(2πx), which is a smooth and in fact an entire function. Let us therefore now define for any s > 1

[math]E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}[/math]

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

We can clean up this definition to get essentially the same function:

[math]F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}[/math]

This new function has the property that [math]F_s(x) = F_s(0) - E_s(x)[/math], so that all we have done is flip the sign of [math]E_s(x)[/math] and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the Riemann zeta function:

[math]F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)[/math]

If we take exponentials of both sides, then

[math]\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|[/math]

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

Into the critical strip

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 critical strip, when s lies between zero and one. As s approaches the value s=1/2 of the 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 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 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 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 Riemann-Siegel formula since 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 Z function

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 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 Z function.

In order to define the Z function, we need first to define the Riemann-Siegel theta function, and in order to do that, we first need to define the 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

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

where the lower-case gamma is Euler's constant. We now may define the Riemann-Siegel theta function as

[math]\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2[/math]

Another approach is to substitute z = (1 + 2i t)/4 into the series for Log Gamma and take the imaginary part, this yields

[math]\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t + \sum_{n=1}^\infty \left(\frac{t}{2n} - \arctan\left(\frac{2t}{4n+1}\right)\right)[/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 theta 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 online function plotter.

Using the theta and zeta functions, we define the Z function as

[math]Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)[/math]

Since theta 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 zeta 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 zeta 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.

Using the online plotter we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2π/ln(2) x, 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 here.

If you have access to Mathematica, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo:

plot12.png

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 Nedo is N/x; if x is slightly larger than N as here with 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.

For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:

plot270.png

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.

Zeta EDO lists

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 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, ... 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 "stretched octave" tuning 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 ks26 has found), if we allow no octave stretching and thus only look at the record |Z(x)| zeta scores corresponding to exact EDOs with pure octaves, we get 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 in this list not included in the previous are 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631, ... and EDOs not included in this list but included in the previous are 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 and etc EDOs.

Integral of Zeta EDOs

Similarly, if we take the integral of |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 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

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of zeta gap edos. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as OEIS: A117537. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.

Strict Zeta EDOs

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 2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578... . 16808edo is also known to be strict zeta.

Optimal Octave Stretch

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:

    0.00000
    1.12657
    1.97277
    3.05976
    3.90445
    5.03448
    6.95669
   10.00846
   12.02318
   18.94809
   22.02515
   27.08661
   30.97838
   40.98808
   52.99683
   71.95061
   99.04733
  117.96951
  130.00391
  152.05285
  170.99589
  217.02470
  224.00255
  270.01779
  341.97485
  422.05570
  441.01827
  494.01377
  742.01093
  764.01938
  935.03297
  953.94128
 1012.02423
 1105.99972
 1177.96567
 1236.02355
 1394.98350
 1447.97300
 1577.98315
 2459.98488
 2683.99168
 3395.02659
 5585.00172
 6079.01642
 7032.96529
 8268.98378
 8539.00834
11664.01488
14347.99444
16807.99325
28742.01019
34691.00191

Removing primes

The Euler product for the Riemann zeta function is

[math]\zeta(s) = \prod_p (1 - p^{-s})^{-1}[/math]

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.

Along the critical line, |1 - p^(-1/2-i t)| may be written

[math]\sqrt{1 + \frac{1}{p} - \frac{2 \cos(\ln p\ t)}{\sqrt{p}}}[/math]

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 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, the Bohlen-Pierce division of the tritave, but the multiples 26, 39 and 52 also.

The Black Magic Formulas

When 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 asymptotic expansion

[math]\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots[/math]

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.

For an example, consider x = 12, so that r = 12/ln(2) = 17.312. Then r ln(r) - r - 1/8 = 31.927, which rounded to the nearest integer is 32, so n = 32. Then (r + n + 1/8)/ln(r) = 17.338, corresponding to 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 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.

Computing zeta

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

The zeta function has a simple pole at z=1 which forms a barrier against continuing it with its Euler product or Dirichlet series representation. We could subtract off the pole, or multiply by a factor of (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 (1-2^(1-z)), leading to the eta function:

[math]\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots[/math]

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 Euler summation.

Mike Battaglia's Expanded Results

Zeta Yields "Relative Error" Over All Rationals

Above, Gene proves that the zeta function measures the 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.

Here, we strengthen that result to show that the zeta function additionally measures weighted relative error over all 'rational numbers,' relative to the size of the EDO.

Let's dive in!

First, let's take the zeta function, expressed as a Dirichlet series:

[math] \displaystyle \zeta(s) = \sum_n n^{-s}[/math]

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)|2. We get:

[math] \displaystyle \left| \zeta(s) \right|^2 = \left[\sum_n n^{-(\sigma+it)}\right] \cdot \left[\sum_d d^{-(\sigma-it)}\right][/math]

where d is a new variable used internally in the second summation.

Now, let's focus on σ > 1, so that both series are absolutely convergent. The following rearrangement of terms is then justified:

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n,d} \left[n^{-(\sigma+it)} \cdot d^{-(\sigma-it)}\right] = \sum_{n,d} \frac{\left({\tfrac{n}{d}}\right)^{-it}}{(nd)^{\sigma}}[/math]

Now let's do a bit of algebra with the exponential function, and use Euler's identity:

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n,d} \frac{e^{-it \ln\left({\tfrac{n}{d}}\right)}}{(nd)^{\sigma}} = \sum_{n,d} \frac{\cos\left(-t \ln\left({\tfrac{n}{d}}\right)\right) + i\sin\left(-t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} = \sum_{n,d} \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}[/math]

where the last equality makes use of the fact that cos(-x) = cos(x) and sin(-x) = -sin(x).

Now, let's decompose the sum into three parts: n=d, n>d, n<d. Here's what we get:

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n=d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] + \sum_{n\gt d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] + \sum_{n\lt d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right][/math]

We'll deal with each of these separately.

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

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n=d} \left[ \frac{\cos\left( t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] + \sum_{n\gt d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] + \sum_{n\lt d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right][/math]

We will not simplify the cosine term further right now, the reasons for which will become apparent below.

Now, let's handle the two summations on the right. The key thing to note here is that we can pair up every term in the second summation with a corresponding term in the third summation that interchanges n and d. To make this clear, let p and q be two integers, and assume without loss of generality that p>q. The term corresponding to n=p, d=q will then appear in the second summation, and the term n=q, d=p will appear in the third summation. Juxtaposing those together, we get the following:

[math] \displaystyle \frac{\cos\left(t \ln\left({\tfrac{p}{q}}\right)\right) - i\sin\left(t \ln\left({\tfrac{p}{q}}\right)\right)}{(pq)^{\sigma}} + \frac{\cos\left(t \ln\left({\tfrac{q}{p}}\right)\right) - i\sin\left(t \ln\left({\tfrac{q}{p}}\right)\right)}{(pq)^{\sigma}}[/math]

Now, noting that ln(p/q) = -ln(q/p) and that sin is an odd function, we can see that the sin terms cancel out, leaving

[math] \displaystyle \frac{\cos\left(t \ln\left({\tfrac{p}{q}}\right)\right)}{(pq)^{\sigma}} + \frac{\cos\left(t \ln\left({\tfrac{q}{p}}\right)\right)}{(pq)^{\sigma}}[/math]

Now, since every term in these two summations has a pair like this, and since we've done nothing but continued rearrangements of an absolutely convergent series, we can modify the original three-part summation to cancel the sin terms out as follows:

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n=d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] + \sum_{n\gt d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] + \sum_{n\lt d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right][/math]

Putting the whole thing back into one series, we get

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n,d} \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}[/math]

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

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n,d} \frac{\cos\left(2\pi x \log_2\left(\tfrac{n}{d}\right)\right)}{(nd)^{\sigma}}[/math]

Let's take a breather and see what we've got.

Interpretation Of Results: "Cosine Relative Error"

For every strictly positive rational n/d, there is a cosine with period 2π log2(n/d). This cosine peaks at 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 e ≈ 535.49, or what 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 \lt x \lt 0.5[/math] we have a decent enough approximation.

We will call this cosine (relative) error, by analogy with TE (relative) error. It is easy to see that the cosine error is approximately equal to the TE error when the error is small, and only diverges slightly for large errors.

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:

1. 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. 2. Instead of weighting each interval by [math]1/\log(nd)[/math], we weight it by [math]1/(nd)^\sigma[/math]. 3. 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.

There are also a few notes we will only write in passing, for now, perhaps to build on later: 1. If we do want [math]1/\log(nd)[/math] weighting, we can derive this kind of weighting from an antiderivative of the zeta function. 2. 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. 3. 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.

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 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.

Putting this all together, we can take the approach to fix σ, specifying a rolloff, and then let x (or t) vary, specifying an EDO. The resulting function gives us the measured accuracy of EDOs across all unreduced rational numbers with respect to the chosen rolloff. Taking it all together, we get a Tenney-weighted sum of cosine accuracy over all unreduced rationals. QED.

It is extremely noteworthy to mention how "composite" rationals are treated differently than with TE error. In addition to our usual error metric on the primes, we also go to each rational, look for the best "direct" or "patent" mapping of that rational within the EDO, and add 'that' to the EDO's score. In particular, we do this even when the best mapping for some rational doesn't match up with the mapping you'd get from it just looking at the primes.

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 (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 inconsistent for some chord of interest.

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 free group over the strictly positive rationals, which we'll call "meta-JI." The zeta function can then be thought of as yielding an error for all meta-JI 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.

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:

From Unreduced Rationals to Reduced Rationals

Let's go back to this expression here:

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n,d} \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}[/math]

Note that since there's no restriction that n and d be coprime, the "rationals" we're using here don't have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all EDOs.

To see this, let's first note that every "unreduced" rational n/d can be decomposed into the product of a reduced rational n'/d' and a common factor c/c. Furthermore, note that for any reduced rational n'/d', we can generate all unreduced rationals n/d corresponding to it by multiplying it by all such common factors c/c, where c is a strictly positive natural number.

This allows us to change our original summation so that it's over three variables, n', d', and c, where n' and d' are coprime, and c is a strictly positive natural number:

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n',d',c} \frac{\cos\left(t \ln\left({\tfrac{cn'}{cd'}}\right)\right)}{(cn' \cdot cd')^{\sigma}}[/math]

Now, the common factor c/c cancels out inside the log in the numerator. However, in the denominator, we get an extra factor of c2 to contend with. This yields

[math] \displaystyle \left| \zeta(s) \right|^2 = \sum_{n',d',c} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(c^2 \cdot n'd')^{\sigma}} = \sum_{n',d',c} \left[ \frac{1}{c^{2\sigma}} \cdot \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}} \right][/math]

Now, since we're still assuming that σ > 1 and everything is absolutely convergent, we can decompose this into a product of series as follows

[math] \displaystyle \left| \zeta(s) \right|^2 = \left[ \sum_c \frac{1}{c^{2\sigma}} \right] \cdot \left[ \sum_{n',d'} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}} \right][/math]

Finally, we note that on the left summation we simply have another zeta series, yielding

[math] \displaystyle \left| \zeta(s) \right|^2 = \zeta(2\sigma) \cdot \left[ \sum_{n',d'} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}} \right][/math]

[math] \displaystyle \frac{\left| \zeta(s) \right|^2}{\zeta(2\sigma)} = \sum_{n',d'} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}}[/math]

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.

Measuring Error on Harmonics Only

So far we have shown the following:

  • Error on prime powers: [math]\log |\zeta(\sigma+it)|[/math]
  • Error on unreduced rationals: [math]|\zeta(\sigma+it)|^2[/math]
  • Error on reduced rationals: [math]\frac{|\zeta(\sigma+it)|^2}{\zeta(2\sigma)}[/math]

Since the second is a simple monotonic transformation of the first, we can see that the same function basically measures both the relative error on just the prime powers, and also on all unreduced rationals, at least in the sense that EDOs will be ranked identically by both measures. The third function is really just the second function divided by a constant, since we only really care about letting [math]t[/math] vary - we instead typically set [math]\sigma[/math] to some value which represents the weighting "rolloff" on rationals. So, all three of these functions will rank EDOs identically.

We also note that, above, Gene tended to look at things in terms of the [math]Z(t)[/math] function, which is defined so that we have [math]|Z(t)| = |\zeta(t)|[/math]. So, the absolute value of the [math]Z[/math] function is also monotonically equivalent to the above set of expressions, so that any one of these things will produce the same ranking on EDOs.

It turns out that using the same principles of derivation above, we can also derive another expression, this time for the relative error on only the harmonics - i.e. those intervals of the form [math]1/1, 2/1, 3/1, ... n/1, ...[/math]. This was studied in a paper by Peter Buch called "Favored cardinalities of scales". The expression is:

Error on harmonics only: [math]|\textbf{Re}[\zeta(\sigma+it)]|[/math]

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).

Relationship to Harmonic Entropy

The expression

[math]|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}[/math]

is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for [math]N=\infty[/math], where [math]\phi(t)[/math] is the characteristic function (aka Fourier transform) of the spreading distribution and [math]\overline {\phi(t)}[/math] denotes complex conjugation.

Note that in the most common case where the spreading distribution is symmetric (as in the case of the Gaussian and Laplace distributions), the characteristic function is purely real and hence the conjugate is unnecessary. In particular, when the spreading distribution is a Gaussian, the characteristic function is also a Gaussian.

More can be found at the page on Harmonic Entropy, including a generalization to Renyi entropy for arbitrary [math]a[/math].

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