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 intriguing musical interpretation: the zeta function shows how “well” a given equal temperament approximates the no-limit just intonation relative to its size.

As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some 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 a simple expression of the zeta function.

If you look for a filter to quickly sort all the equal temperaments into those that approximate JI well and those that do not, the edo lists below can be useful. The caveat is that it collapses the variety of characteristics of a temperament to a one-dimensional rating, with little capacity to show the nuances of each system. It is therefore best to keep in mind that judging the temperaments by zeta is no replacement for investigating each temperament in detail.

There are other metrics besides zeta for other definitions of “approximating well”, which you can find in: optimised regular temperament tunings.

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.

Terminology

Riemann zeta function (“zeta”) A mathematical function which is tied to the harmonic series and to prime numbers, used in tuning theory as an “edo goodness” function to evaluate how close to JI an edo is.

Zeta record edo An equal tuning that sets some kind of record in regards to the zeta function compared to all smaller equal tunings.

Record zeta peak An equal tuning which is closer to JI than any previous tuning, and is usually an edo with compressed or stretched octaves, evaluated by the absolute “goodness” of the edo according to the zeta function.

Record zeta peak integer A zeta record edo by absolute “goodness”, when compared only to other edos (i.e. ignoring stretched/compressed equivalences).

Zero A point where the Riemann zeta function is equal to zero, such as ~2.759edo, representing an equal tuning that does not represent JI much at all. Edos close to zeroes are called zeta valley edos; all known zeroes are on the “critical line” used to obtain tuning information.

Record z gap A zeta record edo by the size of the gap between its surrounding zeroes, adjusted for the fact that zeroes generally become more dense with larger inputs.

Record zeta integral A zeta record edo by the size of the area enclosed by the shape of the function between the edo’s surrounding zeroes.

Quick info: zeta peak edos

These lists give the best equal divisions of the octave for their size according to the zeta metric.

Zeta peak edos (tempered octaves): 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, …

Zeta peak integer edos (pure octaves): 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, …

See the section below for more information.

Graph links

A link to the graph of zeta can be found at Zeta in Samuelj Plotter. (In the top left menu, make sure that “Enable Checkerboard” is unticked and “Invert Gradient” and “Continuous Gradient” are ticked.) The function has been reoriented to place edo size along the horizontal axis and weight along the vertical axis, and also scaled by ⁠2π/ln(2)⁠ to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line.

You may also view the graph of zeta along the critical line on Desmos: Zeta in Desmos. This makes it easier to see peaks, but only works for σ = ⁠1/2⁠.

Plots

Below are some demonstrative plots of the zeta function (strictly speaking, the Z function) on the critical line.

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 Mathematica-generated plot of Z(⁠2πx/ln(2)⁠) in the region around 12edo:

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an approximation of JI. 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 270edo completely dominates its neighbors:

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.

To generate this plot using the free version of Wolfram Cloud, you can run Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}] and then in the menu select Evaluation > Evaluate Cells. Change “11.9” and “12.1” to whatever values you want, e.g. to view the curve around 15edo you might use the values “14.9” and “15.1”.

You can also view the plot using Zeta in Desmos.

Gene Smith’s original derivation

Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x corresponds to 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, so that it ranges over all possible equal temperaments. 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:

[math]\displaystyle{ \lfloor x \rceil = \left| x - \left\lfloor x + \frac{1}{2} \right\rfloor \right| }[/math]

For example, ⌊8.202⌉ = 0.202, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, ⌊7.95⌉ = 0.05, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning x, or alternatively how much x is detuned from an edo.

For any value of x, we can construct a p-limit val by rounding x log2(q) to the nearest integer for each prime q up to p. (More technically, this corresponds to what’s known as a generalized patent val.) For example, for x = 12, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider the function

[math]\displaystyle{ \displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \ q \text{ prime}}} \left(\frac{\lfloor x \log_2 q \rceil}{\log_2 q}\right)^2 }[/math]

Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime[note 1], so the function represents a p-limit badness metric. Also, for those unfamiliar, squaring the error is commonly done because it solves the flaws of two alternative ways of measuring error. Specifically, if you look only at the maximum error, you miss opportunities to make the tuning much better overall by allowing slightly more damage on the most damaged intervals, while if you look only at the average error, then it may be that you are unnecessarily damaging a few intervals a lot just to get intervals that are already in-tune slightly more in-tune, so both extremes have pathological behaviours, and using the squared error counters both of these behaviours so that it represents a more balanced approach to optimization that is used in a variety of disciplines.

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 ξp 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]\displaystyle{ \displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \ q \text{ prime}}} \frac{\lfloor x \log_2 q \rceil^2}{q^s} }[/math]

Importantly, when s = 1, the weighting is 1/p for a prime p, so that it’s very similar to a ⁠1/p − 1⁠ weighting. This latter weighting is equal to the average number of times prime p occurs as a factor in the harmonic series, counting repetition, so is of interest because it represents how many harmonics will feel damage from this prime being mistuned. Therefore, the weighting 1/p corresponds to under-prioritizing small primes slightly (with the effect being less slight the smaller the prime, so that at the most extreme, at prime 2 we have 1/2 instead of 1/1 weighting and at prime 3 we have 1/3 instead of 1/2 weighting).

Seeing that we notate the power as s, it might become apparent where the Riemann zeta function will eventually show up.

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—in fact, the primary intuition behind Tenney weighting is that it is the weighting pattern that values 25, 27, and 29 approximately evenly in importance despite being different powers. 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]\displaystyle{ \displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\lfloor x \log_2 n \rceil^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]\displaystyle{ \displaystyle E_s(x) = \sum_{n \geq 1} \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]\displaystyle{ \displaystyle F_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s} }[/math]

This new function has the property that Fs(x) = Fs(0) − Es(x), so that all we have done is flip the sign of Es(x) 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. The logarithm of the Riemann zeta function function can be expressed in terms of a Dirichlet series involving the von Mangoldt function:

[math]\displaystyle{ \displaystyle \ln \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\ln(n)},\frac{1}{n^s} }[/math]

We can rewrite the cosine term in [math]\displaystyle{ F_s(x) }[/math] using the real part of an exponential:

[math]\displaystyle{ \displaystyle \cos(2 \pi x \log_2 n) = \mathrm{Re}\left( \exp{(- 2 \pi i x \log_2 n)} \right) = \mathrm{Re}\left( n^{- \frac{2 \pi i}{\ln 2} x} \right) }[/math]

Substituting back into [math]\displaystyle{ F_s(x) }[/math] gives:

[math]\displaystyle{ \displaystyle F_s(x) = \mathrm{Re} \left( \sum_{n=2}^{\infty \frac{\Lambda(n)}{\ln n} \frac{n}{- \frac{2 \pi i}{\ln 2} x}}{n^s} \right) = \mathrm{Re} \left( \ln \zeta \left(s + \frac{2 \pi i}{\ln 2}x \right) \right) }[/math]

If we take exponentials of both sides, then

[math]\displaystyle{ \displaystyle \exp(F_s(x)) = \left| \zeta\left(s + \frac{2 \pi i}{\ln 2}x\right) \right| }[/math]

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

There is an open question as to whether the analytic continuation preserves the properties we are interested in, though empirically it seems to. However, if you do not want to trust the analytic continuation, then the s = 1 line is of interest for the reasons discussed, as though the infinite sum does not converge, every sum for s > 1 does converge, so that we can consider the line at s = 1 as being the “limit” as s approaches 1 from above. This can also be used to sanity-check results at s = 1/2 by seeing where they agree, e.g. as done in the section on #Absolute zeta peak edos, which looks at getting more practical tuning information out of the zeta function via an adjustment for considering absolute mistuning rather than relative error.

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 absolute 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{ \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 [math]\displaystyle{ \zeta(s) }[/math] by its complex conjugate [math]\displaystyle{ \overline{\zeta(s)} }[/math], noting that [math]\displaystyle{ \overline{\zeta(s)}=\zeta(\overline{s}) }[/math] and [math]\displaystyle{ \left| \zeta(s) \right|^{2} = \zeta(s)\overline{\zeta(s)} }[/math]. We get:

[math]\displaystyle{ \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{ \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{ \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, and n < d. Here’s what we get:

[math]\displaystyle{ \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{ \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{ \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{ \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{ \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{ \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{ \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 log2(⁠n/d⁠). This cosine peaks at x = ⁠N/log2(n/d)⁠ for all integers N, or in other words, the N-th 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 e2π ≈ 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 ⁠1 − cos(x)/2⁠, 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 −0.5 < x < 0.5 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 are incorporating them into the result, and what TE is doing:

1. First, the function here is flipped upside down—that is, we are 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 ⁠1/log(nd)⁠, we weight it by ⁠1/(nd)σ⁠.
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 ⁠1/log(nd)⁠ 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 approximation 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 does not match up with the mapping you would get from it just looking at the primes.

So, for instance, in 16edo, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since 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: 16edo’s direct approximation for 9 is useful when playing the chord 4:5:7:9, and the val mapping for 9 is useful when playing the major ninth 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 16edo 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 us go back to this expression here:

[math]\displaystyle{ \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 is no restriction that n and d be coprime, the rationals we are using here do not 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 us 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{ \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{ \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 are still assuming that σ > 1 and everything is absolutely convergent, we can decompose this into a product of series as follows

[math]\displaystyle{ \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{ \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{ \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 are 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]\displaystyle{ \log,\left| \zeta(\sigma+it) \right| }[/math]
• Error on unreduced rationals: [math]\displaystyle{ \left| \zeta(\sigma+it) \right|^2 }[/math]
• Error on reduced rationals: [math]\displaystyle{ \frac{\left| \zeta(\sigma+it) \right|^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 t vary—we instead typically set σ 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 Z(t) function, which is defined so that we have |Z(t)| = |ζ(t)|. So, the absolute value of the Z 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 1/1, 2/1, 3/1, …, n/1, …. This was studied in a paper by Peter Buch called “Favored cardinalities of scales”. The expression is:

Error on harmonics only: [math]\displaystyle{ \mathrm{Re}\left(\zeta(\sigma + it)\right) }[/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]\displaystyle{ \displaystyle{\left| \zeta\left(\frac{1}{2} + it\right) \right|^2 \cdot \overline {\phi(t)}} }[/math]

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

The matter of sigma: the critical strip, zeta peaks, and Gram points

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 critical strip, when 0 < s < 1. As s approaches the value s = ⁠1/2⁠ of the critical line, the “information content” 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; that is, for s > ⁠1/2⁠, 1 − s essentially multiplies the zeta function at s by a fixed, monotonic increasing function. 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.

Introduction to Gram points

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)/2s⁠, 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 + ig) 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.

Gram points and zeta peaks

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 + ig) at the corresponding Gram point should be especially large.

The Z function: a mathematically convenient version of zeta

The absolute value of ζ(⁠1/2⁠ + ig), 1.8 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 + it), 1.8 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 + it) 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, which is defined (in terms of the Riemann-Siegel theta function) as:

[math]\displaystyle{ Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right) }[/math].

The factor of [math]\displaystyle{ \exp(i \theta(t)) }[/math] simply modifies zeta by a complex phase, and so 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 the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −⁠1/2⁠ and ⁠1/2⁠, so is Z. And Z is a real even function of the real variable t, since theta was defined so as to give precisely this property.

Zeta edo lists

Record edos

The prime-approximating strength of an edo can be determined by the magnitude of Z(x). Since a higher |Z(x)| correlates to a stronger tuning, we would like to find a sequence with successively larger |Z(x)|-associated values satisfying some property.

Zeta 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](https://en.xen.wiki/w/14348edo “14348edo”