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

There are other metrics besides zeta for other definitions of "approximating well", such as [[mu badness]] and the various [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments.

'''Zeta record peak:''' A zeta record equal tuning, evaluated by the absolute "goodness" of the edo according to the zeta function. This is a tuning which is closer to JI than any previous tuning, and is usually an EDO with compressed or stretched octaves.

'''Zeta record integer:''' A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring nonoctave tunings and stretched/compressed octaves).

'''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.  

'''Zeta record 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.

'''Zeta record integral:''' A zeta record edo by the size of the area enclosed by the shape of the function between the edo's surrounding zeroes.  

Zeta peak integer EDOs (Pure octaves): {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, ...}}

Zeta peak EDOs (Tempered octaves): {{EDOs|1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, ...}}

A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) 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 <math>\tfrac{2\pi}{\ln(2)}</math> 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: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for sigma=1/2.  

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

For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log₂(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]

Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime, so the function represents a p-limit badness metric.

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."

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 {{sfrac|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 {{w|von Mangoldt function}}, an {{w|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

Of more interest is the fact that it is a known mathematical function.

The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:

= \mathrm{Re}\left( \exp{(2 \pi i x \log_2 n)} \right)

= \mathrm{Re}\left( n^{\frac{2 \pi i}{\ln 2} x} \right)

= \mathrm{Re} \left( \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \frac{n^{\frac{2 \pi i}{\ln 2} x}}{n^s} \right)

So long as {{nowrap|''s'' &ge; 1}}, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when {{nowrap|''s'' &lt; 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 &lt; ''s'' &lt; 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.

 

As {{nowrap|''s'' &gt; 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2^−''z''}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2^−''s''}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +&infin;}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|ln(2)|2^''s''}}, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as ''s'' decreases. The zeta function approaches 1 uniformly as ''s'' increases to infinity, so as ''s'' decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} are real are called "Gram points", after {{w|Jørgen Pedersen Gram}}. We thus have associated pure-octave edos, where ''x'' is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos.

 

Because the value of zeta increased continuously as it made its way from +&infin; to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if −ζ′(''z'') is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to {{w|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 {{w|Riemann–Siegel formula}} since {{w|Carl Ludwig Siegel}} went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large.

 

The absolute value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of {{nowrap|ζ({{frac|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' &gt; {{sfrac|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 {{w|''Z'' function}}.

 

 

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

 

 

<math>\displaystyle\theta(z) = \frac{\Upsilon\left(\frac{1 + 2 i z}{4}\right) - \Upsilon\left(\frac{1 - 2 i z}{4}\right)}{2 i} - \frac{\ln(\pi)}{2} z</math>

 

Another approach is to substitute {{nowrap|''z'' {{=}} {{sfrac|1 + 2''it''|4}}}} into the series for Log Gamma and take the imaginary part, this yields

 

 

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 −{{frac|1|2}} and {{frac|1|2}}. It may be described for real arguments as an odd real analytic function of ''x'', increasing when {{nowrap|{{abs|''x''}} &gt; 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].

 

Using the theta and zeta functions, we define the {{w|Z function}} as

 

 

Since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|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''.

 

 

Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor {{nowrap|''t'' {{=}} {{sfrac|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 [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here].

 

If you have access to {{w|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({{frac|2π''x''|ln(2)}}) in the region around 12edo:

 

[[File:plot12.png|alt=plot12.png|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 ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.

 

 

 

 

 

You can also view the plot using [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos].

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

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

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

In addition to looking at {{nowrap|{{abs|Z(x)}}}} maxima, we can also look at {{nowrap|{{abs|Z(x)}}}} ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}} … These tunings tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.

 

Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that {{nowrap|{{abs|Z(x)}}}} grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.

Note that "tempered-octave zeta valley edos" would simply be any zero of Z(x).

Here is a list of successively higher zeta peaks, taken to five decimal places:

 

 

For all edos 1 through 100, see [[table of zeta-stretched edos]].

These octave-stretched edos are not the only tunings which can be produced from zeta peaks. They are only one type of tuning within a larger family of equal-step tunings called zeta peak indexes. They have their own article here, with a table of the first 500 or so: [[ZPI|zeta peak index (ZPI)]].

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

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 {{nowrap|''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 {{nowrap|(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, {{nowrap|(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:

\left| 1 - p^{-\frac{1}{2} - it} \right| = \sqrt{1 + \frac{1}{p} - \frac{2 \cos(t \ln p)}{\sqrt{p}}}

 

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.

 

 

\displaystyle Z\left(\frac{2\pi}{\ln(3)}t\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}t\right)}

</math>

 

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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} is real. This implies that exp(''i''θ(''g'')) is real, so that {{frac|θ(''g'')|π}} is an integer. Theta has an {{w|asymptotic expansion}}

\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 {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''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'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{cent}}}}.

Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{abs|ζ}} {{=}} {{abs|Z}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} ⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋}} is the nearest integer to {{sfrac|θ(2π''r'')|π}}, then we may set {{nowrap|''r''⁺ {{=}} {{sfrac|''r'' + ''n'' + {{frac|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 {{nowrap|''x'' {{=}} 12}}, so that {{nowrap|''r'' {{=}} {{sfrac|12|ln(2)}} {{=}} 17.312}}. Then {{nowrap|''r'' ln(''r'') − ''r'' − {{sfrac|1|8}} {{=}} 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|''n'' {{=}} 32}}. Then {{nowrap|{{sfrac|''r'' + ''n'' + {{frac|1|8}}|ln(''r'')}} {{=}} 17.338}}, corresponding to {{nowrap|''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 {{nowrap|θ(2π''r'') / π}}, which was 31.927. Then {{nowrap|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 {{nowrap|⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋ − ''r'' ln(''r'') + ''r'' + {{frac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''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 {{w|Dirichlet eta function}} which was introduced to mathematics by {{w|Johann Peter Gustav Lejeune Dirichlet}}, who despite his name was a German and the brother-in-law of {{w|Felix Mendelssohn}}.

 

The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|''z'' {{=}} 1}} which forms a barrier against continuing it with its {{w|Euler product}} or {{w|Dirichlet series}} representation. We could subtract off the pole, or multiply by a factor of {{nowrap|''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 {{nowrap|1 − 2^{1&#x200A;−&#x200A;''z''}}}, leading to the eta function:

 

 

The Dirichlet series for the zeta function is absolutely convergent when {{nowrap|''s'' &gt; 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 {{nowrap|1 + {{sfrac|2π''i''|ln(2)}}x}} corresponding to pure octave divisions along the line {{nowrap|''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 {{w|Euler summation}}.