The Riemann zeta function and tuning: Difference between revisions
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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: | 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== | ==Removing primes== | ||
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\zeta(s) = \sum_n n^{-s}</math> | \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)|<span style="font-size: 90%; vertical-align: super;">2 | Now let's do two things: we're going to expand s = σ+it, and we're going to multiply ζ(s) by its conjugate ζ(s)', noting that ζ(s)' = ζ(s') and ζ(s)·ζ(s)' = |ζ(s)|<span style="font-size: 90%; vertical-align: super;">2. We get: | ||
<math> \displaystyle | <math> \displaystyle | ||
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\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> | \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> | ||
<span style="line-height: 1.5;">Now let's do a bit of algebra with the exponential function, and use Euler's identity: | <span style="line-height: 1.5;">Now let's do a bit of algebra with the exponential function, and use Euler's identity: | ||
<math> \displaystyle | <math> \displaystyle | ||
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== Interpretation Of Results: "Cosine Relative Error" == | == Interpretation Of Results: "Cosine Relative Error" == | ||
For every strictly positive rational n/d, there is a cosine with period 2π log<span style="font-size: 90%; vertical-align: sub;">2 | For every strictly positive rational n/d, there is a cosine with period 2π log<span style="font-size: 90%; vertical-align: sub;">2(n/d). This cosine peaks at x=N/log<span style="font-size: 11.6999998092651px; vertical-align: sub;">2(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<span style="font-size: 90%; vertical-align: super;">2π | 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<span style="font-size: 90%; vertical-align: super;">2π ≈ 535.49, or what [[Keenan_Pepper|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 < x < 0.5</math> we have a decent enough approximation. | 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 < x < 0.5</math> we have a decent enough approximation. | ||
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For now, though, we will focus only on the basic zeta result that we have. | 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)<span style="font-size: 90%; vertical-align: super;">σ | Going back to the infinite summation above, we note that these cosine error (or really "cosine accuracy") functions are being weighted by 1/(nd)<span style="font-size: 90%; vertical-align: super;">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. | 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. | ||
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\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> | \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 c<span style="font-size: 90%; vertical-align: super;">2 | Now, the common factor c/c cancels out inside the log in the numerator. However, in the denominator, we get an extra factor of c<span style="font-size: 90%; vertical-align: super;">2 to contend with. This yields | ||
<math> \displaystyle | <math> \displaystyle | ||