User:Sintel/Zeta working page: Difference between revisions

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The Riemann zeta function is an important function in number theory, most famous for its connection to the unsolved Riemann hypothesis. Beyond its mathematical significance, however, the zeta function has a surprising connection to music theory: it provides a quantitative measure of how effectively a given [[equal temperament]] approximates [[just intonation]], relative to its size.

The zeta function provides a way to measure how well different [[equal divisions of the octave]] (EDOs) approximate the infinite harmonic series ~~found in just intonation~~. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement.

The zeta function provides a way to measure how well different [[equal divisions of the octave]] (EDOs) approximate the infinite [[harmonic series]]. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement.

When you look at the lists of [[#Zeta peak ~~edos~~|"zeta peak EDOs"]] below, you're seeing equal temperaments that the zeta function identifies as particularly good approximations of just intonation. While these mathematical ratings provide a quick way to identify promising tuning systems, they don't tell the whole story. Each temperament has unique musical qualities that can only be discovered through listening and exploration.

When you look at the lists of "[[#Zeta peak EDOs|zeta peak EDOs]]" below, you're seeing equal temperaments that the zeta function identifies as particularly good approximations of just intonation. While these mathematical ratings provide a quick way to identify promising tuning systems, they don't tell the whole story. Each temperament has unique musical qualities that can only be discovered through listening and exploration.

Other methods for evaluating temperaments exist, such as various [[:Category:Regular temperament tuning|optimized regular temperament tunings]]. The zeta function approach presented here was developed primarily through the work of [[Gene Ward Smith]], with additional contributions from [[Mike Battaglia]].

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[[File:Zeta_first_harmonics.png|700px|thumb|none|Plot of the cosine error function for the first 5 harmonics. Note how the peaks line up for some popular EDOs.]]

Ideally, we would like to extend this function to sum over the entire harmonic series:

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{{todo|inline=1|text=add the original lists back here, and discuss their relevance.}}

== 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{{c}}.

For all edos 1 through 100, and for a list of successively higher zeta peaks, taken to five decimal places, 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 indices. They have their own article here, with a table of the first 500 or so: [[ZPI|zeta peak index (ZPI)]].

== Removing primes ==

An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function [[The Riemann zeta function and tuning/Appendix#1. Euler product expression for the zeta function|can be easily derived]]:

:<math>\displaystyle

\zeta(s) = \prod_p \left(1 - p^{-s}\right)^{-1}

</math>

where the product is over all primes ''p''. Like the original sum, this product converges for values <math>\mathrm{Re}(s) = \sigma > 1</math>.

We can remove a finite list of primes from consideration by multiplying <math>\zeta(s)</math> by the corresponding factors <math>(1 - p^{-s})</math> for each prime ''p'' we wish to remove.

After doing this, the smallest prime remaining will dominate peak values for ''s''.

In particular if we remove the prime 2, <math>\left(1 - p^{-2}\right)\zeta(s)</math> is now dominated by 3, and the large peak values occur near [[edt|equal divisions of the tritave]] (3/1).

Along any line of constant <math>\sigma</math>, [[The Riemann zeta function and tuning/Appendix#1b. Conversion factor for removing primes|it can be shown that]]:

:<math>\displaystyle

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

</math>;

in particular, on the critical line,

:<math>\displaystyle

\left| 1 - p^{-\frac{1}{2} - it} \right| = \sqrt{1 + \frac{1}{p} - \frac{2 \cos(t \ln p)}{\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. Note that multiplying this factor is technically only accurate for sums whose result is related to the Z function rather than the real part of the zeta function.

For example, if we want to find zeta peak [[EDT]]s (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave")—noting that here we must substitute <math>t = \frac{2\pi x}{\ln(3)}</math> instead of <math>\frac{2\pi x}{\ln(2)}</math>—in the no-twos subgroup, our modified zeta function is:

:<math>\displaystyle

\zeta \left( \sigma + \frac{2 \pi i}{\ln(3)}x \right)

\sqrt{ \frac{3}{2} - \sqrt{2} \cos \left( \frac{2\pi\ln(2)}{\ln(3)} x \right) }

</math>.

Removing 2 leads to increasing adjusted peak values corresponding to EDTs into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.

== Open problems ==

* Are there metrics similar to zeta metrics, but for ~~edos~~' performance at approximating arbitrary [[delta-rational]] chords?

* Are there metrics similar to zeta metrics, but for an EDO's performance at approximating arbitrary [[delta-rational]] chords?

== Further information ==

@@ Line 2: / Line 2: @@
 The Riemann zeta function is an important function in number theory, most famous for its connection to the unsolved Riemann hypothesis. Beyond its mathematical significance, however, the zeta function has a surprising connection to music theory: it provides a quantitative measure of how effectively a given [[equal temperament]] approximates [[just intonation]], relative to its size.
-The zeta function provides a way to measure how well different [[equal divisions of the octave]] (EDOs) approximate the infinite harmonic series found in just intonation. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement.
+The zeta function provides a way to measure how well different [[equal divisions of the octave]] (EDOs) approximate the infinite [[harmonic series]]. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement.
-When you look at the lists of [[#Zeta peak edos|"zeta peak EDOs"]] below, you're seeing equal temperaments that the zeta function identifies as particularly good approximations of just intonation. While these mathematical ratings provide a quick way to identify promising tuning systems, they don't tell the whole story. Each temperament has unique musical qualities that can only be discovered through listening and exploration.
+When you look at the lists of "[[#Zeta peak EDOs|zeta peak EDOs]]" below, you're seeing equal temperaments that the zeta function identifies as particularly good approximations of just intonation. While these mathematical ratings provide a quick way to identify promising tuning systems, they don't tell the whole story. Each temperament has unique musical qualities that can only be discovered through listening and exploration.
 Other methods for evaluating temperaments exist, such as various [[:Category:Regular temperament tuning|optimized regular temperament tunings]]. The zeta function approach presented here was developed primarily through the work of [[Gene Ward Smith]], with additional contributions from [[Mike Battaglia]].
@@ Line 51: / Line 51: @@
 [[File:Zeta_first_harmonics.png|700px|thumb|none|Plot of the cosine error function for the first 5 harmonics. Note how the peaks line up for some popular EDOs.]]
 Ideally, we would like to extend this function to sum over the entire harmonic series:
@@ Line 118: / Line 117: @@
 {{todo|inline=1|text=add the original lists back here, and discuss their relevance.}}
+== 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{{c}}.
+For all edos 1 through 100, and for a list of successively higher zeta peaks, taken to five decimal places, 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 indices. They have their own article here, with a table of the first 500 or so: [[ZPI|zeta peak index (ZPI)]].
+== Removing primes ==
+An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function [[The Riemann zeta function and tuning/Appendix#1. Euler product expression for the zeta function|can be easily derived]]:
+:<math>\displaystyle
+     \zeta(s) = \prod_p \left(1 - p^{-s}\right)^{-1}
+</math>
+where the product is over all primes ''p''. Like the original sum, this product converges for values <math>\mathrm{Re}(s) = \sigma > 1</math>.
+We can remove a finite list of primes from consideration by multiplying <math>\zeta(s)</math> by the corresponding factors <math>(1 - p^{-s})</math> for each prime ''p'' we wish to remove.
+After doing this, the smallest prime remaining will dominate peak values for ''s''.
+In particular if we remove the prime 2, <math>\left(1 - p^{-2}\right)\zeta(s)</math> is now dominated by 3, and the large peak values occur near [[edt|equal divisions of the tritave]] (3/1).
+Along any line of constant <math>\sigma</math>, [[The Riemann zeta function and tuning/Appendix#1b. Conversion factor for removing primes|it can be shown that]]:
+:<math>\displaystyle
+     \left| 1 - p^{-\sigma - it} \right| = \sqrt{1 + \frac{1}{p^{2\sigma}} - \frac{2 \cos(t \ln p)}{p^\sigma}}
+</math>;
+in particular, on the critical line,
+:<math>\displaystyle
+     \left| 1 - p^{-\frac{1}{2} - it} \right| = \sqrt{1 + \frac{1}{p} - \frac{2 \cos(t \ln p)}{\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. Note that multiplying this factor is technically only accurate for sums whose result is related to the Z function rather than the real part of the zeta function.
+For example, if we want to find zeta peak [[EDT]]s (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave")—noting that here we must substitute <math>t = \frac{2\pi x}{\ln(3)}</math> instead of <math>\frac{2\pi x}{\ln(2)}</math>—in the no-twos subgroup, our modified zeta function is:
+:<math>\displaystyle
+     \zeta \left( \sigma + \frac{2 \pi i}{\ln(3)}x \right)
+     \sqrt{ \frac{3}{2} - \sqrt{2} \cos \left( \frac{2\pi\ln(2)}{\ln(3)} x \right) }
+</math>.
+Removing 2 leads to increasing adjusted peak values corresponding to EDTs into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.
 == Open problems ==
-* Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?
+* Are there metrics similar to zeta metrics, but for an EDO's performance at approximating arbitrary [[delta-rational]] chords?
 == Further information ==