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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-step tuning]]s (including [[EDO]]s) 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; and being a continuous function, the zeta function also is able to highlight [[nonoctave]] and tempered-octave tunings. However, zeta invariably prioritizes lower primes over higher primes (though in a "minimal" way provided the requirement of converging in the infinite limit) and therefore fails to capture systems with efficacy in certain [[subgroup]]s.

~~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~~.

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 [[#Zeta peak EDOs|zeta peak EDOs]] 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.

~~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]].

There are other metrics besides zeta for other definitions of "approximating well", such as the various [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments. The zeta function approach presented here was developed primarily through the work of [[Gene Ward Smith]], with additional contributions from [[Mike Battaglia]].

== Zeta peak EDOs ==

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See the [[#Zeta lists|section below]] for more information.

== Intuitive ~~Explanation~~ ==

== Intuitive explanation ==

When we talk about how well an equal temperament (ET) approximates just intonation, we're essentially asking: "How accurately can this system represent the harmonic series?"

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We then sum these weighted scores across the entire harmonic series.

The resulting value tells us how well a given equal temperament approximates the harmonic series. A higher score indicates that many important harmonics are well-represented by the system. Remarkably, this final scoring function turns out to be equivalent to the famous Riemann zeta function.

The resulting value tells us how well a given equal temperament approximates the harmonic series. A higher score indicates that many important harmonics are well-represented by the system. Remarkably, this final scoring function (given certain mathematical choices) turns out to be equivalent to the famous Riemann zeta function.

== Derivation ==

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If <math>x \log_2(n)</math> is very close to an integer, it means our equal temperament system can closely approximate that harmonic. If it's far from an integer, the approximation will be poor.

To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric where values close to 1 indicate good approximations and values close to -1 indicate poor ones.

To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric for the [[Relative interval error|relative error]], where values close to 1 indicate good approximations and values close to -1 indicate poor ones.

Let's begin by considering only the first few harmonics and look at <math>f(x \log_2 2)</math>, <math>f(x \log_2 3)</math>, <math>f(x \log_2 4)</math>, <math>f(x \log_2 5)</math>.

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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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== Choice of sigma ==

{{todo|inline=1|text=This section is incomplete! Discuss symmetry and add some conclusion}}

{{todo|complete section|inline=1|text=This section is incomplete! Discuss symmetry and add some conclusion}}

The infinite sum derived above converges only when <math>\mathrm{Re}(s) = \sigma > 1</math>. However, mathematicians found that the zeta function can be "extended" or "continued" to other values of s where the original sum doesn't converge. This mathematical technique is called {{w|analytic continuation}}.

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== Zeta lists ==

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

{{todo|complete list|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#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 - 2^{-s}\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#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 ==

For the interested reader, more information and derivations can be found in the [[The Riemann zeta function and tuning/Appendix|~~Appendix~~]].

For the interested reader, more information and derivations can be found in the [[The Riemann zeta function and tuning/Appendix|appendix]].

== Links ==

@@ 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-step tuning]]s (including [[EDO]]s) 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; and being a continuous function, the zeta function also is able to highlight [[nonoctave]] and tempered-octave tunings. However, zeta invariably prioritizes lower primes over higher primes (though in a "minimal" way provided the requirement of converging in the infinite limit) and therefore fails to capture systems with efficacy in certain [[subgroup]]s.
-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.
+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 [[#Zeta peak EDOs|zeta peak EDOs]] 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.
-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]].
+There are other metrics besides zeta for other definitions of "approximating well", such as the various [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments. The zeta function approach presented here was developed primarily through the work of [[Gene Ward Smith]], with additional contributions from [[Mike Battaglia]].
 == Zeta peak EDOs ==
@@ Line 17: / Line 17: @@
 See the [[#Zeta lists|section below]] for more information.
-== Intuitive Explanation ==
+== Intuitive explanation ==
 When we talk about how well an equal temperament (ET) approximates just intonation, we're essentially asking: "How accurately can this system represent the harmonic series?"
@@ Line 29: / Line 29: @@
 We then sum these weighted scores across the entire harmonic series.
-The resulting value tells us how well a given equal temperament approximates the harmonic series. A higher score indicates that many important harmonics are well-represented by the system. Remarkably, this final scoring function turns out to be equivalent to the famous Riemann zeta function.
+The resulting value tells us how well a given equal temperament approximates the harmonic series. A higher score indicates that many important harmonics are well-represented by the system. Remarkably, this final scoring function (given certain mathematical choices) turns out to be equivalent to the famous Riemann zeta function.
 == Derivation ==
@@ Line 40: / Line 40: @@
 If <math>x \log_2(n)</math> is very close to an integer, it means our equal temperament system can closely approximate that harmonic. If it's far from an integer, the approximation will be poor.
-To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric where values close to 1 indicate good approximations and values close to -1 indicate poor ones.
+To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric for the [[Relative interval error|relative error]], where values close to 1 indicate good approximations and values close to -1 indicate poor ones.
 Let's begin by considering only the first few harmonics and look at <math>f(x \log_2 2)</math>, <math>f(x \log_2 3)</math>, <math>f(x \log_2 4)</math>, <math>f(x \log_2 5)</math>.
@@ 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 106: / Line 105: @@
 == Choice of sigma ==
-{{todo|inline=1|text=This section is incomplete! Discuss symmetry and add some conclusion}}
+{{todo|complete section|inline=1|text=This section is incomplete! Discuss symmetry and add some conclusion}}
 The infinite sum derived above converges only when <math>\mathrm{Re}(s) = \sigma > 1</math>. However, mathematicians found that the zeta function can be "extended" or "continued" to other values of s where the original sum doesn't converge. This mathematical technique is called {{w|analytic continuation}}.
@@ Line 117: / Line 116: @@
 == Zeta lists ==
-{{todo|inline=1|text=add the original lists back here, and discuss their relevance.}}
+{{todo|complete list|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#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 - 2^{-s}\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#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 ==
-For the interested reader, more information and derivations can be found in the [[The Riemann zeta function and tuning/Appendix|Appendix]].
+For the interested reader, more information and derivations can be found in the [[The Riemann zeta function and tuning/Appendix|appendix]].
 == Links ==