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.

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]].

== Zeta peak EDOs ==

These lists give [[equal divisions of the octave]] that are better than any before them according to the zeta metric.

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, ...}}

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

== 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?"

These harmonics occur at whole-number ratios (1:1, 2:1, 3:1, 4:1, 5:1, etc.), which are not equally spaced. In contrast, equal temperaments divide the octave into equal steps, which is convenient but introduces slight deviations from these ratios.

The zeta function essentially measures these deviations across the entire harmonic series, and gives more weight to lower harmonics, which are typically more audible and musically significant than higher ones.

For each harmonic in the series, we check how closely our equal temperament can approximate it.

When a harmonic falls nearly exactly on an ET step, it receives a positive score. When it falls halfway between steps (requiring significant compromises in tuning), it receives a negative score.

Lower harmonics (which make up simple intervals like the perfect fifth or major third) count more toward the final score than higher harmonics.

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.

== Derivation ==

Our goal is to derive a function that quantifies how effectively an [[equal division of the octave]] (EDO) approximates [[just intonation]] (JI). We will use the variable ''x'' to denote an EDO where x=12 represents [[12edo]].

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[[File:Zeta_real_simple.png|700px|thumb|none|Real part of zeta function for sigma = 1.]]

== Choice of sigma ==

{{todo|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}}.

The extended zeta function is defined everywhere in the complex plane except at s = 1 (where there is a pole). This continuation allows us to evaluate the zeta function at values like <math>s = 1/2 + it</math>, which lies on what's called the "critical line."

When we use the zeta function to evaluate equal temperaments, we're interested in how well these temperaments approximate the harmonic series. The question becomes: which weight (value of s) gives us the most musically relevant information?

When s > 1, lower harmonics (especially the octave) dominate the evaluation. When s approaches 0, higher harmonics dominate. Neither extreme provides a balanced assessment of how well a temperament approximates the full harmonic series.

== Zeta lists ==

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

== Open problems ==

* Are there metrics similar to zeta metrics, but for edos' 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]].

== Links ==

* [https://arxiv.org/abs/math/0309433 X-Ray of Riemann zeta-function] by Juan Arias-de-Reyna

* [http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/ Selberg's limit theorem] by Terence Tao [http://www.webcitation.org/5xrvgjW6T Permalink]

* [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch

* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|ζ({{frac|1|2}} + ''it'')}}] by Tadej Kotnik

* [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]

* [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]

* [https://www.lmfdb.org/zeros/zeta/?N=1&t=&limit=100 Zeros of Zeta]

[[Category:Zeta| ]]

[[Category:Math]]

[[Category:Tuning]]

[[Category:Number theory]]

[[Category:Pages with proofs]]

[[Category:Pages with open problems]]

@@ Line 1: / Line 1: @@
+{{Wikipedia|Riemann zeta function}}
+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.
+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]].
+== Zeta peak EDOs ==
+These lists give [[equal divisions of the octave]] that are better than any before them according to the zeta metric.
+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, ...}}
+See the [[#Zeta lists|section below]] for more information.
+== 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?"
+These harmonics occur at whole-number ratios (1:1, 2:1, 3:1, 4:1, 5:1, etc.), which are not equally spaced. In contrast, equal temperaments divide the octave into equal steps, which is convenient but introduces slight deviations from these ratios.
+The zeta function essentially measures these deviations across the entire harmonic series, and gives more weight to lower harmonics, which are typically more audible and musically significant than higher ones.
+For each harmonic in the series, we check how closely our equal temperament can approximate it.
+When a harmonic falls nearly exactly on an ET step, it receives a positive score. When it falls halfway between steps (requiring significant compromises in tuning), it receives a negative score.
+Lower harmonics (which make up simple intervals like the perfect fifth or major third) count more toward the final score than higher harmonics.
+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.
 == Derivation ==
 Our goal is to derive a function that quantifies how effectively an [[equal division of the octave]] (EDO) approximates [[just intonation]] (JI). We will use the variable ''x'' to denote an EDO where x=12 represents [[12edo]].
@@ Line 71: / Line 103: @@
 [[File:Zeta_real_simple.png|700px|thumb|none|Real part of zeta function for sigma = 1.]]
+== Choice of sigma ==
+{{todo|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}}.
+The extended zeta function is defined everywhere in the complex plane except at s = 1 (where there is a pole). This continuation allows us to evaluate the zeta function at values like <math>s = 1/2 + it</math>, which lies on what's called the "critical line."
+When we use the zeta function to evaluate equal temperaments, we're interested in how well these temperaments approximate the harmonic series. The question becomes: which weight (value of s) gives us the most musically relevant information?
+When s > 1, lower harmonics (especially the octave) dominate the evaluation. When s approaches 0, higher harmonics dominate. Neither extreme provides a balanced assessment of how well a temperament approximates the full harmonic series.
+== Zeta lists ==
+{{todo|inline=1|text=add the original lists back here, and discuss their relevance.}}
+== Open problems ==
+* Are there metrics similar to zeta metrics, but for edos' 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]].
+== Links ==
+* [https://arxiv.org/abs/math/0309433 X-Ray of Riemann zeta-function] by Juan Arias-de-Reyna
+* [http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/ Selberg's limit theorem] by Terence Tao [http://www.webcitation.org/5xrvgjW6T Permalink]
+* [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch
+* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|ζ({{frac|1|2}} + ''it'')}}] by Tadej Kotnik
+* [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]
+* [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]
+* [https://www.lmfdb.org/zeros/zeta/?N=1&t=&limit=100 Zeros of Zeta]
+[[Category:Zeta| ]] <!-- Main article -->
+[[Category:Math]]
+[[Category:Tuning]]
+[[Category:Number theory]]
+[[Category:Pages with proofs]]
+[[Category:Pages with open problems]]