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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 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 | 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. | ||
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. | |||
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 == | == Zeta peak EDOs == | ||
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See the [[#Zeta lists|section below]] for more information. | See the [[#Zeta lists|section below]] for more information. | ||
== Intuitive | == 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?" | 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. | 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 == | == 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. | 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>. | 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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== Choice of sigma == | == 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}}. | 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 == | == 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 == | == Optimal octave stretch == | ||
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== Removing primes == | == Removing primes == | ||
An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function [[The Riemann zeta function and tuning/Appendix# | 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 | :<math>\displaystyle | ||
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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). | 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# | 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 | :<math>\displaystyle | ||
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== Further information == | == Further information == | ||
For the interested reader, more information and derivations can be found in the [[The Riemann zeta function and tuning/Appendix| | For the interested reader, more information and derivations can be found in the [[The Riemann zeta function and tuning/Appendix|appendix]]. | ||
== Links == | == Links == | ||