The Riemann zeta function and tuning: Difference between revisions

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The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an ~~incredible~~ musical interpretation ~~as measuring the "harmonicity" of an [[equal temperament]]. Put simply,~~ the zeta function shows~~, in a certain sense~~, how ~~well~~ a given equal temperament approximates the no-limit [[just intonation]] relative to its size.

The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows, how ‘well’ a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is ~~ever-~~present in the background of tuning ~~theory—the~~ [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as an simple expression of the zeta function.

As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory — the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function.

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 edo lists|edo lists]] 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 [[mu badness]] and the various [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments.

Much of the below is thanks to the insights of [[Gene Ward Smith]]. Below is the original derivation as he presented it, followed by a different derivation from [[Mike Battaglia]] below which extends some of the results.

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 {{Wikipedia|Riemann zeta function}}
-The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an incredible musical interpretation as measuring the "harmonicity" of an [[equal temperament]]. Put simply, the zeta function shows, in a certain sense, how well a given equal temperament approximates the no-limit [[just intonation]] relative to its size.
+The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows, how ‘well’ a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.
-As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is ever-present in the background of tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as an simple expression of the zeta function.
+As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory — the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function.
 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 edo lists|edo lists]] 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 [[mu badness]] and the various [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments.
 Much of the below is thanks to the insights of [[Gene Ward Smith]]. Below is the original derivation as he presented it, followed by a different derivation from [[Mike Battaglia]] below which extends some of the results.