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 harmonic series, and indeed ''all'' rational numbers, even up to "infinite-limit JI."

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 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 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.

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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 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 harmonic series, and indeed ''all'' rational numbers, even up to "infinite-limit JI."
-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 &mdash; the [[harmonic entropy]] model 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 ever-present in the background of tuning theory &mdash; 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.
 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.