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." | 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." | ||
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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. | 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. | ||
== Gene Smith's original derivation == | == Gene Smith's original derivation == | ||