The Riemann zeta function and tuning: Difference between revisions

Line 1:

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.

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 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.

Line 10:

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.

== Intuitive explanation ==

We start with the basic concept of the mu function, which takes a cyclic function (a zigzag, in the case of mu) and sums up the values of that function scaled and weighted (canonically by a quadratic factor 1/n^2, but can be varied) to match each harmonic n. Technically, any cyclic function can be used this way. Specifically, complex exponentials are especially useful as a) they are analytic and b) they package the "weight" and the cyclic function together into a single function, where the weight corresponds to the real value of the input s, and so equal tunings are the imaginary part of the input. The sum of complex exponentials corresponding to each harmonic is, by definition, the zeta function, and it turns out that the weight 1/sqrt(n) gives the optimal information on higher primes, corresponding to the critical line where all known zeroes lie (intuitively, smaller weights lead to higher primes dominating too much and obscuring lower primes, and larger weights lead to the converse).

== Gene Smith's original derivation ==

@@ Line 1: / Line 1: @@
 {{Texops}}
 {{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 intriguing musical interpretation: the zeta function shows, 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 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.
@@ Line 10: / Line 10: @@
 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.
+== Intuitive explanation ==
+We start with the basic concept of the mu function, which takes a cyclic function (a zigzag, in the case of mu) and sums up the values of that function scaled and weighted (canonically by a quadratic factor 1/n^2, but can be varied) to match each harmonic n. Technically, any cyclic function can be used this way. Specifically, complex exponentials are especially useful as a) they are analytic and b) they package the "weight" and the cyclic function together into a single function, where the weight corresponds to the real value of the input s, and so equal tunings are the imaginary part of the input. The sum of complex exponentials corresponding to each harmonic is, by definition, the zeta function, and it turns out that the weight 1/sqrt(n) gives the optimal information on higher primes, corresponding to the critical line where all known zeroes lie (intuitively, smaller weights lead to higher primes dominating too much and obscuring lower primes, and larger weights lead to the converse).
 == Gene Smith's original derivation ==