User:Sintel/Zeta working page: Difference between revisions

(3 intermediate revisions by 2 users not shown)

Line 6:

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.

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]].

~~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~~.

== Zeta peak EDOs ==

Line 32:

Line 29:

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 ==

Line 43:

Line 40:

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>.

Line 108:

Line 105:

== 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}}.

Line 119:

Line 116:

== 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 ==

@@ Line 6: / Line 6: @@
 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.
+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]].
-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.
 == Zeta peak EDOs ==
@@ Line 32: / Line 29: @@
 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 ==
@@ Line 43: / Line 40: @@
 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>.
@@ Line 108: / Line 105: @@
 == 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}}.
@@ Line 119: / Line 116: @@
 == 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 ==