Zeta peak index: Difference between revisions

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A '''zeta peak index''' ('''ZPI''' or '''zpi''') is an [[equal-step tuning]] obtained from one of the peaks of the [[~~The Riemann zeta function and tuning|~~Riemann zeta function]]. The peaks provided are for the common value of σ = 0.5.

A '''zeta peak index''' ('''ZPI''' or '''zpi''') is an [[equal-step tuning]] obtained from one of the peaks of the [[Riemann zeta function]]. The peaks provided are for the common value of σ = 0.5.

For instance, the closest zeta peak of 12edo, which has a value of 12.023edo, is the 34th peak of the Riemann zeta function: this tuning is 34zpi.

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The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, ~~such as [[mu badness]] and the various~~ [[:Category:Regular temperament tuning|optimised regular temperament tunings]] ~~when applied to [[rank]]-1 (i.e. equal) temperaments. These can give different answers~~.

Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, which you can find in: [[:Category:Regular temperament tuning|optimised regular temperament tunings]].

Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[~~The~~ Riemann zeta function ~~and tuning~~]] for a fuller explanation of how zeta peaks are arrived at.

Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[Riemann zeta function]] for a fuller explanation of how zeta peaks are arrived at.

== Sigma ==

=== Sigma = 1/2 ===

Peaks provided here are for sigma = 1/2.

Other sigma choices are possible.

=== Sigma = 1 ===

* [[User:Contribution/Gallery of Zeta Peak Indexes at sigma = 1|Zeta Peak Indexes at sigma = 1]]

== Gallery of ZPIs ==

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-A '''zeta peak index''' ('''ZPI''' or '''zpi''') is an [[equal-step tuning]] obtained from one of the peaks of the [[The Riemann zeta function and tuning|Riemann zeta function]]. The peaks provided are for the common value of σ = 0.5.
+A '''zeta peak index''' ('''ZPI''' or '''zpi''') is an [[equal-step tuning]] obtained from one of the peaks of the [[Riemann zeta function]]. The peaks provided are for the common value of σ = 0.5.
 For instance, the closest zeta peak of 12edo, which has a value of 12.023edo, is the 34th peak of the Riemann zeta function: this tuning is 34zpi.
@@ Line 8: / Line 8: @@
 The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.
-Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, such as [[mu badness]] and the various [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments. These can give different answers.
+Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, which you can find in: [[:Category:Regular temperament tuning|optimised regular temperament tunings]].
-Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.
+Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[Riemann zeta function]] for a fuller explanation of how zeta peaks are arrived at.
+== Sigma ==
+=== Sigma = 1/2 ===
+Peaks provided here are for sigma = 1/2.
+Other sigma choices are possible.
+=== Sigma = 1 ===
+* [[User:Contribution/Gallery of Zeta Peak Indexes at sigma = 1|Zeta Peak Indexes at sigma = 1]]
 == Gallery of ZPIs ==