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]].

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.

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.

ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.

== What are zeta peaks? ==

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, 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.

== Gallery of ZPIs ==

~~This is a gallery of notable ZPIs. No such gallery could possibly be complete (as there are so many possible zeta peaks), so please add tunings of interest as you see fit.~~

=== ZPIs with dedicated pages ===

* [[:Category:Zeta peak indexes]]''

=== Table of ZPIs up to 100 steps/octave ===

== ~~Gallery~~ of the first 10 000 ZPIs ==

=== Table of the first 10 000 ZPIs ===

[[:User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]]

* [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]] (may take a long time to load)

~~[[Category:Equal-step tuning]]~~

[[Category:Zeta peak indexes| ]]

[[Category:Zeta]]

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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]].
+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.
 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.
 ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.
+== What are zeta peaks? ==
+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, 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.
 == Gallery of ZPIs ==
-This is a gallery of notable ZPIs. No such gallery could possibly be complete (as there are so many possible zeta peaks), so please add tunings of interest as you see fit.
-{{:User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}
+=== ZPIs with dedicated pages ===
+* [[:Category:Zeta peak indexes]]''
+=== Table of ZPIs up to 100 steps/octave ===
+{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}
-== Gallery of the first 10 000 ZPIs ==
+=== Table of the first 10 000 ZPIs ===
-[[:User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]]
+* [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]] (may take a long time to load)
-[[Category:Equal-step tuning]]
+[[Category:Zeta peak indexes| ]] <!-- main article -->
-[[Category:Zeta]]