Zeta peak index: Difference between revisions

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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.

== 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 ===

~~{{main|~~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:Zeta peak indexes| ]]

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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.
 == 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 ===
 {{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}
-== Gallery of the first 10 000 ZPIs ==
+=== Table of the first 10 000 ZPIs ===
-{{main|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:Zeta peak indexes| ]] <!-- main article -->