Zeta peak index: Difference between revisions
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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. | 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 == | == 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. | 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, | 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 [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at. | ||
== Gallery of ZPIs == | == Gallery of ZPIs == | ||
=== 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)}} | {{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}} | ||
== | === Table of the first 10 000 ZPIs === | ||
* [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]] (may take a long time to load) | |||
[[Category:Zeta peak indexes| ]] <!-- main article --> | [[Category:Zeta peak indexes| ]] <!-- main article --> | ||