User:Sintel/Zeta working page: Difference between revisions

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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 ==

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 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 ==