The Riemann zeta function and tuning: 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 edo lists|edo lists]] 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 [[mu badness]] and 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 [[mu badness]], [[target tuning|least squares tuning]], [[target tuning|minimax tuning]], and the many [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments.

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.

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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 edo lists|edo lists]] 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 [[mu badness]] and 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 [[mu badness]], [[target tuning|least squares tuning]], [[target tuning|minimax tuning]], and the many [[:Category:Regular temperament tuning|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments.
 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.