User:Sintel/Zeta working page: Difference between revisions

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The Riemann zeta function is an important function in number theory, most famous for its connection to the unsolved Riemann hypothesis. Beyond its mathematical significance, however, the zeta function has a surprising connection to music theory: it provides a quantitative measure of how effectively a given [[equal temperament]] approximates [[just intonation]], relative to its size.

The zeta function provides a way to measure how well different [[equal ~~divisions of the octave~~]] (~~EDOs~~) approximate the infinite [[harmonic series]]. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement.

The zeta function provides a way to measure how well different [[equal-step tunings]] (including [[EDO]]s) approximate the infinite [[harmonic series]]. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement; and being a continuous function, the zeta function also is able to highlight [[nonoctave]] and tempered-octave tunings. However, zeta invariably prioritizes lower primes over higher primes (though in a "minimal" way provided the requirement of converging in the infinite limit) and therefore fails to capture systems with efficacy in certain [[subgroup]]s.

~~When~~ you look at the ~~lists of "~~[[#Zeta peak EDOs|zeta peak EDOs]]" below, ~~you're seeing equal temperaments~~ that the zeta ~~function identifies~~ as ~~particularly good approximations~~ of ~~just intonation~~. ~~While these mathematical ratings provide~~ a ~~quick way to identify promising tuning systems, they don't tell~~ the ~~whole story. Each temperament has unique musical qualities that can only be discovered through listening and exploration~~.

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.

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.

Other methods for evaluating temperaments exist, such as various [[:Category:Regular temperament tuning|optimized regular temperament tunings]]. The zeta function approach presented here was developed primarily through the work of [[Gene Ward Smith]], with additional contributions from [[Mike Battaglia]].

== Zeta peak EDOs ==

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 The Riemann zeta function is an important function in number theory, most famous for its connection to the unsolved Riemann hypothesis. Beyond its mathematical significance, however, the zeta function has a surprising connection to music theory: it provides a quantitative measure of how effectively a given [[equal temperament]] approximates [[just intonation]], relative to its size.
-The zeta function provides a way to measure how well different [[equal divisions of the octave]] (EDOs) approximate the infinite [[harmonic series]]. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement.
+The zeta function provides a way to measure how well different [[equal-step tunings]] (including [[EDO]]s) approximate the infinite [[harmonic series]]. Unlike other evaluation methods that must select a specific [[prime limit]] as a cutoff, the zeta function approach incorporates all harmonics into its calculations, effectively functioning as an "infinite limit" measurement; and being a continuous function, the zeta function also is able to highlight [[nonoctave]] and tempered-octave tunings. However, zeta invariably prioritizes lower primes over higher primes (though in a "minimal" way provided the requirement of converging in the infinite limit) and therefore fails to capture systems with efficacy in certain [[subgroup]]s.
-When you look at the lists of "[[#Zeta peak EDOs|zeta peak EDOs]]" below, you're seeing equal temperaments that the zeta function identifies as particularly good approximations of just intonation. While these mathematical ratings provide a quick way to identify promising tuning systems, they don't tell the whole story. Each temperament has unique musical qualities that can only be discovered through listening and exploration.
+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.
+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.
-Other methods for evaluating temperaments exist, such as various [[:Category:Regular temperament tuning|optimized regular temperament tunings]]. The zeta function approach presented here was developed primarily through the work of [[Gene Ward Smith]], with additional contributions from [[Mike Battaglia]].
 == Zeta peak EDOs ==