The Riemann zeta function and tuning: Difference between revisions

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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.

== Terminology ==

'''Riemann zeta function ("zeta"):''' A mathematical function which is tied to the harmonic series and to prime numbers, used in tuning theory as an "EDO goodness" function to evaluate how close to JI an EDO is.

'''Zeta record edo:''' An equal tuning that sets some kind of record in regards to the zeta function compared to all smaller equal tunings.

'''Zeta peak:''' A zeta record equal tuning, evaluated by the absolute "goodness" of the edo according to the zeta function. This is a tuning which is closer to JI than any previous tuning, and is usually an EDO with compressed or stretched octaves.

'''Zeta peak integer:''' A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring nonoctave tunings and stretched/compressed octaves).

'''Zero:''' A point where the Riemann zeta function is equal to zero, such as ~2.759edo, representing an equal tuning that does not represent JI much at all. EDOs close to zeroes are called zeta valley EDOs.

'''Zeta gap:''' A zeta record edo by the size of the gap between its surrounding zeroes, adjusted for the fact that zeroes generally become more dense with larger inputs.

'''Zeta integral:''' A zeta record edo by the size of the area enclosed by the shape of the function between the edo's surrounding zeroes.

== Quick info: zeta peak edos ==

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 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.
+== Terminology ==
+'''Riemann zeta function ("zeta"):''' A mathematical function which is tied to the harmonic series and to prime numbers, used in tuning theory as an "EDO goodness" function to evaluate how close to JI an EDO is.
+'''Zeta record edo:''' An equal tuning that sets some kind of record in regards to the zeta function compared to all smaller equal tunings.
+'''Zeta peak:''' A zeta record equal tuning, evaluated by the absolute "goodness" of the edo according to the zeta function. This is a tuning which is closer to JI than any previous tuning, and is usually an EDO with compressed or stretched octaves.
+'''Zeta peak integer:''' A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring nonoctave tunings and stretched/compressed octaves).
+'''Zero:''' A point where the Riemann zeta function is equal to zero, such as ~2.759edo, representing an equal tuning that does not represent JI much at all. EDOs close to zeroes are called zeta valley EDOs.
+'''Zeta gap:''' A zeta record edo by the size of the gap between its surrounding zeroes, adjusted for the fact that zeroes generally become more dense with larger inputs.
+'''Zeta integral:''' A zeta record edo by the size of the area enclosed by the shape of the function between the edo's surrounding zeroes.
 == Quick info: zeta peak edos ==