The Riemann zeta function and tuning: Difference between revisions

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'''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 record~~ 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.

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

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

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

'''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; all known zeroes are on the "critical line" used to obtain tuning information.

'''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''; all known zeroes are on the "critical line" used to obtain tuning information.

'''~~Zeta record~~ 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.

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

'''Record 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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 '''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 record 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.
+'''Record zeta peak:''' An equal tuning which is closer to JI than any previous tuning, and is usually an EDO with compressed or stretched octaves, evaluated by the absolute "goodness" of the edo according to the zeta function.
-'''Zeta record integer:''' A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring nonoctave tunings and stretched/compressed octaves).
+'''Record zeta peak integer:''' A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring stretched/compressed equivalences).
-'''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; all known zeroes are on the "critical line" used to obtain tuning information.
+'''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''; all known zeroes are on the "critical line" used to obtain tuning information.
-'''Zeta record 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.
+'''Record z 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 record integral:''' A zeta record edo by the size of the area enclosed by the shape of the function between the edo's surrounding zeroes.
+'''Record 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 ==