The Riemann zeta function and tuning: Difference between revisions

Line 566:

=== Anti-record edos ===

==== Zeta valley edos ====

In addition to looking at {{nowrap|{{abs|Z(x)}}}} maxima, we can also look at {{nowrap|{{abs|Z(x)}}}} ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}} … ~~These~~ tunings tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.

In addition to looking at {{nowrap|{{abs|Z(x)}}}} maxima, we can also look at {{nowrap|{{abs|Z(x)}}}} ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… Zeta valley EDOs can be though of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley EDOs are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.

Notice ~~the sudden~~ jump from [[79edo]] to [[5941edo]]. We know that {{nowrap|{{abs|Z(x)}}}} ~~grows~~ logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.

Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.

~~Note that "tempered-octave zeta valley edos" would simply be any zero of Z(x)~~.

=== ''k''-ary-peak edos ===

@@ Line 566: / Line 566: @@
 === Anti-record edos ===
 ==== Zeta valley edos ====
-In addition to looking at {{nowrap|{{abs|Z(x)}}}} maxima, we can also look at {{nowrap|{{abs|Z(x)}}}} ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}} … These tunings tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.
+In addition to looking at {{nowrap|{{abs|Z(x)}}}} maxima, we can also look at {{nowrap|{{abs|Z(x)}}}} ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… Zeta valley EDOs can be though of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley EDOs are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.
-Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that {{nowrap|{{abs|Z(x)}}}} grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.
+Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.
-Note that "tempered-octave zeta valley edos" would simply be any zero of Z(x).
 === ''k''-ary-peak edos ===