The Riemann zeta function and tuning: Difference between revisions

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Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that |Z(x)| grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval [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 ~~make no sense, since~~ any zero of Z(x) ~~would qualify for a |Z(x)| minimum~~.

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

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

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{{EDOs|6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87, 111, 140, 183, 243, 301, 311, 460, 472, 525, 571, 581, 814, 836, 882, 1205}}...

We can then remove those secondary peaks again to get '''~~Grothendieck~~ edos'''.

We can then remove those secondary peaks again to get '''tertiary-peak edos'''.

'''~~Grothendieck~~ edos'''

'''Tertiary-peak edos'''

Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo~~. Named after the "Grothendieck prime" (the number 57), another reference to the "almost" nature of these edos~~.

Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo.

@@ Line 308: / Line 308: @@
 Notice the sudden jump from [[79edo]] to [[5941edo]]. We know that |Z(x)| grows logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval [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 make no sense, since any zero of Z(x) would qualify for a |Z(x)| minimum.
+Note that ''tempered-octave'' zeta valley edos would simply be any zero of Z(x).
 === ''k''-ary peak edos ===
@@ Line 322: / Line 322: @@
 {{EDOs|6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87, 111, 140, 183, 243, 301, 311, 460, 472, 525, 571, 581, 814, 836, 882, 1205}}...
-We can then remove those secondary peaks again to get '''Grothendieck edos'''.
+We can then remove those secondary peaks again to get '''tertiary-peak edos'''.
-'''Grothendieck edos'''
+'''Tertiary-peak edos'''
-Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo. Named after the "Grothendieck prime" (the number 57), another reference to the "almost" nature of these edos.
+Non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak ''or'' Parker edo.
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