The Riemann zeta function and tuning: Difference between revisions

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Note that since there is no restriction that ''n'' and ''d'' be coprime, the rationals we are using here do not have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all edos.

To see this, let us first note that every unreduced rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''′|''d''{{-'}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''′|''d''{{-'}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.

To see this, let us first note that every unreduced rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.

This allows us to change our original summation so that it's over three variables, ''n''′, ''d''′, and ''c''′, where ''n''′ and ''d''′ are coprime, and ''c'' is a strictly positive natural number:

This allows us to change our original summation so that it's over three variables, ''n''{{``}}, ''d''{{-`}}, and ''c''{{-`}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is a strictly positive natural number:

<math> \displaystyle

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=== Anti-record edos ===

==== Zeta valley edos ====

Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. ~~These correspond to ''zeta valley edos'', and we get~~ {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… Zeta valley edos can be thought 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.

Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. This gives us {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… which correspond to ''zeta valley edos''. Zeta valley edos can be thought 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 (essentially, tuning systems that avoid ''all'' ''p''-limit JI as much as possible) would not contain octaves at all.

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.

=== Other lists ===

{{Idiosyncratic terms|"~~''k''-ary-~~peak edos" was coined by {{u|~~Akselai~~}}, "~~absolute zeta~~ peak edos" was coined by {{u|~~Godtone~~}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}

{{Idiosyncratic terms|"Absolute zeta peak edos" was coined by {{u|Godtone}}, "''k''-ary-peak edos" was coined by {{u|Akselai}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}

==== Absolute zeta peak edos ====

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Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.

~~== Open problems ==~~

~~# Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?~~

== Further information ==

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[[Category:Number theory]]

[[Category:Pages with proofs]]

~~[[Category:Pages with open problems]]~~

@@ Line 266: / Line 266: @@
 Note that since there is no restriction that ''n'' and ''d'' be coprime, the rationals we are using here do not have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all edos.
-To see this, let us first note that every unreduced rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''′|''d''{{-'}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''′|''d''{{-'}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.
+To see this, let us first note that every unreduced rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.
-This allows us to change our original summation so that it's over three variables, ''n''′, ''d''′, and ''c''′, where ''n''′ and ''d''′ are coprime, and ''c'' is a strictly positive natural number:
+This allows us to change our original summation so that it's over three variables, ''n''{{``}}, ''d''{{-`}}, and ''c''{{-`}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is a strictly positive natural number:
 <math> \displaystyle
@@ Line 551: / Line 551: @@
 === Anti-record edos ===
 ==== Zeta valley edos ====
-Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. These correspond to ''zeta valley edos'', and we get {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… Zeta valley edos can be thought 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.
+Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. This gives us {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… which correspond to ''zeta valley edos''. Zeta valley edos can be thought 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 (essentially, tuning systems that avoid ''all'' ''p''-limit JI as much as possible) would not contain octaves at all.
 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.
 === Other lists ===
-{{Idiosyncratic terms|"''k''-ary-peak edos" was coined by {{u|Akselai}}, "absolute zeta peak edos" was coined by {{u|Godtone}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}
+{{Idiosyncratic terms|"Absolute zeta peak edos" was coined by {{u|Godtone}}, "''k''-ary-peak edos" was coined by {{u|Akselai}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}
 ==== Absolute zeta peak edos ====
@@ Line 645: / Line 645: @@
 Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.
-== Open problems ==
-# Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?
 == Further information ==
@@ Line 670: / Line 667: @@
 [[Category:Number theory]]
 [[Category:Pages with proofs]]
-[[Category:Pages with open problems]]
 {{Todo| increase applicability | simplify }}