The Riemann zeta function and tuning - Revision history

Squib: /* Zeta peak integer edos */

2026-06-24T15:52:06Z

Zeta peak integer edos

← Older revision		Revision as of 15:52, 24 June 2026
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	==== Zeta peak integer edos ====		==== Zeta peak integer edos ====
	Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap\|{{abs\|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs\| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs\| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs\| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing ~~the strength of 53 in~~ that 72 does not improve on 53's peak. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."		Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap\|{{abs\|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs\| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs\| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs\| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing that pure-octave 72 does not improve on 53's peak while stretched 72 does. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."

	==== Zeta integral edos ====		==== Zeta integral edos ====

ArrowHead294 at 23:31, 9 March 2026

2026-03-09T23:31:22Z

← Older revision		Revision as of 23:31, 9 March 2026
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	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		<math> \displaystyle

ArrowHead294: update shortcut

2026-02-08T14:52:38Z

update shortcut

← Older revision		Revision as of 14:52, 8 February 2026
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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.		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		<math> \displaystyle

ArrowHead294 at 19:25, 26 January 2026

2026-01-26T19:25:31Z

← Older revision		Revision as of 19:25, 26 January 2026
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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.		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		<math> \displaystyle

ArrowHead294: /* From unreduced rationals to reduced rationals */ Use actual prime symbol

2026-01-26T02:12:34Z

From unreduced rationals to reduced rationals: Use actual prime symbol

← Older revision		Revision as of 02:12, 26 January 2026
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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.		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		<math> \displaystyle

Inthar: /* Notes */

2026-01-17T17:42:49Z

Notes

← Older revision		Revision as of 17:42, 17 January 2026
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	[[Category:Number theory]]		[[Category:Number theory]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	~~[[Category:Pages with open problems]]~~
	{{Todo\| increase applicability \| simplify }}		{{Todo\| increase applicability \| simplify }}

Inthar: /* Open problems */

2026-01-17T17:42:30Z

Open problems

← Older revision		Revision as of 17:42, 17 January 2026
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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.		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 ==		== Further information ==

ArrowHead294 at 21:40, 15 January 2026

2026-01-15T21:40:14Z

← Older revision		Revision as of 21:40, 15 January 2026
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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.		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:

ArrowHead294: /* Anti-record edos */

2025-09-19T14:23:57Z

Anti-record edos

← Older revision		Revision as of 14:23, 19 September 2025
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	=== Anti-record edos ===		=== Anti-record edos ===
	==== Zeta valley 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.		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.

BudjarnLambeth: /* Other lists */

2025-09-18T23:02:58Z

Other lists

← Older revision		Revision as of 23:02, 18 September 2025
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	=== Other lists ===		=== 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 ====		==== Absolute zeta peak edos ====