Riemann zeta function - Revision history

Fredg999: Fredg999 moved page The Riemann zeta function and tuning to Riemann zeta function: No need to specify "tuning" (this is the Xen Wiki), WP:THE

2026-07-16T18:30:57Z

Fredg999 moved page The Riemann zeta function and tuning to Riemann zeta function: No need to specify "tuning" (this is the Xen Wiki), WP:THE

← Older revision	Revision as of 18:30, 16 July 2026
(No difference)

Fredg999: Removing todo:increase applicability and todo:simplify — this is wishful thinking: it is an advanced math topic and its applicability is already presented, namely through zeta edo lists and optimal octave stretch

2026-07-16T06:57:03Z

Removing todo:increase applicability and todo:simplify — this is wishful thinking: it *is* an advanced math topic and its applicability is already presented, namely through zeta edo lists and optimal octave stretch

← Older revision		Revision as of 06:57, 16 July 2026
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	{{Texops}}		{{Texops}}
	{{Wikipedia\|Riemann zeta function}}		{{Wikipedia\|Riemann zeta function}}
	The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.		The '''Riemann zeta function''' is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

	As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function.		As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function.
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	[[Category:Number theory]]		[[Category:Number theory]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	~~{{Todo\| increase applicability \| simplify }}~~

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: