Zeta peak index - Revision history

Contribution at 17:29, 28 August 2025

2025-08-28T17:29:08Z

← Older revision		Revision as of 17:29, 28 August 2025
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	Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.		Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.

			== Sigma ==

			=== Sigma = 1/2 ===
			Peaks provided here are for sigma = 1/2.

			Other sigma choices are possible.

			=== Sigma = 1 ===

			* [[User:Contribution/Gallery of Zeta Peak Indexes at sigma = 1\|Zeta Peak Indexes at sigma = 1]]

	== Gallery of ZPIs ==		== Gallery of ZPIs ==

BudjarnLambeth at 05:51, 17 August 2025

2025-08-17T05:51:21Z

← Older revision		Revision as of 05:51, 17 August 2025
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	The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a 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 mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

	Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, ~~such as [[mu badness]], [[target tuning\|least squares tuning]], [[target tuning\|minimax tuning]], and the many~~ [[:Category:Regular temperament tuning\|optimised regular temperament tunings]] ~~when applied to [[rank]]-1 (i.e. equal) temperaments~~.		Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, which you can find in: [[:Category:Regular temperament tuning\|optimised regular temperament tunings]].

	Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.		Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.

BudjarnLambeth: /* What are zeta peaks? */ mention target tuning

2025-08-17T05:35:10Z

What are zeta peaks?: mention target tuning

← Older revision		Revision as of 05:35, 17 August 2025
Line 8:		Line 8:
	The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a 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 mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

	Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, such as [[mu badness]] and the ~~various~~ [[:Category:Regular temperament tuning\|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments~~. These can give different answers~~.		Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, such as [[mu badness]], [[target tuning\|least squares tuning]], [[target tuning\|minimax tuning]], and the many [[:Category:Regular temperament tuning\|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments.

	Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.		Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.

BudjarnLambeth: /* Table of the ZPIs up to 100 steps/octave */

2025-08-16T06:53:17Z

Table of the ZPIs up to 100 steps/octave

← Older revision		Revision as of 06:53, 16 August 2025
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	* [[:Category:Zeta peak indexes]]''		* [[:Category:Zeta peak indexes]]''

	=== Table of ~~the~~ ZPIs up to 100 steps/octave ===		=== Table of ZPIs up to 100 steps/octave ===
	{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}		{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}

BudjarnLambeth: /* Table of the first 574 ZPIs */

2025-08-16T06:53:04Z

Table of the first 574 ZPIs

← Older revision		Revision as of 06:53, 16 August 2025
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	* [[:Category:Zeta peak indexes]]''		* [[:Category:Zeta peak indexes]]''

	=== Table of the ~~first 574~~ ZPIs ===		=== Table of the ZPIs up to 100 steps/octave ===
	{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}		{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}

BudjarnLambeth: /* Table of the first 10 000 ZPIs */

2025-08-16T06:52:16Z

Table of the first 10 000 ZPIs

← Older revision		Revision as of 06:52, 16 August 2025
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	* [[:Category:Zeta peak indexes]]''		* [[:Category:Zeta peak indexes]]''

	=== Table of the first ~~10 000~~ ZPIs ===		=== Table of the first 574 ZPIs ===
	{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}		{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}

BudjarnLambeth at 06:51, 16 August 2025

2025-08-16T06:51:58Z

← Older revision		Revision as of 06:51, 16 August 2025
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	== Gallery of ZPIs ==		== Gallery of ZPIs ==
	~~''See also: [[:Category:Zeta peak indexes]]''~~

			=== ZPIs with dedicated pages ===
			* [[:Category:Zeta peak indexes]]''

	~~This is a gallery~~ of ~~notable~~ ZPIs~~. No such gallery could possibly be complete (as there are so many possible zeta peaks), so please add tunings of interest as you see fit.~~		=== Table of the first 10 000 ZPIs ===

	{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}		{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}

	== ~~Gallery~~ of the first 10 000 ZPIs ==		=== Table of the first 10 000 ZPIs ===
	~~{{main\|~~User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)}}		* [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]] (may take a long time to load)

	[[Category:Zeta peak indexes\| ]] <!-- main article -->		[[Category:Zeta peak indexes\| ]] <!-- main article -->

BudjarnLambeth: /* Gallery of ZPIs */

2025-08-16T06:49:17Z

Gallery of ZPIs

← Older revision		Revision as of 06:49, 16 August 2025
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	== Gallery of ZPIs ==		== Gallery of ZPIs ==
			''See also: [[:Category:Zeta peak indexes]]''


	This is a gallery of notable ZPIs. No such gallery could possibly be complete (as there are so many possible zeta peaks), so please add tunings of interest as you see fit.		This is a gallery of notable ZPIs. No such gallery could possibly be complete (as there are so many possible zeta peaks), so please add tunings of interest as you see fit.

BudjarnLambeth: /* What are zeta peaks */

2025-08-15T23:42:06Z

What are zeta peaks

← Older revision		Revision as of 23:42, 15 August 2025
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	ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.		ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.

	== What are zeta peaks ==		== What are zeta peaks? ==
	The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a 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 mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

	Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, such as [[mu badness]] and the various [[:Category:Regular temperament tuning\|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments. These can give different answers.		Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, such as [[mu badness]] and the various [[:Category:Regular temperament tuning\|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments. These can give different answers.

	Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.		Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.

	== Gallery of ZPIs ==		== Gallery of ZPIs ==

BudjarnLambeth: Add a section based on this Discord message from another editor: "My issue with the zpi page is that theres no explanation for any of the strength metrics or a link to a relevant section on the zeta page or something. Ignoring my bias against zeta, it makes no sense"

2025-08-15T23:41:54Z

Add a section based on this Discord message from another editor: "My issue with the zpi page is that theres no explanation for any of the strength metrics or a link to a relevant section on the zeta page or something. Ignoring my bias against zeta, it makes no sense"

← Older revision		Revision as of 23:41, 15 August 2025
Line 4:		Line 4:

	ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.		ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.

			== What are zeta peaks ==
			The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.

			Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, such as [[mu badness]] and the various [[:Category:Regular temperament tuning\|optimised regular temperament tunings]] when applied to [[rank]]-1 (i.e. equal) temperaments. These can give different answers.

			Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.

	== Gallery of ZPIs ==		== Gallery of ZPIs ==