764edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"

2026-03-13T13:34:36Z

Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"

← Older revision		Revision as of 13:34, 13 March 2026
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	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|}		\|}
	<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms\|minimal form]] in parentheses if distinct

	[[Category:Abigail]]		[[Category:Abigail]]

FloraC: Cleanup

2025-10-27T13:35:17Z

Cleanup

← Older revision		Revision as of 13:35, 27 October 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|764\|columns=15}}		{{Harmonics in equal\|764\|columns=11}}
			{{Harmonics in equal\|764\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 764edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 764 factors into {{~~factorization~~\|~~764~~}}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo\|22\16808]]).		Since 764 factors into primes as {{nowrap\| 2<sup>2</sup> × 191 }}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo\|22\16808]]).

	== Regular temperament properties ==		== Regular temperament properties ==
Line 19:		Line 20:
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
Line 26:		Line 27:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 1211 -764 }}		\| {{Monzo\| 1211 -764 }}
	\| {{~~mapping~~\| 764 1211 }}		\| {{Mapping\| 764 1211 }}
	\| −0.0439		\| −0.0439
	\| 0.0439		\| 0.0439
Line 33:		Line 34:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}		\| {{Monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}
	\| {{~~mapping~~\| 764 1211 1774 }}		\| {{Mapping\| 764 1211 1774 }}
	\| −0.0399		\| −0.0399
	\| 0.0363		\| 0.0363
Line 41:		Line 42:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}		\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}
	\| {{~~mapping~~\| 764 1211 1774 2145 }}		\| {{Mapping\| 764 1211 1774 2145 }}
	\| −0.0552		\| −0.0552
	\| 0.0412		\| 0.0412
Line 48:		Line 49:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041		\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
	\| {{~~mapping~~\| 764 1211 1774 2145 2643 }}		\| {{Mapping\| 764 1211 1774 2145 2643 }}
	\| −0.0436		\| −0.0436
	\| 0.0435		\| 0.0435
Line 55:		Line 56:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875
	\| {{~~mapping~~\| 764 1211 1774 2145 2643 2827 }}		\| {{Mapping\| 764 1211 1774 2145 2643 2827 }}
	\| −0.0267		\| −0.0267
	\| 0.0548		\| 0.0548
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913		\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913
	\| {{~~mapping~~\| 764 1211 1774 2145 2643 2827 3123 }}		\| {{Mapping\| 764 1211 1774 2145 2643 2827 3123 }}
	\| −0.0327		\| −0.0327
	\| 0.0528		\| 0.0528
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	\| 2.3.5.7.11.13.17.23		\| 2.3.5.7.11.13.17.23
	\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095		\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
	\| {{~~mapping~~\| 764 1211 1774 2145 2643 2827 3123 3456 }}		\| {{Mapping\| 764 1211 1774 2145 2643 2827 3123 3456 }}
	\| −0.0286		\| −0.0286
	\| 0.0506		\| 0.0506
Line 81:		Line 82:
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 97:		Line 98:
	\| 435.08		\| 435.08
	\| 9/7		\| 9/7
	\| [[Supermajor]]		\| [[Supermajor (temperament)\|Supermajor]]
	\|-		\|-
	\| 2		\| 2
Line 106:		Line 107:
	\|-		\|-
	\| 2		\| 2
	\| 277\764<br />(105\764)		\| 277\764<br>(105\764)
	\| 435.08<br />(164.92)		\| 435.08<br>(164.92)
	\| 9/7<br />(11/10)		\| 9/7<br>(11/10)
	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[Normal ~~lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms\|minimal form]] in parentheses if distinct

	[[Category:Abigail]]		[[Category:Abigail]]

ArrowHead294 at 16:14, 20 February 2025

2025-02-20T16:14:00Z

← Older revision		Revision as of 16:14, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|764~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:58, 15 January 2025

2025-01-15T18:58:37Z

← Older revision		Revision as of 18:58, 15 January 2025
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	\| {{monzo\| 1211 -764 }}		\| {{monzo\| 1211 -764 }}
	\| {{mapping\| 764 1211 }}		\| {{mapping\| 764 1211 }}
	\| ~~−0~~.0439		\| −0.0439
	\| 0.0439		\| 0.0439
	\| 2.80		\| 2.80
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	\| {{monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}		\| {{monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}
	\| {{mapping\| 764 1211 1774 }}		\| {{mapping\| 764 1211 1774 }}
	\| ~~−0~~.0399		\| −0.0399
	\| 0.0363		\| 0.0363
	\| 2.31		\| 2.31
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	\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}		\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}
	\| {{mapping\| 764 1211 1774 2145 }}		\| {{mapping\| 764 1211 1774 2145 }}
	\| ~~−0~~.0552		\| −0.0552
	\| 0.0412		\| 0.0412
	\| 2.62		\| 2.62
Line 49:		Line 49:
	\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041		\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
	\| {{mapping\| 764 1211 1774 2145 2643 }}		\| {{mapping\| 764 1211 1774 2145 2643 }}
	\| ~~−0~~.0436		\| −0.0436
	\| 0.0435		\| 0.0435
	\| 2.77		\| 2.77
Line 56:		Line 56:
	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875
	\| {{mapping\| 764 1211 1774 2145 2643 2827 }}		\| {{mapping\| 764 1211 1774 2145 2643 2827 }}
	\| ~~−0~~.0267		\| −0.0267
	\| 0.0548		\| 0.0548
	\| 3.49		\| 3.49
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	\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913		\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913
	\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 }}		\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 }}
	\| ~~−0~~.0327		\| −0.0327
	\| 0.0528		\| 0.0528
	\| 3.36		\| 3.36
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	\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095		\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
	\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 3456 }}		\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 3456 }}
	\| ~~−0~~.0286		\| −0.0286
	\| 0.0506		\| 0.0506
	\| 3.22		\| 3.22
Line 111:		Line 111:
	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:Abigail]]		[[Category:Abigail]]

ArrowHead294: Partial undo

2024-11-16T12:19:51Z

Partial undo

← Older revision		Revision as of 12:19, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.0506		\| 0.0506
	\| 3.22		\| 3.22
	~~{{comma basis end}~~}		\|}
	* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo\|684]] and is only bettered by [[935edo\|935]]. In the 17-limit it beats [[742edo\|742]] and is only bettered by [[814edo\|814]].		* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo\|684]] and is only bettered by [[935edo\|935]]. In the 17-limit it beats [[742edo\|742]] and is only bettered by [[814edo\|814]].
	* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo\|494]] and before [[1578edo\|1578]].		* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo\|494]] and before [[1578edo\|1578]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
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	\| 9/7<br />(11/10)		\| 9/7<br />(11/10)
	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	[[Category:Abigail]]		[[Category:Abigail]]

ArrowHead294 at 05:01, 16 November 2024

2024-11-16T05:01:23Z

← Older revision		Revision as of 05:01, 16 November 2024
Line 14:		Line 14:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{monzo\| 1211 -764 }}		\| {{monzo\| 1211 -764 }}
	\| {{mapping\| 764 1211 }}		\| {{mapping\| 764 1211 }}
	\| -0.0439		\| −0.0439
	\| 0.0439		\| 0.0439
	\| 2.80		\| 2.80
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	\| {{monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}		\| {{monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}
	\| {{mapping\| 764 1211 1774 }}		\| {{mapping\| 764 1211 1774 }}
	\| -0.0399		\| −0.0399
	\| 0.0363		\| 0.0363
	\| 2.31		\| 2.31
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	\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}		\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}
	\| {{mapping\| 764 1211 1774 2145 }}		\| {{mapping\| 764 1211 1774 2145 }}
	\| -0.0552		\| −0.0552
	\| 0.0412		\| 0.0412
	\| 2.62		\| 2.62
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	\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041		\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
	\| {{mapping\| 764 1211 1774 2145 2643 }}		\| {{mapping\| 764 1211 1774 2145 2643 }}
	\| -0.0436		\| −0.0436
	\| 0.0435		\| 0.0435
	\| 2.77		\| 2.77
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	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875
	\| {{mapping\| 764 1211 1774 2145 2643 2827 }}		\| {{mapping\| 764 1211 1774 2145 2643 2827 }}
	\| -0.0267		\| −0.0267
	\| 0.0548		\| 0.0548
	\| 3.49		\| 3.49
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	\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913		\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913
	\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 }}		\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 }}
	\| -0.0327		\| −0.0327
	\| 0.0528		\| 0.0528
	\| 3.36		\| 3.36
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	\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095		\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
	\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 3456 }}		\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 3456 }}
	\| -0.0286		\| −0.0286
	\| 0.0506		\| 0.0506
	\| 3.22		\| 3.22
	\|}		{{comma basis end}}
	* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo\|684]] and is only bettered by [[935edo\|935]]. In the 17-limit it beats [[742edo\|742]] and is only bettered by [[814edo\|814]].		* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo\|684]] and is only bettered by [[935edo\|935]]. In the 17-limit it beats [[742edo\|742]] and is only bettered by [[814edo\|814]].
	* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo\|494]] and before [[1578edo\|1578]].		* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo\|494]] and before [[1578edo\|1578]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 2		\| 2
	\| 277\764<br>(105\764)		\| 277\764<br />(105\764)
	\| 435.08<br>(164.92)		\| 435.08<br />(164.92)
	\| 9/7<br>(11/10)		\| 9/7<br />(11/10)
	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	[[Category:Abigail]]		[[Category:Abigail]]

FloraC: /* Regular temperament properties */ no-19 23-limit data

2024-07-15T15:59:22Z

Regular temperament properties: no-19 23-limit data

← Older revision		Revision as of 15:59, 15 July 2024
Line 65:		Line 65:
	\| 0.0528		\| 0.0528
	\| 3.36		\| 3.36
			\|-
			\| 2.3.5.7.11.13.17.23
			\| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
			\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 3456 }}
			\| -0.0286
			\| 0.0506
			\| 3.22
	\|}		\|}
	* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo\|684]] and is only bettered by [[935edo\|935]]. In the 17-limit it beats [[742edo\|742]] and is only bettered by [[814edo\|814]].		* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo\|684]] and is only bettered by [[935edo\|935]]. In the 17-limit it beats [[742edo\|742]] and is only bettered by [[814edo\|814]].
			* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo\|494]] and before [[1578edo\|1578]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: /* Theory */ 41-odd-limit

2024-07-15T15:26:55Z

Theory: 41-odd-limit

← Older revision		Revision as of 15:26, 15 July 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	764edo is a very strong 17-limit system [[~~consistency\|distinctly~~ consistent]] to the [[17-odd-limit]]~~, and~~ is the fourteenth [[zeta integral edo]]. In the 5-limit it [[tempering out\|tempers out]] the hemithirds comma, {{monzo\| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.		764edo is a very strong 17-limit system, [[consistent]] to the [[17-odd-limit]] or the no-19 no-29 [[41-odd-limit]]. It is the fourteenth [[zeta integral edo]]. In the 5-limit it [[tempering out\|tempers out]] the hemithirds comma, {{monzo\| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.

	In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.		In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|764\|columns=12}}		{{Harmonics in equal\|764\|columns=15}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 764 factors into {{factorization\|764}}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo\|22\16808]]).		Since 764 factors into {{factorization\|764}}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo\|22\16808]]).

	== Regular temperament properties ==		== Regular temperament properties ==

FloraC: Adopt template: Factorization; misc. cleanup

2023-11-02T12:14:39Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 12:14, 2 November 2023
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	== Theory ==		== Theory ==
	764edo is a very strong 17-limit system ~~distinctly~~ [[consistent]] to the 17-odd-limit, and is the fourteenth [[~~The Riemann zeta function and tuning #Zeta EDO lists\|~~zeta integral edo]]. In the 5-limit it tempers out the hemithirds comma, {{monzo\| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.		764edo is a very strong 17-limit system [[consistency\|distinctly consistent]] to the [[17-odd-limit]], and is the fourteenth [[zeta integral edo]]. In the 5-limit it [[tempering out\|tempers out]] the hemithirds comma, {{monzo\| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.

	In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.		In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 764 factors into ~~2<sup>2</sup> × 191~~, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s.		Since 764 factors into {{factorization\|764}}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo\|22\16808]]).

	== Regular temperament properties ==		== Regular temperament properties ==
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	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-

FloraC: Cleanup; clarify the title row of the rank-2 temp table

2023-10-20T12:03:34Z

Cleanup; clarify the title row of the rank-2 temp table

← Older revision		Revision as of 12:03, 20 October 2023
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	== Theory ==		== Theory ==
	764edo is a very strong 17-limit system distinctly [[consistent]] to the 17-odd-limit, and is the fourteenth [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta integral edo]]. In the 5-limit it tempers out the hemithirds comma, {{monzo\| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit 2431/2430, [[2500/2499]], 4914/4913 and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.		764edo is a very strong 17-limit system distinctly [[consistent]] to the 17-odd-limit, and is the fourteenth [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta integral edo]]. In the 5-limit it tempers out the hemithirds comma, {{monzo\| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.

	In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.		In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.
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	\| 2.3		\| 2.3
	\| {{monzo\| 1211 -764 }}		\| {{monzo\| 1211 -764 }}
	\| [{{~~val~~\| 764 1211 }}]		\| {{mapping\| 764 1211 }}
	\| -0.0439		\| -0.0439
	\| 0.0439		\| 0.0439
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	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}		\| {{monzo\| 38 -2 -15 }}, {{monzo\| 25 -48 22 }}
	\| [{{~~val~~\| 764 1211 1774 }}]		\| {{mapping\| 764 1211 1774 }}
	\| -0.0399		\| -0.0399
	\| 0.0363		\| 0.0363
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	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}		\| 4375/4374, 52734375/52706752, {{monzo\| 31 -6 -2 -6 }}
	\| [{{~~val~~\| 764 1211 1774 2145 }}]		\| {{mapping\| 764 1211 1774 2145 }}
	\| -0.0552		\| -0.0552
	\| 0.0412		\| 0.0412
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041		\| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
	\| [{{~~val~~\| 764 1211 1774 2145 2643 }}]		\| {{mapping\| 764 1211 1774 2145 2643 }}
	\| -0.0436		\| -0.0436
	\| 0.0435		\| 0.0435
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875
	\| [{{~~val~~\| 764 1211 1774 2145 2643 2827 }}]		\| {{mapping\| 764 1211 1774 2145 2643 2827 }}
	\| -0.0267		\| -0.0267
	\| 0.0548		\| 0.0548
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913		\| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913
	\| [{{~~val~~\| 764 1211 1774 2145 2643 2827 3123 }}]		\| {{mapping\| 764 1211 1774 2145 2643 2827 3123 }}
	\| -0.0327		\| -0.0327
	\| 0.0528		\| 0.0528
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	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio
	! Temperaments		! Temperaments
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	\| [[Semisupermajor]]		\| [[Semisupermajor]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	[[Category:Abigail]]		[[Category:Abigail]]