422edo - Revision history

FloraC: Countritonic -> ragitritonic

2026-05-20T11:11:06Z

Countritonic -> ragitritonic

← Older revision		Revision as of 11:11, 20 May 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency\|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. As an equal temperament, it [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[~~hemigamera~~]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor~~]], and [[countritonic~~]].		422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency\|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. As an equal temperament, it [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s and provides the [[optimal patent val]]s for [[gamera]] in the 7-limit, [[hemigamera]] in the 13-limit, and [[ragitritonic]] in the 11- and 13-limit. Other notable temperaments it supports are [[vishnu]] and [[semisupermajor]].

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|422}}		{{Harmonics in equal\|422\|columns=11}}
	{{Harmonics in equal\|422\|start=12\|collapsed=1\|title=Approximation of prime harmonics in 422edo (continued)}}		{{Harmonics in equal\|422\|columns=11\|start=12\|collapsed=1\|title=Approximation of prime harmonics in 422edo (continued)}}
	{{Harmonics in equal\|422\|start=23\|collapsed=1\|title=Approximation of prime harmonics in 422edo (continued)}}		{{Harmonics in equal\|422\|columns=11\|start=23\|collapsed=1\|title=Approximation of prime harmonics in 422edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
Line 115:		Line 115:
	\| 588.63		\| 588.63
	\| 128/91		\| 128/91
	\| [[~~Countritonic~~]]		\| [[Ragitritonic]]
	\|-		\|-
	\| 2		\| 2
Line 137:		Line 137:
	<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|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-generator form\|minimal form]] in parentheses if distinct

	[[Category:~~Countritonic~~]]		[[Category:Ragitritonic]]
	[[Category:Gamera]]		[[Category:Gamera]]
	[[Category:Vishnu]]		[[Category:Vishnu]]

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:30:01Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:30, 13 March 2026
Line 135:		Line 135:
	\| [[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-generator form\|minimal form]] in parentheses if distinct

	[[Category:Countritonic]]		[[Category:Countritonic]]
	[[Category:Gamera]]		[[Category:Gamera]]
	[[Category:Vishnu]]		[[Category:Vishnu]]

FloraC: Cleanup

2025-10-27T13:37:45Z

Cleanup

← Older revision		Revision as of 13:37, 27 October 2025
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].		Since 422 factors into primes as {{nowrap\| 2 × 211 }}, 422edo has subset edos [[2edo]] and [[211edo]].

	== Regular temperament properties ==		== Regular temperament properties ==
Line 19:		Line 19:
	! 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 26:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 669 -422 }}		\| {{Monzo\| 669 -422 }}
	\| {{~~mapping~~\| 422 669 }}		\| {{Mapping\| 422 669 }}
	\| −0.1308		\| −0.1308
	\| 0.1308		\| 0.1308
Line 33:		Line 33:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 23 6 -14 }}, {{monzo\| 33 -34 9 }}		\| {{Monzo\| 23 6 -14 }}, {{monzo\| 33 -34 9 }}
	\| {{~~mapping~~\| 422 669 980 }}		\| {{Mapping\| 422 669 980 }}
	\| −0.1469		\| −0.1469
	\| 0.1092		\| 0.1092
Line 41:		Line 41:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 589824/588245, 29360128/29296875		\| 4375/4374, 589824/588245, 29360128/29296875
	\| {{~~mapping~~\| 422 669 980 1185 }}		\| {{Mapping\| 422 669 980 1185 }}
	\| −0.1852		\| −0.1852
	\| 0.1155		\| 0.1155
Line 48:		Line 48:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 5632/5625, 589824/588245		\| 3025/3024, 4375/4374, 5632/5625, 589824/588245
	\| {{~~mapping~~\| 422 669 980 1185 1460 }}		\| {{Mapping\| 422 669 980 1185 1460 }}
	\| −0.1679		\| −0.1679
	\| 0.1090		\| 0.1090
Line 55:		Line 55:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625		\| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
	\| {{~~mapping~~\| 422 669 980 1185 1460 1562 }}		\| {{Mapping\| 422 669 980 1185 1460 1562 }}
	\| −0.1930		\| −0.1930
	\| 0.1142		\| 0.1142
Line 62:		Line 62:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
	\| {{~~mapping~~\| 422 669 980 1185 1460 1562 1725 }}		\| {{Mapping\| 422 669 980 1185 1460 1562 1725 }}
	\| −0.1744		\| −0.1744
	\| 0.1151		\| 0.1151
Line 69:		Line 69:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197		\| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
	\| {{~~mapping~~\| 422 669 980 1185 1460 1562 1725 1793 }}		\| {{Mapping\| 422 669 980 1185 1460 1562 1725 1793 }}
	\| −0.1839		\| −0.1839
	\| 0.1106		\| 0.1106
Line 76:		Line 76:
	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715		\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
	\| {{~~mapping~~\| 422 669 980 1185 1460 1562 1725 1793 1909 }}		\| {{Mapping\| 422 669 980 1185 1460 1562 1725 1793 1909 }}
	\| −0.1675		\| −0.1675
	\| 0.1142		\| 0.1142
Line 87:		Line 87:
	\|+ 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 109:		Line 109:
	\| 435.07		\| 435.07
	\| 9/7		\| 9/7
	\| [[Supermajor]]		\| [[Supermajor (temperament)\|Supermajor]]
	\|-		\|-
	\| 1		\| 1
Line 130:		Line 130:
	\|-		\|-
	\| 2		\| 2
	\| 153\422<br />(58\422)		\| 153\422<br>(58\422)
	\| 435.07<br />(164.93)		\| 435.07<br>(164.93)
	\| 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:Countritonic]]		[[Category:Countritonic]]
	[[Category:Gamera]]		[[Category:Gamera]]
	[[Category:Vishnu]]		[[Category:Vishnu]]

Overthink: /* Prime harmonics */ Added higher primes as many are approximated well

2025-10-20T00:39:45Z

Prime harmonics: Added higher primes as many are approximated well

← Older revision		Revision as of 00:39, 20 October 2025
Line 7:		Line 7:
	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|422}}		{{Harmonics in equal\|422}}
			{{Harmonics in equal\|422\|start=12\|collapsed=1\|title=Approximation of prime harmonics in 422edo (continued)}}
			{{Harmonics in equal\|422\|start=23\|collapsed=1\|title=Approximation of prime harmonics in 422edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

ArrowHead294 at 14:55, 20 February 2025

2025-02-20T14:55:32Z

← Older revision		Revision as of 14:55, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|422~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T16:15:53Z

Theory: cleanup

← Older revision		Revision as of 16:15, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. It [[tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].		422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency\|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. As an equal temperament, it [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 422 factors into ~~{{factorization\|422}}~~, 422edo has subset edos [[2edo]] and [[211edo]].		Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].

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

ArrowHead294 at 14:02, 16 January 2025

2025-01-16T14:02:01Z

← Older revision		Revision as of 14:02, 16 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[~~consistency\|~~distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. ~~The equal temperament~~ [[~~tempering out\|~~tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].		422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. It [[tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 26:		Line 26:
	\| {{monzo\| 669 -422 }}		\| {{monzo\| 669 -422 }}
	\| {{mapping\| 422 669 }}		\| {{mapping\| 422 669 }}
	\| ~~−0~~.1308		\| −0.1308
	\| 0.1308		\| 0.1308
	\| 4.60		\| 4.60
Line 33:		Line 33:
	\| {{monzo\| 23 6 -14 }}, {{monzo\| 33 -34 9 }}		\| {{monzo\| 23 6 -14 }}, {{monzo\| 33 -34 9 }}
	\| {{mapping\| 422 669 980 }}		\| {{mapping\| 422 669 980 }}
	\| ~~−0~~.1469		\| −0.1469
	\| 0.1092		\| 0.1092
	\| 3.84		\| 3.84
Line 40:		Line 40:
	\| 4375/4374, 589824/588245, 29360128/29296875		\| 4375/4374, 589824/588245, 29360128/29296875
	\| {{mapping\| 422 669 980 1185 }}		\| {{mapping\| 422 669 980 1185 }}
	\| ~~−0~~.1852		\| −0.1852
	\| 0.1155		\| 0.1155
	\| 4.06		\| 4.06
Line 47:		Line 47:
	\| 3025/3024, 4375/4374, 5632/5625, 589824/588245		\| 3025/3024, 4375/4374, 5632/5625, 589824/588245
	\| {{mapping\| 422 669 980 1185 1460 }}		\| {{mapping\| 422 669 980 1185 1460 }}
	\| ~~−0~~.1679		\| −0.1679
	\| 0.1090		\| 0.1090
	\| 3.83		\| 3.83
Line 54:		Line 54:
	\| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625		\| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
	\| {{mapping\| 422 669 980 1185 1460 1562 }}		\| {{mapping\| 422 669 980 1185 1460 1562 }}
	\| ~~−0~~.1930		\| −0.1930
	\| 0.1142		\| 0.1142
	\| 4.02		\| 4.02
Line 61:		Line 61:
	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
	\| {{mapping\| 422 669 980 1185 1460 1562 1725 }}		\| {{mapping\| 422 669 980 1185 1460 1562 1725 }}
	\| ~~−0~~.1744		\| −0.1744
	\| 0.1151		\| 0.1151
	\| 4.05		\| 4.05
Line 68:		Line 68:
	\| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197		\| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
	\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 }}		\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 }}
	\| ~~−0~~.1839		\| −0.1839
	\| 0.1106		\| 0.1106
	\| 3.89		\| 3.89
Line 75:		Line 75:
	\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715		\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
	\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 1909 }}		\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 1909 }}
	\| ~~−0~~.1675		\| −0.1675
	\| 0.1142		\| 0.1142
	\| 4.02		\| 4.02
Line 133:		Line 133:
	\| [[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:Countritonic]]		[[Category:Countritonic]]
	[[Category:Gamera]]		[[Category:Gamera]]
	[[Category:Vishnu]]		[[Category:Vishnu]]

ArrowHead294 at 12:17, 16 November 2024

2024-11-16T12:17:04Z

← Older revision		Revision as of 12:17, 16 November 2024
Line 12:		Line 12:

	== 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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	\| 0.1142		\| 0.1142
	\| 4.02		\| 4.02
	~~{{comma basis end}~~}		\|}
	* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo\|400]] and is bettered by [[460edo\|460]]. In the 23-limit it beats [[373edo\|373g]] and is bettered by [[525edo\|525]].		* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo\|400]] and is bettered by [[460edo\|460]]. In the 23-limit it beats [[373edo\|373g]] and is bettered by [[525edo\|525]].

	=== 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:Countritonic]]		[[Category:Countritonic]]
	[[Category:Gamera]]		[[Category:Gamera]]
	[[Category:Vishnu]]		[[Category:Vishnu]]

ArrowHead294 at 05:07, 16 November 2024

2024-11-16T05:07:40Z

← Older revision		Revision as of 05:07, 16 November 2024
Line 12:		Line 12:

	== 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\| 669 -422 }}		\| {{monzo\| 669 -422 }}
	\| {{mapping\| 422 669 }}		\| {{mapping\| 422 669 }}
	\| -0.1308		\| −0.1308
	\| 0.1308		\| 0.1308
	\| 4.60		\| 4.60
Line 32:		Line 24:
	\| {{monzo\| 23 6 -14 }}, {{monzo\| 33 -34 9 }}		\| {{monzo\| 23 6 -14 }}, {{monzo\| 33 -34 9 }}
	\| {{mapping\| 422 669 980 }}		\| {{mapping\| 422 669 980 }}
	\| -0.1469		\| −0.1469
	\| 0.1092		\| 0.1092
	\| 3.84		\| 3.84
Line 39:		Line 31:
	\| 4375/4374, 589824/588245, 29360128/29296875		\| 4375/4374, 589824/588245, 29360128/29296875
	\| {{mapping\| 422 669 980 1185 }}		\| {{mapping\| 422 669 980 1185 }}
	\| -0.1852		\| −0.1852
	\| 0.1155		\| 0.1155
	\| 4.06		\| 4.06
Line 46:		Line 38:
	\| 3025/3024, 4375/4374, 5632/5625, 589824/588245		\| 3025/3024, 4375/4374, 5632/5625, 589824/588245
	\| {{mapping\| 422 669 980 1185 1460 }}		\| {{mapping\| 422 669 980 1185 1460 }}
	\| -0.1679		\| −0.1679
	\| 0.1090		\| 0.1090
	\| 3.83		\| 3.83
Line 53:		Line 45:
	\| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625		\| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
	\| {{mapping\| 422 669 980 1185 1460 1562 }}		\| {{mapping\| 422 669 980 1185 1460 1562 }}
	\| -0.1930		\| −0.1930
	\| 0.1142		\| 0.1142
	\| 4.02		\| 4.02
Line 60:		Line 52:
	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
	\| {{mapping\| 422 669 980 1185 1460 1562 1725 }}		\| {{mapping\| 422 669 980 1185 1460 1562 1725 }}
	\| -0.1744		\| −0.1744
	\| 0.1151		\| 0.1151
	\| 4.05		\| 4.05
Line 67:		Line 59:
	\| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197		\| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
	\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 }}		\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 }}
	\| -0.1839		\| −0.1839
	\| 0.1106		\| 0.1106
	\| 3.89		\| 3.89
Line 74:		Line 66:
	\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715		\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
	\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 1909 }}		\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 1909 }}
	\| -0.1675		\| −0.1675
	\| 0.1142		\| 0.1142
	\| 4.02		\| 4.02
	\|}		{{comma basis end}}
	* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19- and 23-limit. In the 17- and 19-limit it beats [[400edo\|400]] and is bettered by [[460edo\|460]]. In the 23-limit it beats [[373edo\|373g]] and is bettered by [[525edo\|525]].		* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo\|400]] and is bettered by [[460edo\|460]]. In the 23-limit it beats [[373edo\|373g]] and is bettered by [[525edo\|525]].

	=== 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
Line 126:		Line 112:
	\|-		\|-
	\| 2		\| 2
	\| 153\422<br>(58\422)		\| 153\422<br />(58\422)
	\| 435.07<br>(164.93)		\| 435.07<br />(164.93)
	\| 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:Countritonic]]
	[[Category:Gamera]]		[[Category:Gamera]]
	[[Category:Vishnu]]		[[Category:Vishnu]]
	~~[[Category:Countritonic]]~~

FloraC: +23-limit data

2024-10-12T13:55:34Z

+23-limit data

← Older revision		Revision as of 13:55, 12 October 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency\|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and 589824/588245 in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]], and [[2432/2431]] in the 19-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].		422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency\|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]], {{monzo\| 23 6 -14 }} and the countritonic comma, {{monzo\| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 70:		Line 70:
	\| 0.1106		\| 0.1106
	\| 3.89		\| 3.89
			\|-
			\| 2.3.5.7.11.13.17.19.23
			\| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
			\| {{mapping\| 422 669 980 1185 1460 1562 1725 1793 1909 }}
			\| -0.1675
			\| 0.1142
			\| 4.02
	\|}		\|}
	* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19- and 23-limit. In the 17- and 19-limit it beats [[400edo\|400]] and is bettered by [[460edo\|460]]. In the 23-limit it beats [[373edo\|373g]] and is bettered by [[525edo\|525]].		* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19- and 23-limit. In the 17- and 19-limit it beats [[400edo\|400]] and is bettered by [[460edo\|460]]. In the 23-limit it beats [[373edo\|373g]] and is bettered by [[525edo\|525]].