414edo - Revision history

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"

2026-03-13T13:32:14Z

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 13:32, 13 March 2026
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	\| [[Semihemiennealimmal]]		\| [[Semihemiennealimmal]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|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

	== Music ==		== Music ==

ArrowHead294 at 22:44, 20 February 2025

2025-02-20T22:44:11Z

← Older revision		Revision as of 22:44, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 22:41, 20 February 2025

2025-02-20T22:41:15Z

← Older revision		Revision as of 22:41, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro~~\|414~~}}		{{EDO intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T16:15:05Z

Theory: cleanup

← Older revision		Revision as of 16:15, 17 January 2025
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	== Theory ==		== Theory ==
	414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1\|5]]~~. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system~~. It [[tempers out]] {{monzo\| -36 11 8 }} (submajor comma) and {{monzo\| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo\| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].		414edo is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the [[harmonic]]s, making for a good full [[17-limit]] system. It is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1\|5]]. It [[tempering out\|tempers out]] {{monzo\| -36 11 8 }} (submajor comma) and {{monzo\| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo\| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 414 factors into ~~{{factorization\|414}}~~, 414edo has subset edos {{EDOs\| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.		Since 414 factors into 2 × 3<sup>2</sup> × 23, 414edo has subset edos {{EDOs\| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.

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

ArrowHead294 at 19:46, 15 January 2025

2025-01-15T19:46:47Z

← Older revision		Revision as of 19:46, 15 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1\|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. ~~The equal temperament~~ [[~~tempering out\|~~tempers out]] {{monzo\| -36 11 8 }} (submajor comma) and {{monzo\| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo\| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].		414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1\|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. It [[tempers out]] {{monzo\| -36 11 8 }} (submajor comma) and {{monzo\| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo\| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	\| [[Semihemiennealimmal]]		\| [[Semihemiennealimmal]]
	\|}		\|}
	<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

	== Music ==		== Music ==

ArrowHead294: Partial undo

2024-11-16T12:28:36Z

Partial undo

← Older revision		Revision as of 12:28, 16 November 2024
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 414 factors into {{factorization\|414}}, 414edo has subset edos {{EDOs\| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.		Since 414 factors into {{factorization\|414}}, 414edo has subset edos {{EDOs\| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.

	== 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]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.1329		\| 0.1329
	\| 4.58		\| 4.58
	~~{{comma basis end}~~}		\|}

	=== 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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	\| 1053/800<br>(1287/1280)		\| 1053/800<br>(1287/1280)
	\| [[Semihemiennealimmal]]		\| [[Semihemiennealimmal]]
	~~{{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

	== Music ==		== Music ==

ArrowHead294 at 04:36, 16 November 2024

2024-11-16T04:36:21Z

← Older revision		Revision as of 04:36, 16 November 2024
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	== 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]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.1329		\| 0.1329
	\| 4.58		\| 4.58
	\|}		{{comma basis end}}

	=== 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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	\| 1053/800<br>(1287/1280)		\| 1053/800<br>(1287/1280)
	\| [[Semihemiennealimmal]]		\| [[Semihemiennealimmal]]
	\|}		{{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}}

	== Music ==		== Music ==
	; [[No Clue Music]]		; [[No Clue Music]]
	* [https://www.youtube.com/watch?v=j6KPW-Hr1sI ''DISconnectioN''] (2024)		* [https://www.youtube.com/watch?v=j6KPW-Hr1sI ''DISconnectioN''] (2024)

			[[Category:Listen]]

FloraC: /* Regular temperament properties */ fix a link

2024-06-19T11:30:39Z

Regular temperament properties: fix a link

← Older revision		Revision as of 11:30, 19 June 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \| [Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|Comma List]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]

Francium: +music

2024-06-19T07:18:22Z

+music

← Older revision		Revision as of 07:18, 19 June 2024
Line 98:		Line 98:
	\|}		\|}
	<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct		<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

			== Music ==
			; [[No Clue Music]]
			* [https://www.youtube.com/watch?v=j6KPW-Hr1sI ''DISconnectioN''] (2024)

FloraC: Adopt template: EDO intro; cleanup; +subsets and supersets

2023-11-06T14:43:35Z

Adopt template: EDO intro; cleanup; +subsets and supersets

← Older revision		Revision as of 14:43, 6 November 2023
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	{{Infobox ET}}		{{Infobox ET}}
	~~The '''414 equal divisions of the octave''' ('''414edo'''), or the '''414(-tone) equal temperament''' ('''414tet''', '''414et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO\|~~equal division of the octave]] into~~ 414 ~~parts of about 2.90 [[cent]]s each.~~		{{EDO intro\|414}}

	== Theory ==		== Theory ==
	414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo\| -36 11 8 }} (submajor comma) and {{monzo\|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo\| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].		414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1\|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. The equal temperament [[tempering out\|tempers out]] {{monzo\| -36 11 8 }} (submajor comma) and {{monzo\| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo\| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|414}}		{{Harmonics in equal\|414}}

			=== Subsets and supersets ===
			Since 414 factors into {{factorization\|414}}, 414edo has subset edos {{EDOs\| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \| Subgroup		! rowspan="2" \| [Subgroup]]
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list\|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
	\|-		\|-
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
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	\| 2.3.5		\| 2.3.5
	\| {{monzo\| -36 11 8 }}, {{monzo\| 1 -27 18 }}		\| {{monzo\| -36 11 8 }}, {{monzo\| 1 -27 18 }}
	\| [{{~~val~~\| 414 656 961 }}]		\| {{mapping\| 414 656 961 }}
	\| +0.2222		\| +0.2222
	\| 0.1575		\| 0.1575
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	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 4375/4374, {{monzo\| -36 11 8 }}		\| 2401/2400, 4375/4374, {{monzo\| -36 11 8 }}
	\| [{{~~val~~\| 414 656 961 1162 }}]		\| {{mapping\| 414 656 961 1162 }}
	\| +0.2299		\| +0.2299
	\| 0.1371		\| 0.1371
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944		\| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944
	\| [{{~~val~~\| 414 656 961 1162 1432 }}]		\| {{mapping\| 414 656 961 1162 1432 }}
	\| +0.2182		\| +0.2182
	\| 0.1248		\| 0.1248
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400		\| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400
	\| [{{~~val~~\| 414 656 961 1162 1432 1532 }}]		\| {{mapping\| 414 656 961 1162 1432 1532 }}
	\| +0.1795		\| +0.1795
	\| 0.1431		\| 0.1431
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197		\| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197
	\| [{{~~val~~\| 414 656 961 1162 1432 1532 1692 }}]		\| {{mapping\| 414 656 961 1162 1432 1532 1692 }}
	\| +0.1751		\| +0.1751
	\| 0.1329		\| 0.1329
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	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per ~~octave~~		! Periods<br>per 8ve
	! Generator~~<br>(reduced)~~		! Generator*
	! Cents~~<br>(reduced)~~		! Cents*
	! Associated<br>~~ratio~~		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[Semihemiennealimmal]]		\| [[Semihemiennealimmal]]
	\|}		\|}
			<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:Equal divisions of~~ the octave\|~~###~~]] ~~<!-- 3-digit number -->~~

← Older revision		Revision as of 11:30, 19 June 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \| [Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|Comma List]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]