221edo - 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"

2026-03-13T13:31:37Z

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 13:31, 13 March 2026
Line 102:		Line 102:
	\| [[Tritriple]] (221e)		\| [[Tritriple]] (221e)
	\|}		\|}
	<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

ArrowHead294 at 19:31, 20 February 2025

2025-02-20T19:31:01Z

← Older revision		Revision as of 19:31, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|221~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:28, 15 January 2025

2025-01-15T19:28:26Z

← Older revision		Revision as of 19:28, 15 January 2025
Line 102:		Line 102:
	\| [[Tritriple]] (221e)		\| [[Tritriple]] (221e)
	\|}		\|}
	<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

ArrowHead294: Partial undo

2024-11-16T12:41:51Z

Partial undo

← Older revision		Revision as of 12:41, 16 November 2024
Line 16:		Line 16:

	== 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
Line 45:		Line 54:
	\| 0.3618		\| 0.3618
	\| 6.66		\| 6.66
	~~{{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
Line 85:		Line 101:
	\| 864/625		\| 864/625
	\| [[Tritriple]] (221e)		\| [[Tritriple]] (221e)
	~~{{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

ArrowHead294 at 04:09, 16 November 2024

2024-11-16T04:09:23Z

← Older revision		Revision as of 04:09, 16 November 2024
Line 16:		Line 16:

	== 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
Line 53:		Line 45:
	\| 0.3618		\| 0.3618
	\| 6.66		\| 6.66
	\|}		{{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
Line 99:		Line 85:
	\| 864/625		\| 864/625
	\| [[Tritriple]] (221e)		\| [[Tritriple]] (221e)
	\|}		{{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}}

FloraC: /* Regular temperament properties */

2024-04-02T07:00:29Z

Regular temperament properties

← Older revision		Revision as of 07:00, 2 April 2024
Line 49:		Line 49:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 385/384, 441/440, 19683/19600, 235298/234375		\| 385/384, 441/440, 19683/19600, 235298/234375
	\| {{mapping\| 221 350 513 620 764 }}		\| {{mapping\| 221 350 513 620 764 }} (221e)
	\| +0.5904		\| +0.5904
	\| 0.3618		\| 0.3618

FloraC: /* Regular temperament properties */ +11-limit; +triwell

2024-04-02T07:00:08Z

Regular temperament properties: +11-limit; +triwell

← Older revision		Revision as of 07:00, 2 April 2024
Line 29:		Line 29:
	\| {{monzo\| -350 221 }}		\| {{monzo\| -350 221 }}
	\| {{mapping\| 221 350 }}		\| {{mapping\| 221 350 }}
	\| 0.4740		\| +0.4740
	\| 0.4742		\| 0.4742
	\| 8.73		\| 8.73
Line 36:		Line 36:
	\| {{monzo\| -21 3 7 }}, {{monzo\| -11 26 -13 }}		\| {{monzo\| -21 3 7 }}, {{monzo\| -11 26 -13 }}
	\| {{mapping\| 221 350 513 }}		\| {{mapping\| 221 350 513 }}
	\| 0.4299		\| +0.4299
	\| 0.3921		\| 0.3921
	\| 7.22		\| 7.22
Line 43:		Line 43:
	\| 1029/1024, 19683/19600, 235298/234375		\| 1029/1024, 19683/19600, 235298/234375
	\| {{mapping\| 221 350 513 620 }}		\| {{mapping\| 221 350 513 620 }}
	\| 0.5282		\| +0.5282
	\| 0.3799		\| 0.3799
	\| 7.00		\| 7.00
			\|-
			\| 2.3.5.7.11
			\| 385/384, 441/440, 19683/19600, 235298/234375
			\| {{mapping\| 221 350 513 620 764 }}
			\| +0.5904
			\| 0.3618
			\| 6.66
	\|}		\|}

Line 62:		Line 69:
	\| 75/64		\| 75/64
	\| [[Orson]]		\| [[Orson]]
			\|-
			\| 1
			\| 57\221
			\| 309.50
			\| 448/375
			\| [[Triwell]] (221e)
	\|-		\|-
	\| 1		\| 1

FloraC: Rework

2024-04-02T06:53:16Z

Rework

← Older revision		Revision as of 06:53, 2 April 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|221}}		{{EDO intro\|221}}

	== Theory ==		== Theory ==
	~~221et~~ tempers out 2109375/2097152 (semicomma) and ~~2541865828329/2500000000000~~ in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the [[optimal patent val]] for the 7-limit [[~~Gamelismic clan\|~~hemiseven temperament]].		221edo has a flat tendency, with [[harmonic]]s [[3/1\|3]], [[5/1\|5]], and [[7/1\|7]] all tuned flat. The equal temperament [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| -11 26 -13 }} in the 5-limit; [[1029/1024]], [[19683/19600]], and [[235298/234375]] in the 7-limit, so that it provides the [[optimal patent val]] for the 7-limit [[hemiseven]] temperament.

			Using the 221ef val, which does the best into the 17-limit, it tempers out [[385/384]], [[441/440]], 24057/24010, and 43923/43750 in the 11-limit; [[351/350]], [[676/675]], [[1287/1280]], [[1573/1568]], and 14641/14625 in the 13-limit; [[273/272]], [[561/560]], [[715/714]], [[833/832]], [[2187/2176]], and 10648/10625 in the 17-limit, supporting 17-limit hemiseven and 11-limit [[triwell]].

	Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit.		Using the [[patent val]], it tempers out [[540/539]], 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; [[364/363]], [[625/624]], 1701/1690, and [[2200/2197]] in the 13-limit.

	Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit [[Semicomma family\|triwell]].
	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|221}}		{{Harmonics in equal\|221}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	221 factors into 13 × 17, ~~with its subset edos~~ [[13edo]] and [[17edo]].		Since 221 factors into 13 × 17, 221edo has [[13edo]] and [[17edo]] as its subsets.
	==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]]
	! 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]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\|2.3		\| 2.3
	\|{{monzo\|-350 221}}		\| {{monzo\| -350 221 }}
	\|{{~~val~~\|221 350}}		\| {{mapping\| 221 350 }}
	\| 0.4740		\| 0.4740
	\| 0.4742		\| 0.4742
	\| 8.73		\| 8.73
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|-21 3 7}}, {{monzo\|-11 26 -13}}		\| {{monzo\| -21 3 7 }}, {{monzo\| -11 26 -13 }}
	\|{{~~val~~\|221 350 513}}		\| {{mapping\| 221 350 513 }}
	\| 0.4299		\| 0.4299
	\| 0.3921		\| 0.3921
	\| 7.22		\| 7.22
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|1029/1024, 19683/19600, 235298/234375		\| 1029/1024, 19683/19600, 235298/234375
	\|{{~~val~~\|221 350 513 620}}		\| {{mapping\| 221 350 513 620 }}
	\| 0.5282		\| 0.5282
	\| 0.3799		\| 0.3799
	\| 7.00		\| 7.00
	\|}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| 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 8ve		! Periods<br>per 8ve
	! Generator~~<br>(reduced)~~		! Generator*
	! Cents~~<br>(reduced)~~		! Cents*
	! Associated<br>~~ratio~~		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|50\221		\| 50\221
	\|271.49		\| 271.49
	\|75/64		\| 75/64
	\|[[Orson]]		\| [[Orson]]
	\|-		\|-
	\|1		\| 1
	\|84\221		\| 84\221
	\|456.11		\| 456.11
	\|125/96		\| 125/96
	\|[[Qak]]		\| [[Qak]]
	\|-		\|-
	\|1		\| 1
	\|89\221		\| 89\221
	\|483.26		\| 483.26
	\|320/243		\| 320/243
	\|[[Hemiseven]]		\| [[Hemiseven]] (221ef)
	\|-		\|-
	\|1		\| 1
	\|93\221		\| 93\221
	\|504.98		\| 504.98
	\|104976/78125		\| 104976/78125
	\|[[Countermeantone]]		\| [[Countermeantone]]
	\|-		\|-
	\|1		\| 1
	\|103\221		\| 103\221
	\|559.28		\| 559.28
	\|864/625		\| 864/625
	\|[[Tritriple]]		\| [[Tritriple]] (221e)
	\|}		\|}
			<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 -->~~

Francium at 19:30, 28 October 2023

2023-10-28T19:30:48Z

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''221edo''' is the [[EDO|equal division of the octave]] into 221 parts of 5.4299 [[cent]]s each.
+{{EDO intro|221}}
+== Theory ==
-It tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the [[optimal patent val]] for the 7-limit [[Gamelismic clan|hemiseven temperament]].
+et tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the [[optimal patent val]] for the 7-limit [[Gamelismic clan|hemiseven temperament]].
 Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit.
 Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit [[Semicomma family|triwell]].
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

BudjarnLambeth: Added "harmonics in equal" table

2023-06-24T03:39:12Z

Added "harmonics in equal" table

← Older revision		Revision as of 03:39, 24 June 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	'''221edo''' is the [[EDO\|equal division of the octave]] into 221 parts of 5.4299 [[cent]]s each. It tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the [[Optimal_patent_val\|optimal patent val]] for the 7-limit [[Gamelismic clan\|hemiseven temperament]]. Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit. Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit [[Semicomma family\|triwell]].		'''221edo''' is the [[EDO\|equal division of the octave]] into 221 parts of 5.4299 [[cent]]s each.

			It tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the [[optimal patent val]] for the 7-limit [[Gamelismic clan\|hemiseven temperament]].

			Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit.

			Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit [[Semicomma family\|triwell]].

			{{Harmonics in equal\|221}}
	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->