357edo - 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"

2026-03-13T13:32:16Z

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
Line 113:		Line 113:
	\| [[Pnict]]		\| [[Pnict]]
	\|}		\|}
	<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

Francium: changed EDO intro to ED intro

2025-02-21T06:30:58Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:30, 21 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|357~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T15:57:18Z

Theory: cleanup

← Older revision		Revision as of 15:57, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	While not highly accurate for its size, 357et is the point where a few important temperaments meet. It [[tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo\| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit.		While not highly accurate for its size, 357et is the point where a few important temperaments meet. It [[tempering out\|tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo\| 61 4 -29 }} (squarschimidt comma) in the [[5-limit]]; 10976/10935 ([[hemimage comma]]), 235298/234375 ([[triwellisma]]), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the [[7-limit]]; [[3025/3024]], [[5632/5625]], [[12005/11979]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the [[13-limit]].

	It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[~~Avicenna~~ (temperament)\|avicenna]], but [[270edo]] is better suited for this purpose.		It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[avicenna (temperament)\|avicenna]], but [[270edo]] is better suited for this purpose.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 357 factors into ~~{{factorization\|357}}~~, 357edo has subset edos {{EDOs\| 3, 7, 17, 21, 51, and 119 }}.		Since 357 factors into 3 × 7 × 17, 357edo has subset edos {{EDOs\| 3, 7, 17, 21, 51, and 119 }}.

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

ArrowHead294 at 18:56, 15 January 2025

2025-01-15T18:56:05Z

← Older revision		Revision as of 18:56, 15 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	While not highly accurate for its size, 357et is the point where a few important temperaments meet. ~~The equal temperament~~ [[~~tempering out\|~~tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo\| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit.		While not highly accurate for its size, 357et is the point where a few important temperaments meet. It [[tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo\| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit.

	It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[Avicenna (temperament)\|avicenna]], but [[270edo]] is better suited for this purpose.		It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[Avicenna (temperament)\|avicenna]], but [[270edo]] is better suited for this purpose.
Line 28:		Line 28:
	\| {{monzo\| 566 -357 }}		\| {{monzo\| 566 -357 }}
	\| {{mapping\| 357 566 }}		\| {{mapping\| 357 566 }}
	\| ~~−0~~.1786		\| −0.1786
	\| 0.1785		\| 0.1785
	\| 5.31		\| 5.31
Line 35:		Line 35:
	\| 1600000/1594323, {{monzo\| 61 4 -29 }}		\| 1600000/1594323, {{monzo\| 61 4 -29 }}
	\| {{mapping\| 357 566 829 }}		\| {{mapping\| 357 566 829 }}
	\| ~~−0~~.1536		\| −0.1536
	\| 0.1500		\| 0.1500
	\| 4.46		\| 4.46
Line 42:		Line 42:
	\| 10976/10935, 235298/234375, 2100875/2097152		\| 10976/10935, 235298/234375, 2100875/2097152
	\| {{mapping\| 357 566 829 1002 }}		\| {{mapping\| 357 566 829 1002 }}
	\| ~~−0~~.0477		\| −0.0477
	\| 0.2248		\| 0.2248
	\| 6.69		\| 6.69
Line 49:		Line 49:
	\| 3025/3024, 5632/5625, 10976/10935, 102487/102400		\| 3025/3024, 5632/5625, 10976/10935, 102487/102400
	\| {{mapping\| 357 566 829 1002 1235 }}		\| {{mapping\| 357 566 829 1002 1235 }}
	\| ~~−0~~.0348		\| −0.0348
	\| 0.2027		\| 0.2027
	\| 6.03		\| 6.03
Line 56:		Line 56:
	\| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935		\| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
	\| {{mapping\| 357 566 829 1002 1235 1321 }}		\| {{mapping\| 357 566 829 1002 1235 1321 }}
	\| ~~−0~~.0204		\| −0.0204
	\| 0.1879		\| 0.1879
	\| 5.59		\| 5.59
Line 113:		Line 113:
	\| [[Pnict]]		\| [[Pnict]]
	\|}		\|}
	<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

2024-11-16T12:38:23Z

Partial undo

← Older revision		Revision as of 12:38, 16 November 2024
Line 14:		Line 14:

	== 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 50:		Line 59:
	\| 0.1879		\| 0.1879
	\| 5.59		\| 5.59
	~~{{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 96:		Line 112:
	\| 192/175		\| 192/175
	\| [[Pnict]]		\| [[Pnict]]
	~~{{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 04:14, 16 November 2024

2024-11-16T04:14:50Z

← Older revision		Revision as of 04:14, 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\| 566 -357 }}		\| {{monzo\| 566 -357 }}
	\| {{mapping\| 357 566 }}		\| {{mapping\| 357 566 }}
	\| -0.1786		\| −0.1786
	\| 0.1785		\| 0.1785
	\| 5.31		\| 5.31
Line 34:		Line 26:
	\| 1600000/1594323, {{monzo\| 61 4 -29 }}		\| 1600000/1594323, {{monzo\| 61 4 -29 }}
	\| {{mapping\| 357 566 829 }}		\| {{mapping\| 357 566 829 }}
	\| -0.1536		\| −0.1536
	\| 0.1500		\| 0.1500
	\| 4.46		\| 4.46
Line 41:		Line 33:
	\| 10976/10935, 235298/234375, 2100875/2097152		\| 10976/10935, 235298/234375, 2100875/2097152
	\| {{mapping\| 357 566 829 1002 }}		\| {{mapping\| 357 566 829 1002 }}
	\| -0.0477		\| −0.0477
	\| 0.2248		\| 0.2248
	\| 6.69		\| 6.69
Line 48:		Line 40:
	\| 3025/3024, 5632/5625, 10976/10935, 102487/102400		\| 3025/3024, 5632/5625, 10976/10935, 102487/102400
	\| {{mapping\| 357 566 829 1002 1235 }}		\| {{mapping\| 357 566 829 1002 1235 }}
	\| -0.0348		\| −0.0348
	\| 0.2027		\| 0.2027
	\| 6.03		\| 6.03
Line 55:		Line 47:
	\| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935		\| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
	\| {{mapping\| 357 566 829 1002 1235 1321 }}		\| {{mapping\| 357 566 829 1002 1235 1321 }}
	\| -0.0204		\| −0.0204
	\| 0.1879		\| 0.1879
	\| 5.59		\| 5.59
	\|}		{{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 110:		Line 96:
	\| 192/175		\| 192/175
	\| [[Pnict]]		\| [[Pnict]]
	\|}		{{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: Adopt template: Factorization

2023-11-10T09:10:13Z

Adopt template: Factorization

← Older revision		Revision as of 09:10, 10 November 2023
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 357 factors into ~~3 × 5 × 7~~, 357edo has subset edos {{EDOs\| 3, 7, 17, 21, 51, and 119 }}.		Since 357 factors into {{factorization\|357}}, 357edo has subset edos {{EDOs\| 3, 7, 17, 21, 51, and 119 }}.

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

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

2023-11-10T09:07:40Z

Rework; cleanup; clarify the title row of the rank-2 temp table

← Older revision		Revision as of 09:07, 10 November 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament tempers out 1600000/1594323 ([[amity comma]]), and {{monzo\| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit~~; notably~~ [[support]]~~ing~~ [[amity]], [[chromat]], ~~and~~ [[Avicenna (temperament)\|avicenna]].		While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament [[tempering out\|tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo\| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit.

			It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[Avicenna (temperament)\|avicenna]], but [[270edo]] is better suited for this purpose.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 24:		Line 26:
	\| 2.3		\| 2.3
	\| {{monzo\| 566 -357 }}		\| {{monzo\| 566 -357 }}
	\| [{{~~val~~\| 357 566 }}]		\| {{mapping\| 357 566 }}
	\| -0.1786		\| -0.1786
	\| 0.1785		\| 0.1785
Line 31:		Line 33:
	\| 2.3.5		\| 2.3.5
	\| 1600000/1594323, {{monzo\| 61 4 -29 }}		\| 1600000/1594323, {{monzo\| 61 4 -29 }}
	\| [{{~~val~~\| 357 566 829 }}]		\| {{mapping\| 357 566 829 }}
	\| -0.1536		\| -0.1536
	\| 0.1500		\| 0.1500
Line 38:		Line 40:
	\| 2.3.5.7		\| 2.3.5.7
	\| 10976/10935, 235298/234375, 2100875/2097152		\| 10976/10935, 235298/234375, 2100875/2097152
	\| [{{~~val~~\| 357 566 829 1002 }}]		\| {{mapping\| 357 566 829 1002 }}
	\| -0.0477		\| -0.0477
	\| 0.2248		\| 0.2248
Line 45:		Line 47:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 5632/5625, 10976/10935, 102487/102400		\| 3025/3024, 5632/5625, 10976/10935, 102487/102400
	\| [{{~~val~~\| 357 566 829 1002 1235 }}]		\| {{mapping\| 357 566 829 1002 1235 }}
	\| -0.0348		\| -0.0348
	\| 0.2027		\| 0.2027
Line 52:		Line 54:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935		\| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
	\| [{{~~val~~\| 357 566 829 1002 1235 1321 }}]		\| {{mapping\| 357 566 829 1002 1235 1321 }}
	\| -0.0204		\| -0.0204
	\| 0.1879		\| 0.1879
Line 62:		Line 64:
	\|+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
	\|-		\|-
Line 109:		Line 111:
	\| [[Pnict]]		\| [[Pnict]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

FloraC: /* Regular temperament properties */ +hemichromat, +calamity

2023-07-10T08:10:24Z

Regular temperament properties: +hemichromat, +calamity

← Older revision		Revision as of 08:10, 10 July 2023
Line 78:		Line 78:
	\| 44/35		\| 44/35
	\| [[Squarschmidt]]		\| [[Squarschmidt]]
			\|-
			\| 1
			\| 163\357
			\| 547.90
			\| 48/35
			\| [[Calamity]]
	\|-		\|-
	\| 3		\| 3
	\| 48\357		\| 9\357
	\| ~~161~~.34		\| 30.25
	\| ~~192~~/~~175~~		\| 55/54
	\| [[~~Pnict~~]]		\| [[Hemichromat]]
	\|-		\|-
	\| 3		\| 3
Line 96:		Line 102:
	\| 13/12		\| 13/12
	\| [[Avicenna (temperament)\|Avicenna]]		\| [[Avicenna (temperament)\|Avicenna]]
			\|-
			\| 3
			\| 48\357
			\| 161.34
			\| 192/175
			\| [[Pnict]]
	\|}		\|}

FloraC: Created page with "{{Infobox ET}} {{EDO intro|357}} == Theory == While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament temper..."

2023-07-10T06:51:34Z

Created page with "{{Infobox ET}} {{EDO intro|357}} == Theory == While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament temper..."

New page

{{Infobox ET}}
{{EDO intro|357}}

== Theory ==
While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament tempers out 1600000/1594323 ([[amity comma]]), and {{monzo| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit; notably [[support]]ing [[amity]], [[chromat]], and [[Avicenna (temperament)|avicenna]].

=== Prime harmonics ===
{{Harmonics in equal|357}}

=== Subsets and supersets ===
Since 357 factors into 3 × 5 × 7, 357edo has subset edos {{EDOs| 3, 7, 17, 21, 51, and 119 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 566 -357 }}
| [{{val| 357 566 }}]
| -0.1786
| 0.1785
| 5.31
|-
| 2.3.5
| 1600000/1594323, {{monzo| 61 4 -29 }}
| [{{val| 357 566 829 }}]
| -0.1536
| 0.1500
| 4.46
|-
| 2.3.5.7
| 10976/10935, 235298/234375, 2100875/2097152
| [{{val| 357 566 829 1002 }}]
| -0.0477
| 0.2248
| 6.69
|-
| 2.3.5.7.11
| 3025/3024, 5632/5625, 10976/10935, 102487/102400
| [{{val| 357 566 829 1002 1235 }}]
| -0.0348
| 0.2027
| 6.03
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
| [{{val| 357 566 829 1002 1235 1321 }}]
| -0.0204
| 0.1879
| 5.59
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods per 8ve
! Generator (Reduced)
! Cents (Reduced)
! Associated Ratio
! Temperaments
|-
| 1
| 101\357
| 339.50
| 243/200
| [[Amity]] (5-limit)
|-
| 1
| 118\357
| 396.64
| 44/35
| [[Squarschmidt]]
|-
| 3
| 48\357
| 161.34
| 192/175
| [[Pnict]]
|-
| 3
| 18\357
| 60.50
| 28/27
| [[Chromat]] (7-limit)
|-
| 3
| 41\357
| 137.82
| 13/12
| [[Avicenna (temperament)|Avicenna]]
|}