Hemimage family - Revision history

FloraC: + intro to hemimage

2026-06-29T05:43:56Z

+ intro to hemimage

← Older revision		Revision as of 05:43, 29 June 2026
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	The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the [[hemimage comma]] ({{monzo\|legend=1\| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).		The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the [[hemimage comma]] ({{monzo\|legend=1\| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).

	== Hemimage ==		== Hemimage ==
			Hemimage is generated by a [[3/2\|perfect fifth]] and a [[~]][[28/27]] subminor second. Two of the subminor seconds make ~[[15/14]], and three make ~[[10/9]]. The ubiquitous presence of this little interval gives rise to the highly intense [[hemimage bleeding chord]]s.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: + optimal tunings

2026-06-29T04:57:54Z

+ optimal tunings

@@ Line 9: / Line 9: @@
 {{Mapping|legend=1| 1 0 -1 -2 | 0 1 2 3 | 0 0 3 1 }}
 : mapping generators: ~2, ~3, ~28/27
 [[Minimax tuning]]:
@@ Line 30: / Line 36: @@
 {{Mapping|legend=1| 1 0 -1 -2 5 | 0 1 2 3 -1 | 0 0 3 1 1 }}
 [[Minimax tuning]]:
@@ Line 46: / Line 58: @@
 Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}
 {{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
@@ Line 57: / Line 73: @@
 Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}
 {{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
@@ Line 68: / Line 88: @@
 {{Mapping|legend=1| 1 0 -1 -2 -6 | 0 1 2 3 6 | 0 0 3 1 -1 }}
 [[Minimax tuning]]:

FloraC: Switch to Sintel's badness

2026-06-29T04:49:12Z

Switch to Sintel's badness

← Older revision		Revision as of 04:49, 29 June 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''hemimage family''' of [[rank-3 temperament~~\|rank-3~~]] [[~~regular temperament~~\|~~temperaments~~]] ~~tempers out~~ the [[hemimage comma]] ({{monzo\|legend=1\| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).		The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the [[hemimage comma]] ({{monzo\|legend=1\| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).

	== Hemimage ==		== Hemimage ==
Line 7:		Line 7:
	[[Comma list]]: 10976/10935		[[Comma list]]: 10976/10935

	{{Mapping\|legend=1\| 1 0 2 -1 \| 0 1 2 3 \| 0 0 -3 -1 }}		{{Mapping\|legend=1\| 1 0 -1 -2 \| 0 1 2 3 \| 0 0 3 1 }}
	: mapping generators: ~2, ~3, ~27~~/14~~		: mapping generators: ~2, ~3, ~28/27

	[[Minimax tuning]]:		[[Minimax tuning]]:
Line 16:		Line 16:
	* [[9-odd-limit]]		* [[9-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}
	: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7

	{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}

	[[Badness]] (~~Smith~~): 0.~~199 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 0.878

	[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7		[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7
Line 29:		Line 29:
	[[Comma list]]: 540/539, 896/891		[[Comma list]]: 540/539, 896/891

	{{Mapping\|legend=1\| 1 0 2 -1 6 \| 0 1 2 3 -1 \| 0 0 -3 -1 -1 }}		{{Mapping\|legend=1\| 1 0 -1 -2 5 \| 0 1 2 3 -1 \| 0 0 3 1 1 }}

	[[Minimax tuning]]:		[[Minimax tuning]]:
Line 38:		Line 38:
	{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}

	[[Badness]] (~~Smith~~): 0.~~702 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 0.844

	=== 13-limit ===		=== 13-limit ===
Line 45:		Line 45:
	Comma list: 352/351, 364/363, 540/539		Comma list: 352/351, 364/363, 540/539

	Mapping: {{mapping\| 1 0 2 -1 ~~6 11~~ \| 0 1 2 3 -1 -4 \| 0 0 -3 -1 -1 -1 }}		Mapping: {{mapping\| 1 0 -1 -2 5 10 \| 0 1 2 3 -1 -4 \| 0 0 3 1 1 1 }}

	{{Optimal ET sequence\|legend=0\| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}		{{Optimal ET sequence\|legend=0\| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}

	Badness (~~Smith~~): 1.~~06 × 10<sup>-3</sup>~~		Badness (Sintel): 0.993

	=== Tin ===		=== Tin ===
Line 56:		Line 56:
	Comma list: 144/143, 196/195, 729/728		Comma list: 144/143, 196/195, 729/728

	Mapping: {{mapping\| 1 0 2 -1 6 -2 \| 0 1 2 3 -1 3 \| 0 0 -3 -1 -1 1 }}		Mapping: {{mapping\| 1 0 -1 -2 5 -1 \| 0 1 2 3 -1 3 \| 0 0 3 1 1 -1 }}

	{{Optimal ET sequence\|legend=0\| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}		{{Optimal ET sequence\|legend=0\| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}

	Badness (~~Smith~~): 1.~~07 × 10<sup>-3</sup>~~		Badness (Sintel): 1.00

	== Turan ==		== Turan ==
Line 67:		Line 67:
	[[Comma list]]: 441/440, 10976/10935		[[Comma list]]: 441/440, 10976/10935

	{{Mapping\|legend=1\| 1 0 2 -1 -7 \| 0 1 2 3 6 \| 0 0 -3 -1 1 }}		{{Mapping\|legend=1\| 1 0 -1 -2 -6 \| 0 1 2 3 6 \| 0 0 3 1 -1 }}

	[[Minimax tuning]]:		[[Minimax tuning]]:
Line 74:		Line 74:
	: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 36, ~~39d~~, 41, 58, 99e, 118, 159, 217, 335 }}		{{Optimal ET sequence\|legend=1\| 17c, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335 }}

	[[Badness]] (~~Smith~~): 0.~~936 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 1.13

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Hemimage family\| ]] <!-- main article -->		[[Category:Hemimage family\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: Units, misc. cleanup, - redundant category

2026-04-28T18:49:33Z

Units, misc. cleanup, - redundant category

← Older revision		Revision as of 18:49, 28 April 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''hemimage family''' of ~~temperaments are~~ [[rank-3 temperament]]~~s tempering~~ out [[10976/10935]].		The '''hemimage family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] tempers out the [[hemimage comma]] ({{monzo\|legend=1\| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).

	== Hemimage ==		== Hemimage ==
Line 8:		Line 8:

	{{Mapping\|legend=1\| 1 0 2 -1 \| 0 1 2 3 \| 0 0 -3 -1 }}		{{Mapping\|legend=1\| 1 0 2 -1 \| 0 1 2 3 \| 0 0 -3 -1 }}

	: mapping generators: ~2, ~3, ~27/14		: mapping generators: ~2, ~3, ~27/14

Line 14:		Line 13:
	* [[7-odd-limit]]		* [[7-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}
	: [[~~Eigenmonzo~~ basis\|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
	* [[9-odd-limit]]		* [[9-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}
	: [[~~Eigenmonzo~~ basis\|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5

	{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}

	[[Badness]]: 0.199 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>

	[[Projection pair]]: 5 10976/2187 to 2.3.7		[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7

	== Uni ==		== Uni ==
Line 35:		Line 34:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]
	: [[~~Eigenmonzo~~ basis\|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}

	[[Badness]]: 0.702 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>

	=== 13-limit ===		=== 13-limit ===
Line 48:		Line 47:
	Mapping: {{mapping\| 1 0 2 -1 6 11 \| 0 1 2 3 -1 -4 \| 0 0 -3 -1 -1 -1 }}		Mapping: {{mapping\| 1 0 2 -1 6 11 \| 0 1 2 3 -1 -4 \| 0 0 -3 -1 -1 -1 }}

	{{Optimal ET sequence\|legend=1\| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}		{{Optimal ET sequence\|legend=0\| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}

	Badness: 1.06 × 10<sup>-3</sup>		Badness (Smith): 1.06 × 10<sup>-3</sup>

	=== Tin ===		=== Tin ===
Line 59:		Line 58:
	Mapping: {{mapping\| 1 0 2 -1 6 -2 \| 0 1 2 3 -1 3 \| 0 0 -3 -1 -1 1 }}		Mapping: {{mapping\| 1 0 2 -1 6 -2 \| 0 1 2 3 -1 3 \| 0 0 -3 -1 -1 1 }}

	{{Optimal ET sequence\|legend=1\| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}		{{Optimal ET sequence\|legend=0\| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}

	Badness: 1.07 × 10<sup>-3</sup>		Badness (Smith): 1.07 × 10<sup>-3</sup>

	== Turan ==		== Turan ==
Line 73:		Line 72:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]
	: [[~~Eigenmonzo~~ basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}

	[[Badness]]: 0.936 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.936 × 10<sup>-3</sup>

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Hemimage family\| ]] <!-- main article -->		[[Category:Hemimage family\| ]] <!-- main article -->
	~~[[Category:Hemimage\| ]] <!-- key article -->~~
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:31:51Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:31, 27 November 2025
Line 80:		Line 80:

	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Hemimage family\| ]] <!-- main article -->		[[Category:Hemimage family\| ]] <!-- main article -->
	[[Category:Hemimage\| ]] <!-- key article -->		[[Category:Hemimage\| ]] <!-- key article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:32:56Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:32, 24 June 2025
Line 80:		Line 80:

	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Hemimage family\| ]] <!-- main article -->		[[Category:Hemimage family\| ]] <!-- main article -->
	[[Category:Hemimage\| ]] <!-- key article -->		[[Category:Hemimage\| ]] <!-- key article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

Lériendil: Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "

2025-06-21T06:58:11Z

Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "

← Older revision		Revision as of 06:58, 21 June 2025
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	* [[7-odd-limit]]		* [[7-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5/3.7		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
	* [[9-odd-limit]]		* [[9-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7.9/5		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5

	{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
Line 35:		Line 35:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.11/9		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
Line 73:		Line 73:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5.11/9		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}

Lériendil: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:50:48Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:50, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''hemimage family''' of temperaments are [[rank-3 temperament]]s tempering out [[10976/10935]].		The '''hemimage family''' of temperaments are [[rank-3 temperament]]s tempering out [[10976/10935]].

BudjarnLambeth: {{Technical data page}}

2025-04-13T11:40:27Z

{{Technical data page}}<br><br>

← Older revision		Revision as of 11:40, 13 April 2025
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	The '''hemimage family''' of temperaments are rank-3 ~~temperaments~~ tempering out [[10976/10935]].		{{Technical data page}}<br><br>
			The '''hemimage family''' of temperaments are [[rank-3 temperament]]s tempering out [[10976/10935]].

	== Hemimage ==		== Hemimage ==

FloraC: Update keys

2023-09-11T07:34:26Z

Update keys

← Older revision		Revision as of 07:34, 11 September 2023
Line 2:		Line 2:

	== Hemimage ==		== Hemimage ==
	Subgroup: 2.3.5.7		[[Subgroup]]: 2.3.5.7

	[[Comma list]]: 10976/10935		[[Comma list]]: 10976/10935

	~~[[Mapping]]: [~~{{~~val~~\| 1 0 2 -1 ~~}}, {{val~~\| 0 1 2 3 ~~}}, {{val~~\| 0 0 -3 -1 }}]		{{Mapping\|legend=1\| 1 0 2 -1 \| 0 1 2 3 \| 0 0 -3 -1 }}

	~~Mapping~~ generators: ~2, ~3, ~27/14		: mapping generators: ~2, ~3, ~27/14

	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit]]		* [[7-odd-limit]]
	: [{{monzo\| 1 0 0 0 ~~}}, {{monzo~~\| 5/8 1/8 -1/8 3/8 ~~}}, {{monzo~~\| 5/8 -7/8 7/8 3/8 ~~}}, {{monzo~~\| 0 0 0 1 }}]		: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}
	: [[Eigenmonzo~~]]s~~ (unchanged-~~intervals~~): 2~~, 8~~/7~~, 6/5~~		: [[Eigenmonzo basis\|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7
	* [[9-odd-limit]]		* [[9-odd-limit]]
	: [{{monzo\| 1 0 0 0 ~~}}, {{monzo~~\| 5/9 2/9 -1/9 1/3 ~~}}, {{monzo~~\| 10/9 -14/9 7/9 2/3 ~~}}, {{monzo~~\| 0 0 0 1 }}]		: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}
	: [[Eigenmonzo~~]]s~~ (unchanged-~~intervals~~): 2~~, 10/~~9~~, 8~~/7		: [[Eigenmonzo basis\|eigenmonzo (unchanged-interval) basis]]: 2.7.9/5

	{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
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	[[Projection pair]]: 5 10976/2187 to 2.3.7		[[Projection pair]]: 5 10976/2187 to 2.3.7

	== Uni ==		== Uni ==
	Subgroup: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

	[[Comma list]]: 540/539, 896/891		[[Comma list]]: 540/539, 896/891

	~~[[Mapping]]: [~~{{~~val~~\| 1 0 2 -1 6 ~~}}, {{val~~\| 0 1 2 3 -1 ~~}}, {{val~~\| 0 0 -3 -1 -1 }}]		{{Mapping\|legend=1\| 1 0 2 -1 6 \| 0 1 2 3 -1 \| 0 0 -3 -1 -1 }}

	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]
	: [[Eigenmonzo~~]]s~~ (unchanged-~~intervals~~): 2, 11/9~~, 5/4~~		: [[Eigenmonzo basis\|eigenmonzo (unchanged-interval) basis]]: 2.5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
Line 45:		Line 45:
	Comma list: 352/351, 364/363, 540/539		Comma list: 352/351, 364/363, 540/539

	Mapping: [{{~~val~~\| 1 0 2 -1 6 11 ~~}}, {{val~~\| 0 1 2 3 -1 -4 ~~}}, {{val~~\| 0 0 -3 -1 -1 -1 }}]		Mapping: {{mapping\| 1 0 2 -1 6 11 \| 0 1 2 3 -1 -4 \| 0 0 -3 -1 -1 -1 }}

	{{Optimal ET sequence\|legend=1\| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}		{{Optimal ET sequence\|legend=1\| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
Line 51:		Line 51:
	Badness: 1.06 × 10<sup>-3</sup>		Badness: 1.06 × 10<sup>-3</sup>

	=== Tin ===		=== Tin ===
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: 144/143, 196/195, 729/728		Comma list: 144/143, 196/195, 729/728

	Mapping: [{{~~val~~\| 1 0 2 -1 6 -2 ~~}}, {{val~~\| 0 1 2 3 -1 3 ~~}}, {{val~~\| 0 0 -3 -1 -1 1 }}]		Mapping: {{mapping\| 1 0 2 -1 6 -2 \| 0 1 2 3 -1 3 \| 0 0 -3 -1 -1 1 }}

	{{Optimal ET sequence\|legend=1\| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
Line 62:		Line 62:
	Badness: 1.07 × 10<sup>-3</sup>		Badness: 1.07 × 10<sup>-3</sup>

	== Turan ==		== Turan ==
	Subgroup: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

	[[Comma list]]: 441/440, 10976/10935		[[Comma list]]: 441/440, 10976/10935

	~~[[Mapping]]: [~~{{~~val~~\| 1 0 2 -1 -7 ~~}}, {{val~~\| 0 1 2 3 6 ~~}}, {{val~~\| 0 0 -3 -1 1 }}]		{{Mapping\|legend=1\| 1 0 2 -1 -7 \| 0 1 2 3 6 \| 0 0 -3 -1 1 }}

	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]
	: [[Eigenmonzo~~]]s~~ (unchanged-~~intervals~~): 2~~, 10~~/9, 11/9		: [[Eigenmonzo basis\|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}
Line 79:		Line 79:

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Hemimage]]		[[Category:Hemimage family\| ]] <!-- main article -->
			[[Category:Hemimage\| ]] <!-- key article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

← Older revision		Revision as of 06:58, 21 June 2025
Line 14:		Line 14:
	* [[7-odd-limit]]		* [[7-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/8 1/8 -1/8 3/8 \| 5/8 -7/8 7/8 3/8 \| 0 0 0 1 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5/3.7		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
	* [[9-odd-limit]]		* [[9-odd-limit]]
	: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}		: {{monzo list\| 1 0 0 0 \| 5/9 2/9 -1/9 1/3 \| 10/9 -14/9 7/9 2/3 \| 0 0 0 1 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7.9/5		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5

	{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
Line 35:		Line 35:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 16/11 6/11 1/11 0 -3/11 }}, {{monzo\| 0 0 1 0 0 }}, {{monzo\| 19/11 14/11 6/11 0 -7/11 }}, {{monzo\| 32/11 -10/11 2/11 0 5/11 }}]
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.11/9		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
Line 73:		Line 73:
	* [[11-odd-limit]]		* [[11-odd-limit]]
	: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]		: [{{monzo\| 1 0 0 0 0 }}, {{monzo\| 19/12 -2/3 1/12 0 1/4 }}, {{monzo\| 19/6 -10/3 7/6 0 1/2 }}, {{monzo\| 37/12 -8/3 7/12 0 3/4 }}, {{monzo\| 19/6 -10/3 1/6 0 3/2 }}]
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5.11/9		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

	{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}		{{Optimal ET sequence\|legend=1\| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}