Keemic family - Revision history

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

2025-11-27T12:27:35Z

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

← Older revision		Revision as of 12:27, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Keemic family\| ]] <!-- main article -->		[[Category:Keemic family\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: Supermagic -> keemic, per discussion on Discord

2025-10-12T10:35:13Z

Supermagic -> keemic, per discussion on Discord

← Older revision		Revision as of 10:35, 12 October 2025
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	The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2 temperament]]s, see [[keemic temperaments]].		The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2 temperament]]s, see [[keemic temperaments]].

	== ~~Supermagic~~ ==		== Keemic ==
	~~Supermagic~~ has the same lattice structure as [[5-limit]] JI, and [[7/4]] is found by a stack of three [[~]][[6/5]]'s.		Keemic, also known as ''supermagic'' in earlier materials, has the same lattice structure as [[5-limit]] JI, and [[7/4]] is found by a stack of three [[~]][[6/5]]'s.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	Scales: [[supermagic15]]		Scales: [[supermagic15]]

	== Undecimal ~~supermagic~~ ==		== Undecimal keemic ==
	{{See also\| Ptolemismic clan #~~Supermagic~~ }}		{{See also\| Ptolemismic clan #Keemic }}

	~~Supermagic~~ is naturally an [[11-limit]] temperament due to the identity 875/864 = ([[100/99]])⋅([[385/384]]). This identifies a stack of two ~6/5's as ~[[16/11]].		Keemic is naturally an [[11-limit]] temperament due to the identity 875/864 = ([[100/99]])⋅([[385/384]]). This identifies a stack of two ~6/5's as ~[[16/11]].

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	[[Category:Pages with mostly numerical content]]		[[Category:Pages with mostly numerical content]]
	[[Category:Keemic family\| ]] <!-- main article -->		[[Category:Keemic family\| ]] <!-- main article -->
	~~[[Category:Supermagic]]~~
	[[Category:Rank 3]]		[[Category:Rank 3]]

Lériendil at 13:57, 10 October 2025

2025-10-10T13:57:19Z

← Older revision		Revision as of 13:57, 10 October 2025
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	{{Technical data page}}		{{Technical data page}}
	The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2]] ~~temperaments~~, see [[keemic temperaments]].		The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2 temperament]]s, see [[keemic temperaments]].

	== Supermagic ==		== Supermagic ==

Lériendil at 13:56, 10 October 2025

2025-10-10T13:56:35Z

← Older revision		Revision as of 13:56, 10 October 2025
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	{{Technical data page}}		{{Technical data page}}
	The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864).		The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2]] temperaments, see [[keemic temperaments]].

	== Supermagic ==		== Supermagic ==

Lériendil: linkage

2025-10-10T13:55:05Z

linkage

← Older revision		Revision as of 13:55, 10 October 2025
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	== Supermagic ==		== Supermagic ==
	Supermagic has the same lattice structure as 5-limit JI, and 7/4 is found by a stack of three ~6/5's.		Supermagic has the same lattice structure as [[5-limit]] JI, and [[7/4]] is found by a stack of three [[~]][[6/5]]'s.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	{{See also\| Ptolemismic clan #Supermagic }}		{{See also\| Ptolemismic clan #Supermagic }}

	Supermagic is naturally an 11-limit temperament due to the identity 875/864 = (100/99)⋅(385/384). This identifies a stack of two ~6/5's as ~16/11.		Supermagic is naturally an [[11-limit]] temperament due to the identity 875/864 = ([[100/99]])⋅([[385/384]]). This identifies a stack of two ~6/5's as ~[[16/11]].

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	== Supernatural ==		== Supernatural ==
	Supernatural uses the same mapping as 7-limit [[magic]] with an independent generator for prime 11.		Supernatural uses the same mapping as [[7-limit]] [[magic]] with an independent generator for prime 11.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

Lériendil: /* Supernatural */

2025-10-10T13:53:26Z

Supernatural

← Older revision		Revision as of 13:53, 10 October 2025
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	== Supernatural ==		== Supernatural ==
			Supernatural uses the same mapping as 7-limit [[magic]] with an independent generator for prime 11.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

FloraC: + intro

2025-10-10T13:12:46Z

+ intro

← Older revision		Revision as of 13:12, 10 October 2025
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	== Supermagic ==		== Supermagic ==
			Supermagic has the same lattice structure as 5-limit JI, and 7/4 is found by a stack of three ~6/5's.

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

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	== Undecimal supermagic ==		== Undecimal supermagic ==
	{{See also\| Ptolemismic clan #Supermagic }}		{{See also\| Ptolemismic clan #Supermagic }}

			Supermagic is naturally an 11-limit temperament due to the identity 875/864 = (100/99)⋅(385/384). This identifies a stack of two ~6/5's as ~16/11.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}

	{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}		{{Optimal ET sequence\|legend=0\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}

	Badness (Sintel): 1.07		Badness (Sintel): 1.07

FloraC: - POTE tunings

2025-10-10T13:03:42Z

- POTE tunings

← Older revision		Revision as of 13:03, 10 October 2025
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}
	: error map: {{val\| 0.000 +0.851 -5.260 -3.569 }}		: error map: {{val\| 0.000 +0.851 -5.260 -3.569 }}
	~~<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8733{{c}}, ~5/4 = 380.4684{{c}} -->~~

	[[Minimax tuning]]:		[[Minimax tuning]]:
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1922{{c}}, ~5/4 = 381.5339{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1922{{c}}, ~5/4 = 381.5339{{c}}
	: error map: {{val\| 0.000 +2.237 -4.780 -0.851 +3.365 }}		: error map: {{val\| 0.000 +2.237 -4.780 -0.851 +3.365 }}
	~~<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1985{{c}}, ~5/4 = 381.5394{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 7, 15, 19, 22, 37, 41, 104 }}		{{Optimal ET sequence\|legend=1\| 7, 15, 19, 22, 37, 41, 104 }}
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}, ~11/8 = 548.8962{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}, ~11/8 = 548.8962{{c}}
	: error map: {{val\| 0.000 +0.333 -5.856 -3.335 -2.422 }}		: error map: {{val\| 0.000 +0.333 -5.856 -3.335 -2.422 }}
	~~<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.3520{{c}}, ~11/8 = 547.5758{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde }} *		{{Optimal ET sequence\|legend=1\| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde }} *
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	* WE: ~2 = 1201.3389{{c}}, ~5/4 = 380.4641{{c}}, ~11/8 = 547.2772{{c}}		* WE: ~2 = 1201.3389{{c}}, ~5/4 = 380.4641{{c}}, ~11/8 = 547.2772{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}
	~~<!-- * POTE: ~2 = 1200.0000{{c}}, ~5/4 = 380.0401{{c}}, ~11/8 = 546.6672{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}		{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}

FloraC: Switch to Sintel's badness, WE & CWE tunings, per community consensus

2025-10-10T13:03:15Z

Switch to Sintel's badness, WE & CWE tunings, per community consensus

← Older revision		Revision as of 13:03, 10 October 2025
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	: Angle (6/5, 3/2) = 77.834		: Angle (6/5, 3/2) = 77.834

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.8733{{c}}, ~5/4 = 380.4684{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1201.0490{{c}}, ~3/2 = 702.4868{{c}}, ~5/4 = 380.8010{{c}}
			: [[error map]]: {{val\| +1.049 +1.581 -3.415 -1.670 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}
			: error map: {{val\| 0.000 +0.851 -5.260 -3.569 }}
			<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8733{{c}}, ~5/4 = 380.4684{{c}} -->

	[[Minimax tuning]]:		[[Minimax tuning]]:
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	{{Optimal ET sequence\|legend=1\| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}		{{Optimal ET sequence\|legend=1\| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}

	[[Badness]] (~~Smith~~): 0.~~212 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 0.937

	[[Projection pair]]: 7 864/125		[[Projection pair]]: 7 864/125
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	{{Mapping\|legend=1\| 1 0 0 5 2 \| 0 1 0 3 -2 \| 0 0 1 -3 2 }}		{{Mapping\|legend=1\| 1 0 0 5 2 \| 0 1 0 3 -2 \| 0 0 1 -3 2 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1985{{c}}, ~5/4 = 381.5394{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9868{{c}}, ~3/2 = 704.1908{{c}}, ~5/4 = 381.5352{{c}}
			: [[error map]]: {{val\| -0.013 +2.223 -4.805 -0.885 +3.318 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1922{{c}}, ~5/4 = 381.5339{{c}}
			: error map: {{val\| 0.000 +2.237 -4.780 -0.851 +3.365 }}
			<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1985{{c}}, ~5/4 = 381.5394{{c}} -->

	{{Optimal ET sequence\|legend=1\| 7, 15, 19, 22, 37, 41, 104 }}		{{Optimal ET sequence\|legend=1\| 7, 15, 19, 22, 37, 41, 104 }}

	[[Badness]] (~~Smith~~): 0.~~641 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 0.770

	Scales: [[supermagic15]]		Scales: [[supermagic15]]
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	: mapping generators: ~2, ~5, ~11		: mapping generators: ~2, ~5, ~11

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 380.3520{{c}}, ~11/8 = 547.5758{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}, ~11/8 = 548.0690{{c}}
			: [[error map]]: {{val\| +1.079 +1.514 -3.463 -1.578 -0.013 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}, ~11/8 = 548.8962{{c}}
			: error map: {{val\| 0.000 +0.333 -5.856 -3.335 -2.422 }}
			<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.3520{{c}}, ~11/8 = 547.5758{{c}} -->

	{{Optimal ET sequence\|legend=1\| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde * }}		{{Optimal ET sequence\|legend=1\| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde }} *

	<nowiki>*</nowiki> [[optimal patent val]]: [[104edo\|104]]		<nowiki>*</nowiki> [[optimal patent val]]: [[104edo\|104]]

	[[Badness]] (~~Smith~~): 0.~~888 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 1.07

	=== 13-limit ===		=== 13-limit ===
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	Mapping: {{mapping\| 1 0 2 -1 0 -2 \| 0 5 1 12 0 18 \| 0 0 0 0 1 0 }}		Mapping: {{mapping\| 1 0 2 -1 0 -2 \| 0 5 1 12 0 18 \| 0 0 0 0 1 0 }}

	Optimal ~~tuning (~~POTE): ~2 = 1200.0000{{c}}, ~5/4 = 380.0401{{c}}, ~11/8 = 546.6672{{c}}		Optimal tunings:
			* WE: ~2 = 1201.3389{{c}}, ~5/4 = 380.4641{{c}}, ~11/8 = 547.2772{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}
			<!-- * POTE: ~2 = 1200.0000{{c}}, ~5/4 = 380.0401{{c}}, ~11/8 = 546.6672{{c}} -->

	{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}		{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}

	Badness (~~Smith~~): 1.~~14 × 10<sup>-3</sup>~~		Badness (Sintel): 1.07

	[[Category:Temperament families]]		[[Category:Temperament families]]

FloraC: Units

2025-10-10T12:51:50Z

Units

← Older revision		Revision as of 12:51, 10 October 2025
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	{{Technical data page}}		{{Technical data page}}
	The '''keemic family''' is a family of [[rank-3 temperament]]s which [[temper out]] the [[keema]]~~, 875/864 =~~ {{monzo\| -5 -3 3 1 }}.		The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out\|temper out]] the [[keema]] ({{monzo\|legend=1\| -5 -3 3 1 }}, [[ratio]]: 875/864).

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

	: mapping generators: ~2, ~3, ~5		: mapping generators: ~2, ~3, ~5

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	: Angle (6/5, 3/2) = 77.834		: Angle (6/5, 3/2) = 77.834

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 701.8733, ~5/4 = 380.4684		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.8733{{c}}, ~5/4 = 380.4684{{c}}

	[[Minimax tuning]]:		[[Minimax tuning]]:
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	{{Optimal ET sequence\|legend=1\| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}		{{Optimal ET sequence\|legend=1\| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}

	[[Badness]]: 0.212 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.212 × 10<sup>-3</sup>

	[[Projection pair]]: 7 864/125		[[Projection pair]]: 7 864/125
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	{{Mapping\|legend=1\| 1 0 0 5 2 \| 0 1 0 3 -2 \| 0 0 1 -3 2 }}		{{Mapping\|legend=1\| 1 0 0 5 2 \| 0 1 0 3 -2 \| 0 0 1 -3 2 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 704.1985, ~5/4 = 381.5394		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1985{{c}}, ~5/4 = 381.5394{{c}}

	{{Optimal ET sequence\|legend=1\| 7, 15, 19, 22, 37, 41, 104 }}		{{Optimal ET sequence\|legend=1\| 7, 15, 19, 22, 37, 41, 104 }}

	[[Badness]]: 0.641 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.641 × 10<sup>-3</sup>

	Scales: [[supermagic15]]		Scales: [[supermagic15]]
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	{{Mapping\|legend=1\| 1 0 2 -1 0 \| 0 5 1 12 0 \| 0 0 0 0 1 }}		{{Mapping\|legend=1\| 1 0 2 -1 0 \| 0 5 1 12 0 \| 0 0 0 0 1 }}

	: mapping generators: ~2, ~5, ~11		: mapping generators: ~2, ~5, ~11

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~5/4 = 380.3520, ~11/8 = 547.5758		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 380.3520{{c}}, ~11/8 = 547.5758{{c}}

	{{Optimal ET sequence\|legend=1\| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde * }}		{{Optimal ET sequence\|legend=1\| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde * }}
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	<nowiki>*</nowiki> [[optimal patent val]]: [[104edo\|104]]		<nowiki>*</nowiki> [[optimal patent val]]: [[104edo\|104]]

	[[Badness]]: 0.888 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.888 × 10<sup>-3</sup>

	=== 13-limit ===		=== 13-limit ===
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	Mapping: {{mapping\| 1 0 2 -1 0 -2 \| 0 5 1 12 0 18 \| 0 0 0 0 1 0 }}		Mapping: {{mapping\| 1 0 2 -1 0 -2 \| 0 5 1 12 0 18 \| 0 0 0 0 1 0 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~5/4 = 380.0401, ~11/8 = 546.6672		Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~5/4 = 380.0401{{c}}, ~11/8 = 546.6672{{c}}

	{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}		{{Optimal ET sequence\|legend=1\| 19, 22f, 38df, 41, 60e, 79d, 101cd }}

	Badness: 1.14 × 10<sup>-3</sup>		Badness (Smith): 1.14 × 10<sup>-3</sup>

	[[Category:Temperament families]]		[[Category:Temperament families]]