Sengic family - Revision history

FloraC: + overview to extensions

2026-04-11T10:31:34Z

+ overview to extensions

← Older revision		Revision as of 10:31, 11 April 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''sengic family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the senga a.k.a. sengic comma, [[686/675]].		The '''sengic family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the senga a.k.a. sengic comma, [[686/675]].

	~~Temperaments considered below are demeter, krypton, and sensigh.~~

	== Sengic ==		== Sengic ==
	Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 ({{S\|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[169/168]])⋅(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.		Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It was discovered and named in 2005.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5		[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5

			=== Overview to extensions ===
			First noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>, sengic is naturally a [[2.3.5.7.13 subgroup\|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 = ([[169/168]])⋅([[196/195]])<sup>2</sup>, as we can see from its [[S-expression]], S13⋅S14<sup>2</sup>. This identifies the last generator as [[13/12]]~[[14/13]]~15/14. This extension is considered immediately below.

			11-limit temperaments considered below are demeter, krypton, and sensigh.

	=== 2.3.5.7.13 subgroup ===		=== 2.3.5.7.13 subgroup ===

FloraC: Move sensigh here

2026-04-11T10:05:36Z

Move sensigh here

@@ Line 2: / Line 2: @@
 The '''sengic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the senga a.k.a. sengic comma, [[686/675]].
-Temperament discussed elsewhere include [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered below are demeter and krypton.
+Temperaments considered below are demeter, krypton, and sensigh.
 == Sengic ==
-Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 ({{S|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[169/168]])⋅(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group | ''It's the "thirds", stupid!'']</ref>.
+Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 ({{S|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[169/168]])⋅(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group | ''It's the "thirds", stupid!'']</ref>.
 [[Subgroup]]: 2.3.5.7
@@ Line 135: / Line 135: @@
 Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11
 == References ==

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

2025-11-27T12:27:10Z

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:Sengic family\| ]] <!-- main article -->		[[Category:Sengic family\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: - redundant category

2025-11-26T17:02:57Z

- redundant category

← Older revision		Revision as of 17:02, 26 November 2025
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	[[Category:Pages with mostly numerical content]]		[[Category:Pages with mostly numerical content]]
	[[Category:Sengic family\| ]] <!-- main article -->		[[Category:Sengic family\| ]] <!-- main article -->
	~~[[Category:Sengic\| ]] <!-- key article -->~~
	[[Category:Rank 3]]		[[Category:Rank 3]]

Lériendil: /* Sengic */

2025-10-10T14:16:32Z

Sengic

← Older revision		Revision as of 14:16, 10 October 2025
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	== Sengic ==		== Sengic ==
	Sengic is generated by a perfect fifth and a wide semitone of ~15/14, two of which make ~7/6 and three make ~5/4. It is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (91/90)⋅(196/195) and 91/90 = (169/168)⋅(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.		Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 ({{S\|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[169/168]])⋅(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.

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

FloraC: /* Sengic */ intro to the 7-limit temp

2025-10-10T12:38:47Z

Sengic: intro to the 7-limit temp

← Older revision		Revision as of 12:38, 10 October 2025
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	== Sengic ==		== Sengic ==
	Sengic is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.		Sengic is generated by a perfect fifth and a wide semitone of ~15/14, two of which make ~7/6 and three make ~5/4. It is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (91/90)⋅(196/195) and 91/90 = (169/168)⋅(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.

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

FloraC: - CTE & POTE tunings

2025-10-10T12:32:34Z

- CTE & POTE tunings

← Older revision		Revision as of 12:32, 10 October 2025
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9671{{c}}, ~15/14 = 129.7330{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9671{{c}}, ~15/14 = 129.7330{{c}}
	: error map: {{val\| 0.000 +2.012 +2.885 -5.393 }}		: error map: {{val\| 0.000 +2.012 +2.885 -5.393 }}
	~~<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7873{{c}}, ~15/14 = 129.6451{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46 }}		{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46 }}
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	* WE: ~2 = 1200.3448{{c}}, ~3/2 = 704.1998{{c}}, ~14/13 = 129.5253{{c}}		* WE: ~2 = 1200.3448{{c}}, ~3/2 = 704.1998{{c}}, ~14/13 = 129.5253{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2688{{c}}, ~14/13 = 129.6115{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2688{{c}}, ~14/13 = 129.6115{{c}}
	~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}		{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2789{{c}}, ~15/14 = 129.8330{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2789{{c}}, ~15/14 = 129.8330{{c}}
	: error map: {{val\| 0.000 +3.324 +3.185 -3.881 -0.369 }}		: error map: {{val\| 0.000 +3.324 +3.185 -3.881 -0.369 }}
	~~<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.518{{c}}, ~15/14 = 130.039{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 46, 102, 148 }}		{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 46, 102, 148 }}
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	* WE: ~2 = 1200.0393{{c}}, ~3/2 = 705.1360{{c}}, ~14/13 = 129.6770{{c}}		* WE: ~2 = 1200.0393{{c}}, ~3/2 = 705.1360{{c}}, ~14/13 = 129.6770{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1277{{c}}, ~14/13 = 129.6854{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1277{{c}}, ~14/13 = 129.6854{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}		{{Optimal ET sequence\|legend=0\| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}
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	* WE: ~2 = 1200.0255{{c}}, ~3/2 = 705.1616{{c}}, ~14/13 = 129.7024{{c}}		* WE: ~2 = 1200.0255{{c}}, ~3/2 = 705.1616{{c}}, ~14/13 = 129.7024{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1553{{c}}, ~14/13 = 129.7071{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1553{{c}}, ~14/13 = 129.7071{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}		{{Optimal ET sequence\|legend=0\| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6964{{c}}, ~15/14 = 131.8166{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6964{{c}}, ~15/14 = 131.8166{{c}}
	: error map: {{val\| 0.000 +1.741 +9.136 -1.496 +20.562 }}		: error map: {{val\| 0.000 +1.741 +9.136 -1.496 +20.562 }}
	~~<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.978{{c}}, ~12/11 = 132.544{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 63cee }} *		{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 63cee }} *
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	* WE: ~2 = 1197.6484{{c}}, ~3/2 = 704.6450{{c}}, ~14/13 = 132.1686{{c}}		* WE: ~2 = 1197.6484{{c}}, ~3/2 = 704.6450{{c}}, ~14/13 = 132.1686{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1616{{c}}, ~14/13 = 131.8445{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1616{{c}}, ~14/13 = 131.8445{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27e }} *		{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27e }} *

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

2025-10-10T12:32:01Z

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

← Older revision		Revision as of 12:32, 10 October 2025
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	: mapping generators: ~2, ~3, ~15/14		: mapping generators: ~2, ~3, ~15/14

	[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.7873{{c}}, ~15/14 = 129.6451{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.7533{{c}}, ~3/2 = 704.0092{{c}}, ~15/14 = 129.7976{{c}}
			: [[error map]]: {{val\| -0.247 +1.808 +2.586 -5.715 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9671{{c}}, ~15/14 = 129.7330{{c}}
			: error map: {{val\| 0.000 +2.012 +2.885 -5.393 }}
			<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7873{{c}}, ~15/14 = 129.6451{{c}} -->

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46 }}		{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46 }}

	[[Badness]] (~~Smith~~): 0.~~320 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 1.41

	[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5		[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
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	Comma list: 91/90, 169/168		Comma list: 91/90, 169/168

	~~Sval~~ mapping: {{mapping\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 1 }}		Subgroup-val mapping: {{mapping\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 1 }}

	Optimal ~~tuning (~~CTE): ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}}		Optimal tunings:
			* WE: ~2 = 1200.3448{{c}}, ~3/2 = 704.1998{{c}}, ~14/13 = 129.5253{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2688{{c}}, ~14/13 = 129.6115{{c}}
			<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}} -->

	{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}		{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}

	Badness (~~Smith~~): 0.~~320 × 10<sup>-3</sup>~~		Badness (Sintel): 0.425

	== Demeter ==		== Demeter ==
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	{{Mapping\|legend=1\| 1 0 2 1 -3 \| 0 1 0 1 4 \| 0 0 3 2 1 }}		{{Mapping\|legend=1\| 1 0 2 1 -3 \| 0 1 0 1 4 \| 0 0 3 2 1 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.518{{c}}, ~15/14 = 130.039{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.4002{{c}}, ~3/2 = 705.1652{{c}}, ~15/14 = 129.9738{{c}}
			: [[error map]]: {{val\| -0.600 +2.610 +2.408 -4.913 -1.283 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2789{{c}}, ~15/14 = 129.8330{{c}}
			: error map: {{val\| 0.000 +3.324 +3.185 -3.881 -0.369 }}
			<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.518{{c}}, ~15/14 = 130.039{{c}} -->

	{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 46, 102, 148 }}		{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 46, 102, 148 }}

	[[Badness]] (~~Smith~~): 1.~~32 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 1.58

	[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11		[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11
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	Mapping: {{mapping\| 1 0 2 1 -3 2 \| 0 1 0 1 4 1 \| 0 0 3 2 1 1 }}		Mapping: {{mapping\| 1 0 2 1 -3 2 \| 0 1 0 1 4 1 \| 0 0 3 2 1 1 }}

	Optimal ~~tuning (~~POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}}		Optimal tunings:
			* WE: ~2 = 1200.0393{{c}}, ~3/2 = 705.1360{{c}}, ~14/13 = 129.6770{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1277{{c}}, ~14/13 = 129.6854{{c}}
			<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}} -->

	{{Optimal ET sequence\|legend=0\| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}		{{Optimal ET sequence\|legend=0\| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}

	Badness (~~Smith~~): 0.~~977 × 10<sup>-3</sup>~~		Badness (Sintel): 0.913

	Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11		Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
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	Mapping: {{mapping\| 1 0 2 1 -3 2 -1 \| 0 1 0 1 4 1 3 \| 0 0 3 2 1 1 3 }}		Mapping: {{mapping\| 1 0 2 1 -3 2 -1 \| 0 1 0 1 4 1 3 \| 0 0 3 2 1 1 3 }}

	Optimal ~~tuning (~~POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}}		Optimal tunings:
			* WE: ~2 = 1200.0255{{c}}, ~3/2 = 705.1616{{c}}, ~14/13 = 129.7024{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1553{{c}}, ~14/13 = 129.7071{{c}}
			<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}} -->

	{{Optimal ET sequence\|legend=0\| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}		{{Optimal ET sequence\|legend=0\| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}

	Badness (~~Smith~~): 0.~~830 × 10<sup>-3</sup>~~		Badness (Sintel): 0.789

	Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11		Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11
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	{{Mapping\|legend=1\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 -1 }}		{{Mapping\|legend=1\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 -1 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.978{{c}}, ~12/11 = 132.544{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1197.0576{{c}}, ~3/2 = 704.2467{{c}}, ~15/14 = 132.2189{{c}}
			: [[error map]]: {{val\| -2.942 -0.651 +4.458 -6.026 +11.883 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6964{{c}}, ~15/14 = 131.8166{{c}}
			: error map: {{val\| 0.000 +1.741 +9.136 -1.496 +20.562 }}
			<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 705.978{{c}}, ~12/11 = 132.544{{c}} -->

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 36 }}		{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 63cee }} *

	[[Badness]] (~~Smith~~): 0.~~856 × 10<sup>-3</sup>~~		<nowiki/>* [[optimal patent val]]: [[36edo\|36]]

			[[Badness]] (Sintel): 1.03

	[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11		[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11
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	Mapping: {{mapping\| 1 0 2 1 2 2 \| 0 1 0 1 1 1 \| 0 0 3 2 -1 1 }}		Mapping: {{mapping\| 1 0 2 1 2 2 \| 0 1 0 1 1 1 \| 0 0 3 2 -1 1 }}

	Optimal ~~tuning (~~POTE): ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}}		Optimal tunings:
			* WE: ~2 = 1197.6484{{c}}, ~3/2 = 704.6450{{c}}, ~14/13 = 132.1686{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1616{{c}}, ~14/13 = 131.8445{{c}}
			<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}} -->

			{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27e }} *

	~~{{Optimal ET sequence~~\|~~legend=0\| 8d, 9, 10, 17c, 19, 27e,~~ 36 }}		<nowiki/>* optimal patent val: [[36edo\|36]]

	Badness (~~Smith~~): 0.~~727 × 10<sup>-3</sup>~~		Badness (Sintel): 0.680

	Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11		Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11

FloraC: Units

2025-10-10T12:10:33Z

Units

← Older revision		Revision as of 12:10, 10 October 2025
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	== Sengic ==		== Sengic ==
	Sengic is naturally a 2.3.5.7.13 subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.		Sengic is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group \| ''It's the "thirds", stupid!'']</ref>.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	{{Mapping\|legend=1\| 1 0 2 1 \| 0 1 0 1 \| 0 0 3 2 }}		{{Mapping\|legend=1\| 1 0 2 1 \| 0 1 0 1 \| 0 0 3 2 }}

	: mapping generators: ~2, ~3, ~15/14		: mapping generators: ~2, ~3, ~15/14

	[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 703.7873, ~15/14 = 129.6451		[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.7873{{c}}, ~15/14 = 129.6451{{c}}

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46 }}		{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46 }}

	[[Badness]]: 0.320 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.320 × 10<sup>-3</sup>

	[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5		[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
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	Sval mapping: {{mapping\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 1 }}		Sval mapping: {{mapping\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 1 }}

	Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 704.5918, ~14/13 = 129.7585		Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}}

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}		{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}

	Badness: 0.320 × 10<sup>-3</sup>		Badness (Smith): 0.320 × 10<sup>-3</sup>

	== Demeter ==		== Demeter ==
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	{{Mapping\|legend=1\| 1 0 2 1 -3 \| 0 1 0 1 4 \| 0 0 3 2 1 }}		{{Mapping\|legend=1\| 1 0 2 1 -3 \| 0 1 0 1 4 \| 0 0 3 2 1 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 705.518, ~15/14 = 130.039		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.518{{c}}, ~15/14 = 130.039{{c}}

	{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 46, 102, 148 }}		{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 46, 102, 148 }}

	[[Badness]]: 1.32 × 10<sup>-3</sup>		[[Badness]] (Smith): 1.32 × 10<sup>-3</sup>

	[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11		[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11
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	Mapping: {{mapping\| 1 0 2 1 -3 2 \| 0 1 0 1 4 1 \| 0 0 3 2 1 1 }}		Mapping: {{mapping\| 1 0 2 1 -3 2 \| 0 1 0 1 4 1 \| 0 0 3 2 1 1 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~3/2 = 705.113, ~14/13 = 129.673		Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}}

	{{Optimal ET sequence\|legend=1\| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}		{{Optimal ET sequence\|legend=0\| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}

	Badness: 0.977 × 10<sup>-3</sup>		Badness (Smith): 0.977 × 10<sup>-3</sup>

	Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11		Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
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	Mapping: {{mapping\| 1 0 2 1 -3 2 -1 \| 0 1 0 1 4 1 3 \| 0 0 3 2 1 1 3 }}		Mapping: {{mapping\| 1 0 2 1 -3 2 -1 \| 0 1 0 1 4 1 3 \| 0 0 3 2 1 1 3 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~3/2 = 705.147, ~14/13 = 129.700		Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}}

	{{Optimal ET sequence\|legend=1\| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}		{{Optimal ET sequence\|legend=0\| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}

	Badness: 0.830 × 10<sup>-3</sup>		Badness (Smith): 0.830 × 10<sup>-3</sup>

	Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11		Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11
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	{{Mapping\|legend=1\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 -1 }}		{{Mapping\|legend=1\| 1 0 2 1 2 \| 0 1 0 1 1 \| 0 0 3 2 -1 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 705.978, ~12/11 = 132.544		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.978{{c}}, ~12/11 = 132.544{{c}}

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 36 }}		{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 36 }}

	[[Badness]]: 0.856 × 10<sup>-3</sup>		[[Badness]] (Smith): 0.856 × 10<sup>-3</sup>

	[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11		[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11
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	Mapping: {{mapping\| 1 0 2 1 2 2 \| 0 1 0 1 1 1 \| 0 0 3 2 -1 1 }}		Mapping: {{mapping\| 1 0 2 1 2 2 \| 0 1 0 1 1 1 \| 0 0 3 2 -1 1 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~3/2 = 706.029, ~14/13 = 132.428		Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}}

	{{Optimal ET sequence\|legend=1\| 8d, 9, 10, 17c, 19, 27e, 36 }}		{{Optimal ET sequence\|legend=0\| 8d, 9, 10, 17c, 19, 27e, 36 }}

	Badness: 0.727 × 10<sup>-3</sup>		Badness (Smith): 0.727 × 10<sup>-3</sup>

	Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11		Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11

	== ~~Notes~~ ==		== References ==

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

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

2025-06-24T00:41:36Z

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

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