Sengic family: Difference between revisions

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'''~~Sengic~~ family''' ~~is a collection~~ of ~~the~~ [[~~planar~~ temperament]]s tempering out the senga comma, [[686/675]].

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.

== 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>.

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~3/2 = ~~704~~.~~154~~, ~15/14 = 129.~~824~~

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

[[Badness]]: 0.320 × 10-3

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

=== 2.3.5.7.13 subgroup ===

Subgroup: 2.3.5.7.13

Comma list: 91/90, 169/168

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

Badness: 0.320 × 10-3

== Demeter ==

Named by [[Graham Breed]] in 2011, demeter was found to be locally efficient in the 17-limit among all rank-3 temperaments<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>.

[[Subgroup]]: 2.3.5.7.11

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

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

[[Badness]]: 1.32 × 10-3

[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11

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

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

Badness: 0.977 × 10-3

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11

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

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

Badness: 0.830 × 10-3

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11

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

== Notes ==

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Sengic family| ]]

[[Category:Sengic| ]]

[[Category:Rank 3]]

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-'''Sengic family''' is a collection of the [[planar temperament]]s tempering out the senga comma, [[686/675]].
+{{Technical data page}}
+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.
 == 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>.
 [[Subgroup]]: 2.3.5.7
@@ Line 12: / Line 15: @@
 : mapping generators: ~2, ~3, ~15/14
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.154, ~15/14 = 129.824
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 703.7873, ~15/14 = 129.6451
-{{Optimal ET sequence|legend=1| 9, 10, 18, 19, 27, 46 }}
+{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }}
 [[Badness]]: 0.320 × 10<sup>-3</sup>
 [[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
+=== 2.3.5.7.13 subgroup ===
+Subgroup: 2.3.5.7.13
+Comma list: 91/90, 169/168
+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 ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
+Badness: 0.320 × 10<sup>-3</sup>
 == Demeter ==
+Named by [[Graham Breed]] in 2011, demeter was found to be locally efficient in the 17-limit among all rank-3 temperaments<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 29: / Line 47: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.518, ~15/14 = 130.039
-{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
+{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }}
+[[Badness]]: 1.32 × 10<sup>-3</sup>
 [[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11
@@ Line 42: / Line 62: @@
 Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.113, ~14/13 = 129.673
-{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
+{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}
+Badness: 0.977 × 10<sup>-3</sup>
 Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
@@ Line 55: / Line 77: @@
 Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.147, ~14/13 = 129.700
-{{Optimal ET sequence|legend=1| 10, 17cg, 19eg, 27eg, 29g, 46 }}
+{{Optimal ET sequence|legend=1| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}
+Badness: 0.830 × 10<sup>-3</sup>
 Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11
@@ Line 88: / Line 112: @@
 Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11
+== Notes ==
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Sengic family| ]] <!-- main article -->
 [[Category:Sengic| ]] <!-- key article -->
 [[Category:Rank 3]]