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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]].  


[[Comma]]: 686/675
Temperament discussed elsewhere include [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered below are demeter and krypton.


[[Map]]: {{lbrack}} {{val| 1 0 2 1 }}, {{val| 0 1 0 1 }}, {{val| 0 0 3 2 }} {{rbrack}}
== 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>.


[[POTE tuning|POTE generators]]: 2, ~3/2 = 704.154, ~15/14 = 129.824
[[Subgroup]]: 2.3.5.7


[[EDO]]s: {{Vals| 9, 10, 18, 19, 27, 46 }}
[[Comma list]]: [[686/675]]


[[Badness]]: 0.000320
{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}


[[Projection pair]]s: 5 3375/686 7 675/98 to 2.3.7/5
: mapping generators: ~2, ~3, ~15/14


== Demeter  ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 703.7873, ~15/14 = 129.6451
Commas: 441/440, 686/675


Map: {{lbrack}} {{val| 1 0 2 1 -3 }}, {{val| 0 1 0 1 4 }}, {{val| 0 0 3 2 1 }} {{rbrack}}
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }}


POTE generators: 2, ~3/2 = 705.518, ~15/14 = 130.039
[[Badness]]: 0.320 × 10<sup>-3</sup>


EDOs: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}
[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5


=== 13-limit  ===
=== 2.3.5.7.13 subgroup ===
Commas: 91/90, 169/168, 352/351
Subgroup: 2.3.5.7.13


Map: {{lbrack}} {{val| 1 0 2 1 -3 2 }}, {{val| 0 1 0 1 4 1 }}, {{val| 0 0 3 2 1 1 }} {{rbrack}}
Comma list: 91/90, 169/168


POTE generators: 2, ~3/2 = 705.113, ~13/12 = 129.673
Sval mapping: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }}


EDOs: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 704.5918, ~14/13 = 129.7585


=== 17-limit  ===
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
Commas: 91/90, 136/135, 154/153, 169/168


Map: {{lbrack}} {{val| 1 0 2 1 -3 2 -1 }}, {{val| 0 1 0 1 4 1 3 }}, {{val| 0 0 3 2 1 1 3 }} {{rbrack}}
Badness: 0.320 × 10<sup>-3</sup>


POTE generators: 2, ~3/2 = 705.147, ~13/12 = 129.700
== 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>.


EDOs: {{Vals| 10, 17cg, 19eg, 27eg, 29g, 46 }}
[[Subgroup]]: 2.3.5.7.11


== Krypton  ==
[[Comma list]]: 441/440, 686/675
Commas: 56/55, 540/539


Map: {{lbrack}} {{val| 1 0 2 1 2 }}, {{val| 0 1 0 1 1 }}, {{val| 0 0 3 2 -1 }} {{rbrack}}
{{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }}


POTE generators: 2, ~3/2 = 705.978, ~12/11 = 132.544
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.518, ~15/14 = 130.039


EDOs: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }}


Badness: 0.000856
[[Badness]]: 1.32 × 10<sup>-3</sup>


Projection pairs: 5 6912/1331 7 854/121 to 2.3.11
[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 78/77, 91/90
Subgroup: 2.3.5.7.11.13


Map: {{lbrack}} {{val| 1 0 2 1 2 2 }}, {{val| 0 1 0 1 1 1 }}, {{val| 0 0 3 2 -1 1 }} {{rbrack}}
Comma list: 91/90, 169/168, 352/351


POTE generators: 2, ~3/2 = 706.029, ~12/11 = 132.428
Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }}


EDOs: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.113, ~14/13 = 129.673


Badness: 0.000727
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}


Projection pairs: 5 6912/1331 7 854/121 13 144/11 to 2.3.11
Badness: 0.977 × 10<sup>-3</sup>


[[Category:Theory]]
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
[[Category:Temperament family]]
 
[[Category:Sengic]]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168
 
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 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
 
== Krypton ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 540/539
 
{{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 ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
 
[[Badness]]: 0.856 × 10<sup>-3</sup>
 
[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 91/90
 
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 ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
 
Badness: 0.727 × 10<sup>-3</sup>
 
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]]
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