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'''Sengic family''' is a collection of the [[Planar_Temperament|planar temperament]]s tempering out the [[senga|senga]], [[686/675|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: [<1 0 2 1|, <0 1 0 1|, <0 0 3 2|]
== 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>.


EDOs: [[9edo|9]], [[10edo|10]], [[18edo|18]], [[19edo|19]], [[27edo|27]], [[46edo|46]]
[[Subgroup]]: 2.3.5.7


[[Badness|Badness]]: 0.000320
[[Comma list]]: [[686/675]]


[[Projection_pair|Projection pair]]s: 5 3375/686 7 675/98 to 2.3.7/5
{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}


==Krypton==
: mapping generators: ~2, ~3, ~15/14
Commas: 56/55, 540/539


Map: [&lt;1 0 2 1 2|, &lt;0 1 0 1 1| &lt;0 0 3 2 -1|]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 703.7873, ~15/14 = 129.6451


EDOs: 9, 10, 19, [[36edo|36]]
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }}


Badness: 0.000856
[[Badness]]: 0.320 × 10<sup>-3</sup>


[[Projection_pair|Projection pair]]s: 5 6912/1331 7 854/121 to 2.3.11
[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5


===13-limit===
=== 2.3.5.7.13 subgroup ===
Commas: 56/55, 78/77, 91/90
Subgroup: 2.3.5.7.13


Map: [&lt;1 0 2 1 2 2|, &lt;0 1 0 1 1 1| &lt;0 0 3 2 -1 1|]
Comma list: 91/90, 169/168


EDOs: 9, 10, 19, 36
Sval mapping: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }}


Badness: 0.000727
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 704.5918, ~14/13 = 129.7585


[[Projection_pair|Projection pair]]s: 5 6912/1331 7 854/121 13 144/11 to 2.3.11
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
[[Category:family]]
 
[[Category:overview]]
Badness: 0.320 × 10<sup>-3</sup>
[[Category:sengic]]
 
== 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
 
[[Comma list]]: 441/440, 686/675
 
{{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 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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 169/168, 352/351
 
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 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
 
=== 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]]

Latest revision as of 00:41, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The sengic family of rank-3 temperaments tempers out the senga a.k.a. sengic comma, 686/675.

Temperament discussed elsewhere include 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[1].

Subgroup: 2.3.5.7

Comma list: 686/675

Mapping: [⟨1 0 2 1], ⟨0 1 0 1], ⟨0 0 3 2]]

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

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

Optimal ET sequence: 8d, 9, 10, 17c, 19, 27, 46

Badness: 0.320 × 10-3

Projection pairs: ~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: [⟨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: 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[2].

Subgroup: 2.3.5.7.11

Comma list: 441/440, 686/675

Mapping: [⟨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 ET sequence: 10, 17c, 19e, 27e, 46, 102, 148

Badness: 1.32 × 10-3

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

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 352/351

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 ET sequence: 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

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 136/135, 154/153, 169/168

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: 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

Krypton

Subgroup: 2.3.5.7.11

Comma list: 56/55, 540/539

Mapping: [⟨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: 8d, 9, 10, 17c, 19, 27e, 36

Badness: 0.856 × 10-3

Projection pairs: ~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: [⟨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: 8d, 9, 10, 17c, 19, 27e, 36

Badness: 0.727 × 10-3

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

Notes

  1. ↑ Yahoo! Tuning Group | It's the "thirds", stupid!
  2. ↑ Yahoo! Tuning Group | Artemis and friends
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