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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
== 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 was discovered and named in 2005.


[[Map]]: {{lbrack}} {{val| 1 0 2 1 }}, {{val| 0 1 0 1 }}, {{val| 0 0 3 2 }} {{rbrack}}
[[Subgroup]]: 2.3.5.7


[[POTE tuning|POTE generators]]: 2, ~3/2 = 704.154, ~15/14 = 129.824
[[Comma list]]: [[686/675]]


[[EDO]]s: {{Vals| 9, 10, 18, 19, 27, 46 }}
{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}
: mapping generators: ~2, ~3, ~15/14


[[Badness]]: 0.000320
[[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 }}


[[Projection pair]]s: 5 3375/686 7 675/98 to 2.3.7/5
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }}


= Demeter =
[[Badness]] (Sintel): 1.41
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}}
[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5


POTE generators: 2, ~3/2 = 705.518, ~15/14 = 130.039
=== 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.


EDOs: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}
11-limit temperaments considered below are demeter, krypton, and sensigh.


== 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
Subgroup-val 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 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}}


= Krypton =
{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
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}}
Badness (Sintel): 0.425


POTE generators: 2, ~3/2 = 705.978, ~12/11 = 132.544
== 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| 8d, 9, 10, 17c, 19, 27e, 36 }}
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000856
[[Comma list]]: 441/440, 686/675


Projection pairs: 5 6912/1331 7 854/121 to 2.3.11
{{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }}


== 13-limit ==
[[Optimal tuning]]s:
Commas: 56/55, 78/77, 91/90
* [[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 }}


Map: {{lbrack}} {{val| 1 0 2 1 2 2 }}, {{val| 0 1 0 1 1 1 }}, {{val| 0 0 3 2 -1 1 }} {{rbrack}}
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }}


POTE generators: 2, ~3/2 = 706.029, ~12/11 = 132.428
[[Badness]] (Sintel): 1.58


EDOs: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}
[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11


Badness: 0.000727
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Projection pairs: 5 6912/1331 7 854/121 13 144/11 to 2.3.11
Comma list: 91/90, 169/168, 352/351


[[Category:Theory]]
Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }}
[[Category:Temperament family]]
 
[[Category:Sengic]]
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}}
 
{{Optimal ET sequence|legend=0| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}
 
Badness (Sintel): 0.913
 
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 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}}
 
{{Optimal ET sequence|legend=0| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}
 
Badness (Sintel): 0.789
 
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]]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 }}
 
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 63cee }} *
 
<nowiki/>* [[optimal patent val]]: [[36edo|36]]
 
[[Badness]] (Sintel): 1.03
 
[[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 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}}
 
{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27e }} *
 
<nowiki/>* optimal patent val: [[36edo|36]]
 
Badness (Sintel): 0.680
 
Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11
 
== Sensigh ==
Sensigh uses the same mapping as 7-limit [[sensi]] with an independent generator for prime 11.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 245/243
 
{{Mapping|legend=1| 1 -1 -1 -2 3 | 0 7 9 13 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~9/7, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}, ~11/8 = 552.1736{{c}}
: [[error map]]: {{val| -0.2919 +1.2608 +3.4518 -5.6691 -0.0202 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}, ~11/8 = 551.8069{{c}}
: error map: {{val| 0.0000 +1.4899 +3.8297 -5.2854 +0.4890 }}
 
{{Optimal ET sequence|legend=1| 27e, 38df, 46, 111d }}
 
[[Badness]] (Sintel): 1.48
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168
 
Mapping: {{mapping| 1 -1 -1 -2 3 0 | 0 7 9 13 0 10 | 0 0 0 0 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}
 
Badness (Sintel): 0.878
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 126/125, 154/153, 169/168
 
Mapping: {{mapping| 1 -1 -1 -2 3 0 4 | 0 7 9 13 0 10 -1 | 0 0 0 0 1 0 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}
 
Badness (Sintel): 0.917
 
== References ==
 
[[Category:Temperament families]]
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Rank 3]]

Latest revision as of 10:31, 11 April 2026

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.

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 was discovered and named in 2005.

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 tunings:

  • WE: ~2 = 1199.7533 ¢, ~3/2 = 704.0092 ¢, ~15/14 = 129.7976 ¢
error map: ⟨-0.247 +1.808 +2.586 -5.715]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9671 ¢, ~15/14 = 129.7330 ¢
error map: ⟨0.000 +2.012 +2.885 -5.393]

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

Badness (Sintel): 1.41

Projection pairs: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5

Overview to extensions

First noted by Keenan Pepper in 2011[1], sengic is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (169/168)⋅(196/195)2, as we can see from its S-expression, S13⋅S142. 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

Subgroup: 2.3.5.7.13

Comma list: 91/90, 169/168

Subgroup-val mapping: [⟨1 0 2 1 2], ⟨0 1 0 1 1], ⟨0 0 3 2 1]]

Optimal tunings:

  • WE: ~2 = 1200.3448 ¢, ~3/2 = 704.1998 ¢, ~14/13 = 129.5253 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2688 ¢, ~14/13 = 129.6115 ¢

Optimal ET sequence: 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df

Badness (Sintel): 0.425

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 tunings:

  • WE: ~2 = 1199.4002 ¢, ~3/2 = 705.1652 ¢, ~15/14 = 129.9738 ¢
error map: ⟨-0.600 +2.610 +2.408 -4.913 -1.283]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.2789 ¢, ~15/14 = 129.8330 ¢
error map: ⟨0.000 +3.324 +3.185 -3.881 -0.369]

Optimal ET sequence: 10, 17c, 19e, 27e, 46, 102, 148

Badness (Sintel): 1.58

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 tunings:

  • WE: ~2 = 1200.0393 ¢, ~3/2 = 705.1360 ¢, ~14/13 = 129.6770 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.1277 ¢, ~14/13 = 129.6854 ¢

Optimal ET sequence: 10, 17c, 19e, 27e, 29, 46, 102, 148f

Badness (Sintel): 0.913

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 tunings:

  • WE: ~2 = 1200.0255 ¢, ~3/2 = 705.1616 ¢, ~14/13 = 129.7024 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.1553 ¢, ~14/13 = 129.7071 ¢

Optimal ET sequence: 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f

Badness (Sintel): 0.789

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 tunings:

  • WE: ~2 = 1197.0576 ¢, ~3/2 = 704.2467 ¢, ~15/14 = 132.2189 ¢
error map: ⟨-2.942 -0.651 +4.458 -6.026 +11.883]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.6964 ¢, ~15/14 = 131.8166 ¢
error map: ⟨0.000 +1.741 +9.136 -1.496 +20.562]

Optimal ET sequence: 8d, 9, 10, 17c, 19, 27e, 63cee *

* optimal patent val: 36

Badness (Sintel): 1.03

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 tunings:

  • WE: ~2 = 1197.6484 ¢, ~3/2 = 704.6450 ¢, ~14/13 = 132.1686 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1616 ¢, ~14/13 = 131.8445 ¢

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

* optimal patent val: 36

Badness (Sintel): 0.680

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

Sensigh

Sensigh uses the same mapping as 7-limit sensi with an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/243

Mapping: [⟨1 -1 -1 -2 3], ⟨0 7 9 13 0], ⟨0 0 0 0 1]]

mapping generators: ~2, ~9/7, ~11

Optimal tunings:

  • WE: ~2 = 1199.7081 ¢, ~9/7 = 443.2748 ¢, ~11/8 = 552.1736 ¢
error map: ⟨-0.2919 +1.2608 +3.4518 -5.6691 -0.0202]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3493 ¢, ~11/8 = 551.8069 ¢
error map: ⟨0.0000 +1.4899 +3.8297 -5.2854 +0.4890]

Optimal ET sequence: 27e, 38df, 46, 111d

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168

Mapping: [⟨1 -1 -1 -2 3 0], ⟨0 7 9 13 0 10], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~9/7 = 443.4379 ¢, ~11/8 = 550.3462 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3581 ¢, ~11/8 = 550.7092 ¢

Optimal ET sequence: 27e, 38df, 46, 111df

Badness (Sintel): 0.878

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168

Mapping: [⟨1 -1 -1 -2 3 0 4], ⟨0 7 9 13 0 10 -1], ⟨0 0 0 0 1 0 1]]

Optimal tunings:

  • WE: ~2 = 1200.2286 ¢, ~9/7 = 443.4291 ¢, ~11/8 = 549.2790 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3707 ¢, ~11/8 = 549.5775 ¢

Optimal ET sequence: 27eg, 38df, 46

Badness (Sintel): 0.917

References

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