Sengic family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus
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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
Line 27: Line 32:
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 ==
Line 44: Line 52:
{{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
Line 59: Line 72:
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
Line 74: Line 90:
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
Line 89: Line 108:
{{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
Line 104: Line 130:
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

Revision as of 12:32, 10 October 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 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

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

References

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