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Sengic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''sengic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the senga a.k.a. sengic 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 ==
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>.  
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
[[Subgroup]]: 2.3.5.7
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9671{{c}}, ~15/14 = 129.7330{{c}}
* [[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 }}
: 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 }}
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[[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
=== 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.
11-limit temperaments considered below are demeter, krypton, and sensigh.


=== 2.3.5.7.13 subgroup ===
=== 2.3.5.7.13 subgroup ===
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* WE: ~2 = 1200.3448{{c}}, ~3/2 = 704.1998{{c}}, ~14/13 = 129.5253{{c}}
* 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}}
* 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 }}
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2789{{c}}, ~15/14 = 129.8330{{c}}
* [[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 }}
: 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 }}
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* WE: ~2 = 1200.0393{{c}}, ~3/2 = 705.1360{{c}}, ~14/13 = 129.6770{{c}}
* 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}}
* 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 }}
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* WE: ~2 = 1200.0255{{c}}, ~3/2 = 705.1616{{c}}, ~14/13 = 129.7024{{c}}
* 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}}
* 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 }}
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6964{{c}}, ~15/14 = 131.8166{{c}}
* [[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 }}
: 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, 63cee }} *
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 63cee }} *
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* WE: ~2 = 1197.6484{{c}}, ~3/2 = 704.6450{{c}}, ~14/13 = 132.1686{{c}}
* 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}}
* 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 }} *
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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
== 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 ==
== References ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Sengic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Sengic_family"