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Sengic family: Difference between revisions

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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]].  


Temperament discussed elsewhere include [[sensigh]]. Considered below are demeter and krypton.  
== 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.  


== Sengic ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: [[686/675]]
[[Comma list]]: [[686/675]]


[[Mapping]]: [{{val| 1 0 2 1 }}, {{val| 0 1 0 1 }}, {{val| 0 0 3 2 }}]
{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}
: mapping generators: ~2, ~3, ~15/14
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }}


Mapping generators: ~2, ~3, ~15/14
[[Badness]] (Sintel): 1.41


[[POTE generator]]s: ~3/2 = 704.154, ~15/14 = 129.824
[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5


{{Val list|legend=1| 9, 10, 18, 19, 27, 46 }}
=== 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.


[[Badness]]: 0.320 × 10<sup>-3</sup>
11-limit temperaments considered below are demeter, krypton, and sensigh.  


[[Projection pair]]s: ~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: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
 
Badness (Sintel): 0.425


== Demeter ==
== Demeter ==
Subgroup: 2.3.5.7.11
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
[[Comma list]]: 441/440, 686/675


[[Mapping]]: [{{val| 1 0 2 1 -3 }}, {{val| 0 1 0 1 4 }}, {{val| 0 0 3 2 1 }}]
{{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }}


[[POTE generator]]s: ~3/2 = 705.518, ~15/14 = 130.039
[[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 }}


{{Val list|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }}
 
[[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 38: Line 70:
Comma list: 91/90, 169/168, 352/351
Comma list: 91/90, 169/168, 352/351


Mapping: [{{val| 1 0 2 1 -3 2 }}, {{val| 0 1 0 1 4 1 }}, {{val| 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 }}


POTE generators: ~3/2 = 705.113, ~14/13 = 129.673
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 GPV sequence: {{Val list| 10, 17c, 19e, 27e, 29, 46 }}
{{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
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
Line 51: Line 87:
Comma list: 91/90, 136/135, 154/153, 169/168
Comma list: 91/90, 136/135, 154/153, 169/168


Mapping: [{{val| 1 0 2 1 -3 2 -1 }}, {{val| 0 1 0 1 4 1 3 }}, {{val| 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 }}


POTE generators: ~3/2 = 705.147, ~14/13 = 129.700
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 GPV sequence: {{Val list| 10, 17cg, 19eg, 27eg, 29g, 46 }}
{{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
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11


== Krypton ==
== Krypton ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 56/55, 540/539
[[Comma list]]: 56/55, 540/539


[[Mapping]]: [{{val| 1 0 2 1 2 }}, {{val| 0 1 0 1 1 }}, {{val| 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]]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 generator]]s: ~3/2 = 705.978, ~12/11 = 132.544
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 63cee }} *


{{Val list|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
<nowiki/>* [[optimal patent val]]: [[36edo|36]]


[[Badness]]: 0.856 × 10<sup>-3</sup>
[[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 79: Line 125:
Comma list: 56/55, 78/77, 91/90
Comma list: 56/55, 78/77, 91/90


Mapping: [{{val| 1 0 2 1 2 2 }}, {{val| 0 1 0 1 1 1 }}, {{val| 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 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 generators: ~3/2 = 706.029, ~14/13 = 132.428
{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27e }} *


Optimal GPV sequence: {{Val list| 8d, 9, 10, 17c, 19, 27e, 36 }}
<nowiki/>* optimal patent val: [[36edo|36]]


Badness: 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
== 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:Temperament families]]
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Sengic_family"