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== Trisected ==
== Trisected ==
{{Main|Trisected}}
{{Main|Trisected}}
{{See also|Subgroup temperaments #Trisect}}
Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap| 15 & 36 }} temperament.
Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap| 15 & 36 }} temperament.


It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.
It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.
{{Todo|review|comment=Last digits in ~8/7 for WE may be wrong. Also, comma lists could potentially be simplified.}}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 128/125, 1029/1000
[[Comma list]]: 128/125, 1029/1000


{{Mapping|legend=1| 3 3 7 9 | 0 3 0 -1 }}
{{Mapping|legend=1| 3 0 7 10 | 0 3 0 -1 }}
 
: mapping generators: ~5/4, ~10/7
: mapping generators: ~5/4, ~8/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 399.520{{c}}, ~8/7 = 234.195{{c}}
* [[WE]]: ~5/4 = 399.5199{{c}}, ~10/7 = 633.7146{{c}}
: [[error map]]: {{val| -1.440 -0.811 10.325 -7.342 }}
: [[error map]]: {{val| -1.440 -0.811 +10.325 -7.342 }}
 
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~10/7 = 634.3393{{c}}
* [[CWE]]: ~5/4 = 400.000{{c}}, ~8/7 = 234.339{{c}}
: error map: {{val| 0.000 +1.063 +13.686 -3.165 }}
: error map: {{val| 0.000 +1.063 +13.686 -3.165 }}


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=== 11-limit ===
=== 11-limit ===
This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Comma list: 56/55, 128/125, 1029/1000
Comma list: 56/55, 128/125, 1029/1000


Mapping: {{Mapping| 3 3 7 9 11 | 0 3 0 -1 -1 }}
Mapping: {{Mapping| 3 0 7 10 12 | 0 3 0 -1 -1 }}
 
: mapping generators: ~5/4, ~8/7


Optimal tunings:  
Optimal tunings:  
* WE: ~5/4 = 399.305{{c}}, ~8/7 = 234.484{{c}}
* WE: ~5/4 = 399.3049{{c}}, ~10/7 = 633.7891{{c}}
 
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.7695{{c}}
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.769{{c}}


{{Optimal ET sequence|legend=1| 15, 36, 51ce }}
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}


[[Badness]] (Sintel): 1.60
Badness (Sintel): 1.60


=== 13-limit ===
=== 13-limit ===
This extension is natural since it equates the [[10/7]], which is 1/3 of [[3/1]] since 1029/1000 is tempered out, with [[13/9]], thus tempering out [[91/90]] and [[2197/2187]].


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
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Comma list: 56/55, 91/90, 128/125, 1029/1000
Comma list: 56/55, 91/90, 128/125, 1029/1000


Mapping: {{Mapping| 3 3 7 9 11 7 | 0 3 0 -1 -1 7 }}
Mapping: {{Mapping| 3 0 7 10 12 0 | 0 3 0 -1 -1 7 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.319{{c}}, ~8/7 = 234.745{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.991{{c}}
 
{{Optimal ET sequence|legend=1| 15, 36, 51ce }}
 
[[Badness]] (Sintel): 1.70
 
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 56/55, 91/90, 128/125, 136/135, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 | 0 3 0 -1 -1 7 9 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.317{{c}}, ~8/7 = 234.609{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.83
 
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 10 | 0 3 0 -1 -1 7 9 3 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.328{{c}}, ~8/7 = 234.614{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.62
 
=== 23-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 10 13 | 0 3 0 -1 -1 7 9 3 1 }}
 
: mapping generators: ~5/4, ~8/7


Optimal tunings:  
Optimal tunings:  
* WE: ~5/4 = 399.353{{c}}, ~8/7 = 234.601{{c}}
* WE: ~5/4 = 399.3194{{c}}, ~10/7 = 634.0637{{c}}
 
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.9907{{c}}
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.843{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.54
 
=== 29-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 203/200, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 10 13 14 | 0 3 0 -1 -1 7 9 3 1 1 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.388{{c}}, ~8/7 = 234.581{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.819{{c}}


{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}


[[Badness]] (Sintel): 1.52
Badness (Sintel): 1.70


== Niner ==
== Niner ==
Retrieved from "https://en.xen.wiki/w/Augmented_family"