Augmented family: Difference between revisions

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@@ Line 574: / Line 574: @@
 Badness (Sintel): 2.67
+== 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.
+It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 128/125, 1029/1000
+{{Mapping|legend=1| 3 0 7 10 | 0 3 0 -1 }}
+: mapping generators: ~5/4, ~10/7
+[[Optimal tuning]]s:
+* [[WE]]: ~5/4 = 399.5199{{c}}, ~10/7 = 633.7146{{c}}
+: [[error map]]: {{val| -1.440 -0.811 +10.325 -7.342 }}
+* [[CWE]]: ~5/4 = 400.0000{{c}}, ~10/7 = 634.3393{{c}}
+: error map: {{val| 0.000 +1.063 +13.686 -3.165 }}
+{{Optimal ET sequence|legend=1| 15, 36, 51c }}
+[[Badness]] (Sintel): 2.61
+=== 11-limit ===
+This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]].
+Subgroup: 2.3.5.7.11
+Comma list: 56/55, 128/125, 1029/1000
+Mapping: {{Mapping| 3 0 7 10 12 | 0 3 0 -1 -1 }}
+Optimal tunings:
+* WE: ~5/4 = 399.3049{{c}}, ~10/7 = 633.7891{{c}}
+* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.7695{{c}}
+{{Optimal ET sequence|legend=0| 15, 36, 51ce }}
+Badness (Sintel): 1.60
+=== 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
+Comma list: 56/55, 91/90, 128/125, 1029/1000
+Mapping: {{Mapping| 3 0 7 10 12 0 | 0 3 0 -1 -1 7 }}
+Optimal tunings:
+* WE: ~5/4 = 399.3194{{c}}, ~10/7 = 634.0637{{c}}
+* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.9907{{c}}
+{{Optimal ET sequence|legend=0| 15, 36, 51ce }}
+Badness (Sintel): 1.70
 == Niner ==
@@ Line 644: / Line 702: @@
 {{Optimal ET sequence|legend=1| 3b, 9bd, 12 }}
+[[Badness]] (Sintel): 3.50
 === 11-limit ===
-{{Todo|unify precision}}
 Subgroup: 2.3.5.7.11
@@ Line 653: / Line 712: @@
 Mapping: {{Mapping| 3 1 7 6 8 | 0 3 0 2 2 }}
-: mapping generators: ~5/4, ~15/11
+Optimal tunings:
+* WE: ~5/4 = 397.4954{{c}}, ~15/11 = 499.0962{{c}} (~15/14 = 101.6008{{c}})
+* CWE: ~5/4 = 400.0000{{c}}, ~15/11 = 500.7745{{c}} (~15/14 = 100.7745{{c}})
+{{Optimal ET sequence|legend=0| 3b, 9bde, 12 }}
+Badness (Sintel): 2.60
+== Subgroup extensions ==
+=== Augmented (2.3.5.19 subgroup) ===
+Augmented works on the [[2.3.5.19 subgroup]] very well, where the 1/3-octave period stands in for ~5/4, ~19/15, and ~24/19.
+[[Subgroup]]: 2.3.5.19
-Optimal tunings:
+[[Comma list]]: 76/75, 96/95
-* WE: ~5/4 = 397.495{{c}}, ~15/11 = 499.096{{c}} (~15/14 = 101.601{{c}})
-* CWE: ~5/4 = 400.000{{c}}, ~15/11 = 500.775{{c}} (~15/14 = 100.775{{c}})
+{{Mapping|legend=2| 3 0 7 8 | 0 1 0 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~5/4 = 399.0754{{c}}, ~3/2 = 705.0615{{c}} (~19/18 = 93.0893{{c}})
+* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.1072{{c}} (~19/18 = 94.8928{{c}})
-{{Optimal ET sequence|legend=1| 3b, 9bde, 12 }}
+{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cch }}
-[[Badness]] (Sintel): 2.60
+[[Badness]] (Sintel): 0.264
 == External links ==