Augmented family: Difference between revisions

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-{{Interwiki
-| en = Augmented family
-| de = Übermässige Temperaturen
-}}
 {{Technical data page}}
-The '''augmented family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[2/1|octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].
+The '''augmented family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].
 == Augmented ==
@@ Line 574: / Line 570: @@
 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 ==
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 Mapping: {{Mapping| 3 1 7 6 8 | 0 3 0 2 2 }}
-: mapping generators: ~5/4, ~15/11
 Optimal tunings:
-* WE: ~5/4 = 397.495{{c}}, ~15/11 = 499.096{{c}} (~12/11 = 101.601{{c}})
+* WE: ~5/4 = 397.4954{{c}}, ~15/11 = 499.0962{{c}} (~15/14 = 101.6008{{c}})
-* CWE: ~5/4 = 400.000{{c}}, ~15/11 = 500.775{{c}} (~12/11 = 100.775{{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 }}
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 == Subgroup extensions ==
-=== 2.3.5.19 subgroup (3 & 12) ===
+=== Augmented (2.3.5.19 subgroup) ===
-Augmented temperament works on the 2.3.5.19 subgroup very well, tempering out both [[76/75]] and [[96/95]], alongside the augmented comma.
+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
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 {{Mapping|legend=2| 3 0 7 8 | 0 1 0 1 }}
-: sval mapping generators: ~5/4, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~5/4 = 399.075{{c}}, ~3/2 = 705.062{{c}} (~19/16 = 305.986{{c}})
+* [[WE]]: ~5/4 = 399.0754{{c}}, ~3/2 = 705.0615{{c}} (~19/18 = 93.0893{{c}})
-* [[CWE]]: ~5/4 = 400.000{{c}}, ~3/2 = 705.107{{c}} (~19/16 = 305.107{{c}})
+* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.1072{{c}} (~19/18 = 94.8928{{c}})
-{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39 }}
+{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cch }}
 [[Badness]] (Sintel): 0.264
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 * [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
+[[Category:Augmented family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Augmented family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]