@@ Line 4: / Line 4: @@
 }}
 {{Technical data page}}
-The 5-limit parent comma for the '''augmented family''' is [[128/125]], the diesis. The [[period]] is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.
+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]].
 == Augmented ==
+{{Main| Augmented (temperament) }}
+The [[period]] is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third. Its [[ploidacot]] is triploid monocot.
 [[Subgroup]]: 2.3.5
@@ Line 26: / Line 30: @@
 === Overview to extensions ===
-The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds [[36/35]], augene [[64/63]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal augmented adds [[64/63]], august [[36/35]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones. All of these can be extended to the [[11-limit]] by adding [[56/55]] and [[176/175]], which sum to 128/125, equating the 400{{c}} major third to [[14/11]] as well as 5/4.
-== August ==
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 36/35, 128/125
-{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}
-[[Optimal tuning]]s:
-* [[WE]]: ~5/4 = 399.1036{{c}}, ~3/2 = 694.4509{{c}} (~16/15 = 103.7564{{c}})
-: [[error map]]: {{val| -2.689 -10.193 +7.412 +15.594 }}
-* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 694.6812{{c}} (~16/15 = 105.3188{{c}})
-: error map: {{val| 0.000 -7.274 +13.686 +20.537 }}
-{{Optimal ET sequence|legend=1| 9, 12, 45cd }}
-[[Badness]] (Sintel): 0.670
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 36/35, 45/44, 56/55
-Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
-Optimal tunings:
-* WE: ~5/4 = 398.9225{{c}}, ~3/2 = 690.6486{{c}} (~16/15 = 107.1966{{c}})
-* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 690.8519{{c}} (~16/15 = 109.1481{{c}})
-{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}
-Badness (Sintel): 0.668
+== Septimal augmented (augene) ==
+{{Main| Augmented (temperament) }}
-==== 13-limit ====
+Septimal augmented, a.k.a. augene, tempers out [[64/63]] and [[126/125]]. It may be described as the {{nowrap| 12 & 15 }} temperament. [[27edo]] and [[39edo]] in the 39d val make for good tunings. In the 11-limit, it tempers together [[7/5]] and [[11/8]], and 27edo in the 27e val may be recommended as a tuning.
-Subgroup: 2.3.5.7.11.13
-Comma list: 27/26, 36/35, 45/44, 56/55
-Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
-Optimal tunings:
-* WE: ~5/4 = 399.0956{{c}}, ~3/2 = 687.2261{{c}} (~16/15 = 110.9651{{c}})
-* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 687.5057{{c}} (~16/15 = 112.4943{{c}})
-{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}
-Badness (Sintel): 0.762
-==== Augustus ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 26/25, 36/35, 45/44, 56/55
-Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
-Optimal tunings:
-* WE: ~5/4 = 400.4230{{c}}, ~3/2 = 686.0809{{c}} (~16/15 = 114.7650{{c}})
-* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 685.8446{{c}} (~16/15 = 114.1554{{c}})
-{{Optimal ET sequence|legend=0| 9, 12 }}
-Badness (Sintel): 0.919
-== Augene ==
-{{Main| Augene }}
 [[Subgroup]]: 2.3.5.7
@@ Line 183: / Line 127: @@
 Badness (Sintel): 1.18
+== August ==
+August tempers out 36/35 and 225/224. It may be described as the {{nowrap| 9 & 12 }} temperament. Unlike augene, august calls for a flat tuning of the fifth, and besides [[12edo]], [[21edo]] is among the possible tunings.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 36/35, 128/125
+{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~5/4 = 399.1036{{c}}, ~3/2 = 694.4509{{c}} (~16/15 = 103.7564{{c}})
+: [[error map]]: {{val| -2.689 -10.193 +7.412 +15.594 }}
+* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 694.6812{{c}} (~16/15 = 105.3188{{c}})
+: error map: {{val| 0.000 -7.274 +13.686 +20.537 }}
+{{Optimal ET sequence|legend=1| 9, 12, 45cd }}
+[[Badness]] (Sintel): 0.670
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 36/35, 45/44, 56/55
+Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
+Optimal tunings:
+* WE: ~5/4 = 398.9225{{c}}, ~3/2 = 690.6486{{c}} (~16/15 = 107.1966{{c}})
+* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 690.8519{{c}} (~16/15 = 109.1481{{c}})
+{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}
+Badness (Sintel): 0.668
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 27/26, 36/35, 45/44, 56/55
+Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
+Optimal tunings:
+* WE: ~5/4 = 399.0956{{c}}, ~3/2 = 687.2261{{c}} (~16/15 = 110.9651{{c}})
+* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 687.5057{{c}} (~16/15 = 112.4943{{c}})
+{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}
+Badness (Sintel): 0.762
+==== Augustus ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 26/25, 36/35, 45/44, 56/55
+Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
+Optimal tunings:
+* WE: ~5/4 = 400.4230{{c}}, ~3/2 = 686.0809{{c}} (~16/15 = 114.7650{{c}})
+* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 685.8446{{c}} (~16/15 = 114.1554{{c}})
+{{Optimal ET sequence|legend=0| 9, 12 }}
+Badness (Sintel): 0.919
 == Inflated ==
@@ Line 249: / Line 257: @@
 == Hexe ==
+Hexe tempers out 50/49 and may be described as {{nowrap| 6 & 12 }}, viewed as [[6edo|6et]] with an independent generator for prime 3. Its ploidacot is hexaploid monocot.
 [[Subgroup]]: 2.3.5.7
@@ Line 299: / Line 309: @@
 == Triforce ==
 [[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of triforce]]
+Triforce tempers out 49/48 and may be described as {{nowrap| 9 & 15 }}. Its ploidacot is triploid alpha-dicot. [[24edo]] and [[39edo]] are among the possible tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 334: / Line 346: @@
 ; Music
-* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)''] by [[Igliashon Jones]] (2018)
+* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)'']{{dead link}} by [[Igliashon Jones]] (2018)
-* [http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ ''2-2-1-2-2-1-2-2-1 mode of 15 edo''] [http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]]
+* [https://www.chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ ''2-2-1-2-2-1-2-2-1 mode of 15 edo''] [https://web.archive.org/web/20201127015017/http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]] (2013)
 ==== 13-limit ====
@@ Line 356: / Line 368: @@
 ==== Semitriforce ====
+This extension splits the period into 1/6-octave for ~44/39. Its ploidacot is hexaploid dicot.
 Subgroup: 2.3.5.7.11.13
@@ Line 373: / Line 387: @@
 == Hemiaug ==
+Hemiaug tempers out 245/243 and may be described as {{nowrap| 24 & 27 }}. The generator may be taken as ~14/9, but also a neutral third or a neutral second that stand in for 11/9~16/13 and 12/11~13/12 in the higher limits, respectively. Hemiaug's ploidacot is triploid dicot. [[27edo]] makes for a recommendable tuning in the 7-limit, but [[51edo]] serves better in the higher limits.
 [[Subgroup]]: 2.3.5.7
@@ Line 409: / Line 425: @@
 Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 91/90, 128/125, 245/243
+Comma list: 56/55, 91/90, 128/125, 243/242
 Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}
@@ Line 422: / Line 438: @@
 == Hemiug ==
+Hemiug tempers out 1323/1250 and may be described as {{nowrap| 21 & 24 }}. The generator is a similar interval but for ~32/21 instead of ~14/9, and the ploidacot is triploid dicot, the same as hemiaug.
 [[Subgroup]]: 2.3.5.7
@@ Line 471: / Line 489: @@
 == Oodako ==
+Oodako tempers out 2401/2400 and may be described as {{nowrap| 21 & 27 }}. It is generated by a quarter of a fifth, which stands in for ~28/25. Its ploidacot is triploid tetracot.
 [[Subgroup]]: 2.3.5.7
@@ Line 520: / Line 540: @@
 == Hemisemiaug ==
+Hemisemiaug tempers out 12005/11664 and splits both the period and generator of augmented in two. Its ploidacot is hexaploid alpha-dicot.
 [[Subgroup]]: 2.3.5.7
@@ Line 552: / 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 ==
-Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.
+Niner tempers out 686/675 and may be described as the {{nowrap| 9 & 27 }} temperament. Its ploidacot is enneaploid monocot. It gives 9 as the complexity of a [[harmonic seventh chord]], tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads. 27edo, [[36edo]] and [[63edo]] in the 63c val are among the possible tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 605: / Line 685: @@
 == Trug ==
+Trug tempers out 360/343. It is generated by an interval of ~48/35, tuned very close to a perfect fourth, but the perfect fourth is mapped to three generator steps and a period. Its ploidacot is triploid alpha-tricot. 12edo is about as accurate as it can be tuned.
 [[Subgroup]]: 2.3.5.7
@@ Line 622: / Line 704: @@
 [[Badness]] (Sintel): 3.50
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 56/55, 128/125, 360/343
+Mapping: {{Mapping| 3 1 7 6 8 | 0 3 0 2 2 }}
+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
+[[Comma list]]: 76/75, 96/95
+{{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| 3, 9, 12, 27, 39, 51c, 90cch }}
+[[Badness]] (Sintel): 0.264
 == External links ==
-* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament] {{dead link}}
+* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Augmented family| ]] <!-- main article -->
 [[Category:Rank 2]]

Augmented family: Difference between revisions