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__FORCETOC__
{{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 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo]] is {{Monzo| 7 0 -3 }}, and flipping that yields <<3 0 -7|| for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for a 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.
== Augmented ==
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.  


== Seven limit children ==
[[Subgroup]]: 2.3.5
The second comma of the [[Normal lists|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 hemis 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quarter-tones instead of semitones.


== August ==
[[Comma list]]: 128/125
[[Comma]]s: 36/35, 128/125
 
{{Mapping|legend=1| 3 0 7 | 0 1 0 }}
 
: mapping generators: ~5/4, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.0128{{c}}, ~3/2 = 704.8937{{c}} (~16/15 = 93.1320{{c}})
: [[error map]]: {{val| -2.962 -0.023 +6.776 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.0691{{c}} (~16/15 = 94.9309{{c}})
: error map: {{val| 0.000 +3.114 +13.686 }}
 
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}
 
[[Badness]] (Sintel): 0.523
 
=== 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. Augene 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.
 
== Augene ==
{{Main| Augene }}
 
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.
 
[[Subgroup]]: 2.3.5.7


[[POTE tuning|POTE generator]]: 696.011
[[Comma list]]: 64/63, 126/125


[[Map]]: [<3 0 7 -1|, <0 1 0 2|]
{{Mapping|legend=1| 3 0 7 18 | 0 1 0 -2 }}


[[Wedgie]]: <<3 0 6 -7 1 14||
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 398.7461{{c}}, ~3/2 = 707.0335{{c}} (~21/20 = 90.4587{{c}})
: [[error map]]: {{val| -3.762 +1.317 +4.909 +2.060 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 709.3249{{c}} (~21/20 = 90.6751{{c}})
: error map: {{val| 0.000 +7.370 +13.686 +12.524 }}


[[EDO|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}


[[Badness]]: 0.0265
[[Badness]] (Sintel): 0.628


=== 11-limit ===
=== 11-limit ===
Commas: 36/35, 45/44, 56/55
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 692.514
Comma list: 56/55, 64/63, 100/99


Map: [<3 0 7 -1 1|, <0 1 0 2 2|]
Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}


EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
Optimal tunings:  
* WE: ~5/4 = 398.4962{{c}}, ~3/2 = 708.5030{{c}} (~21/20 = 88.4895{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 711.6031{{c}} (~21/20 = 88.3969{{c}})


Badness: 0.0202
{{Optimal ET sequence|legend=0| 12, 15, 27e }}


=== 13-limit ===
Badness (Sintel): 0.648
Commas: 27/26, 36/35, 45/44, 56/55
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 56/55, 64/63, 66/65
 
Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}
 
Optimal tunings:
* WE: ~5/4 = 398.0488{{c}}, ~3/2 = 708.5402{{c}} (~21/20 = 87.5574{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.6704{{c}} (~21/20 = 87.3296{{c}})
 
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}
 
Badness (Sintel): 0.859
 
==== Ogene ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 688.783
Comma list: 56/55, 64/63, 91/90, 100/99


Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]
Mapping: {{mapping| 3 0 7 18 20 -8 | 0 1 0 -2 -2 4 }}


EDOs: 9, 12, [[21edo|21]], [[33edo|33ef]], [[54edo|54bcef]]
Optimal tunings:  
* WE: ~5/4 = 398.6473{{c}}, ~3/2 = 710.1987{{c}} (~21/20 = 87.0959{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.5057{{c}} (~21/20 = 87.4943{{c}})


Badness: 0.0184
{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}


=== Augustus ===
Badness (Sintel): 0.946
Commas: 26/25, 36/35, 45/44, 56/55


POTE generator: ~3/2 = 685.356
==== Agene ====
Subgroup: 2.3.5.7.11.13


Map: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]
Comma list: 56/55, 64/63, 78/77, 100/99


EDOs: 9, 12, 21f
Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}


Badness: 0.0222
Optimal tunings:  
* WE: ~5/4 = 398.5229{{c}}, ~3/2 = 707.0562{{c}} (~21/20 = 89.9897{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 710.1903{{c}} (~21/20 = 89.8097{{c}})


== Augene ==
{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}
Commas: 64/63, 126/125


[[POTE_tuning|POTE generator]]: 709.257
Badness (Sintel): 0.955


Map: [<3 0 7 18|, <0 1 0 -2|]
=== Eugene ===
Subgroup: 2.3.5.7.11


Wedgie: <<3 0 -6 -7 -18 -14||
Comma list: 55/54, 64/63, 77/75


EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]
Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}


Badness: 0.0248
Optimal tunings:  
* WE: ~5/4 = 399.1743{{c}}, ~3/2 = 712.6763{{c}} (~21/20 = 85.6723{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 713.9414{{c}} (~21/20 = 86.0586{{c}})


=== 11-limit ===
{{Optimal ET sequence|legend=0| 12e, 15, 27, 42 }}
Commas: 56/55, 64/63, 100/99


POTE generator: ~3/2 = 711.177
Badness (Sintel): 1.18


Map: [<3 0 7 18 20|, <0 1 0 -2 -2|]
== 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.


EDOs: 12, 15, 27e
[[Subgroup]]: 2.3.5.7


Badness: 0.0196
[[Comma list]]: 36/35, 128/125


=== 13-limit ===
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}
Commas: 40/39, 56/55, 64/63, 66/65


POTE generator: ~3/2 = 712.013
[[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 }}


Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}


EDOs: 12f, 15, 27ef
[[Badness]] (Sintel): 0.670


Badness: 0.0208
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== Ogene ===
Comma list: 36/35, 45/44, 56/55
Commas: 56/55, 64/63, 91/90, 100/99


POTE generator: ~3/2 = 712.609
Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}


Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]
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}})


EDOs: 12, 15, 27e, 69bcef
{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}


Badness: 0.0229
Badness (Sintel): 0.668


=== Agene ===
==== 13-limit ====
Commas: 56/55, 64/63, 78/77, 100/99
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 709.677
Comma list: 27/26, 36/35, 45/44, 56/55


Map: [<3 0 7 18 20 35|, <0 1 0 -2 -2 -5|]
Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}


EDOs: 12f, 27e
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}})


Badness: 0.0231
{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}


=== Eugene ===
Badness (Sintel): 0.762
Commas: 55/54, 64/63, 77/75


POTE generator: ~3/2 = 714.150
==== Augustus ====
Subgroup: 2.3.5.7.11.13


Map: [<3 0 7 18 -4|, <0 1 0 -2 3|]
Comma list: 26/25, 36/35, 45/44, 56/55


EDOs: 15, 27, 42
Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}


Badness: 0.0356
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}})


* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]
{{Optimal ET sequence|legend=0| 9, 12 }}


* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor]
Badness (Sintel): 0.919


== Inflated ==
== Inflated ==
Commas: 28/27, 128/125
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 722.719
[[Comma list]]: 28/27, 128/125


Map: [<3 0 7 -6|, <0 1 0 3|]
{{Mapping|legend=1| 3 0 7 -6 | 0 1 0 3 }}


Wedgie: <<3 0 9 -7 6 21||
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 398.4023{{c}}, ~3/2 = 719.8327{{c}} (~25/24 = 76.9719{{c}})
: [[error map]]: {{val| -3.762 +1.317 +4.909 +2.060 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 721.0196{{c}} (~25/24 = 78.9804{{c}})
: error map: {{val| 0.000 +19.065 +13.686 -5.767 }}


EDOs: 15, 48bc, 63bc, 78bc
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}


Badness: 0.0547
[[Badness]] (Sintel): 1.39


=== 11-limit ===
=== 11-limit ===
Commas: 28/27, 55/54, 128/125
Subgroup: 2.3.5.7.11
 
Comma list: 28/27, 55/54, 128/125


POTE generator: ~3/2 = 722.663
Mapping: {{mapping| 3 0 7 -6 -4 | 0 1 0 3 3 }}


Map: [<3 0 7 -6|, <0 1 0 3|]
Optimal tunings:  
* WE: ~5/4 = 398.4016{{c}}, ~3/2 = 719.7758{{c}} (~25/24 = 77.0275{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 720.9386{{c}} (~25/24 = 79.0614{{c}})


EDOs: 15, 48bce, 63bce, 78bce
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}


Badness: 0.0312
Badness (Sintel): 1.03


== Deflated ==
== Deflated ==
Commas: 21/20, 128/125
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 21/20, 128/125
 
{{Mapping|legend=1| 3 0 7 13 | 0 1 0 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 401.9566{{c}}, ~3/2 = 684.9634{{c}} (~16/15 = 118.9497{{c}})
: [[error map]]: {{val| +5.870 -11.122 +27.382 -34.224 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 682.2587{{c}} (~16/15 = 117.7413{{c}})
: error map: {{val| 0.000 -19.696 +13.686 -51.085 }}
 
{{Optimal ET sequence|legend=1| 3, 6b, 9 }}
 
[[Badness]] (Sintel): 1.50


POTE generator: ~3/2 = 681.629
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<3 0 7 13|, <0 1 0 -1|]
Comma list: 21/20, 33/32, 128/125


Wedgie: <<3 0 -3 -7 -13 -7||
Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}


EDOs: 3, 9, 18bd, 21d, 30bd
Optimal tunings:  
* WE: ~5/4 = 402.1799{{c}}, ~3/2 = 683.7477{{c}} (~16/15 = 120.6120{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 680.0162{{c}} (~16/15 = 119.9838{{c}})


Badness: 0.0591
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}
 
Badness (Sintel): 1.23


== Hexe ==
== Hexe ==
Commas: 50/49, 128/125
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
 
[[Comma list]]: 50/49, 128/125


[[POTE_tuning|POTE generator]]: 710.963
{{Mapping|legend=1| 6 0 14 17 | 0 1 0 0 }}


Map: [<6 0 14 17|, <0 1 0 0|]
: mapping generators: ~28/25, ~3


Wedgie: <<6 0 0 -14 -17 0||
[[Optimal tuning]]s:  
* [[WE]]: ~28/25 = 199.0488{{c}}, ~3/2 = 707.5815{{c}} (~25/24 = 88.6137{{c}})
: [[error map]]: {{val| +5.870 -11.122 +27.382 -34.224 }}
* [[CWE]]: ~28/25 = 200.0000{{c}}, ~3/2 = 708.6907{{c}} (~25/24 = 91.3093{{c}})
: error map: {{val| 0.000 +6.735 +13.686 +31.174 }}


EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[42edo|42d]], [[54edo|54cd]]
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}


Badness: 0.0577
[[Badness]] (Sintel): 1.46


=== 11-limit ===
=== 11-limit ===
Commas: 50/49, 56/55, 125/121
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 56/55, 125/121


POTE generator: ~3/2 = 714.304
Mapping: {{mapping| 6 0 14 17 21 | 0 1 0 0 0 }}


Map: [<6 0 14 17 21|, <0 1 0 0 0|]
Optimal tunings:  
* WE: ~28/25 = 198.6942{{c}}, ~3/2 = 709.6404{{c}} (~25/24 = 85.1362{{c}})
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 711.8043{{c}} (~25/24 = 88.1957{{c}})


EDOs: 6, 12, 30de
{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}


Badness: 0.0384
Badness (Sintel): 1.27


=== 13-limit ===
=== 13-limit ===
Commas: 50/49, 56/55, 66/65, 105/104
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 710.005
Comma list: 50/49, 56/55, 66/65, 105/104


Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]
Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}


EDOs: 12f
Optimal tunings:  
* WE: ~28/25 = 198.4492{{c}}, ~3/2 = 704.4994{{c}} (~25/24 = 89.2973{{c}})
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 706.6050{{c}} (~16/15 = 93.3950{{c}})


Badness: 0.0359
{{Optimal ET sequence|legend=0| 6f, 12f }}


== Hemiaug ==
Badness (Sintel): 1.49
Commas: 128/125, 245/243
 
== 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
 
[[Comma list]]: 49/48, 128/125


[[POTE_tuning|POTE generator]]: ~28/27 = 45.118
{{Mapping|legend=1| 3 0 7 6 | 0 2 0 1 }}


Map: [<3 1 7 -1|, <0 2 0 5|]
: mapping generators: ~5/4, ~7/4


Wedgie: <<6 0 15 -14 7 35||
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 399.7480{{c}}, ~7/4 = 952.3507{{c}} (~35/32 = 152.8547{{c}})
: [[error map]]: {{val| -0.756 +2.746 +11.922 -17.987 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7463{{c}} (~35/32 = 152.7463{{c}})
: error map: {{val| 0.000 +3.538 +13.686 -16.080 }}


EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}


Badness: 0.0705
[[Badness]] (Sintel): 1.39


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 128/125, 245/243
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 56/55, 77/75
 
Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}
 
Optimal tunings:
* WE: ~5/4 = 399.7654{{c}}, ~7/4 = 952.3730{{c}} (~12/11 = 152.8421{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7447{{c}} (~12/11 = 152.7447{{c}})
 
{{Optimal ET sequence|legend=0| 6, 9, 15, 24, 39 }}


POTE generator: ~28/27 = 45.788
Badness (Sintel): 0.865


Map: [<3 1 7 -1 1|, <0 2 0 5 5|]
; Music
* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)'']{{dead link}} by [[Igliashon Jones]] (2018)
* [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)


EDOs: 24, 27e, 51ce, 78ce
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0382
Comma list: 49/48, 56/55, 66/65, 77/75


=== 13-limit ===
Mapping: {{mapping| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}
Commas: 56/55, 91/90, 128/125, 245/243
 
Optimal tunings:
* WE: ~5/4 = 399.7107{{c}}, ~7/4 = 950.9983{{c}} (~12/11 = 151.5768{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 951.4465{{c}} (~12/11 = 151.4465{{c}})
 
{{Optimal ET sequence|legend=0| 6f, 9, 15, 24 }}
 
Badness (Sintel): 0.837


POTE generator: ~28/27 = 46.250
; Scales
* [[triphi]], Triforce[9] with L:s = phi


Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]
==== Semitriforce ====
This extension splits the period into 1/6-octave for ~44/39. Its ploidacot is hexaploid dicot.


EDOs: 24, 27e, 51ce, 78ce
Subgroup: 2.3.5.7.11.13


Badness: 0.0302
Comma list: 49/48, 56/55, 77/75, 507/500


== Triforce ==
Mapping: {{mapping| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}
Commas: 49/48, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.049
: mapping generators: ~44/39, ~7/4


Map: [<3 0 7 6|, <0 2 0 1|]
Optimal tunings:  
* WE: ~44/39 = 199.8321{{c}}, ~7/4 = 952.5580{{c}} (~40/39 = 46.6024{{c}})
* CWE: ~44/39 = 200.0000{{c}}, ~7/4 = 953.2005{{c}} (~40/39 = 46.7995{{c}})


Wedgie: <<6 0 3 -14 -12 7||
{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}


EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
Badness (Sintel): 2.44


Badness: 0.0202
== 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.  


[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 128/125, 245/243
Commas: 56/55, 77/75, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
{{Mapping|legend=1| 3 1 7 -1 | 0 2 0 5 }}


Map: [<3 0 7 6 8|, <0 2 0 1 1|]
: mapping generators: ~5/4, ~14/9


EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.9278{{c}}, ~14/9 = 752.8583{{c}} (~36/35 = 44.9973{{c}})
: [[error map]]: {{val| -3.217 +2.689 +6.181 -3.462 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~14/9 = 754.2078{{c}} (~36/35 = 45.7922{{c}})
: error map: {{val| 0.000 +6.461 +13.686 +2.213 }}


Badness: 0.0262
{{Optimal ET sequence|legend=1| 24, 27 }}


==== Musical example ====
[[Badness]] (Sintel): 1.78
* [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones


=== 13-limit ===
=== 11-limit ===
Commas: 49/48, 56/55, 66/65, 77/75
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~7/6 = 248.313
Comma list: 56/55, 128/125, 243/242


Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]
Mapping: {{mapping| 3 1 7 -1 1 | 0 2 0 5 5 }}


EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
Optimal tunings:  
* WE: ~5/4 = 398.8946{{c}}, ~14/9 = 752.1272{{c}} (~36/35 = 45.6619{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.5000{{c}} (~36/35 = 46.5000{{c}})


Badness: 0.0202
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


[[triphi|Triforce(9) with L:s = phi]]
Badness (Sintel): 1.26


=== Semitriforce ===
=== 13-limit ===
Commas: 49/48, 56/55, 77/75, 507/500
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/6 = 246.642
Comma list: 56/55, 91/90, 128/125, 243/242


Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]
Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}


EDOs: 6, 24, 54cd, 78cd, 102cdf
Optimal tunings:  
* WE: ~5/4 = 399.1053{{c}}, ~14/9 = 752.0643{{c}} (~36/35 = 46.1463{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.3806{{c}} (~36/35 = 46.6194{{c}})


Badness: 0.0592
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


Badness (Sintel): 1.25


== Hemiug ==
== Hemiug ==
Commas: 128/125, 1323/1250
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.


POTE generator: ~21/20 = 52.093
[[Subgroup]]: 2.3.5.7


Map: [<3 1 7 14|, <0 2 0 -3|]
[[Comma list]]: 128/125, 1323/1250


Wedgie: <<6 0 -9 -14 -31 -21||
{{Mapping|legend=1| 3 1 7 14 | 0 2 0 -3 }}


EDOs: 21, 24, 45c, 69cd
: mapping generators: ~5/4, ~32/21


Badness: 0.1378
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 400.1805{{c}}, ~32/21 = 748.2436{{c}} (~21/20 = 52.1174{{c}})
: [[error map]]: {{val| +0.542 -5.287 +14.950 -11.030 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~32/21 = 747.9138{{c}} (~21/20 = 52.0862{{c}})
: error map: {{val| 0.000 -6.127 +13.686 -12.567 }}
 
{{Optimal ET sequence|legend=1| 21, 24, 45c }}
 
[[Badness]] (Sintel): 3.49


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 128/125, 1323/1250
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 51.655
Comma list: 56/55, 128/125, 1323/1250


Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]
Mapping: {{mapping| 3 1 7 14 16 | 0 2 0 -3 -3 }}


EDOs: 21, 24, 69cd, 93cd
Optimal tunings:  
* WE: ~5/4 = 400.0637{{c}}, ~32/21 = 748.4638{{c}} (~33/32 = 51.6637{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.3383{{c}} (~33/32 = 51.6617{{c}})


Badness: 0.0681
{{Optimal ET sequence|legend=0| 21, 24 }}
 
Badness (Sintel): 2.25


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 66/65, 105/104, 507/500
Subgroup: 2.3.5.7.11.13
 
POTE generator: ~21/20 = 51.548


Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]
Comma list: 56/55, 66/65, 105/104, 507/500


EDOs: 21, 24, 69cdf, 93cdf
Mapping: {{mapping| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}


Badness: 0.0423
Optimal tunings:  
* WE: ~5/4 = 399.8855{{c}}, ~32/21 = 748.2378{{c}} (~33/32 = 51.5332{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.4655{{c}} (~33/32 = 51.5345{{c}})


== Trug ==
{{Optimal ET sequence|legend=0| 21, 24 }}
Commas: 128/125, 360/343


POTE generator: ~48/35 = 501.980
Badness (Sintel): 1.75


Map: [<3 1 7 6|, <0 3 0 2|]
== 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.


EDOs: 9bd, 12
[[Subgroup]]: 2.3.5.7


Badness: 0.1383
[[Comma list]]: 128/125, 2401/2400


== Oodako ==
{{Mapping|legend=1| 3 3 7 8 | 0 4 0 1 }}
Commas: 128/125, 2401/2400


POTE generator: ~8/7 = 223.3540873874507317
: mapping generators: ~5/4, ~28/25


Map: [<3 3 7 8|, <0 4 0 1|]
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 399.0296{{c}}, ~28/25 = 176.2174{{c}} (~49/48 = 46.5949{{c}})
: [[error map]]: {{val| -2.911 +0.004 +6.894 -0.371 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~28/25 = 176.2984{{c}} (~49/48 = 47.4031{{c}})
: error map: {{val| 0.000 +3.239 +13.686 +7.473 }}


Wedgie: <<12 0 3 -28 -29 7||
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}


EDOs: 6, 21, 27, 75c, 102cd, 129bcd
[[Badness]] (Sintel): 2.86


Badness: 0.1132
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit ===
Comma list: 56/55, 128/125, 2401/2400
Commas: 56/55, 128/125, 3773/3750


POTE generator: ~8/7 = 223.019
Mapping: {{mapping| 3 3 7 8 10 | 0 4 0 1 1 }}


Map: [<3 3 7 8 10|, <0 4 0 1 1|]
Optimal tunings:  
* WE: ~5/4 = 398.6615{{c}}, ~11/10 = 176.3886{{c}} (~49/48 = 45.8843{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.5471{{c}} (~49/48 = 46.9059{{c}})


EDOs: 6, 21, 27e
{{Optimal ET sequence|legend=0| 6, 21, 27e }}


Badness: 0.0592
Badness (Sintel): 1.96


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 78/77, 128/125, 507/500
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 128/125, 507/500


POTE generator: ~8/7 = 223.449
Mapping: {{mapping| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}


Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]
Optimal tunings:  
* WE: ~5/4 = 398.8612{{c}}, ~11/10 = 176.0486{{c}} (~49/48 = 46.7640{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.3326{{c}} (~49/48 = 47.3348{{c}})


EDOs: 6, 21, 27e
{{Optimal ET sequence|legend=0| 6, 21, 27e }}


Badness: 0.0423
Badness (Sintel): 1.75


== Hemisemiaug ==
== Hemisemiaug ==
Commas: 128/125, 12005/11664
Hemisemiaug tempers out 12005/11664 and splits both the period and generator of augmented in two. Its ploidacot is hexaploid alpha-dicot.


POTE generator: ~15/14 = 144.515
[[Subgroup]]: 2.3.5.7


Map: [<6 1 14 4|, <0 2 0 3|]
[[Comma list]]: 128/125, 12005/11664


Wedgie: <<12 0 18 -28 -5 42||
{{Mapping|legend=1| 6 1 14 4 | 0 2 0 3 }}


Edos: 18, 24, 42, 66c, 108bc
: mapping generators: ~54/49, ~45/28


Badness: 0.2110
[[Optimal tuning]]s:
* [[WE]]: ~54/49 = 199.5469{{c}}, ~45/28 = 853.5468{{c}} (~36/35 = 55.3594{{c}})
: [[error map]]: {{val| -2.719 +4.686 +7.342 -9.998 }}
* [[CWE]]: ~54/49 = 200.0000{{c}}, ~45/28 = 854.7144{{c}} (~36/35 = 54.7144{{c}})
: error map: {{val| 0.000 +7.474 +13.686 -4.683 }}
 
{{Optimal ET sequence|legend=1| 18, 24, 42 }}
 
[[Badness]] (Sintel): 5.34


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 128/125, 3773/3645
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 144.780
Comma list: 56/55, 128/125, 3773/3645


Map: [<6 1 14 4 8|, <0 2 0 3 3|]
Mapping: {{mapping| 6 1 14 4 8 | 0 2 0 3 3 }}


EDOs: 24, 42e, 66ce, 108bce
Optimal tunings:  
* WE: ~54/49 = 199.5188{{c}}, ~18/11 = 853.1623{{c}} (~36/35 = 55.0872{{c}})
* CWE: ~54/49 = 200.0000{{c}}, ~18/11 = 854.3545{{c}} (~36/35 = 54.3545{{c}})


Badness: 0.0807
{{Optimal ET sequence|legend=0| 18e, 24, 42e, 66ce, 108bccee }}
 
Badness (Sintel): 2.67


== Niner ==
== 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


Commas: 128/125, 686/675
[[Comma list]]: 128/125, 686/675


POTE generator: ~3/2 = 707.167
{{Mapping|legend=1| 9 0 21 11 | 0 1 0 1 }}


Map: [<9 0 21 11|, <0 1 0 1|]
: mapping generators: ~49/45, ~3


Wedgie: <<9 0 9 -21 -11 21||
[[Optimal tuning]]s:  
* [[WE]]: ~49/45 = 133.0272{{c}}, ~3/2 = 705.5438{{c}} (~36/35 = 40.4075{{c}})
: [[error map]]: {{val| -2.755 +0.834 +7.259 -2.737 }}
* [[CWE]]: ~49/45 = 133.3333{{c}}, ~3/2 = 705.5157{{c}} (~36/35 = 38.8490{{c}})
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}


EDOs: 9, 18, 27, 63c, 90c
{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}


Badness: 0.0672
[[Badness]] (Sintel): 1.70


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 128/125, 540/539
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 128/125, 540/539


POTE generator: ~3/2 = 706.726
Mapping: {{mapping| 9 0 21 11 17 | 0 1 0 1 1 }}


Map: [<9 0 21 11 17|, <0 1 0 1 1|]
Optimal tunings:  
* WE: ~12/11 = 132.9553{{c}}, ~3/2 = 704.7217{{c}} (~36/35 = 39.9453{{c}})
* CWE: ~12/11 = 133.3333{{c}}, ~3/2 = 704.5723{{c}} (~36/35 = 37.9056{{c}})


EDOs: 9, 27e, 36, 63ce
{{Optimal ET sequence|legend=0| 9, 18e, 27e, 63cee }}


Badness: 0.0349
Badness (Sintel): 1.15


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 78/77, 91/90, 128/125
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 91/90, 128/125
 
Mapping: {{mapping| 9 0 21 11 17 19 | 0 1 0 1 1 1 }}
 
Optimal tunings:
* WE: ~14/13 = 133.0143{{c}}, ~3/2 = 705.1969{{c}} (~36/35 = 40.1256{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~3/2 = 705.0176{{c}} (~36/35 = 38.3510{{c}})
 
{{Optimal ET sequence|legend=0| 9, 18e, 27e }}
 
Badness (Sintel): 0.998
 
== 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
 
[[Comma list]]: 128/125, 360/343


POTE generator: ~3/2 = 706.889
{{Mapping|legend=1| 3 1 7 6 | 0 3 0 2 }}


Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]
: mapping generators: ~5/4, ~48/35


EDOs: 9, 27e, 36, 63ce
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 398.2337{{c}}, ~48/35 = 499.7635{{c}} (~15/14 = 101.5299{{c}})
: [[error map]]: {{val| -2.755 +0.834 +7.259 -2.737 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~48/35 = 500.9654{{c}} (~15/14 = 100.9654{{c}})
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}


Badness: 0.0241
{{Optimal ET sequence|legend=1| 3b, 9bd, 12 }}


== Music ==
[[Badness]] (Sintel): 3.50
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]


[[Category:Theory]]
== External links ==
[[Category:Temperament family]]
* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
[[Category:Augmented]]


<!-- interwiki links -->
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[[Category:Augmented family| ]] <!-- main article -->
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