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{{Interwiki
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.
| 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]].  


== Seven limit children ==
== Augmented ==
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.
{{Main| Augmented (temperament) }}


== August ==
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.
[[Comma]]s: 36/35, 128/125
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 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 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.
 
== Septimal augmented (augene) ==
{{Main| Augmented (temperament) }}
 
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


[[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
 
Comma list: 56/55, 64/63, 100/99
 
Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 12, 15, 27e }}
 
Badness (Sintel): 0.648
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 56/55, 64/63, 66/65


POTE generator: ~3/2 = 692.514
Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}


Map: [<3 0 7 -1 1|, <0 1 0 2 2|]
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}})


EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}


Badness: 0.0202
Badness (Sintel): 0.859


=== 13-limit ===
==== Ogene ====
Commas: 27/26, 36/35, 45/44, 56/55
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


POTE generator: ~3/2 = 722.663
Comma list: 28/27, 55/54, 128/125


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


EDOs: 15, 48bce, 63bce, 78bce
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}})


Badness: 0.0312
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}
 
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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 681.629
Comma list: 21/20, 33/32, 128/125


Map: [<3 0 7 13|, <0 1 0 -1|]
Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}


Wedgie: <<3 0 -3 -7 -13 -7||
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}})


EDOs: 3, 9, 18bd, 21d, 30bd
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}


Badness: 0.0591
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
 
Comma list: 50/49, 56/55, 66/65, 105/104
 
Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 6f, 12f }}
 
Badness (Sintel): 1.49
 
== 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
 
{{Mapping|legend=1| 3 0 7 6 | 0 2 0 1 }}
 
: mapping generators: ~5/4, ~7/4
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
 
[[Badness]] (Sintel): 1.39


POTE generator: ~3/2 = 710.005
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]
Comma list: 49/48, 56/55, 77/75


EDOs: 12f
Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}


Badness: 0.0359
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 }}
 
Badness (Sintel): 0.865
 
; 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)
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 66/65, 77/75
 
Mapping: {{mapping| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}
 
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
 
; Scales
* [[triphi]], Triforce[9] with L:s = phi
 
==== 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
 
Comma list: 49/48, 56/55, 77/75, 507/500
 
Mapping: {{mapping| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}
 
: mapping generators: ~44/39, ~7/4
 
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}})
 
{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}
 
Badness (Sintel): 2.44


== Hemiaug ==
== Hemiaug ==
Commas: 128/125, 245/243
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.


[[POTE_tuning|POTE generator]]: ~28/27 = 45.118
[[Subgroup]]: 2.3.5.7


Map: [<3 1 7 -1|, <0 2 0 5|]
[[Comma list]]: 128/125, 245/243


Wedgie: <<6 0 15 -14 7 35||
{{Mapping|legend=1| 3 1 7 -1 | 0 2 0 5 }}


EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
: mapping generators: ~5/4, ~14/9


Badness: 0.0705
[[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 }}
 
{{Optimal ET sequence|legend=1| 24, 27 }}
 
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 128/125, 245/243
Subgroup: 2.3.5.7.11


POTE generator: ~28/27 = 45.788
Comma list: 56/55, 128/125, 243/242


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


EDOs: 24, 27e, 51ce, 78ce
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: 0.0382
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
 
Badness (Sintel): 1.26


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 91/90, 128/125, 245/243
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 128/125, 243/242
 
Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}


POTE generator: ~28/27 = 46.250
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}})


Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


EDOs: 24, 27e, 51ce, 78ce
Badness (Sintel): 1.25


Badness: 0.0302
== 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.  


== Triforce ==
[[Subgroup]]: 2.3.5.7
Commas: 49/48, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.049
[[Comma list]]: 128/125, 1323/1250


Map: [<3 0 7 6|, <0 2 0 1|]
{{Mapping|legend=1| 3 1 7 14 | 0 2 0 -3 }}


Wedgie: <<6 0 3 -14 -12 7||
: mapping generators: ~5/4, ~32/21


EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
[[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 }}


Badness: 0.0202
{{Optimal ET sequence|legend=1| 21, 24, 45c }}


[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
[[Badness]] (Sintel): 3.49


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 77/75, 128/125
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
Comma list: 56/55, 128/125, 1323/1250


Map: [<3 0 7 6 8|, <0 2 0 1 1|]
Mapping: {{mapping| 3 1 7 14 16 | 0 2 0 -3 -3 }}


EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
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: 0.0262
{{Optimal ET sequence|legend=0| 21, 24 }}


==== Musical example ====
Badness (Sintel): 2.25
* [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones


=== 13-limit ===
=== 13-limit ===
Commas: 49/48, 56/55, 66/65, 77/75
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 66/65, 105/104, 507/500
 
Mapping: {{mapping| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}
 
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}})


[[POTE_tuning|POTE generator]]: ~7/6 = 248.313
{{Optimal ET sequence|legend=0| 21, 24 }}


Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]
Badness (Sintel): 1.75


EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
== 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.


Badness: 0.0202
[[Subgroup]]: 2.3.5.7


[[triphi|Triforce(9) with L:s = phi]]
[[Comma list]]: 128/125, 2401/2400


=== Semitriforce ===
{{Mapping|legend=1| 3 3 7 8 | 0 4 0 1 }}
Commas: 49/48, 56/55, 77/75, 507/500


POTE generator: ~7/6 = 246.642
: mapping generators: ~5/4, ~28/25


Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -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 }}


EDOs: 6, 24, 54cd, 78cd, 102cdf
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}


Badness: 0.0592
[[Badness]] (Sintel): 2.86


=== 11-limit ===
Subgroup: 2.3.5.7.11


== Hemiug ==
Comma list: 56/55, 128/125, 2401/2400
Commas: 128/125, 1323/1250
 
Mapping: {{mapping| 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}})
 
{{Optimal ET sequence|legend=0| 6, 21, 27e }}
 
Badness (Sintel): 1.96
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 128/125, 507/500


POTE generator: ~21/20 = 52.093
Mapping: {{mapping| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}


Map: [<3 1 7 14|, <0 2 0 -3|]
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}})


Wedgie: <<6 0 -9 -14 -31 -21||
{{Optimal ET sequence|legend=0| 6, 21, 27e }}


EDOs: 21, 24, 45c, 69cd
Badness (Sintel): 1.75


Badness: 0.1378
== Hemisemiaug ==
Hemisemiaug tempers out 12005/11664 and splits both the period and generator of augmented in two. Its ploidacot is hexaploid alpha-dicot.  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Commas: 56/55, 128/125, 1323/1250


POTE generator: ~21/20 = 51.655
[[Comma list]]: 128/125, 12005/11664


Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]
{{Mapping|legend=1| 6 1 14 4 | 0 2 0 3 }}


EDOs: 21, 24, 69cd, 93cd
: mapping generators: ~54/49, ~45/28


Badness: 0.0681
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 18, 24, 42 }}
Commas: 56/55, 66/65, 105/104, 507/500


POTE generator: ~21/20 = 51.548
[[Badness]] (Sintel): 5.34


Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 21, 24, 69cdf, 93cdf
Comma list: 56/55, 128/125, 3773/3645


Badness: 0.0423
Mapping: {{mapping| 6 1 14 4 8 | 0 2 0 3 3 }}


== Trug ==
Optimal tunings:
Commas: 128/125, 360/343
* 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}})


POTE generator: ~48/35 = 501.980
{{Optimal ET sequence|legend=0| 18e, 24, 42e, 66ce, 108bccee }}


Map: [<3 1 7 6|, <0 3 0 2|]
Badness (Sintel): 2.67


EDOs: 9bd, 12
== 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.


Badness: 0.1383
It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.


== Oodako ==
[[Subgroup]]: 2.3.5.7
Commas: 128/125, 2401/2400


POTE generator: ~8/7 = 223.3540873874507317
[[Comma list]]: 128/125, 1029/1000


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


Wedgie: <<12 0 3 -28 -29 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 }}


EDOs: 6, 21, 27, 75c, 102cd, 129bcd
{{Optimal ET sequence|legend=1| 15, 36, 51c }}


Badness: 0.1132
[[Badness]] (Sintel): 2.61


=== 11-limit ===
=== 11-limit ===
Commas: 56/55, 128/125, 3773/3750
This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]].


POTE generator: ~8/7 = 223.019
Subgroup: 2.3.5.7.11


Map: [<3 3 7 8 10|, <0 4 0 1 1|]
Comma list: 56/55, 128/125, 1029/1000


EDOs: 6, 21, 27e
Mapping: {{Mapping| 3 0 7 10 12 | 0 3 0 -1 -1 }}


Badness: 0.0592
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 ===
=== 13-limit ===
Commas: 56/55, 78/77, 128/125, 507/500
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


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


Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]
Optimal tunings:  
* WE: ~5/4 = 399.3194{{c}}, ~10/7 = 634.0637{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.9907{{c}}


EDOs: 6, 21, 27e
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}


Badness: 0.0423
Badness (Sintel): 1.70


== Hemisemiaug ==
== Niner ==
Commas: 128/125, 12005/11664
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
 
[[Comma list]]: 128/125, 686/675


POTE generator: ~15/14 = 144.515
{{Mapping|legend=1| 9 0 21 11 | 0 1 0 1 }}


Map: [<6 1 14 4|, <0 2 0 3|]
: mapping generators: ~49/45, ~3


Wedgie: <<12 0 18 -28 -5 42||
[[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: 18, 24, 42, 66c, 108bc
{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}


Badness: 0.2110
[[Badness]] (Sintel): 1.70


=== 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, 540/539


Map: [<6 1 14 4 8|, <0 2 0 3 3|]
Mapping: {{mapping| 9 0 21 11 17 | 0 1 0 1 1 }}


EDOs: 24, 42e, 66ce, 108bce
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}})


Badness: 0.0807
{{Optimal ET sequence|legend=0| 9, 18e, 27e, 63cee }}


== Niner ==
Badness (Sintel): 1.15
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.
 
=== 13-limit ===
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


Commas: 128/125, 686/675
[[Comma list]]: 128/125, 360/343


POTE generator: ~3/2 = 707.167
{{Mapping|legend=1| 3 1 7 6 | 0 3 0 2 }}


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


Wedgie: <<9 0 9 -21 -11 21||
[[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 }}


EDOs: 9, 18, 27, 63c, 90c
{{Optimal ET sequence|legend=1| 3b, 9bd, 12 }}


Badness: 0.0672
[[Badness]] (Sintel): 3.50


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


POTE generator: ~3/2 = 706.726
Comma list: 56/55, 128/125, 360/343


Map: [<9 0 21 11 17|, <0 1 0 1 1|]
Mapping: {{Mapping| 3 1 7 6 8 | 0 3 0 2 2 }}


EDOs: 9, 27e, 36, 63ce
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}})


Badness: 0.0349
{{Optimal ET sequence|legend=0| 3b, 9bde, 12 }}


=== 13-limit ===
Badness (Sintel): 2.60
Commas: 56/55, 78/77, 91/90, 128/125
 
== 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


POTE generator: ~3/2 = 706.889
[[Comma list]]: 76/75, 96/95


Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]
{{Mapping|legend=2| 3 0 7 8 | 0 1 0 1 }}


EDOs: 9, 27e, 36, 63ce
[[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}})


Badness: 0.0241
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cch }}


== Music ==
[[Badness]] (Sintel): 0.264
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]


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


<!-- interwiki links -->
[[Category:Temperament families]]
[[de:Übermässige Temperaturen]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 22:09, 19 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The augmented family of temperaments 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

Main article: 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

Comma list: 128/125

Mapping: [⟨3 0 7], ⟨0 1 0]]

mapping generators: ~5/4, ~3

Optimal tunings:

  • WE: ~5/4 = 399.0128 ¢, ~3/2 = 704.8937 ¢ (~16/15 = 93.1320 ¢)
error map: ⟨-2.962 -0.023 +6.776]
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 705.0691 ¢ (~16/15 = 94.9309 ¢)
error map: ⟨0.000 +3.114 +13.686]

Optimal ET sequence: 3, 9, 12, 27, 39, 51c, 90cc

Badness (Sintel): 0.523

Overview to extensions

The second comma of the 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 ¢ major third to 14/11 as well as 5/4.

Septimal augmented (augene)

Main article: Augmented (temperament)

Septimal augmented, a.k.a. augene, tempers out 64/63 and 126/125. It may be described as the 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

Comma list: 64/63, 126/125

Mapping: [⟨3 0 7 18], ⟨0 1 0 -2]]

Optimal tunings:

  • WE: ~5/4 = 398.7461 ¢, ~3/2 = 707.0335 ¢ (~21/20 = 90.4587 ¢)
error map: ⟨-3.762 +1.317 +4.909 +2.060]
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 709.3249 ¢ (~21/20 = 90.6751 ¢)
error map: ⟨0.000 +7.370 +13.686 +12.524]

Optimal ET sequence: 12, 27, 39d, 66cd

Badness (Sintel): 0.628

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 100/99

Mapping: [⟨3 0 7 18 20], ⟨0 1 0 -2 -2]]

Optimal tunings:

  • WE: ~5/4 = 398.4962 ¢, ~3/2 = 708.5030 ¢ (~21/20 = 88.4895 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 711.6031 ¢ (~21/20 = 88.3969 ¢)

Optimal ET sequence: 12, 15, 27e

Badness (Sintel): 0.648

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 64/63, 66/65

Mapping: [⟨3 0 7 18 20 16], ⟨0 1 0 -2 -2 -1]]

Optimal tunings:

  • WE: ~5/4 = 398.0488 ¢, ~3/2 = 708.5402 ¢ (~21/20 = 87.5574 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 712.6704 ¢ (~21/20 = 87.3296 ¢)

Optimal ET sequence: 12f, 15, 27eff

Badness (Sintel): 0.859

Ogene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 91/90, 100/99

Mapping: [⟨3 0 7 18 20 -8], ⟨0 1 0 -2 -2 4]]

Optimal tunings:

  • WE: ~5/4 = 398.6473 ¢, ~3/2 = 710.1987 ¢ (~21/20 = 87.0959 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 712.5057 ¢ (~21/20 = 87.4943 ¢)

Optimal ET sequence: 12, 15, 27e, 69bceef

Badness (Sintel): 0.946

Agene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 100/99

Mapping: [⟨3 0 7 18 20 35], ⟨0 1 0 -2 -2 -5]]

Optimal tunings:

  • WE: ~5/4 = 398.5229 ¢, ~3/2 = 707.0562 ¢ (~21/20 = 89.9897 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 710.1903 ¢ (~21/20 = 89.8097 ¢)

Optimal ET sequence: 12f, 27e, 66cdeeef

Badness (Sintel): 0.955

Eugene

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 77/75

Mapping: [⟨3 0 7 18 -4], ⟨0 1 0 -2 3]]

Optimal tunings:

  • WE: ~5/4 = 399.1743 ¢, ~3/2 = 712.6763 ¢ (~21/20 = 85.6723 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 713.9414 ¢ (~21/20 = 86.0586 ¢)

Optimal ET sequence: 12e, 15, 27, 42

Badness (Sintel): 1.18

August

August tempers out 36/35 and 225/224. It may be described as the 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: [⟨3 0 7 -1], ⟨0 1 0 2]]

Optimal tunings:

  • WE: ~5/4 = 399.1036 ¢, ~3/2 = 694.4509 ¢ (~16/15 = 103.7564 ¢)
error map: ⟨-2.689 -10.193 +7.412 +15.594]
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 694.6812 ¢ (~16/15 = 105.3188 ¢)
error map: ⟨0.000 -7.274 +13.686 +20.537]

Optimal ET sequence: 9, 12, 45cd

Badness (Sintel): 0.670

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 56/55

Mapping: [⟨3 0 7 -1 1], ⟨0 1 0 2 2]]

Optimal tunings:

  • WE: ~5/4 = 398.9225 ¢, ~3/2 = 690.6486 ¢ (~16/15 = 107.1966 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 690.8519 ¢ (~16/15 = 109.1481 ¢)

Optimal ET sequence: 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: [⟨3 0 7 -1 1 -3], ⟨0 1 0 2 2 3]]

Optimal tunings:

  • WE: ~5/4 = 399.0956 ¢, ~3/2 = 687.2261 ¢ (~16/15 = 110.9651 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 687.5057 ¢ (~16/15 = 112.4943 ¢)

Optimal ET sequence: 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: [⟨3 0 7 -1 1 11], ⟨0 1 0 2 2 0]]

Optimal tunings:

  • WE: ~5/4 = 400.4230 ¢, ~3/2 = 686.0809 ¢ (~16/15 = 114.7650 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 685.8446 ¢ (~16/15 = 114.1554 ¢)

Optimal ET sequence: 9, 12

Badness (Sintel): 0.919

Inflated

Subgroup: 2.3.5.7

Comma list: 28/27, 128/125

Mapping: [⟨3 0 7 -6], ⟨0 1 0 3]]

Optimal tunings:

  • WE: ~5/4 = 398.4023 ¢, ~3/2 = 719.8327 ¢ (~25/24 = 76.9719 ¢)
error map: ⟨-3.762 +1.317 +4.909 +2.060]
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 721.0196 ¢ (~25/24 = 78.9804 ¢)
error map: ⟨0.000 +19.065 +13.686 -5.767]

Optimal ET sequence: 3d, 12d, 15

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 55/54, 128/125

Mapping: [⟨3 0 7 -6 -4], ⟨0 1 0 3 3]]

Optimal tunings:

  • WE: ~5/4 = 398.4016 ¢, ~3/2 = 719.7758 ¢ (~25/24 = 77.0275 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 720.9386 ¢ (~25/24 = 79.0614 ¢)

Optimal ET sequence: 3de, 12de, 15

Badness (Sintel): 1.03

Deflated

Subgroup: 2.3.5.7

Comma list: 21/20, 128/125

Mapping: [⟨3 0 7 13], ⟨0 1 0 -1]]

Optimal tunings:

  • WE: ~5/4 = 401.9566 ¢, ~3/2 = 684.9634 ¢ (~16/15 = 118.9497 ¢)
error map: ⟨+5.870 -11.122 +27.382 -34.224]
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 682.2587 ¢ (~16/15 = 117.7413 ¢)
error map: ⟨0.000 -19.696 +13.686 -51.085]

Optimal ET sequence: 3, 6b, 9

Badness (Sintel): 1.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 33/32, 128/125

Mapping: [⟨3 0 7 13 15], ⟨0 1 0 -1 -1]]

Optimal tunings:

  • WE: ~5/4 = 402.1799 ¢, ~3/2 = 683.7477 ¢ (~16/15 = 120.6120 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 680.0162 ¢ (~16/15 = 119.9838 ¢)

Optimal ET sequence: 3, 6b, 9

Badness (Sintel): 1.23

Hexe

Hexe tempers out 50/49 and may be described as 6 & 12, viewed as 6et with an independent generator for prime 3. Its ploidacot is hexaploid monocot.

Subgroup: 2.3.5.7

Comma list: 50/49, 128/125

Mapping: [⟨6 0 14 17], ⟨0 1 0 0]]

mapping generators: ~28/25, ~3

Optimal tunings:

  • WE: ~28/25 = 199.0488 ¢, ~3/2 = 707.5815 ¢ (~25/24 = 88.6137 ¢)
error map: ⟨+5.870 -11.122 +27.382 -34.224]
  • CWE: ~28/25 = 200.0000 ¢, ~3/2 = 708.6907 ¢ (~25/24 = 91.3093 ¢)
error map: ⟨0.000 +6.735 +13.686 +31.174]

Optimal ET sequence: 6, 12, 30d, 42dd, 54cdd

Badness (Sintel): 1.46

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 125/121

Mapping: [⟨6 0 14 17 21], ⟨0 1 0 0 0]]

Optimal tunings:

  • WE: ~28/25 = 198.6942 ¢, ~3/2 = 709.6404 ¢ (~25/24 = 85.1362 ¢)
  • CWE: ~28/25 = 200.0000 ¢, ~3/2 = 711.8043 ¢ (~25/24 = 88.1957 ¢)

Optimal ET sequence: 6, 12, 30dee, 42ddeee

Badness (Sintel): 1.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 56/55, 66/65, 105/104

Mapping: [⟨6 0 14 17 21 13], ⟨0 1 0 0 0 1]]

Optimal tunings:

  • WE: ~28/25 = 198.4492 ¢, ~3/2 = 704.4994 ¢ (~25/24 = 89.2973 ¢)
  • CWE: ~28/25 = 200.0000 ¢, ~3/2 = 706.6050 ¢ (~16/15 = 93.3950 ¢)

Optimal ET sequence: 6f, 12f

Badness (Sintel): 1.49

Triforce

triforce9.jpg
Lattice of triforce

Triforce tempers out 49/48 and may be described as 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

Mapping: [⟨3 0 7 6], ⟨0 2 0 1]]

mapping generators: ~5/4, ~7/4

Optimal tunings:

  • WE: ~5/4 = 399.7480 ¢, ~7/4 = 952.3507 ¢ (~35/32 = 152.8547 ¢)
error map: ⟨-0.756 +2.746 +11.922 -17.987]
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 952.7463 ¢ (~35/32 = 152.7463 ¢)
error map: ⟨0.000 +3.538 +13.686 -16.080]

Optimal ET sequence: 6, 9, 15, 24, 39

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 77/75

Mapping: [⟨3 0 7 6 8], ⟨0 2 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 399.7654 ¢, ~7/4 = 952.3730 ¢ (~12/11 = 152.8421 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 952.7447 ¢ (~12/11 = 152.7447 ¢)

Optimal ET sequence: 6, 9, 15, 24, 39

Badness (Sintel): 0.865

Music

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 7 6 8 4], ⟨0 2 0 1 1 3]]

Optimal tunings:

  • WE: ~5/4 = 399.7107 ¢, ~7/4 = 950.9983 ¢ (~12/11 = 151.5768 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~7/4 = 951.4465 ¢ (~12/11 = 151.4465 ¢)

Optimal ET sequence: 6f, 9, 15, 24

Badness (Sintel): 0.837

Scales
  • triphi, Triforce[9] with L:s = phi

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

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

Mapping: [⟨6 0 14 12 16 27], ⟨0 2 0 1 1 -1]]

mapping generators: ~44/39, ~7/4

Optimal tunings:

  • WE: ~44/39 = 199.8321 ¢, ~7/4 = 952.5580 ¢ (~40/39 = 46.6024 ¢)
  • CWE: ~44/39 = 200.0000 ¢, ~7/4 = 953.2005 ¢ (~40/39 = 46.7995 ¢)

Optimal ET sequence: 6, 18bd, 24

Badness (Sintel): 2.44

Hemiaug

Hemiaug tempers out 245/243 and may be described as 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

Comma list: 128/125, 245/243

Mapping: [⟨3 1 7 -1], ⟨0 2 0 5]]

mapping generators: ~5/4, ~14/9

Optimal tunings:

  • WE: ~5/4 = 398.9278 ¢, ~14/9 = 752.8583 ¢ (~36/35 = 44.9973 ¢)
error map: ⟨-3.217 +2.689 +6.181 -3.462]
  • CWE: ~5/4 = 400.0000 ¢, ~14/9 = 754.2078 ¢ (~36/35 = 45.7922 ¢)
error map: ⟨0.000 +6.461 +13.686 +2.213]

Optimal ET sequence: 24, 27

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 243/242

Mapping: [⟨3 1 7 -1 1], ⟨0 2 0 5 5]]

Optimal tunings:

  • WE: ~5/4 = 398.8946 ¢, ~14/9 = 752.1272 ¢ (~36/35 = 45.6619 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~14/9 = 753.5000 ¢ (~36/35 = 46.5000 ¢)

Optimal ET sequence: 24, 27e, 51ce

Badness (Sintel): 1.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 128/125, 243/242

Mapping: [⟨3 1 7 -1 1 13], ⟨0 2 0 5 5 -1]]

Optimal tunings:

  • WE: ~5/4 = 399.1053 ¢, ~14/9 = 752.0643 ¢ (~36/35 = 46.1463 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~14/9 = 753.3806 ¢ (~36/35 = 46.6194 ¢)

Optimal ET sequence: 24, 27e, 51ce

Badness (Sintel): 1.25

Hemiug

Hemiug tempers out 1323/1250 and may be described as 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

Comma list: 128/125, 1323/1250

Mapping: [⟨3 1 7 14], ⟨0 2 0 -3]]

mapping generators: ~5/4, ~32/21

Optimal tunings:

  • WE: ~5/4 = 400.1805 ¢, ~32/21 = 748.2436 ¢ (~21/20 = 52.1174 ¢)
error map: ⟨+0.542 -5.287 +14.950 -11.030]
  • CWE: ~5/4 = 400.0000 ¢, ~32/21 = 747.9138 ¢ (~21/20 = 52.0862 ¢)
error map: ⟨0.000 -6.127 +13.686 -12.567]

Optimal ET sequence: 21, 24, 45c

Badness (Sintel): 3.49

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 1323/1250

Mapping: [⟨3 1 7 14 16], ⟨0 2 0 -3 -3]]

Optimal tunings:

  • WE: ~5/4 = 400.0637 ¢, ~32/21 = 748.4638 ¢ (~33/32 = 51.6637 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~32/21 = 748.3383 ¢ (~33/32 = 51.6617 ¢)

Optimal ET sequence: 21, 24

Badness (Sintel): 2.25

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 105/104, 507/500

Mapping: [⟨3 1 7 14 16 13], ⟨0 2 0 -3 -3 -1]]

Optimal tunings:

  • WE: ~5/4 = 399.8855 ¢, ~32/21 = 748.2378 ¢ (~33/32 = 51.5332 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~32/21 = 748.4655 ¢ (~33/32 = 51.5345 ¢)

Optimal ET sequence: 21, 24

Badness (Sintel): 1.75

Oodako

Oodako tempers out 2401/2400 and may be described as 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

Comma list: 128/125, 2401/2400

Mapping: [⟨3 3 7 8], ⟨0 4 0 1]]

mapping generators: ~5/4, ~28/25

Optimal tunings:

  • WE: ~5/4 = 399.0296 ¢, ~28/25 = 176.2174 ¢ (~49/48 = 46.5949 ¢)
error map: ⟨-2.911 +0.004 +6.894 -0.371]
  • CWE: ~5/4 = 400.0000 ¢, ~28/25 = 176.2984 ¢ (~49/48 = 47.4031 ¢)
error map: ⟨0.000 +3.239 +13.686 +7.473]

Optimal ET sequence: 6, 21, 27, 75c, 102ccd, 129bccd

Badness (Sintel): 2.86

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 2401/2400

Mapping: [⟨3 3 7 8 10], ⟨0 4 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 398.6615 ¢, ~11/10 = 176.3886 ¢ (~49/48 = 45.8843 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~11/10 = 176.5471 ¢ (~49/48 = 46.9059 ¢)

Optimal ET sequence: 6, 21, 27e

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 128/125, 507/500

Mapping: [⟨3 3 7 8 10 12], ⟨0 4 0 1 1 -2]]

Optimal tunings:

  • WE: ~5/4 = 398.8612 ¢, ~11/10 = 176.0486 ¢ (~49/48 = 46.7640 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~11/10 = 176.3326 ¢ (~49/48 = 47.3348 ¢)

Optimal ET sequence: 6, 21, 27e

Badness (Sintel): 1.75

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

Comma list: 128/125, 12005/11664

Mapping: [⟨6 1 14 4], ⟨0 2 0 3]]

mapping generators: ~54/49, ~45/28

Optimal tunings:

  • WE: ~54/49 = 199.5469 ¢, ~45/28 = 853.5468 ¢ (~36/35 = 55.3594 ¢)
error map: ⟨-2.719 +4.686 +7.342 -9.998]
  • CWE: ~54/49 = 200.0000 ¢, ~45/28 = 854.7144 ¢ (~36/35 = 54.7144 ¢)
error map: ⟨0.000 +7.474 +13.686 -4.683]

Optimal ET sequence: 18, 24, 42

Badness (Sintel): 5.34

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 3773/3645

Mapping: [⟨6 1 14 4 8], ⟨0 2 0 3 3]]

Optimal tunings:

  • WE: ~54/49 = 199.5188 ¢, ~18/11 = 853.1623 ¢ (~36/35 = 55.0872 ¢)
  • CWE: ~54/49 = 200.0000 ¢, ~18/11 = 854.3545 ¢ (~36/35 = 54.3545 ¢)

Optimal ET sequence: 18e, 24, 42e, 66ce, 108bccee

Badness (Sintel): 2.67

Trisected

Main article: 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 15 & 36 temperament.

It was named by Overthink in 2026, since it splits every Pythagorean interval in three.

Subgroup: 2.3.5.7

Comma list: 128/125, 1029/1000

Mapping: [⟨3 0 7 10], ⟨0 3 0 -1]]

mapping generators: ~5/4, ~10/7

Optimal tunings:

  • WE: ~5/4 = 399.5199 ¢, ~10/7 = 633.7146 ¢
error map: ⟨-1.440 -0.811 +10.325 -7.342]
  • CWE: ~5/4 = 400.0000 ¢, ~10/7 = 634.3393 ¢
error map: ⟨0.000 +1.063 +13.686 -3.165]

Optimal ET sequence: 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: [⟨3 0 7 10 12], ⟨0 3 0 -1 -1]]

Optimal tunings:

  • WE: ~5/4 = 399.3049 ¢, ~10/7 = 633.7891 ¢
  • CWE: ~5/4 = 400.0000 ¢, ~10/7 = 634.7695 ¢

Optimal ET sequence: 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: [⟨3 0 7 10 12 0], ⟨0 3 0 -1 -1 7]]

Optimal tunings:

  • WE: ~5/4 = 399.3194 ¢, ~10/7 = 634.0637 ¢
  • CWE: ~5/4 = 400.0000 ¢, ~10/7 = 634.9907 ¢

Optimal ET sequence: 15, 36, 51ce

Badness (Sintel): 1.70

Niner

Niner tempers out 686/675 and may be described as the 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

Comma list: 128/125, 686/675

Mapping: [⟨9 0 21 11], ⟨0 1 0 1]]

mapping generators: ~49/45, ~3

Optimal tunings:

  • WE: ~49/45 = 133.0272 ¢, ~3/2 = 705.5438 ¢ (~36/35 = 40.4075 ¢)
error map: ⟨-2.755 +0.834 +7.259 -2.737]
  • CWE: ~49/45 = 133.3333 ¢, ~3/2 = 705.5157 ¢ (~36/35 = 38.8490 ¢)
error map: ⟨0.000 +3.561 +13.686 +3.356]

Optimal ET sequence: 9, 18, 27, 63c, 90cc

Badness (Sintel): 1.70

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 540/539

Mapping: [⟨9 0 21 11 17], ⟨0 1 0 1 1]]

Optimal tunings:

  • WE: ~12/11 = 132.9553 ¢, ~3/2 = 704.7217 ¢ (~36/35 = 39.9453 ¢)
  • CWE: ~12/11 = 133.3333 ¢, ~3/2 = 704.5723 ¢ (~36/35 = 37.9056 ¢)

Optimal ET sequence: 9, 18e, 27e, 63cee

Badness (Sintel): 1.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 128/125

Mapping: [⟨9 0 21 11 17 19], ⟨0 1 0 1 1 1]]

Optimal tunings:

  • WE: ~14/13 = 133.0143 ¢, ~3/2 = 705.1969 ¢ (~36/35 = 40.1256 ¢)
  • CWE: ~14/13 = 133.3333 ¢, ~3/2 = 705.0176 ¢ (~36/35 = 38.3510 ¢)

Optimal ET sequence: 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

Mapping: [⟨3 1 7 6], ⟨0 3 0 2]]

mapping generators: ~5/4, ~48/35

Optimal tunings:

  • WE: ~5/4 = 398.2337 ¢, ~48/35 = 499.7635 ¢ (~15/14 = 101.5299 ¢)
error map: ⟨-2.755 +0.834 +7.259 -2.737]
  • CWE: ~5/4 = 400.0000 ¢, ~48/35 = 500.9654 ¢ (~15/14 = 100.9654 ¢)
error map: ⟨0.000 +3.561 +13.686 +3.356]

Optimal ET sequence: 3b, 9bd, 12

Badness (Sintel): 3.50

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 360/343

Mapping: [⟨3 1 7 6 8], ⟨0 3 0 2 2]]

Optimal tunings:

  • WE: ~5/4 = 397.4954 ¢, ~15/11 = 499.0962 ¢ (~15/14 = 101.6008 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~15/11 = 500.7745 ¢ (~15/14 = 100.7745 ¢)

Optimal ET sequence: 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

Subgroup-val mapping: [⟨3 0 7 8], ⟨0 1 0 1]]

Optimal tunings:

  • WE: ~5/4 = 399.0754 ¢, ~3/2 = 705.0615 ¢ (~19/18 = 93.0893 ¢)
  • CWE: ~5/4 = 400.0000 ¢, ~3/2 = 705.1072 ¢ (~19/18 = 94.8928 ¢)

Optimal ET sequence: 3, 9, 12, 27, 39, 51c, 90cch

Badness (Sintel): 0.264

External links

Retrieved from "https://en.xen.wiki/index.php?title=Augmented_family&oldid=226427"