Xenharmonic Wiki

Augmented family: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled
4,516 edits
mNo edit summary
Intro to some of these temps
 
(41 intermediate revisions by 12 users not shown)
Line 1: Line 1:
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 {{multival|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|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.
{{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]].  


== Augmented ==
== Augmented ==
Subgroup: 2.3.5
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]]: 128/125
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val|3 0 7}}, {{val|0 1 0}}]
[[Comma list]]: 128/125


[[POTE generator]]: ~3/2 = 706.638
{{Mapping|legend=1| 3 0 7 | 0 1 0 }}


{{Val list|legend=1| 3, 12, 27, 39, 51c, 90cc }}
: mapping generators: ~5/4, ~3


[[Badness]]: 0.022315
[[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 }}


=== Seven-limit children ===
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}
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 triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.


== August ==
[[Badness]] (Sintel): 0.523
Subgroup: 2.3.5.7
 
=== 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.  


[[Comma list]]: 36/35, 128/125
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|3 0 7 -1}}, {{val|0 1 0 2}}]
[[Comma list]]: 64/63, 126/125


{{Multival|legend=1|3 0 6 -7 1 14}}
{{Mapping|legend=1| 3 0 7 18 | 0 1 0 -2 }}


[[POTE generator]]: ~3/2 = 696.011
[[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 }}


{{Val list|legend=1| 9, 12, 45cd, 57cd, 69cdd }}
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}


[[Badness]]: 0.026459
[[Badness]] (Sintel): 0.628


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


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


Mapping: [{{val|3 0 7 -1 1}}, {{val|0 1 0 2 2}}]
Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}


POTE generator: ~3/2 = 692.514
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}})


Vals: {{Val list| 9, 12, 21, 33e, 45cde }}
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}


Badness: 0.020191
Badness (Sintel): 0.859


=== 13-limit ===
==== Ogene ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 36/35, 45/44, 56/55
Comma list: 56/55, 64/63, 91/90, 100/99


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


POTE generator: ~3/2 = 688.783
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}})


Vals: {{Val list| 9, 12f, 21, 33ef, 54bceeff }}
{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}


Badness: 0.018448
Badness (Sintel): 0.946


=== Augustus ===
==== Agene ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 36/35, 45/44, 56/55
Comma list: 56/55, 64/63, 78/77, 100/99
 
Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}
 
Badness (Sintel): 0.955
 
=== Eugene ===
Subgroup: 2.3.5.7.11


Mapping: [{{val|3 0 7 -1 1 11}}, {{val|0 1 0 2 2 0}}]
Comma list: 55/54, 64/63, 77/75


POTE generator: ~3/2 = 685.356
Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}


Vals: {{Val list| 9, 12, 21f }}
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}})


Badness: 0.022236
{{Optimal ET sequence|legend=0| 12e, 15, 27, 42 }}


== Augene ==
Badness (Sintel): 1.18
Subgroup: 2.3.5.7


[[Comma list]]: 64/63, 126/125
== 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.


[[Mapping]]: [{{val|3 0 7 18}}, {{val|0 1 0 -2}}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1|3 0 -6 -7 -18 -14}}
[[Comma list]]: 36/35, 128/125


[[POTE generator]]: ~3/2 = 709.257
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}


{{Val list|legend=1| 12, 27, 39d, 66cd }}
[[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 }}


[[Badness]]: 0.024816
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}
 
[[Badness]] (Sintel): 0.670


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


Comma list: 56/55, 64/63, 100/99
Comma list: 36/35, 45/44, 56/55


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


POTE generator: ~3/2 = 711.177
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}})


Vals: {{Val list| 12, 15, 27e }}
{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}


Badness: 0.019613
Badness (Sintel): 0.668


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 56/55, 64/63, 66/65
Comma list: 27/26, 36/35, 45/44, 56/55


Mapping: [{{val|3 0 7 18 20 16}}, {{val|0 1 0 -2 -2 -1}}]
Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}


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


Vals: {{Val list| 12f, 15, 27ef }}
{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}


Badness: 0.020785
Badness (Sintel): 0.762


==== Ogene ====
==== Augustus ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 64/63, 91/90, 100/99
Comma list: 26/25, 36/35, 45/44, 56/55


Mapping: [{{val|3 0 7 18 20 -8}}, {{val|0 1 0 -2 -2 4}}]
Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}


POTE generator: ~3/2 = 712.609
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}})


Vals: {{Val list| 12, 15, 27e, 69bceef }}
{{Optimal ET sequence|legend=0| 9, 12 }}


Badness: 0.022890
Badness (Sintel): 0.919


==== Agene ====
== Inflated ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 56/55, 64/63, 78/77, 100/99
[[Comma list]]: 28/27, 128/125


Mapping: [{{val|3 0 7 18 20 35}}, {{val|0 1 0 -2 -2 -5}}]
{{Mapping|legend=1| 3 0 7 -6 | 0 1 0 3 }}


POTE generator: ~3/2 = 709.677
[[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 }}


Vals: {{Val list| 12f, 27e }}
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}


Badness: 0.023113
[[Badness]] (Sintel): 1.39


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


Comma list: 55/54, 64/63, 77/75
Comma list: 28/27, 55/54, 128/125
 
Mapping: {{mapping| 3 0 7 -6 -4 | 0 1 0 3 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}})
 
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}
 
Badness (Sintel): 1.03


Mapping: [{{val|3 0 7 18 -4}}, {{val|0 1 0 -2 3}}]
== Deflated ==
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 714.150
[[Comma list]]: 21/20, 128/125


Vals: {{Val list| 15, 27, 42 }}
{{Mapping|legend=1| 3 0 7 13 | 0 1 0 -1 }}


Badness: 0.035554
[[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 }}


* [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=1| 3, 6b, 9 }}


* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor]
[[Badness]] (Sintel): 1.50


== Inflated ==
=== 11-limit ===
Comma list: 28/27, 128/125
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 722.719
Comma list: 21/20, 33/32, 128/125


Map: [<3 0 7 -6|, <0 1 0 3|]
Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}


Wedgie: <<3 0 9 -7 6 21||
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: 3d, 15, 48bc, 63bc, 78bc
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}


Badness: 0.054729
Badness (Sintel): 1.23


=== 11-limit ===
== Hexe ==
Comma list: 28/27, 55/54, 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.


POTE generator: ~3/2 = 722.663
[[Subgroup]]: 2.3.5.7


Map: [<3 0 7 -6 -4|, <0 1 0 3 3|]
[[Comma list]]: 50/49, 128/125


EDOs: 3de, 15, 48bce, 63bce, 78bce
{{Mapping|legend=1| 6 0 14 17 | 0 1 0 0 }}


Badness: 0.031171
: mapping generators: ~28/25, ~3


== Deflated  ==
[[Optimal tuning]]s:
Comma list: 21/20, 128/125
* [[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 }}


POTE generator: ~3/2 = 681.629
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}


Map: [<3 0 7 13|, <0 1 0 -1|]
[[Badness]] (Sintel): 1.46


Wedgie: <<3 0 -3 -7 -13 -7||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 3, 9, 21d, 30bd
Comma list: 50/49, 56/55, 125/121


Badness: 0.059079
Mapping: {{mapping| 6 0 14 17 21 | 0 1 0 0 0 }}


=== 11-limit  ===
Optimal tunings:
Comma list: 21/20, 33/32, 128/125
* 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}})


POTE generator: ~3/2 = 680.042
{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}


Map: [<3 0 7 13 15|, <0 1 0 -1 -1|]
Badness (Sintel): 1.27


EDOs: 3, 9, 21de, 30bde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.037183
Comma list: 50/49, 56/55, 66/65, 105/104


== Hexe  ==
Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}
Comma list: 50/49, 128/125


[[POTE_tuning|POTE generator]]: ~3/2 = 710.963
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}})


Map: [<6 0 14 17|, <0 1 0 0|]
{{Optimal ET sequence|legend=0| 6f, 12f }}


Wedgie: <<6 0 0 -14 -17 0||
Badness (Sintel): 1.49


EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[42edo|42d]], [[54edo|54cd]]
== Triforce ==
[[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of triforce]]


Badness: 0.057730
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.  


=== 11-limit  ===
[[Subgroup]]: 2.3.5.7
Comma list: 50/49, 56/55, 125/121


POTE generator: ~3/2 = 714.304
[[Comma list]]: 49/48, 128/125


Map: [<6 0 14 17 21|, <0 1 0 0 0|]
{{Mapping|legend=1| 3 0 7 6 | 0 2 0 1 }}


EDOs: 6, 12, 30de
: mapping generators: ~5/4, ~7/4


Badness: 0.038412
[[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 }}


=== 13-limit  ===
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
Comma list: 50/49, 56/55, 66/65, 105/104


POTE generator: ~3/2 = 710.005
[[Badness]] (Sintel): 1.39


Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 6f, 12f
Comma list: 49/48, 56/55, 77/75


Badness: 0.035946
Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}


== Hemiaug  ==
Optimal tunings:
Comma list: 128/125, 245/243
* 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}})


[[POTE_tuning|POTE generator]]: ~28/27 = 45.118
{{Optimal ET sequence|legend=0| 6, 9, 15, 24, 39 }}


Map: [<3 1 7 -1|, <0 2 0 5|]
Badness (Sintel): 0.865


Wedgie: <<6 0 15 -14 7 35||
; 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: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.070463
Comma list: 49/48, 56/55, 66/65, 77/75


=== 11-limit  ===
Mapping: {{mapping| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}
Comma list: 56/55, 128/125, 245/243


POTE generator: ~28/27 = 45.788
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}})


Map: [<3 1 7 -1 1|, <0 2 0 5 5|]
{{Optimal ET sequence|legend=0| 6f, 9, 15, 24 }}


EDOs: 24, 27e, 51ce, 78ce
Badness (Sintel): 0.837


Badness: 0.038232
; Scales
* [[triphi]], Triforce[9] with L:s = phi


=== 13-limit  ===
==== Semitriforce ====
Comma list: 56/55, 91/90, 128/125, 245/243
This extension splits the period into 1/6-octave for ~44/39. Its ploidacot is hexaploid dicot.


POTE generator: ~28/27 = 46.250
Subgroup: 2.3.5.7.11.13


Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]
Comma list: 49/48, 56/55, 77/75, 507/500


EDOs: 24, 27e, 51ce, 78ce
Mapping: {{mapping| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}


Badness: 0.030159
: mapping generators: ~44/39, ~7/4


== Triforce  ==
Optimal tunings:
Comma list: 49/48, 128/125
* 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}})


[[POTE_tuning|POTE generator]]: ~7/6 = 247.049
{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}


Map: [<3 0 7 6|, <0 2 0 1|]
Badness (Sintel): 2.44


Wedgie: <<6 0 3 -14 -12 7||
== 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.


EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
[[Subgroup]]: 2.3.5.7


Badness: 0.054988
[[Comma list]]: 128/125, 245/243


[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
{{Mapping|legend=1| 3 1 7 -1 | 0 2 0 5 }}


=== 11-limit  ===
: mapping generators: ~5/4, ~14/9
Comma list: 56/55, 77/75, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
[[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 }}


Map: [<3 0 7 6 8|, <0 2 0 1 1|]
{{Optimal ET sequence|legend=1| 24, 27 }}


EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
[[Badness]] (Sintel): 1.78


Badness: 0.026152
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== Musical example  ====
Comma list: 56/55, 128/125, 243/242
* [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones


=== 13-limit  ===
Mapping: {{mapping| 3 1 7 -1 1 | 0 2 0 5 5 }}
Comma list: 49/48, 56/55, 66/65, 77/75


[[POTE_tuning|POTE generator]]: ~7/6 = 248.313
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}})


Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
Badness (Sintel): 1.26


Badness: 0.020248
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[triphi|Triforce(9) with L:s = phi]]
Comma list: 56/55, 91/90, 128/125, 243/242


=== Semitriforce  ===
Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}
Comma list: 49/48, 56/55, 77/75, 507/500


POTE generator: ~7/6 = 246.642
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: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


EDOs: 6, 24, 54cd, 78cd, 102cdf
Badness (Sintel): 1.25


Badness: 0.059160
== 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


== Hemiug  ==
[[Comma list]]: 128/125, 1323/1250
Comma list: 128/125, 1323/1250


POTE generator: ~21/20 = 52.093
{{Mapping|legend=1| 3 1 7 14 | 0 2 0 -3 }}


Map: [<3 1 7 14|, <0 2 0 -3|]
: mapping generators: ~5/4, ~32/21


Wedgie: <<6 0 -9 -14 -31 -21||
[[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 }}


EDOs: 21, 24, 45c, 69cd
{{Optimal ET sequence|legend=1| 21, 24, 45c }}


Badness: 0.137764
[[Badness]] (Sintel): 3.49
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 128/125, 1323/1250
Comma list: 56/55, 128/125, 1323/1250


POTE generator: ~21/20 = 51.655
Mapping: {{mapping| 3 1 7 14 16 | 0 2 0 -3 -3 }}
 
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}})


Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]
{{Optimal ET sequence|legend=0| 21, 24 }}


EDOs: 21, 24, 69cd, 93cd
Badness (Sintel): 2.25


Badness: 0.068062
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 56/55, 66/65, 105/104, 507/500
Comma list: 56/55, 66/65, 105/104, 507/500


POTE generator: ~21/20 = 51.548
Mapping: {{mapping| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}


Map: [<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}})


EDOs: 21, 24, 69cdf, 93cdf
{{Optimal ET sequence|legend=0| 21, 24 }}


Badness: 0.042289
Badness (Sintel): 1.75


== Trug  ==
== Oodako ==
Comma list: 128/125, 360/343
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.


POTE generator: ~48/35 = 501.980
[[Subgroup]]: 2.3.5.7


Map: [<3 1 7 6|, <0 3 0 2|]
[[Comma list]]: 128/125, 2401/2400


EDOs: 9bd, 12
{{Mapping|legend=1| 3 3 7 8 | 0 4 0 1 }}


Badness: 0.138279
: mapping generators: ~5/4, ~28/25


== Oodako  ==
[[Optimal tuning]]s:
Comma list: 128/125, 2401/2400
* [[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 }}


POTE generator: ~8/7 = 223.3540873874507317
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
 
[[Badness]] (Sintel): 2.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<3 3 7 8|, <0 4 0 1|]
Comma list: 56/55, 128/125, 2401/2400


Wedgie: <<12 0 3 -28 -29 7||
Mapping: {{mapping| 3 3 7 8 10 | 0 4 0 1 1 }}


EDOs: 6, 21, 27, 75c, 102cd, 129bcd
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}})


Badness: 0.113209
{{Optimal ET sequence|legend=0| 6, 21, 27e }}


=== 11-limit  ===
Badness (Sintel): 1.96
Comma list: 56/55, 128/125, 3773/3750


POTE generator: ~8/7 = 223.019
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [<3 3 7 8 10|, <0 4 0 1 1|]
Comma list: 56/55, 78/77, 128/125, 507/500


EDOs: 6, 21, 27e
Mapping: {{mapping| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}


Badness: 0.059192
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}})


=== 13-limit  ===
{{Optimal ET sequence|legend=0| 6, 21, 27e }}
Comma list: 56/55, 78/77, 128/125, 507/500


POTE generator: ~8/7 = 223.449
Badness (Sintel): 1.75


Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]
== Hemisemiaug ==
Hemisemiaug tempers out 12005/11664 and splits both the period and generator of augmented in two. Its ploidacot is hexaploid alpha-dicot.


EDOs: 6, 21, 27e
[[Subgroup]]: 2.3.5.7


Badness: 0.042323
[[Comma list]]: 128/125, 12005/11664


== Hemisemiaug  ==
{{Mapping|legend=1| 6 1 14 4 | 0 2 0 3 }}
Comma list: 128/125, 12005/11664


POTE generator: ~15/14 = 144.515
: mapping generators: ~54/49, ~45/28


Map: [<6 1 14 4|, <0 2 0 3|]
[[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 }}


Wedgie: <<12 0 18 -28 -5 42||
{{Optimal ET sequence|legend=1| 18, 24, 42 }}


Edos: 18, 24, 42, 66c, 108bc
[[Badness]] (Sintel): 5.34


Badness: 0.210984
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 128/125, 3773/3645
Comma list: 56/55, 128/125, 3773/3645


POTE generator: ~15/14 = 144.780
Mapping: {{mapping| 6 1 14 4 8 | 0 2 0 3 3 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 18e, 24, 42e, 66ce, 108bccee }}


Map: [<6 1 14 4 8|, <0 2 0 3 3|]
Badness (Sintel): 2.67


EDOs: 24, 42e, 66ce, 108bce
== Niner ==
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.


Badness: 0.080738
[[Subgroup]]: 2.3.5.7


== Niner  ==
[[Comma list]]: 128/125, 686/675
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.


Comma list: 128/125, 686/675
{{Mapping|legend=1| 9 0 21 11 | 0 1 0 1 }}


POTE generator: ~3/2 = 707.167
: mapping generators: ~49/45, ~3


Map: [<9 0 21 11|, <0 1 0 1|]
[[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 }}


Wedgie: <<9 0 9 -21 -11 21||
{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}


EDOs: 9, 18, 27, 63c, 90c
[[Badness]] (Sintel): 1.70


Badness: 0.067157
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 128/125, 540/539
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: 0.034861
Badness (Sintel): 1.15
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 56/55, 78/77, 91/90, 128/125
Comma list: 56/55, 78/77, 91/90, 128/125


POTE generator: ~3/2 = 706.889
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
 
{{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: 0.024143
{{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 -->
[[Category:Temperament families]]
[[de:Übermässige Temperaturen]]
[[Category:Pages with mostly numerical content]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Augmented_family"