Augmented family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m recat
Intro to some of these temps
 
(48 intermediate revisions by 12 users not shown)
Line 1: Line 1:
__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 -->
[[Category:Temperament families]]
[[de:Übermässige Temperaturen]]
[[Category:Pages with mostly numerical content]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 16:58, 21 August 2025

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

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. 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 article: Augene

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.

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

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

External links

Retrieved from "https://en.xen.wiki/index.php?title=Augmented_family&oldid=207920"