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Augmented family: Difference between revisions

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Intro to some of these temps
 
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| de = Übermässige Temperaturen
| de = Übermässige Temperaturen
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The 5-limit parent comma for the '''augmented family''' is [[128/125]], the 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 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.
{{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 ==
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
[[Subgroup]]: 2.3.5


Line 15: Line 18:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~3/2 = 701.9550
* [[WE]]: ~5/4 = 399.0128{{c}}, ~3/2 = 704.8937{{c}} (~16/15 = 93.1320{{c}})
* [[POTE]]: ~5/4 = 1\3, ~3/2 = 706.638
: [[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, 12, 27, 39, 51c, 90cc }}
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}


[[Badness]]: 0.022315
[[Badness]] (Sintel): 0.523


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds [[36/35]], augene [[64/63]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones.
The second comma of the [[Normal 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.  


== August ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 128/125
[[Comma list]]: 64/63, 126/125


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


{{Multival|legend=1| 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 }}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}
* [[CTE]]: ~5/4 = 1\3, ~3/2 = 692.1237
* [[POTE]]: ~5/4 = 1\3, ~3/2 = 696.011


{{Optimal ET sequence|legend=1| 9, 12, 45cd, 57cd, 69cdd }}
[[Badness]] (Sintel): 0.628
 
[[Badness]]: 0.026459


=== 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 -1 1 | 0 1 0 2 2 }}
Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 687.6853
* WE: ~5/4 = 398.4962{{c}}, ~3/2 = 708.5030{{c}} (~21/20 = 88.4895{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 692.514
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 711.6031{{c}} (~21/20 = 88.3969{{c}})


{{Optimal ET sequence|legend=1| 9, 12, 21, 33e, 45cde }}
{{Optimal ET sequence|legend=0| 12, 15, 27e }}


Badness: 0.020191
Badness (Sintel): 0.648


==== 13-limit ====
==== 13-limit ====
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: 40/39, 56/55, 64/63, 66/65


Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 685.0836
* WE: ~5/4 = 398.0488{{c}}, ~3/2 = 708.5402{{c}} (~21/20 = 87.5574{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 688.783
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.6704{{c}} (~21/20 = 87.3296{{c}})


{{Optimal ET sequence|legend=1| 9, 12f, 21, 33ef, 54bceeff }}
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}


Badness: 0.018448
Badness (Sintel): 0.859


==== Augustus ====
==== Ogene ====
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, 91/90, 100/99


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 687.6853
* WE: ~5/4 = 398.6473{{c}}, ~3/2 = 710.1987{{c}} (~21/20 = 87.0959{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 685.356
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.5057{{c}} (~21/20 = 87.4943{{c}})


{{Optimal ET sequence|legend=1| 9, 12, 21f }}
{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}


Badness: 0.022236
Badness (Sintel): 0.946


== Augene ==
==== Agene ====
{{Main| Augene }}
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 56/55, 64/63, 78/77, 100/99


[[Comma list]]: 64/63, 126/125
Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}


{{Mapping|legend=1| 3 0 7 18 | 0 1 0 -2 }}
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}})


{{Multival|legend=1| 3 0 -6 -7 -18 -14 }}
{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}


[[Optimal tuning]]s:  
Badness (Sintel): 0.955
* [[CTE]]: ~5/4 = 1\3, ~3/2 = 709.5948
* [[POTE]]: ~5/4 = 1\3, ~3/2 = 709.257


{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}
=== Eugene ===
 
[[Badness]]: 0.024816
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 64/63, 100/99
Comma list: 55/54, 64/63, 77/75


Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}
Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}


Wedgie: {{Multival| 3 0 -6 -6 -7 -18 -20 -14 -14 4 }}
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}})


Optimal tunings:
{{Optimal ET sequence|legend=0| 12e, 15, 27, 42 }}
* CTE: ~5/4 = 1\3, ~3/2 = 713.5701
* POTE: ~5/4 = 1\3, ~3/2 = 711.177


{{Optimal ET sequence|legend=1| 12, 15, 27e }}
Badness (Sintel): 1.18


Badness: 0.019613
== 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.  


==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 56/55, 64/63, 66/65
[[Comma list]]: 36/35, 128/125


Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~5/4 = 1\3, ~3/2 = 716.1234
* [[WE]]: ~5/4 = 399.1036{{c}}, ~3/2 = 694.4509{{c}} (~16/15 = 103.7564{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 712.013
: [[error map]]: {{val| -2.689 -10.193 +7.412 +15.594 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 694.6812{{c}} (~16/15 = 105.3188{{c}})
: error map: {{val| 0.000 -7.274 +13.686 +20.537 }}


{{Optimal ET sequence|legend=1| 12f, 15, 27eff }}
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}


Badness: 0.020785
[[Badness]] (Sintel): 0.670


==== Ogene ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 711.9020
* WE: ~5/4 = 398.9225{{c}}, ~3/2 = 690.6486{{c}} (~16/15 = 107.1966{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 712.609
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 690.8519{{c}} (~16/15 = 109.1481{{c}})


{{Optimal ET sequence|legend=1| 12, 15, 27e, 69bceef }}
{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}


Badness: 0.022890
Badness (Sintel): 0.668


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


Comma list: 56/55, 64/63, 78/77, 100/99
Comma list: 27/26, 36/35, 45/44, 56/55


Mapping: {{mapping| 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 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 712.5722
* WE: ~5/4 = 399.0956{{c}}, ~3/2 = 687.2261{{c}} (~16/15 = 110.9651{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 709.677
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 687.5057{{c}} (~16/15 = 112.4943{{c}})


{{Optimal ET sequence|legend=1| 12f, 27e }}
{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}


Badness: 0.023113
Badness (Sintel): 0.762


=== Eugene ===
==== Augustus ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 64/63, 77/75
Comma list: 26/25, 36/35, 45/44, 56/55


Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}
Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 713.0025
* WE: ~5/4 = 400.4230{{c}}, ~3/2 = 686.0809{{c}} (~16/15 = 114.7650{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 714.150
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 685.8446{{c}} (~16/15 = 114.1554{{c}})


{{Optimal ET sequence|legend=1| 12e, 15, 27, 42 }}
{{Optimal ET sequence|legend=0| 9, 12 }}


Badness: 0.035554
Badness (Sintel): 0.919


== Inflated ==
== Inflated ==
Line 189: Line 196:


{{Mapping|legend=1| 3 0 7 -6 | 0 1 0 3 }}
{{Mapping|legend=1| 3 0 7 -6 | 0 1 0 3 }}
{{Multival|legend=1| 3 0 9 -7 6 21 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~3/2 = 717.5172
* [[WE]]: ~5/4 = 398.4023{{c}}, ~3/2 = 719.8327{{c}} (~25/24 = 76.9719{{c}})
* [[POTE]]: ~5/4 = 1\3, ~3/2 = 722.719
: [[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 }}


{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}


[[Badness]]: 0.054729
[[Badness]] (Sintel): 1.39


=== 11-limit ===
=== 11-limit ===
Line 208: Line 215:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 717.5172
* WE: ~5/4 = 398.4016{{c}}, ~3/2 = 719.7758{{c}} (~25/24 = 77.0275{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 722.663
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 720.9386{{c}} (~25/24 = 79.0614{{c}})


{{Optimal ET sequence|legend=1| 3de, 12de, 15 }}
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}


Badness: 0.031171
Badness (Sintel): 1.03


== Deflated ==
== Deflated ==
Line 221: Line 228:


{{Mapping|legend=1| 3 0 7 13 | 0 1 0 -1 }}
{{Mapping|legend=1| 3 0 7 13 | 0 1 0 -1 }}
{{Multival|legend=1| 3 0 -3 -7 -13 -7 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~3/2 = 684.8473
* [[WE]]: ~5/4 = 401.9566{{c}}, ~3/2 = 684.9634{{c}} (~16/15 = 118.9497{{c}})
* [[POTE]]: ~5/4 = 1\3, ~3/2 = 681.629
: [[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, 9 }}
{{Optimal ET sequence|legend=1| 3, 6b, 9 }}


[[Badness]]: 0.059079
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 240: Line 247:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~3/2 = 679.881
* WE: ~5/4 = 402.1799{{c}}, ~3/2 = 683.7477{{c}} (~16/15 = 120.6120{{c}})
* POTE: ~5/4 = 1\3, ~3/2 = 680.042
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 680.0162{{c}} (~16/15 = 119.9838{{c}})


{{Optimal ET sequence|legend=1| 3, 9 }}
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}


Badness: 0.037183
Badness (Sintel): 1.23


== Hexe ==
== Hexe ==
Hexe tempers out 50/49 and may be described as {{nowrap| 6 & 12 }}, viewed as [[6edo|6et]] with an independent generator for prime 3. Its ploidacot is hexaploid monocot.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 255: Line 264:


: mapping generators: ~28/25, ~3
: mapping generators: ~28/25, ~3
{{Multival|legend=1| 6 0 0 -14 -17 0 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~28/25 = 1\6, ~3/2 = 701.9550
* [[WE]]: ~28/25 = 199.0488{{c}}, ~3/2 = 707.5815{{c}} (~25/24 = 88.6137{{c}})
* [[POTE]]: ~28/25 = 1\6, ~3/2 = 710.963
: [[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 }}


{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}


[[Badness]]: 0.057730
[[Badness]] (Sintel): 1.46


=== 11-limit ===
=== 11-limit ===
Line 274: Line 283:


Optimal tunings:  
Optimal tunings:  
* CTE: ~28/25 = 1\6, ~3/2 = 701.9550
* WE: ~28/25 = 198.6942{{c}}, ~3/2 = 709.6404{{c}} (~25/24 = 85.1362{{c}})
* POTE: ~28/25 = 1\6, ~3/2 = 714.304
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 711.8043{{c}} (~25/24 = 88.1957{{c}})


{{Optimal ET sequence|legend=1| 6, 12, 30dee, 42ddeee }}
{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}


Badness: 0.038412
Badness (Sintel): 1.27


=== 13-limit ===
=== 13-limit ===
Line 289: Line 298:


Optimal tunings:  
Optimal tunings:  
* CTE: ~28/25 = 1\6, ~3/2 = 692.4327
* WE: ~28/25 = 198.4492{{c}}, ~3/2 = 704.4994{{c}} (~25/24 = 89.2973{{c}})
* POTE: ~28/25 = 1\6, ~3/2 = 710.005
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 706.6050{{c}} (~16/15 = 93.3950{{c}})


{{Optimal ET sequence|legend=1| 6f, 12f }}
{{Optimal ET sequence|legend=0| 6f, 12f }}


Badness: 0.035946
Badness (Sintel): 1.49


== Triforce ==
== Triforce ==
[[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of 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
[[Subgroup]]: 2.3.5.7
Line 306: Line 317:


: mapping generators: ~5/4, ~7/4
: mapping generators: ~5/4, ~7/4
{{Multival|legend=1| 6 0 3 -14 -12 7 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~7/4 = 952.2948
* [[WE]]: ~5/4 = 399.7480{{c}}, ~7/4 = 952.3507{{c}} (~35/32 = 152.8547{{c}})
* [[POTE]]: ~5/4 = 1\3, ~7/4 = 952.951
: [[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, 63cd, 102ccdd }}
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}


[[Badness]]: 0.054988
[[Badness]] (Sintel): 1.39


=== 11-limit ===
=== 11-limit ===
Line 325: Line 336:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~7/4 = 952.2495
* WE: ~5/4 = 399.7654{{c}}, ~7/4 = 952.3730{{c}} (~12/11 = 152.8421{{c}})
* POTE: ~5/4 = 1\3, ~7/4 = 952.932
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7447{{c}} (~12/11 = 152.7447{{c}})


{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
{{Optimal ET sequence|legend=0| 6, 9, 15, 24, 39 }}


Badness: 0.026152
Badness (Sintel): 0.865


; Music
; Music
* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)''] by [[Igliashon Jones]] (2018)
* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)'']{{dead link}} by [[Igliashon Jones]] (2018)
* [http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ ''2-2-1-2-2-1-2-2-1 mode of 15 edo''] [http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]]
* [https://www.chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ ''2-2-1-2-2-1-2-2-1 mode of 15 edo''] [https://web.archive.org/web/20201127015017/http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]] (2013)


==== 13-limit ====
==== 13-limit ====
Line 344: Line 355:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~7/4 = 950.8049
* WE: ~5/4 = 399.7107{{c}}, ~7/4 = 950.9983{{c}} (~12/11 = 151.5768{{c}})
* POTE: ~5/4 = 1\3, ~7/4 = 951.687
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 951.4465{{c}} (~12/11 = 151.4465{{c}})


{{Optimal ET sequence|legend=1| 6, 9, 15, 24 }}
{{Optimal ET sequence|legend=0| 6f, 9, 15, 24 }}


Badness: 0.020248
Badness (Sintel): 0.837


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


==== Semitriforce ====
==== 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
Subgroup: 2.3.5.7.11.13


Line 364: Line 377:


Optimal tunings:  
Optimal tunings:  
* CTE: ~44/39 = 1\6, ~7/4 = 952.5307
* WE: ~44/39 = 199.8321{{c}}, ~7/4 = 952.5580{{c}} (~40/39 = 46.6024{{c}})
* POTE: ~44/39 = 1\6, ~7/4 = 953.358
* CWE: ~44/39 = 200.0000{{c}}, ~7/4 = 953.2005{{c}} (~40/39 = 46.7995{{c}})


{{Optimal ET sequence|legend=1| 6, 18bd, 24 }}
{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}


Badness: 0.059160
Badness (Sintel): 2.44


== Hemiaug ==
== Hemiaug ==
Hemiaug tempers out 245/243 and may be described as {{nowrap| 24 & 27 }}. The generator may be taken as ~14/9, but also a neutral third or a neutral second that stand in for 11/9~16/13 and 12/11~13/12 in the higher limits, respectively. Hemiaug's ploidacot is triploid dicot. [[27edo]] makes for a recommendable tuning in the 7-limit, but [[51edo]] serves better in the higher limits.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 379: Line 394:


: mapping generators: ~5/4, ~14/9
: mapping generators: ~5/4, ~14/9
{{Multival|legend=1| 6 0 15 -14 7 35 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~14/9 = 752.8335 (~36/35 = 47.1665)
* [[WE]]: ~5/4 = 398.9278{{c}}, ~14/9 = 752.8583{{c}} (~36/35 = 44.9973{{c}})
* [[POTE]]: ~5/4 = 1\3, ~14/9 = 754.882 (~36/35 = 45.118)
: [[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 }}
{{Optimal ET sequence|legend=1| 24, 27 }}


[[Badness]]: 0.070463
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Line 398: Line 413:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~14/9 = 752.0512 (~36/35 = 47.9488)
* WE: ~5/4 = 398.8946{{c}}, ~14/9 = 752.1272{{c}} (~36/35 = 45.6619{{c}})
* POTE: ~5/4 = 1\3, ~14/9 = 754.212 (~36/35 = 45.788)
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.5000{{c}} (~36/35 = 46.5000{{c}})


{{Optimal ET sequence|legend=1| 24, 27e, 51ce }}
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


Badness: 0.038232
Badness (Sintel): 1.26


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


Comma list: 56/55, 91/90, 128/125, 245/243
Comma list: 56/55, 91/90, 128/125, 243/242


Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}
Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~14/9 = 752.1284 (~36/35 = 47.8716)
* WE: ~5/4 = 399.1053{{c}}, ~14/9 = 752.0643{{c}} (~36/35 = 46.1463{{c}})
* POTE: ~5/4 = 1\3, ~14/9 = 753.750 (~36/35 = 46.250)
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.3806{{c}} (~36/35 = 46.6194{{c}})


{{Optimal ET sequence|legend=1| 24, 27e, 51ce }}
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}


Badness: 0.030159
Badness (Sintel): 1.25


== Hemiug ==
== 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
[[Subgroup]]: 2.3.5.7


Line 428: Line 445:


: mapping generators: ~5/4, ~32/21
: mapping generators: ~5/4, ~32/21
{{Multival|legend=1| 6 0 -9 -14 -31 -21 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~32/21 = 747.9484 (~21/20 = 52.0516)
* [[WE]]: ~5/4 = 400.1805{{c}}, ~32/21 = 748.2436{{c}} (~21/20 = 52.1174{{c}})
* [[POTE]]: ~5/4 = 1\3, ~32/21 = 747.907 (~21/20 = 52.093)
: [[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 }}
{{Optimal ET sequence|legend=1| 21, 24, 45c }}


[[Badness]]: 0.137764
[[Badness]] (Sintel): 3.49


=== 11-limit ===
=== 11-limit ===
Line 447: Line 464:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~32/21 = 748.2962 (~33/32 = 51.7038)
* WE: ~5/4 = 400.0637{{c}}, ~32/21 = 748.4638{{c}} (~33/32 = 51.6637{{c}})
* POTE: ~5/4 = 1\3, ~32/21 = 748.345 (~33/32 = 51.655)
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.3383{{c}} (~33/32 = 51.6617{{c}})


{{Optimal ET sequence|legend=1| 21, 24 }}
{{Optimal ET sequence|legend=0| 21, 24 }}


Badness: 0.068062
Badness (Sintel): 2.25


=== 13-limit ===
=== 13-limit ===
Line 462: Line 479:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~32/21 = 748.5255 (~33/32 = 51.4745)
* WE: ~5/4 = 399.8855{{c}}, ~32/21 = 748.2378{{c}} (~33/32 = 51.5332{{c}})
* POTE: ~5/4 = 1\3, ~32/21 = 748.452 (~33/32 = 51.548)
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.4655{{c}} (~33/32 = 51.5345{{c}})


{{Optimal ET sequence|legend=1| 21, 24 }}
{{Optimal ET sequence|legend=0| 21, 24 }}


Badness: 0.042289
Badness (Sintel): 1.75


== Oodako ==
== Oodako ==
Oodako tempers out 2401/2400 and may be described as {{nowrap| 21 & 27 }}. It is generated by a quarter of a fifth, which stands in for ~28/25. Its ploidacot is triploid tetracot.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 477: Line 496:


: mapping generators: ~5/4, ~28/25
: mapping generators: ~5/4, ~28/25
{{Multival|legend=1| 12 0 3 -28 -29 7 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~28/25 = 175.3586
* [[WE]]: ~5/4 = 399.0296{{c}}, ~28/25 = 176.2174{{c}} (~49/48 = 46.5949{{c}})
* [[POTE]]: ~5/4 = 1\3, ~28/25 = 176.646
: [[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 }}


{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}


[[Badness]]: 0.113209
[[Badness]] (Sintel): 2.86


=== 11-limit ===
=== 11-limit ===
Line 496: Line 515:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~11/10 = 175.0533
* WE: ~5/4 = 398.6615{{c}}, ~11/10 = 176.3886{{c}} (~49/48 = 45.8843{{c}})
* POTE: ~5/4 = 1\3, ~11/10 = 176.981
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.5471{{c}} (~49/48 = 46.9059{{c}})


{{Optimal ET sequence|legend=1| 6, 21, 27e }}
{{Optimal ET sequence|legend=0| 6, 21, 27e }}


Badness: 0.059192
Badness (Sintel): 1.96


=== 13-limit ===
=== 13-limit ===
Line 511: Line 530:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 1\3, ~11/10 = 175.2524
* WE: ~5/4 = 398.8612{{c}}, ~11/10 = 176.0486{{c}} (~49/48 = 46.7640{{c}})
* POTE: ~5/4 = 1\3, ~11/10 = 176.551
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.3326{{c}} (~49/48 = 47.3348{{c}})


{{Optimal ET sequence|legend=1| 6, 21, 27e }}
{{Optimal ET sequence|legend=0| 6, 21, 27e }}


Badness: 0.042323
Badness (Sintel): 1.75


== Hemisemiaug ==
== 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
[[Subgroup]]: 2.3.5.7


Line 526: Line 547:


: mapping generators: ~54/49, ~45/28
: mapping generators: ~54/49, ~45/28
{{Multival|legend=1| 12 0 18 -28 -5 42 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~54/49 = 1\6, ~45/28 = 853.1901 (~36/35 = 53.1901)
* [[WE]]: ~54/49 = 199.5469{{c}}, ~45/28 = 853.5468{{c}} (~36/35 = 55.3594{{c}})
* [[POTE]]: ~54/49 = 1\6, ~45/28 = 855.485 (~36/35 = 55.485)
: [[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 }}
{{Optimal ET sequence|legend=1| 18, 24, 42 }}


[[Badness]]: 0.210984
[[Badness]] (Sintel): 5.34


=== 11-limit ===
=== 11-limit ===
Line 545: Line 566:


Optimal tunings:  
Optimal tunings:  
* CTE: ~54/49 = 1\6, ~18/11 = 852.5968 (~36/35 = 52.5968)
* WE: ~54/49 = 199.5188{{c}}, ~18/11 = 853.1623{{c}} (~36/35 = 55.0872{{c}})
* POTE: ~54/49 = 1\6, ~18/11 = 855.220 (~36/35 = 55.220)
* CWE: ~54/49 = 200.0000{{c}}, ~18/11 = 854.3545{{c}} (~36/35 = 54.3545{{c}})


{{Optimal ET sequence|legend=1| 24, 42e, 66ce, 108bccee }}
{{Optimal ET sequence|legend=0| 18e, 24, 42e, 66ce, 108bccee }}


Badness: 0.080738
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
[[Subgroup]]: 2.3.5.7
Line 562: Line 583:


: mapping generators: ~49/45, ~3
: mapping generators: ~49/45, ~3
{{Multival|legend=1| 9 0 9 -21 -11 21 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~49/45 = 1\9, ~3/2 = 702.0044
* [[WE]]: ~49/45 = 133.0272{{c}}, ~3/2 = 705.5438{{c}} (~36/35 = 40.4075{{c}})
* [[POTE]]: ~49/45 = 1\9, ~3/2 = 707.167
: [[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 }}


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


[[Badness]]: 0.067157
[[Badness]] (Sintel): 1.70


=== 11-limit ===
=== 11-limit ===
Line 581: Line 602:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 1\9, ~3/2 = 699.6216
* WE: ~12/11 = 132.9553{{c}}, ~3/2 = 704.7217{{c}} (~36/35 = 39.9453{{c}})
* POTE: ~12/11 = 1\9, ~3/2 = 706.726
* CWE: ~12/11 = 133.3333{{c}}, ~3/2 = 704.5723{{c}} (~36/35 = 37.9056{{c}})


{{Optimal ET sequence|legend=1| 9, 18e, 27e, 63cee }}
{{Optimal ET sequence|legend=0| 9, 18e, 27e, 63cee }}


Badness: 0.034861
Badness (Sintel): 1.15


=== 13-limit ===
=== 13-limit ===
Line 596: Line 617:


Optimal tunings:  
Optimal tunings:  
* CTE: ~14/13 = 1\9, ~3/2 = 700.4330
* WE: ~14/13 = 133.0143{{c}}, ~3/2 = 705.1969{{c}} (~36/35 = 40.1256{{c}})
* POTE: ~14/13 = 1\9, ~3/2 = 706.889
* CWE: ~14/13 = 133.3333{{c}}, ~3/2 = 705.0176{{c}} (~36/35 = 38.3510{{c}})


{{Optimal ET sequence|legend=1| 9, 18e, 27e }}
{{Optimal ET sequence|legend=0| 9, 18e, 27e }}


Badness: 0.024143
Badness (Sintel): 0.998


== Trug ==
== 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
[[Subgroup]]: 2.3.5.7


Line 613: Line 636:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 1\3, ~48/35 = 498.6367
* [[WE]]: ~5/4 = 398.2337{{c}}, ~48/35 = 499.7635{{c}} (~15/14 = 101.5299{{c}})
* [[POTE]]: ~5/4 = 1\3, ~48/35 = 501.980
: [[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}})
{{Optimal ET sequence|legend=1| 9bd, 12 }}
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}
 
[[Badness]]: 0.138279
 
== Ternary ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10/9, 16/15
 
{{Mapping|legend=1| 3 5 7 0 | 0 0 0 1 }}
 
: mapping generators: ~5/4, ~7
 
{{Multival|legend=1| 0 0 3 0 5 7 }}
 
[[Optimal tuning]]s:
* [[POTE]]: ~5/4 = 1\3, ~7/4 = 1034.013
 
{{Optimal ET sequence|legend=1| 3d, 6, 9bd, 15bbd }}
 
[[Badness]]: 0.028673
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 10/9, 16/15, 22/21
 
Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}
 
Optimal tunings:
* POTE: ~5/4 = 1\3, ~7/4 = 1033.153


{{Optimal ET sequence|legend=1| 3de, 6, 15bbdee }}
{{Optimal ET sequence|legend=1| 3b, 9bd, 12 }}


Badness: 0.025592
[[Badness]] (Sintel): 3.50


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


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Augmented_family"