Xenharmonic Wiki

Augmented family: Difference between revisions

← Older edit
VisualWikitext
Switch to Sintel's badness, WE & CWE tunings, per community consensus
Overthink (talk | contribs)
autopatrolled
2,696 edits
→Trisected: + subgroup temperaments
 
(20 intermediate revisions by 3 users not shown)
Line 4: Line 4:
}}
}}
{{Technical data page}}
{{Technical data page}}
The 5-limit parent comma for the '''augmented family''' is [[128/125]], the diesis. 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.
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 ==
{{Main| Augmented (temperament) }}
The [[period]] is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third. Its [[ploidacot]] is triploid monocot.
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 20: Line 24:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.0691{{c}} (~16/15 = 94.9309{{c}})
* [[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 }}
: error map: {{val| 0.000 +3.114 +13.686 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 701.955{{c}} (~16/15 = 98.045{{c}})
: [[error map]]: {{val| 0.000 0.000 +13.686 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 706.638{{c}} (~16/15 = 93.362{{c}})
: error map: {{val| 0.000 +4.683 +13.686 }} -->


{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}
Line 30: Line 30:


=== 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 forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal augmented adds [[64/63]], august [[36/35]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones. All of these can be extended to the [[11-limit]] by adding [[56/55]] and [[176/175]], which sum to 128/125, equating the 400{{c}} major third to [[14/11]] as well as 5/4.
 
== August ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 128/125
 
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}
 
[[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 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 692.124{{c}} (~16/15 = 107.876{{c}})
: [[error map]]: {{val| 0.000 -9.831 +13.686 +15.422 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 696.011{{c}} (~16/15 = 103.989{{c}})
: error map: {{val| 0.000 -5.944 +13.686 +23.195 }} -->
 
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}
 
[[Badness]] (Sintel): 0.670
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 56/55
 
Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
 
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}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 687.685{{c}} (~16/15 = 112.315{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 692.514{{c}} (~16/15 = 107.486{{c}}) -->
 
{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}


Badness (Sintel): 0.668
== Septimal augmented (augene) ==
{{Main| Augmented (temperament) }}


==== 13-limit ====
Septimal augmented, a.k.a. augene, tempers out [[64/63]] and [[126/125]]. It may be described as the {{nowrap| 12 & 15 }} temperament. [[27edo]] and [[39edo]] in the 39d val make for good tunings. In the 11-limit, it tempers together [[7/5]] and [[11/8]], and 27edo in the 27e val may be recommended as a tuning.  
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 56/55
 
Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
 
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}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 685.084{{c}} (~16/15 = 114.916{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 688.783{{c}} (~16/15 = 111.217{{c}}) -->
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
 
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}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 687.685{{c}} (~16/15 = 112.315{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 685.356{{c}} (~16/15 = 114.644{{c}}) -->
 
{{Optimal ET sequence|legend=0| 9, 12 }}
 
Badness (Sintel): 0.919
 
== Augene ==
{{Main| Augene }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 118: Line 48:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 709.3249{{c}} (~21/20 = 90.6751{{c}})
* [[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 }}
: error map: {{val| 0.000 +7.370 +13.686 +12.524 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 709.595{{c}} (~21/20 = 90.405{{c}})
: [[error map]]: {{val| 0.000 +7.640 +13.686 +11.984 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 709.257{{c}} (~21/20 = 90.743{{c}})
: error map: {{val| 0.000 +7.302 +13.686 +12.660 }} -->


{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}
Line 137: Line 63:
* WE: ~5/4 = 398.4962{{c}}, ~3/2 = 708.5030{{c}} (~21/20 = 88.4895{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 711.6031{{c}} (~21/20 = 88.3969{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 713.570{{c}} (~21/20 = 86.430{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 711.177{{c}} (~21/20 = 88.823{{c}}) -->


{{Optimal ET sequence|legend=0| 12, 15, 27e }}
{{Optimal ET sequence|legend=0| 12, 15, 27e }}
Line 154: Line 78:
* WE: ~5/4 = 398.0488{{c}}, ~3/2 = 708.5402{{c}} (~21/20 = 87.5574{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.6704{{c}} (~21/20 = 87.3296{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 716.123{{c}} (~21/20 = 83.877{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 712.013{{c}} (~21/20 = 87.987{{c}}) -->


{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}
Line 171: Line 93:
* WE: ~5/4 = 398.6473{{c}}, ~3/2 = 710.1987{{c}} (~21/20 = 87.0959{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.5057{{c}} (~21/20 = 87.4943{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 711.902{{c}} (~21/20 = 88.098{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 712.609{{c}} (~21/20 = 87.391{{c}}) -->


{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}
{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}
Line 188: Line 108:
* WE: ~5/4 = 398.5229{{c}}, ~3/2 = 707.0562{{c}} (~21/20 = 89.9897{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 710.1903{{c}} (~21/20 = 89.8097{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 712.572{{c}} (~21/20 = 87.428{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 709.677{{c}} (~21/20 = 90.323{{c}}) -->


{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}
{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}
Line 205: Line 123:
* WE: ~5/4 = 399.1743{{c}}, ~3/2 = 712.6763{{c}} (~21/20 = 85.6723{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 713.9414{{c}} (~21/20 = 86.0586{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 713.002{{c}} (~21/20 = 86.998{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 714.150{{c}} (~21/20 = 85.850{{c}}) -->


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


Badness (Sintel): 1.18
Badness (Sintel): 1.18
== 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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 36/35, 128/125
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}
[[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 }}
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}
[[Badness]] (Sintel): 0.670
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 56/55
Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
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}})
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
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}})
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
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}})
{{Optimal ET sequence|legend=0| 9, 12 }}
Badness (Sintel): 0.919


== Inflated ==
== Inflated ==
Line 224: Line 204:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 721.0196{{c}} (~25/24 = 78.9804{{c}})
* [[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 }}
: error map: {{val| 0.000 +19.065 +13.686 -5.767 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 717.517{{c}} (~25/24 = 82.483{{c}})
: [[error map]]: {{val| 0.000 +15.562 +13.686 -16.274 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 722.719{{c}} (~25/24 = 77.281{{c}})
: error map: {{val| 0.000 +20.764 +13.686 -0.668 }} -->


{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}
Line 243: Line 219:
* WE: ~5/4 = 398.4016{{c}}, ~3/2 = 719.7758{{c}} (~25/24 = 77.0275{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 720.9386{{c}} (~25/24 = 79.0614{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 717.382{{c}} (~25/24 = 82.618{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 722.663{{c}} (~25/24 = 77.337{{c}}) -->


{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}
Line 262: Line 236:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 682.2587{{c}} (~16/15 = 117.7413{{c}})
* [[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 }}
: error map: {{val| 0.000 -19.696 +13.686 -51.085 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 684.847{{c}} (~16/15 = 115.153{{c}})
: [[error map]]: {{val| 0.000 -17.108 +13.686 -53.673 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~3/2 = 681.629{{c}} (~16/15 = 118.371{{c}})
: error map: {{val| 0.000 -20.326 +13.686 -50.455 }} -->


{{Optimal ET sequence|legend=1| 3, 6b, 9 }}
{{Optimal ET sequence|legend=1| 3, 6b, 9 }}
Line 281: Line 251:
* WE: ~5/4 = 402.1799{{c}}, ~3/2 = 683.7477{{c}} (~16/15 = 120.6120{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 680.0162{{c}} (~16/15 = 119.9838{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~3/2 = 679.881{{c}} (~16/15 = 120.119{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~3/2 = 680.042{{c}} (~16/15 = 119.958{{c}}) -->


{{Optimal ET sequence|legend=0| 3, 6b, 9 }}
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}
Line 289: Line 257:


== 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 302: Line 272:
* [[CWE]]: ~28/25 = 200.0000{{c}}, ~3/2 = 708.6907{{c}} (~25/24 = 91.3093{{c}})
* [[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 }}
: error map: {{val| 0.000 +6.735 +13.686 +31.174 }}
<!-- * [[CTE]]: ~28/25 = 200.000{{c}}, ~3/2 = 701.955{{c}} (~16/15 = 98.045{{c}})
: [[error map]]: {{val| 0.000 0.000 +13.686 +31.174 }}
* [[POTE]]: ~28/25 = 200.000{{c}}, ~3/2 = 710.963{{c}} (~25/24 = 89.037{{c}})
: error map: {{val| 0.000 +9.008 +13.686 +31.174 }} -->


{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}
Line 321: Line 287:
* WE: ~28/25 = 198.6942{{c}}, ~3/2 = 709.6404{{c}} (~25/24 = 85.1362{{c}})
* 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}})
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 711.8043{{c}} (~25/24 = 88.1957{{c}})
<!-- * CTE: ~28/25 = 200.000{{c}}, ~3/2 = 701.955{{c}} (~16/15 = 98.045{{c}})
* POTE: ~28/25 = 200.000{{c}}, ~3/2 = 714.304{{c}} (~25/24 = 85.696{{c}}) -->


{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}
{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}
Line 338: Line 302:
* WE: ~28/25 = 198.4492{{c}}, ~3/2 = 704.4994{{c}} (~25/24 = 89.2973{{c}})
* 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}})
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 706.6050{{c}} (~16/15 = 93.3950{{c}})
<!-- * CTE: ~28/25 = 200.000{{c}}, ~3/2 = 692.433{{c}} (~16/15 = 92.433{{c}})
* POTE: ~28/25 = 200.000{{c}}, ~3/2 = 710.005{{c}} (~16/15 = 89.995{{c}}) -->


{{Optimal ET sequence|legend=0| 6f, 12f }}
{{Optimal ET sequence|legend=0| 6f, 12f }}
Line 347: Line 309:
== 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 361: Line 325:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7463{{c}} (~35/32 = 152.7463{{c}})
* [[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 }}
: error map: {{val| 0.000 +3.538 +13.686 -16.080 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~7/4 = 952.295{{c}} (~35/32 = 152.295{{c}})
: [[error map]]: {{val| 0.000 +2.635 +13.686 -16.531 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~7/4 = 952.951{{c}} (~35/32 = 152.951{{c}})
: error map: {{val| 0.000 +3.947 +13.686 -15.875 }} -->


{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
Line 380: Line 340:
* WE: ~5/4 = 399.7654{{c}}, ~7/4 = 952.3730{{c}} (~12/11 = 152.8421{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7447{{c}} (~12/11 = 152.7447{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~7/4 = 952.250{{c}} (~12/11 = 152.250{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~7/4 = 952.932{{c}} (~12/11 = 152.932{{c}}) -->


{{Optimal ET sequence|legend=0| 6, 9, 15, 24, 39 }}
{{Optimal ET sequence|legend=0| 6, 9, 15, 24, 39 }}
Line 388: Line 346:


; 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 401: Line 359:
* WE: ~5/4 = 399.7107{{c}}, ~7/4 = 950.9983{{c}} (~12/11 = 151.5768{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 951.4465{{c}} (~12/11 = 151.4465{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~7/4 = 950.805{{c}} (~12/11 = 150.805{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~7/4 = 951.687{{c}} (~12/11 = 151.687{{c}}) -->


{{Optimal ET sequence|legend=0| 6f, 9, 15, 24 }}
{{Optimal ET sequence|legend=0| 6f, 9, 15, 24 }}
Line 412: Line 368:


==== 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 423: Line 381:
* WE: ~44/39 = 199.8321{{c}}, ~7/4 = 952.5580{{c}} (~40/39 = 46.6024{{c}})
* 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}})
* CWE: ~44/39 = 200.0000{{c}}, ~7/4 = 953.2005{{c}} (~40/39 = 46.7995{{c}})
<!-- * CTE: ~44/39 = 200.000{{c}}, ~7/4 = 952.531{{c}} (~40/39 = 47.469{{c}})
* POTE: ~44/39 = 200.000{{c}}, ~7/4 = 953.358{{c}} (~40/39 = 46.642{{c}}) -->


{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}
{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}
Line 431: Line 387:


== 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 444: Line 402:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~14/9 = 754.2078{{c}} (~36/35 = 45.7922{{c}})
* [[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 }}
: error map: {{val| 0.000 +6.461 +13.686 +2.213 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~14/9 = 752.834{{c}} (~36/35 = 47.166{{c}})
: [[error map]]: {{val| 0.000 +3.712 +13.686 -4.658 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~14/9 = 754.882{{c}} (~36/35 = 45.118{{c}})
: error map: {{val| 0.000 +7.808 +13.686 +5.583 }} -->


{{Optimal ET sequence|legend=1| 24, 27 }}
{{Optimal ET sequence|legend=1| 24, 27 }}
Line 463: Line 417:
* WE: ~5/4 = 398.8946{{c}}, ~14/9 = 752.1272{{c}} (~36/35 = 45.6619{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.5000{{c}} (~36/35 = 46.5000{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~14/9 = 752.051{{c}} (~36/35 = 47.949{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~14/9 = 754.212{{c}} (~36/35 = 45.788{{c}}) -->


{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
Line 473: Line 425:
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 }}
Line 480: Line 432:
* WE: ~5/4 = 399.1053{{c}}, ~14/9 = 752.0643{{c}} (~36/35 = 46.1463{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.3806{{c}} (~36/35 = 46.6194{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~14/9 = 752.128{{c}} (~36/35 = 47.872{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~14/9 = 753.750{{c}} (~36/35 = 46.250{{c}}) -->


{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
Line 488: Line 438:


== 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 501: Line 453:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~32/21 = 747.9138{{c}} (~21/20 = 52.0862{{c}})
* [[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 }}
: error map: {{val| 0.000 -6.127 +13.686 -12.567 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~32/21 = 747.948{{c}} (~21/20 = 52.052{{c}})
: [[error map]]: {{val| 0.000 -6.058 +13.686 -12.671 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~32/21 = 747.907{{c}} (~21/20 = 52.093{{c}})
: error map: {{val| 0.000 -6.143 +13.686 -12.544 }} -->


{{Optimal ET sequence|legend=1| 21, 24, 45c }}
{{Optimal ET sequence|legend=1| 21, 24, 45c }}
Line 520: Line 468:
* WE: ~5/4 = 400.0637{{c}}, ~32/21 = 748.4638{{c}} (~33/32 = 51.6637{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.3383{{c}} (~33/32 = 51.6617{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~32/21 = 748.296{{c}} (~33/32 = 51.704{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~32/21 = 748.345{{c}} (~33/32 = 51.655{{c}}) -->


{{Optimal ET sequence|legend=0| 21, 24 }}
{{Optimal ET sequence|legend=0| 21, 24 }}
Line 537: Line 483:
* WE: ~5/4 = 399.8855{{c}}, ~32/21 = 748.2378{{c}} (~33/32 = 51.5332{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.4655{{c}} (~33/32 = 51.5345{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~32/21 = 748.525{{c}} (~33/32 = 51.475{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~32/21 = 748.452{{c}} (~33/32 = 51.548{{c}}) -->


{{Optimal ET sequence|legend=0| 21, 24 }}
{{Optimal ET sequence|legend=0| 21, 24 }}
Line 545: Line 489:


== 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 558: Line 504:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~28/25 = 176.2984{{c}} (~49/48 = 47.4031{{c}})
* [[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 }}
: error map: {{val| 0.000 +3.239 +13.686 +7.473 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~28/25 = 175.359{{c}}
: [[error map]]: {{val| 0.000 -0.521 +13.686 +6.533 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~28/25 = 176.646{{c}}
: error map: {{val| 0.000 +4.629 +13.686 +7.820 }} -->


{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
Line 577: Line 519:
* WE: ~5/4 = 398.6615{{c}}, ~11/10 = 176.3886{{c}} (~49/48 = 45.8843{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.5471{{c}} (~49/48 = 46.9059{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~11/10 = 175.053{{c}}
* POTE: ~5/4 = 400.000{{c}}, ~11/10 = 176.981{{c}} -->


{{Optimal ET sequence|legend=0| 6, 21, 27e }}
{{Optimal ET sequence|legend=0| 6, 21, 27e }}
Line 594: Line 534:
* WE: ~5/4 = 398.8612{{c}}, ~11/10 = 176.0486{{c}} (~49/48 = 46.7640{{c}})
* 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}})
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.3326{{c}} (~49/48 = 47.3348{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~11/10 = 175.252{{c}}
* POTE: ~5/4 = 400.000{{c}}, ~11/10 = 176.551{{c}} -->


{{Optimal ET sequence|legend=0| 6, 21, 27e }}
{{Optimal ET sequence|legend=0| 6, 21, 27e }}
Line 602: Line 540:


== 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 615: Line 555:
* [[CWE]]: ~54/49 = 200.0000{{c}}, ~45/28 = 854.7144{{c}} (~36/35 = 54.7144{{c}})
* [[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 }}
: error map: {{val| 0.000 +7.474 +13.686 -4.683 }}
<!-- * [[CTE]]: ~54/49 = 200.000{{c}}, ~45/28 = 853.190{{c}} (~36/35 = 53.190{{c}})
: [[error map]]: {{val| 0.000 +4.425 +13.686 -9.256 }}
* [[POTE]]: ~54/49 = 200.000{{c}}, ~45/28 = 855.485{{c}} (~36/35 = 55.485{{c}})
: error map: {{val| 0.000 +9.015 +13.686 -2.371 }} -->


{{Optimal ET sequence|legend=1| 18, 24, 42 }}
{{Optimal ET sequence|legend=1| 18, 24, 42 }}
Line 634: Line 570:
* WE: ~54/49 = 199.5188{{c}}, ~18/11 = 853.1623{{c}} (~36/35 = 55.0872{{c}})
* 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}})
* CWE: ~54/49 = 200.0000{{c}}, ~18/11 = 854.3545{{c}} (~36/35 = 54.3545{{c}})
<!-- * CTE: ~54/49 = 200.000{{c}}, ~18/11 = 852.597{{c}} (~36/35 = 52.597{{c}})
* POTE: ~54/49 = 200.000{{c}}, ~18/11 = 855.220{{c}} (~36/35 = 55.220{{c}}) -->


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


Badness (Sintel): 2.67
Badness (Sintel): 2.67
== Trisected ==
{{Main|Trisected}}
{{See also|Subgroup temperaments #Trisect}}
Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap| 15 & 36 }} temperament.
It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 128/125, 1029/1000
{{Mapping|legend=1| 3 0 7 10 | 0 3 0 -1 }}
: mapping generators: ~5/4, ~10/7
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.5199{{c}}, ~10/7 = 633.7146{{c}}
: [[error map]]: {{val| -1.440 -0.811 +10.325 -7.342 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~10/7 = 634.3393{{c}}
: error map: {{val| 0.000 +1.063 +13.686 -3.165 }}
{{Optimal ET sequence|legend=1| 15, 36, 51c }}
[[Badness]] (Sintel): 2.61
=== 11-limit ===
This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]].
Subgroup: 2.3.5.7.11
Comma list: 56/55, 128/125, 1029/1000
Mapping: {{Mapping| 3 0 7 10 12 | 0 3 0 -1 -1 }}
Optimal tunings:
* WE: ~5/4 = 399.3049{{c}}, ~10/7 = 633.7891{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.7695{{c}}
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}
Badness (Sintel): 1.60
=== 13-limit ===
This extension is natural since it equates the [[10/7]], which is 1/3 of [[3/1]] since 1029/1000 is tempered out, with [[13/9]], thus tempering out [[91/90]] and [[2197/2187]].
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 91/90, 128/125, 1029/1000
Mapping: {{Mapping| 3 0 7 10 12 0 | 0 3 0 -1 -1 7 }}
Optimal tunings:
* WE: ~5/4 = 399.3194{{c}}, ~10/7 = 634.0637{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.9907{{c}}
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}
Badness (Sintel): 1.70


== 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 657: Line 649:
* [[CWE]]: ~49/45 = 133.3333{{c}}, ~3/2 = 705.5157{{c}} (~36/35 = 38.8490{{c}})
* [[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 }}
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}
<!-- * [[CTE]]: ~49/45 = 133.333{{c}}, ~3/2 = 702.004{{c}} (~36/35 = 35.338{{c}})
: [[error map]]: {{val| 0.000 +0.049 +13.686 -0.155 }}
* [[POTE]]: ~49/45 = 133.333{{c}}, ~3/2 = 707.167{{c}} (~36/35 = 40.501{{c}})
: error map: {{val| 0.000 +5.212 +13.686 +5.008 }} -->


{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}
{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}
Line 676: Line 664:
* WE: ~12/11 = 132.9553{{c}}, ~3/2 = 704.7217{{c}} (~36/35 = 39.9453{{c}})
* 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}})
* CWE: ~12/11 = 133.3333{{c}}, ~3/2 = 704.5723{{c}} (~36/35 = 37.9056{{c}})
<!-- * CTE: ~12/11 = 133.333{{c}}, ~3/2 = 699.622{{c}} (~36/35 = 32.955{{c}})
* POTE: ~12/11 = 133.333{{c}}, ~3/2 = 706.726{{c}} (~36/35 = 40.059{{c}}) -->


{{Optimal ET sequence|legend=0| 9, 18e, 27e, 63cee }}
{{Optimal ET sequence|legend=0| 9, 18e, 27e, 63cee }}
Line 693: Line 679:
* WE: ~14/13 = 133.0143{{c}}, ~3/2 = 705.1969{{c}} (~36/35 = 40.1256{{c}})
* 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}})
* CWE: ~14/13 = 133.3333{{c}}, ~3/2 = 705.0176{{c}} (~36/35 = 38.3510{{c}})
<!-- * CTE: ~14/13 = 133.333{{c}}, ~3/2 = 700.433{{c}} (~36/35 = 33.766{{c}})
* POTE: ~14/13 = 133.333{{c}}, ~3/2 = 706.889{{c}} (~36/35 = 40.222{{c}}) -->


{{Optimal ET sequence|legend=0| 9, 18e, 27e }}
{{Optimal ET sequence|legend=0| 9, 18e, 27e }}
Line 701: Line 685:


== 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 714: Line 700:
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~48/35 = 500.9654{{c}} (~15/14 = 100.9654{{c}})
* [[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 }}
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~48/35 = 498.637{{c}} (~15/14 = 98.637{{c}})
: [[error map]]: {{val| 0.000 -6.045 +13.686 +28.447 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~48/35 = 501.980{{c}} (~15/14 = 101.980{{c}})
: error map: {{val| 0.000 +3.986 +13.686 +35.134 }} -->


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


[[Badness]] (Sintel): 3.50
[[Badness]] (Sintel): 3.50
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 56/55, 128/125, 360/343
Mapping: {{Mapping| 3 1 7 6 8 | 0 3 0 2 2 }}
Optimal tunings:
* WE: ~5/4 = 397.4954{{c}}, ~15/11 = 499.0962{{c}} (~15/14 = 101.6008{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~15/11 = 500.7745{{c}} (~15/14 = 100.7745{{c}})
{{Optimal ET sequence|legend=0| 3b, 9bde, 12 }}
Badness (Sintel): 2.60
== Subgroup extensions ==
=== Augmented (2.3.5.19 subgroup) ===
Augmented works on the [[2.3.5.19 subgroup]] very well, where the 1/3-octave period stands in for ~5/4, ~19/15, and ~24/19.
[[Subgroup]]: 2.3.5.19
[[Comma list]]: 76/75, 96/95
{{Mapping|legend=2| 3 0 7 8 | 0 1 0 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.0754{{c}}, ~3/2 = 705.0615{{c}} (~19/18 = 93.0893{{c}})
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.1072{{c}} (~19/18 = 94.8928{{c}})
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cch }}
[[Badness]] (Sintel): 0.264


== External links ==
== External links ==
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament] {{dead link}}
* [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"