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== 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.  
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.  


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=== 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 }}
 
{{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 }}
== Septimal augmented (augene) ==
{{Main| Augmented (temperament) }}


Optimal tunings:
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.  
* 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
 
== Augene ==
{{Main| Augene }}


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


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== 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
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==== 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


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== 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


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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 }}
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== 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


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== 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


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== 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


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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
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== 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


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[[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 ==
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[[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"