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Augmented family: Difference between revisions

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{{Interwiki
| en = Augmented family
| de = Übermässige Temperaturen
}}
{{Technical data page}}
{{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 '''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 [[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.  
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.
 
== Septimal augmented (augene) ==
{{Main| Augmented (temperament) }}


== Augene ==
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.
{{Main| Augene }}


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


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


Optimal tunings:  
Optimal tunings:  
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Comma list: 40/39, 56/55, 64/63, 66/65
Comma list: 40/39, 56/55, 64/63, 66/65


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


Optimal tunings:  
Optimal tunings:  
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Comma list: 56/55, 64/63, 91/90, 100/99
Comma list: 56/55, 64/63, 91/90, 100/99


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


Optimal tunings:  
Optimal tunings:  
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Comma list: 56/55, 64/63, 78/77, 100/99
Comma list: 56/55, 64/63, 78/77, 100/99


Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}
{{Mapping|legend=0| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 55/54, 64/63, 77/75
Comma list: 55/54, 64/63, 77/75


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


Optimal tunings:  
Optimal tunings:  
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== August ==
== 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
[[Subgroup]]: 2.3.5.7


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


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


Optimal tunings:  
Optimal tunings:  
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Comma list: 27/26, 36/35, 45/44, 56/55
Comma list: 27/26, 36/35, 45/44, 56/55


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


Optimal tunings:  
Optimal tunings:  
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Comma list: 26/25, 36/35, 45/44, 56/55
Comma list: 26/25, 36/35, 45/44, 56/55


Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
{{Mapping|legend=0| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 28/27, 55/54, 128/125
Comma list: 28/27, 55/54, 128/125


Mapping: {{mapping| 3 0 7 -6 -4 | 0 1 0 3 3 }}
{{Mapping|legend=0| 3 0 7 -6 -4 | 0 1 0 3 3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 21/20, 33/32, 128/125
Comma list: 21/20, 33/32, 128/125


Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}
{{Mapping|legend=0| 3 0 7 13 15 | 0 1 0 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
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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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Comma list: 50/49, 56/55, 125/121
Comma list: 50/49, 56/55, 125/121


Mapping: {{mapping| 6 0 14 17 21 | 0 1 0 0 0 }}
{{Mapping|legend=0| 6 0 14 17 21 | 0 1 0 0 0 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 50/49, 56/55, 66/65, 105/104
Comma list: 50/49, 56/55, 66/65, 105/104


Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}
{{Mapping|legend=0| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}


Optimal tunings:  
Optimal tunings:  
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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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Comma list: 49/48, 56/55, 77/75
Comma list: 49/48, 56/55, 77/75


Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}
{{Mapping|legend=0| 3 0 7 6 8 | 0 2 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 49/48, 56/55, 66/65, 77/75
Comma list: 49/48, 56/55, 66/65, 77/75


Mapping: {{mapping| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}
{{Mapping|legend=0| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}


Optimal tunings:  
Optimal tunings:  
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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


Comma list: 49/48, 56/55, 77/75, 507/500
Comma list: 49/48, 56/55, 77/75, 507/500


Mapping: {{mapping| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}
{{Mapping|legend=0| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}


: mapping generators: ~44/39, ~7/4
: mapping generators: ~44/39, ~7/4
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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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Comma list: 56/55, 128/125, 243/242
Comma list: 56/55, 128/125, 243/242


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


Optimal tunings:  
Optimal tunings:  
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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|legend=0| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}


Optimal tunings:  
Optimal tunings:  
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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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Comma list: 56/55, 128/125, 1323/1250
Comma list: 56/55, 128/125, 1323/1250


Mapping: {{mapping| 3 1 7 14 16 | 0 2 0 -3 -3 }}
{{Mapping|legend=0| 3 1 7 14 16 | 0 2 0 -3 -3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 56/55, 66/65, 105/104, 507/500
Comma list: 56/55, 66/65, 105/104, 507/500


Mapping: {{mapping| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}
{{Mapping|legend=0| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 473: Line 485:


== 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 496: Line 510:
Comma list: 56/55, 128/125, 2401/2400
Comma list: 56/55, 128/125, 2401/2400


Mapping: {{mapping| 3 3 7 8 10 | 0 4 0 1 1 }}
{{Mapping|legend=0| 3 3 7 8 10 | 0 4 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 511: Line 525:
Comma list: 56/55, 78/77, 128/125, 507/500
Comma list: 56/55, 78/77, 128/125, 507/500


Mapping: {{mapping| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}
{{Mapping|legend=0| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 522: Line 536:


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


Mapping: {{mapping| 6 1 14 4 8 | 0 2 0 3 3 }}
{{Mapping|legend=0| 6 1 14 4 8 | 0 2 0 3 3 }}


Optimal tunings:  
Optimal tunings:  
Line 554: Line 570:


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 581: Line 655:
Comma list: 56/55, 128/125, 540/539
Comma list: 56/55, 128/125, 540/539


Mapping: {{mapping| 9 0 21 11 17 | 0 1 0 1 1 }}
{{Mapping|legend=0| 9 0 21 11 17 | 0 1 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 596: Line 670:
Comma list: 56/55, 78/77, 91/90, 128/125
Comma list: 56/55, 78/77, 91/90, 128/125


Mapping: {{mapping| 9 0 21 11 17 19 | 0 1 0 1 1 1 }}
{{Mapping|legend=0| 9 0 21 11 17 19 | 0 1 0 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 607: Line 681:


== 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 624: Line 700:


[[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=3| 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://web.archive.org/web/20211201070113/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:Augmented family| ]] <!-- main article -->
[[Category:Augmented| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]
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