Augmented family: Difference between revisions
- CTE & POTE tunings |
Intro to some of these temps |
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{{Technical data page}} | {{Technical data page}} | ||
The | 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 | ||
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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. | 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 == | == Augene == | ||
{{Main| Augene }} | {{Main| Augene }} | ||
Augene tempers out 64/63 and 126/125. It may be described as the {{nowrap| 12 & 15 }} temperament. [[27edo]] and [[39edo]] in the 39d val make for good tunings. | |||
[[Subgroup]]: 2.3.5.7 | [[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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; 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) | ||
* [ | * [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 ==== | ||
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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, | 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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== Niner == | == Niner == | ||
Niner gives 9 as the complexity of | 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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== 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]] | ||