Augmented family: Difference between revisions
Move augene up as it's the best extension |
Intro to some of these temps |
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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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== 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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== 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, | 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 | ||