Augmented family: Difference between revisions
Promoting augene to the canonical septimal augmented temp |
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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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== Septimal augmented (augene) == | == Septimal augmented (augene) == | ||
{{Main| | {{Main| Augmented (temperament) }} | ||
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. | 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. | ||
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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 == | ||
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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 == | ||