Augmented family: Difference between revisions
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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 [[ | 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 == | ||
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== Trisected == | == Trisected == | ||
{{Main|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. | 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. | It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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[[Comma list]]: 128/125, 1029/1000 | [[Comma list]]: 128/125, 1029/1000 | ||
{{Mapping|legend=1| 3 | {{Mapping|legend=1| 3 0 7 10 | 0 3 0 -1 }} | ||
: mapping generators: ~5/4, ~10/7 | |||
: mapping generators: ~5/4, ~ | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~5/4 = 399. | * [[WE]]: ~5/4 = 399.5199{{c}}, ~10/7 = 633.7146{{c}} | ||
: [[error map]]: {{val| -1.440 -0.811 10.325 -7.342 }} | : [[error map]]: {{val| -1.440 -0.811 +10.325 -7.342 }} | ||
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~10/7 = 634.3393{{c}} | |||
* [[CWE]]: ~5/4 = 400. | |||
: error map: {{val| 0.000 +1.063 +13.686 -3.165 }} | : error map: {{val| 0.000 +1.063 +13.686 -3.165 }} | ||
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=== 11-limit === | === 11-limit === | ||
This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]]. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Comma list: 56/55, 128/125, 1029/1000 | Comma list: 56/55, 128/125, 1029/1000 | ||
Mapping: {{Mapping| 3 | Mapping: {{Mapping| 3 0 7 10 12 | 0 3 0 -1 -1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~5/4 = 399. | * 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 === | === 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 | Subgroup: 2.3.5.7.11.13 | ||
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Comma list: 56/55, 91/90, 128/125, 1029/1000 | Comma list: 56/55, 91/90, 128/125, 1029/1000 | ||
Mapping: {{Mapping| 3 | Mapping: {{Mapping| 3 0 7 10 12 0 | 0 3 0 -1 -1 7 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~5/4 = 399. | * 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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* [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:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||