Alphatricot family: Difference between revisions
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{{Technical data page}} | |||
The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883). | The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883). | ||
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: mapping generators: ~2, ~59049/40960 | : mapping generators: ~2, ~59049/40960 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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[[Badness]] (Smith): 0.046093 | [[Badness]] (Smith): 0.046093 | ||
; Scales | |||
* [[Alphatricot17]] – proper [[2L 15s]] | |||
* [[Alphatricot19]] – improper [[17L 2s]] | |||
=== 2.3.5.13 subgroup === | === 2.3.5.13 subgroup === | ||
{{See also| No-fives subgroup temperaments #Threedic }} | {{See also| No-fives subgroup temperaments #Threedic }} | ||
This extension identifies the generator with [[13/9]] by tempering out the threedie, [[2197/2187]], providing a relatively low-complexity mapping for 13. | |||
Subgroup: 2.3.5.13 | Subgroup: 2.3.5.13 | ||
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Comma list: 2197/2187, 41067/40960 | Comma list: 2197/2187, 41067/40960 | ||
Mapping: {{mapping| 1 0 -13 0 | 0 3 29 7 }} | |||
Optimal tunings: | Optimal tunings: | ||
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{{Optimal ET sequence|legend=0| 17c, 36c, 53 }} | {{Optimal ET sequence|legend=0| 17c, 36c, 53 }} | ||
Badness (Sintel): 1.262 | |||
=== Catatricot === | |||
However, alphatricot in the 5-limit is far more accurate than threedic. Catatricot interprets the generator as ~[[75/52]] instead of 13/9, making the tempering of [[140625/140608]], the catasma, instead of the threedie. It also tempers out [[256000/255879]], the phaotisma. | |||
Subgroup: 2.3.5.13 | |||
Comma list: 140628/140625, 256000/255879 | |||
* | |||
* | Mapping: {{mapping| 1 0 -13 -28 | 0 3 29 60 }} | ||
Optimal tunings: | |||
* CTE: ~2 = 1200.000, ~75/52 = 634.009 | |||
* POTE: ~2 = 1200.000, ~75/52 = 634.0108 | |||
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }} | |||
Badness (Sintel): 0.181 | |||
== Alphatrillium == | == Alphatrillium == | ||
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{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }} | {{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }} | {{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }} | {{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }} | {{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }} | ||
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837 | [[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Alphatricot family| ]] <!-- main article --> | [[Category:Alphatricot family| ]] <!-- main article --> | ||
[[Category:Alphatricot| ]] <!-- key article --> | [[Category:Alphatricot| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||