21st-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|21}} | {{Infobox fractional-octave|21}} | ||
This page | This page collects temperaments with a period of 1/21 of an [[octave]]. | ||
Although 21edo itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs|441, 1407, and 1848}} are. 441 and 1848 are also members of [[zeta]] edo list. | Although [[21edo]] itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs|441, 1407, and 1848}} are. 441 and 1848 are also members of the [[zeta]] edo list. | ||
Temperaments discussed elsewhere include | Temperaments discussed elsewhere include | ||
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{{Mapping|legend=2|21 95}} | {{Mapping|legend=2|21 95}} | ||
: | : Mapping generator: ~529/512 = 1\21 | ||
[[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]] | [[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]] | ||
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{{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }} | {{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }} | ||
: | : Mapping generators: ~403368/390625 = 1\21, ~160/147 | ||
[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305 | [[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305 | ||
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{{Mapping|legend=1| 21 0 59 82 24 111 114 38 95 | 0 13 -4 -9 19 -13 -11 20 0}} | {{Mapping|legend=1| 21 0 59 82 24 111 114 38 95 | 0 13 -4 -9 19 -13 -11 20 0}} | ||
: | : Mapping generators: ~216/209 = 1\21, ~160/147 | ||
[[Optimal tuning]] ([[CTE]]): ~160/147 = 146. | [[Optimal tuning]] ([[CTE]]): ~160/147 = 146.308{{C}} | ||
[[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}} | [[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}} | ||
== Blackmagic == | |||
Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes. It was named by [[User:Overthink|Overthink]] in 2026 as a twist on "blackjack" (which itself already refers to the 21-note [[MOS scale|mos]] of [[miracle]]), as well as because of its higher-limit properties. {{Todo|review}} | |||
Subgroup: 2.3.5.7 | |||
Comma list: [[225/224]], {{Monzo|27 1 1 -11}} | |||
{{Mapping|legend=1| 21 0 82 59 | 0 1 -1 0 }} | |||
: Mapping generators: ~16807/16384 = 1\21, ~3 | |||
[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.120{{C}} | |||
{{Optimal ET sequence|legend=1|21, 63, 84, 147}} | |||
[[Badness]] (Sintel): 5.605 | |||
=== 2.3.5.7.11.13.23.29.31.43 subgroup === | |||
Primes 17 and 19 could be included by mapping them to -1 and 1 generators respectively, though in practice this mapping only works in [[84edo]]. | |||
Subgroup: 2.3.5.7.11.13.23.29.31.43 | |||
Comma list: 155/154, 225/224, [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]] | |||
{{Mapping|legend=1| 21 0 82 59 106 111 95 102 104 114 | 0 1 -1 0 -1 -1 0 0 0 0 }} | |||
: Mapping generators: ~16807/16384 = 1\21, ~3 | |||
Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}} | |||
{{Optimal ET sequence|legend=0|21, 63, 84, 147}} | |||
Badness (Sintel): 1.317 | |||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||