21st-octave temperaments: Difference between revisions
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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 [[zeta]] edo list. | ||
Temperaments discussed elsewhere include | |||
* ''[[Akjayland]]'' → [[Landscape microtemperaments #Akjayland|Landscape microtemperaments]] | |||
== 21-23-commatic == | == 21-23-commatic == | ||
Subgroup: 2.23 | Subgroup: 2.23 | ||
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[[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]] | [[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]] | ||
== Scandium == | == Scandium == | ||
Revision as of 03:55, 14 December 2024
Template:Fractional-octave navigation This page collectes 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 441, 1407, and 1848 are. 441 and 1848 are also members of zeta edo list.
Temperaments discussed elsewhere include
21-23-commatic
Subgroup: 2.23
Comma list: [95 0 0 0 0 0 0 0 -21⟩
Subgroup-val mapping: [⟨21 95]]
- mapping generator: ~529/512 = 1\21
Supporting ETs: 21N, N = 1 to 96, largest: 2016edo
Scandium
Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, Encyclopaedia Britannica entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland or many temperaments in the thousands which contain 3edo as a subset, it is not a landscape system. 39/32 is mapped into 6\21 and 23/16 is, as usual, mapped into 11\21.
Subgroup: 2.3.5.7
Comma list: [47 -7 -7 -7⟩, [-29 0 27 -12⟩
Mapping: [⟨21 0 59 82], ⟨0 13 -4 -9]]
- mapping generators: ~403368/390625 = 1\21, ~160/147 = 146.305
Optimal tuning (CTE): ~160/147 = 146.305
Supporting ETs: 189b, 525, 861, 1050, 1386, 1911, 2436
23-limit
Subgroup: 2.3.5.7.11.13.17.19.21.23
Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092
Mapping: [⟨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 = 146.308
Optimal tuning (CTE): ~160/147 = 146.305