21st-octave temperaments: Difference between revisions
akjayland was duplicated; landscape page had slightly better information |
→Blackmagic: note primes 17 and 19, etymology |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{ | {{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 | ||
| Line 14: | Line 14: | ||
{{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]] | ||
| Line 27: | Line 27: | ||
{{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 | ||
| Line 41: | Line 41: | ||
{{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}} | ||
Latest revision as of 06:21, 15 January 2026
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 441, 1407, and 1848 are. 441 and 1848 are also members of the 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
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
Optimal tuning (CTE): ~160/147 = 146.308 ¢
Supporting ETs: 525, 861h, 1050f, 1911
Blackmagic
Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes. It was named by Overthink in 2026 as a twist on "blackjack" (which itself already refers to the 21-note mos of miracle), as well as because of its higher-limit properties.
Subgroup: 2.3.5.7
Comma list: 225/224, [27 1 1 -11⟩
Mapping: [⟨21 0 82 59], ⟨0 1 -1 0]]
- Mapping generators: ~16807/16384 = 1\21, ~3
Optimal tuning (CWE): ~3/2 = 701.120 ¢
Optimal ET sequence: 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: [⟨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 ¢
Optimal ET sequence: 21, 63, 84, 147
Badness (Sintel): 1.317