21st-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|21}}
{{Infobox fractional-octave|21}}
This page collectes temperaments with a period of 1/21 of an octave.
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 the [[zeta]] edo list.
 
Temperaments discussed elsewhere include
* ''[[Akjayland]]'' → [[Landscape microtemperaments #Akjayland|Landscape microtemperaments]]


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.
== 21-23-commatic ==
== 21-23-commatic ==
Subgroup: 2.23
Subgroup: 2.23
Line 10: Line 14:
{{Mapping|legend=2|21 95}}
{{Mapping|legend=2|21 95}}


: mapping generator: ~529/512 = 1\21
: 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]]


== Akjayland ==
== Scandium ==
{{See also| Landscape microtemperaments #Akjayland }}
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
Subgroup: 2.3.5.7


[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}


[[Mapping]]: [{{val| 21 1 38 102 }}, {{val| 0 3 1 -4 }}]
{{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }}


Mapping generators: ~1323/1280, ~131072/91875
: Mapping generators: ~403368/390625 = 1\21, ~160/147


[[Optimal tuning]] ([[CTE]]): ~131072/91875 = 614.9354
[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
[[Support]]ing [[ET]]s: {{EDOs|189b, 525, 861, 1050, 1386, 1911, 2436}}


[[Badness]]: 0.0309
=== 23-limit ===


=== Vasca ===
Subgroup: 2.3.5.7.11.13.17.19.21.23
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of 21 [[23/16]]'s equal with 11 octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  


Subgroup: 2.3.5.7.11
Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092


Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
{{Mapping|legend=1| 21 0 59 82 24 111 114 38 95 | 0 13 -4 -9 19 -13 -11 20 0}}


Mapping: [{{val| 21 4 39 98 58 }}, {{val| 0 6 2 -8 3 3 }}]
: Mapping generators: ~216/209 = 1\21, ~160/147


Mapping generators: ~1323/1280, ~6615/5632
[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.308{{C}}


Optimal tuning (CTE): ~6615/5632 = 278.8998
[[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
== 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}}


Badness: 0.0949
Subgroup: 2.3.5.7
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
 
Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]
 
Optimal tuning (CTE): ~168/143 = 278.9058
 
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
 
Badness: 0.0551
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
 
Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]
 
Optimal tuning (CTE): ~168/143 = 278.9036
 
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
 
Badness: 0.0319
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
 
Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]
 
Optimal tuning (CTE): ~168/143 = 278.9036
 
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
 
Badness: 0.0270
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
 
Mapping: [{{val| 21 34 49 58 73 77 85 91 95 }}, {{val| 0 -6 -2 8 -3 6 7 -15 0 }}]
 
Optimal tuning (CTE): ~168/143 = 278.8971
 
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
 
Badness: 0.0199


== Scandium ==
Comma list: [[225/224]], {{Monzo|27 1 1 -11}}
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
{{Mapping|legend=1| 21 0 82 59 | 0 1 -1 0 }}


Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}
: Mapping generators: ~16807/16384 = 1\21, ~3


{{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }}
[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.120{{C}}


: mapping generators: ~403368/390625 = 1\21, ~160/147 = 146.305
{{Optimal ET sequence|legend=1|21, 63, 84, 147}}


[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305
[[Badness]] (Sintel): 5.605


[[Support]]ing [[ET]]s: {{EDOs|189b, 525, 861, 1050, 1386, 1911, 2436}}
=== 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]].


=== 23-limit ===
Subgroup: 2.3.5.7.11.13.23.29.31.43


Subgroup: 2.3.5.7.11.13.17.19.21.23
Comma list: 155/154, 225/224, [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]]


Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092
{{Mapping|legend=1| 21 0 82 59 106 111 95 102 104 114 | 0 1 -1 0 -1 -1 0 0 0 0 }}


{{Mapping|legend=1| 21 0 59 82 24 111 114 38 95 | 0 13 -4 -9 19 -13 -11 20 0}}
: Mapping generators: ~16807/16384 = 1\21, ~3


: mapping generators: ~216/209 = 1\21, ~160/147 = 146.308
Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}}


[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305
{{Optimal ET sequence|legend=0|21, 63, 84, 147}}


[[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}}
Badness (Sintel): 1.317


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}

Latest revision as of 06:21, 15 January 2026

← 20th-octave • 21st-octave • 22nd-octave →

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

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