21st-octave temperaments: Difference between revisions

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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 [[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.

== Akjayland ==

Temperaments discussed elsewhere include

''~~Main article:~~ [[Landscape microtemperaments#Akjayland]]''

* ''[[Akjayland]]'' → [[Landscape microtemperaments #Akjayland|Landscape microtemperaments]]

== 21-23-commatic ==

Subgroup: 2.23

Comma list: {{monzo|95 0 0 0 0 0 0 0 -21}}

: Mapping generator: ~529/512 = 1\21

[[Support]]ing [[ET]]s: 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]]: ~~250047~~/~~250000~~, {{~~monzo~~| 43 -1 -~~13 -4~~ }}

Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}

: Mapping generators: ~403368/390625 = 1\21, ~160/147

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

[[Support]]ing [[ET]]s: {{EDOs|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|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 = 146.308{{C}}

[[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 generators: ~16807/16384 = 1\21, ~3

[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.120{{C}}

[[~~Mapping~~]]: ~~[{{val| 21 1 38 102 }}, {{val| 0 3 1 -4 }}]~~

[[Badness]] (Sintel): 5.605

~~Mapping~~ generators~~: ~1323/1280~~, ~~~131072/91875~~

=== 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]].

~~[[POTE generator]]~~: ~~~131072/91875 = 614~~.~~9361~~

Subgroup: 2.3.5.7.11.13.23.29.31.43

[[~~Optimal GPV sequence~~]]~~: {{val list| 84~~, ~~273~~, ~~357~~, ~~441~~, ~~966~~, ~~1407, 1848, 7833, 9681, 11529, 13377c }}~~

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

~~[[Badness]]:~~ 0~~.0309~~

{{Mapping|legend=1| 21 0 82 59 106 111 95 102 104 114 | 0 1 -1 0 -1 -1 0 0 0 0 }}

=~~= Vasca ==~~

: Mapping generators: ~16807/16384 = 1\21, ~3

~~Vasca tempers out the {{Monzo|95 0 0 0 0 0 0 0 -21}} in the 23-limit, and sets a stack of~~ 21 ~~[[23/16]]<nowiki/>s equal with 11 octaves. derives from elements Vanadium (23) and Scandium (21), since this uses the 23rd harmonic~~, ~~which itself is extremely well represented in 21edo. It is defined as a 525 & 357 temperament.~~

~~Subgroup~~: 2.~~3.5.7.11.13.17.19.23~~

Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}}

~~Comma list: 1496/1495, 2376/2375, 3060/3059, 17204/17199, 282625/282624,~~ {{~~Monzo~~|95 0 ~~0 0 0 0 0 0 -~~21}}

~~Mapping~~: ~~[{{Val|21 34 49 58 73 77 85 91 95}}, {{Val|0 -6 -2 8 -3 6 7 -15 0}}]~~

Badness (Sintel): 1.317

~~POTE generator: 6.8162~~

@@ Line 1: / Line 1: @@
-This page collectes temperaments with a period of 1/21 of an octave.
+{{Infobox fractional-octave|21}}
+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.
-== Akjayland ==
+Temperaments discussed elsewhere include
-''Main article: [[Landscape microtemperaments#Akjayland]]''
+* ''[[Akjayland]]'' → [[Landscape microtemperaments #Akjayland|Landscape microtemperaments]]
+== 21-23-commatic ==
+Subgroup: 2.23
+Comma list: {{monzo|95 0 0 0 0 0 0 0 -21}}
+{{Mapping|legend=2|21 95}}
+: Mapping generator: ~529/512 = 1\21
+[[Support]]ing [[ET]]s: 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]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
+Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}
+{{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
+[[Support]]ing [[ET]]s: {{EDOs|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|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 = 146.308{{C}}
+[[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}}
-[[Mapping]]: [{{val| 21 1 38 102 }}, {{val| 0 3 1 -4 }}]
+[[Badness]] (Sintel): 5.605
-Mapping generators: ~1323/1280, ~131072/91875
+=== 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]].
-[[POTE generator]]: ~131072/91875 = 614.9361
+Subgroup: 2.3.5.7.11.13.23.29.31.43
-[[Optimal GPV sequence]]: {{val list| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
+Comma list: 155/154, 225/224, [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]]
-[[Badness]]: 0.0309
+{{Mapping|legend=1| 21 0 82 59 106 111 95 102 104 114 | 0 1 -1 0 -1 -1 0 0 0 0 }}
-== Vasca ==
+: Mapping generators: ~16807/16384 = 1\21, ~3
-Vasca tempers out the {{Monzo|95 0 0 0 0 0 0 0 -21}} in the 23-limit, and sets a stack of 21 [[23/16]]<nowiki/>s equal with 11 octaves. derives from elements Vanadium (23) and Scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo. It is defined as a 525 & 357 temperament.
-Subgroup: 2.3.5.7.11.13.17.19.23
+Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}}
-Comma list: 1496/1495, 2376/2375, 3060/3059, 17204/17199, 282625/282624, {{Monzo|95 0 0 0 0 0 0 0 -21}}
+{{Optimal ET sequence|legend=0|21, 63, 84, 147}}
-Mapping: [{{Val|21 34 49 58 73 77 85 91 95}}, {{Val|0 -6 -2 8 -3 6 7 -15 0}}]
+Badness (Sintel): 1.317
-POTE generator: 6.8162
+{{Navbox fractional-octave}}