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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]]. |
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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 the [[zeta]] edo list. |
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| | Temperaments discussed elsewhere include |
| | * ''[[Akjayland]]'' → [[Landscape microtemperaments #Akjayland|Landscape microtemperaments]] |
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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.
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| == 21-23-commatic == | | == 21-23-commatic == |
| Subgroup: 2.23 | | Subgroup: 2.23 |
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| {{Mapping|legend=2|21 95}} | | {{Mapping|legend=2|21 95}} |
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| : mapping generator: ~529/512 = 1\21 | | : Mapping generator: ~529/512 = 1\21 |
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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]] |
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| == 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. |
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| [[Subgroup]]: 2.3.5.7
| | Subgroup: 2.3.5.7 |
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| [[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
| | Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}} |
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| [[Mapping]]: [{{val| 21 1 38 102 }}, {{val| 0 3 1 -4 }}]
| | {{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }} |
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| Mapping generators: ~1323/1280, ~131072/91875 | | : Mapping generators: ~403368/390625 = 1\21, ~160/147 |
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| [[Optimal tuning]] ([[CTE]]): ~131072/91875 = 614.9354 | | [[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305 |
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| {{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}} |
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| [[Badness]]: 0.0309
| | === 23-limit === |
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| === 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.
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| Subgroup: 2.3.5.7.11
| | Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092 |
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| 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}} |
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| Mapping: [{{val| 21 4 39 98 58 }}, {{val| 0 6 2 -8 3 3 }}] | | : Mapping generators: ~216/209 = 1\21, ~160/147 |
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| Mapping generators: ~1323/1280, ~6615/5632
| | [[Optimal tuning]] ([[CTE]]): ~160/147 = 146.308{{C}} |
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| Optimal tuning (CTE): ~6615/5632 = 278.8998
| | [[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}} |
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| {{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}} |
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| Badness: 0.0949
| | Subgroup: 2.3.5.7 |
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13 | |
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| Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683 | | Comma list: [[225/224]], {{Monzo|27 1 1 -11}} |
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| Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]
| | {{Mapping|legend=1| 21 0 82 59 | 0 1 -1 0 }} |
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| Optimal tuning (CTE): ~168/143 = 278.9058
| | : Mapping generators: ~16807/16384 = 1\21, ~3 |
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| {{Optimal ET sequence|legend=1| 168, 357, 525, 882 }} | | [[Optimal tuning]] ([[CWE]]): ~3/2 = 701.120{{C}} |
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| Badness: 0.0551
| | {{Optimal ET sequence|legend=1|21, 63, 84, 147}} |
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| ==== 17-limit ====
| | [[Badness]] (Sintel): 5.605 |
| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
| | === 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]]. |
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| Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]
| | Subgroup: 2.3.5.7.11.13.23.29.31.43 |
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| Optimal tuning (CTE): ~168/143 = 278.9036
| | Comma list: 155/154, 225/224, [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]] |
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| {{Optimal ET sequence|legend=1| 168, 357, 525, 882 }} | | {{Mapping|legend=1| 21 0 82 59 106 111 95 102 104 114 | 0 1 -1 0 -1 -1 0 0 0 0 }} |
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| Badness: 0.0319
| | : Mapping generators: ~16807/16384 = 1\21, ~3 |
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| ==== 19-limit ====
| | Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}} |
| Subgroup: 2.3.5.7.11.13.17.19
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| Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
| | {{Optimal ET sequence|legend=0|21, 63, 84, 147}} |
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| Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]
| | Badness (Sintel): 1.317 |
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| Optimal tuning (CTE): ~168/143 = 278.9036
| | {{Navbox fractional-octave}} |
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| {{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }} | |
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| Badness: 0.0270
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| ==== 23-limit ====
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| Subgroup: 2.3.5.7.11.13.17.19.23
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| Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
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| Mapping: [{{val| 21 34 49 58 73 77 85 91 95 }}, {{val| 0 -6 -2 8 -3 6 7 -15 0 }}]
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| Optimal tuning (CTE): ~168/143 = 278.8971
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| {{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
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| Badness: 0.0199
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| == Scandium ==
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| 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.
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| Subgroup: 2.3.5.7
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| Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}
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| {{Mapping|legend=1| 21 0 59 82 | 0 13 -4 -9 }}
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| : mapping generators: ~403368/390625 = 1\21, ~160/147 = 146.305
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| [[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305
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| [[Support]]ing [[ET]]s: {{EDOs|189b, 525, 861, 1050, 1386, 1911, 2436}}
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| === 23-limit ===
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| Subgroup: 2.3.5.7.11.13.17.19.21.23
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| Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092
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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}}
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| : mapping generators: ~216/209 = 1\21, ~160/147 = 146.308
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| [[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305
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| [[Support]]ing [[ET]]s: {{EDOs|525, 861h, 1050f, 1911}}
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