22nd-octave temperaments: Difference between revisions
→Major arcana: link to what it is |
→Vigintiduo: to new CWE stuff, switch to Sintel badness, etc. (+also in constrained WE I would literally write 1\period as it is that by definition) |
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== Vigintiduo == | == Vigintiduo == | ||
''Vigintiduo'' | ''Vigintiduo'' sets the [[7-limit]] to that of 22edo and harmonic 11 and the rest are independent generators. | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
| Line 14: | Line 14: | ||
{{Mapping|legend=1| 22 35 51 62 0 | 0 0 0 0 1 }} | {{Mapping|legend=1| 22 35 51 62 0 | 0 0 0 0 1 }} | ||
: mapping generators: ~25/24, ~11/8 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~25/24 = 54.462{{c}}, ~11/8 = 556.713 | |||
: [[error map]]: {{val| -1.830, 4.225, -8.737, 7.836, -0.093 }} | |||
* [[CWE]]: ~25/24 = 1\22, ~11/8 = 556.295 | |||
: error map: {{val| 0.000 -0.068 -0.088 -0.106 -0.375 }} | |||
{{Optimal ET sequence|legend=1| 22, 66de, 88bde, 110bd, 198bcdde }} | {{Optimal ET sequence|legend=1| 22, 66de, 88bde, 110bd, 198bcdde }} | ||
[[Badness]]: | [[Badness]]: 1.599 | ||
== Vigin == | == Vigin == | ||
Revision as of 17:31, 7 October 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
22edo is a notable tuning system, and as such it gives rise to some fractional-octave temperaments. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families.
See also:
Vigintiduo
Vigintiduo sets the 7-limit to that of 22edo and harmonic 11 and the rest are independent generators.
Subgroup: 2.3.5.7.11
Comma list: 50/49, 64/63, 245/243
Mapping: [⟨22 35 51 62 0], ⟨0 0 0 0 1]]
- mapping generators: ~25/24, ~11/8
- WE: ~25/24 = 54.462 ¢, ~11/8 = 556.713
- error map: ⟨-1.830, 4.225, -8.737, 7.836, -0.093]
- CWE: ~25/24 = 1\22, ~11/8 = 556.295
- error map: ⟨0.000 -0.068 -0.088 -0.106 -0.375]
Optimal ET sequence: 22, 66de, 88bde, 110bd, 198bcdde
Badness: 1.599
Vigin
Vigin has the same 11-limit mapping as 22et with harmonic 13 mapped to an independent generator.
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 55/54, 64/63, 99/98
Mapping: [⟨22 35 51 62 76 0], ⟨0 0 0 0 0 1]]
Optimal tuning (POTE): ~33/32 = 1\22, ~13/8 = 844.624
Badness: 0.029849
Major arcana
Major arcana is named after a subset of Tarot cards of which there is 22. Tempers out the [-193 154 -22⟩ comma in the 5-limit. Can be conceptualized as a more precise version of icosidillic in the 5-limit and the 7-limit bearing the same pergen (P8/22, P5), however there's no single way to extend it to the 11-limit due to inconsistency.
Subgroup: 2.3.5
Comma list: [-193 154 -22⟩
Mapping: [⟨22 22 -39], ⟨0 1 7]]
- mapping generators: ~[79 -63 9⟩, ~3/2
Optimal tuning (CTE): ~3/2 = 701.942
Supporting ETs: 176, 330, 506, 836, 1342, 1848, ...
7-limit
Subgroup: 2.3.5.7
Comma list: 14348907/14336000, [39 11 -11 -11⟩
Mapping: [⟨22 0 -193 271], ⟨0 1 7 -6]]
- mapping generators: ~8388608/8103375, ~3/2
Optimal tuning (CTE): ~3/2 = 702.014
Supporting ETs: 176, 506, 682
Titanium
Titanium is named after the 22nd element after the previous temperament called "titanium" was found to be the same thing as beep. It reaches 11/8 and 7/6 in one generator, and as a corollary tempers 96/77 to 7 steps of 22edo.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 2166612408926208/2162550048828125
Mapping: [⟨22 2 10 33 72], ⟨0 8 10 7 1]]
Mapping generators: ~48125/46656 = 1\22, 4296875/3779136 = 224.101
Optimal tuning (CTE): ~4296875/3779136 = 224.101
Supporting ETs: 198, 616, 814, 1012, 1210
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2401/2400, 9801/9800, 67392/67375, 142884/142805
Mapping: [⟨22 2 10 33 72 28], ⟨0 8 10 7 1 13]]
Mapping generators: ~65/63 = 1\22, 338/297 = 224.100
Optimal tuning (CTE): ~338/297 = 224.100