22nd-octave temperaments: Difference between revisions

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== Vigin ==
== Vigin ==
''Vigin'' has the same 11-limit mapping as 22et with harmonic 13 mapped to an independent generator.
''Vigin'' sets the [[11-limit]] to that of 22edo and harmonic 13 and the rest are independent generators. Can be described as 22 & 22f temperament, and the only non-trivial tuning besides 22edo itself is [[44edo]], which is also the optimal patent val.


[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13
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{{Mapping|legend=1| 22 35 51 62 76 0 | 0 0 0 0 0 1 }}
{{Mapping|legend=1| 22 35 51 62 76 0 | 0 0 0 0 0 1 }}
: mapping generators: ~36/35, ~13/8


[[Optimal tuning]] ([[POTE]]): ~33/32 = 1\22, ~13/8 = 844.624
[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 54.494{{c}}, ~13/8 = 843.820{{c}}
: [[error map]]: {{val| -1.141, 5.321, -7.141, 9.777, -9.805, -0.130 }}
* [[CWE]]: ~36/35 = 1\22, ~13/8 = 843.919{{c}}
: error map: {{val| 0.000, 7.136, -4.496, 12.992, -5.863, 3.392 }}


{{Optimal ET sequence|legend=1| 22, 44 }}
{{Optimal ET sequence|legend=1| 22, 44 }}


[[Badness]]: 0.029849
[[Badness]]: 1.233


== Major arcana ==
== Major arcana ==
''Major arcana'' is named after a [[wikipedia:Major Arcana|subset of Tarot cards of which there is 22]]. Tempers out the {{monzo|-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.
''Major arcana'' is named after a [[wikipedia:Major Arcana|subset of Tarot cards of which there is 22]] and it tempers out the {{monzo|-193 154 -22}} comma in the 5-limit.  
 
It is similar to the icosidillic temperament, having the same [[pergen]] (P8/22, P5), however while icosidillic is merely an extension of [[Miscellaneous 5-limit temperaments#Hendecatonic|5-limit hendecatonic]], major arcana stands as a an independent and properly not [[enfactoring|enfactored]] 22nd-octave comma.
 
An extension is possible to the 7-limit as the 506 & 682 temperament, however due to inconsistency the temperament branches at the 11-limit. Although notably, a 2.3.5.11/7 extension of remarkably low badness, described as 1848 & 1342, is prescribed. [[1848edo]] as a whole is also a strong tuning despite being only in the optimal ET sequence for 2.3.5.11/7 subgroup version.


Subgroup: 2.3.5
Subgroup: 2.3.5
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Comma list: {{monzo|-193 154 -22}}
Comma list: {{monzo|-193 154 -22}}


Mapping: [{{val|22 22 -39}}, {{val|0 1 7}}]
{{Mapping|legend=1| 22 0 -193 | 0 1 7}}
 
: mapping generators: ~{{monzo|79 -63 9}}, ~3
: mapping generators: ~{{monzo|79 -63 9}}, ~3/2


Optimal tuning (CTE): ~3/2 = 701.942
[[Optimal tuning]]s:  
* [[WE]]: ~{{monzo|79 -63 9}} = 54.546{{c}}, ~3/2 = 701.942
: [[error map]]: {{val| 0.004, -0.008, 0.003 }}
* [[CWE]]: ~{{monzo|79 -63 9}} = 1\22, ~3/2 = 701.941
: error map: {{val| 0.000, -0.014, -0.003 }}


[[Support]]ing [[ET]]s: {{EDOs|176, 330, 506, 836, 1342, 1848}}, ...
{{Optimal ET sequence|legend=1| 22c, 132cc, 154c, 176, 330, 506, 836, 1342, 4532, 5874, 7216, 8558, 15774 }}


[[Badness]]: 28.189
=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
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Comma list: 14348907/14336000, {{monzo|39 11 -11 -11}}
Comma list: 14348907/14336000, {{monzo|39 11 -11 -11}}


Mapping: [{{val|22 0 -193 271}}, {{val|0 1 7 -6}}]
{{Mapping|legend = 1| 22 0 -193 271 | 0 1 7 -6}}
: mapping generators: ~8388608/8103375, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~8388608/8103375 = 54.546{{c}}, ~3/2 = 701.942
: [[error map]]: {{val| -0.152, -0.165, 0.275, 0.384 }}
* [[CWE]]: ~8388608/8103375 = 1\22, ~3/2 = 701.941
: error map: {{val| 0.000, 0.072, 0.604, 0.829 }}


: mapping generators: ~8388608/8103375, ~3/2
{{Optimal ET sequence|legend=1| 176, 330d, 506, 682 }}


Optimal tuning (CTE): ~3/2 = 702.014
[[Badness]]: 28.987


[[Support]]ing [[ET]]s: {{EDOs|176, 506, 682}}
=== 2.3.5.11/7 subgroup ===
Subgroup: 2.3.5.11/7
 
Comma list: 9801/9800, 2.3.5.11/7 {{monzo|-83 59 -2 -9}} (full 11-limit {{monzo|-83 59 -2 9 -9}})
 
{{Mapping|legend = 1| 22 0 -193 -160 | 0 1 7 5}}
: mapping generators: ~3720087/3604480, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~3720087/3604480 = 54.546{{c}}, ~3/2 = 701.943
: [[error map]]: {{val| 0.002, -0.010, 0.013, -0.053 }}
* [[CWE]]: ~3720087/3604480 = 1\22, ~3/2 = 701.942
: error map: {{val| 0.000, -0.013, 0.010, -0.053 }}
 
{{Optimal ET sequence|legend=1| 22*, 154c, 176, 330, 506, 1342, 1848, 2354, 4202, 6556, 8910c, 15466c }}
 
Badness: 0.999


== Titanium ==
== 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.
''Titanium'' is named after the 22nd chemical 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
Subgroup: 2.3.5.7.11
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Comma list: 2401/2400, 9801/9800, 2166612408926208/2162550048828125
Comma list: 2401/2400, 9801/9800, 2166612408926208/2162550048828125


Mapping: [{{val|22 2 10 33 72}}, {{val|0 8 10 7 1}}]
{{Mapping|legend=1| 22 2 10 33 72 | 0 8 10 7 1 }}
: mapping generators: ~48125/46656, 4296875/3779136


Mapping generators: ~48125/46656 = 1\22, 4296875/3779136 = 224.101
[[Optimal tuning]]s:
* [[WE]]: ~48125/46656 = 54.545{{c}}, ~4296875/3779136 = 224.101
: [[error map]]: {{val| -0.002, -0.056, 0.151, -0.121, 0.051 }}
* [[CWE]]: ~48125/46656 = 1\22, ~4296875/3779136 = 224.101
: error map: {{val| 0.000, -0.054, 0.153, -0.117, 0.056 }}


Optimal tuning (CTE): ~4296875/3779136 = 224.101
{{Optimal ET sequence|legend = 1| 198, 616, 814, 1012, 1826 }}


[[Support]]ing [[ET]]s: {{EDOs|198, 616, 814, 1012, 1210}}
Badness: 2.913


=== 13-limit ===
=== 13-limit ===
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Comma list: 2401/2400, 9801/9800, 67392/67375, 142884/142805
Comma list: 2401/2400, 9801/9800, 67392/67375, 142884/142805


Mapping: [{{val|22 2 10 33 72 28}}, {{val|0 8 10 7 1 13}}]
{{Mapping|legend=1| 22 2 10 33 72 28 | 0 8 10 7 1 13 }}
: mapping generators: ~65/63, 338/297


Mapping generators: ~65/63 = 1\22, 338/297 = 224.100
[[Optimal tuning]]s:
* [[WE]]: ~65/63 = 54.545{{c}}, ~338/297 = 224.101
: [[error map]]: {{val| -0.003, -0.062, 0.142, -0.128, 0.047, 0.045 }}
* [[CWE]]: ~65/63 = 1\22, ~338/297 = 224.101
: error map: {{val| -0.000, -0.060, 0.146, -0.122, 0.055, 0.052 }}


Optimal tuning (CTE): ~338/297 = 224.100
{{Optimal ET sequence|legend=1|198, 616, 814, 1012, 1826, 2838}}


[[Support]]ing [[ET]]s: {{EDOs|198, 616, 814, 1012, 1210}}
Badness: 1.373


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
{{Todo| cleanup }}

Revision as of 18:35, 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.

← 21st-octave • 22nd-octave • 23rd-octave →

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

Optimal tunings:

  • 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 sets the 11-limit to that of 22edo and harmonic 13 and the rest are independent generators. Can be described as 22 & 22f temperament, and the only non-trivial tuning besides 22edo itself is 44edo, which is also the optimal patent val.

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]]

mapping generators: ~36/35, ~13/8

Optimal tunings:

  • WE: ~36/35 = 54.494 ¢, ~13/8 = 843.820 ¢
error map: ⟨-1.141, 5.321, -7.141, 9.777, -9.805, -0.130]
  • CWE: ~36/35 = 1\22, ~13/8 = 843.919 ¢
error map: ⟨0.000, 7.136, -4.496, 12.992, -5.863, 3.392]

Optimal ET sequence: 22, 44

Badness: 1.233

Major arcana

Major arcana is named after a subset of Tarot cards of which there is 22 and it tempers out the [-193 154 -22⟩ comma in the 5-limit.

It is similar to the icosidillic temperament, having the same pergen (P8/22, P5), however while icosidillic is merely an extension of 5-limit hendecatonic, major arcana stands as a an independent and properly not enfactored 22nd-octave comma.

An extension is possible to the 7-limit as the 506 & 682 temperament, however due to inconsistency the temperament branches at the 11-limit. Although notably, a 2.3.5.11/7 extension of remarkably low badness, described as 1848 & 1342, is prescribed. 1848edo as a whole is also a strong tuning despite being only in the optimal ET sequence for 2.3.5.11/7 subgroup version.

Subgroup: 2.3.5

Comma list: [-193 154 -22⟩

Mapping: [⟨22 0 -193], ⟨0 1 7]]

mapping generators: ~[79 -63 9⟩, ~3

Optimal tunings:

  • WE: ~[79 -63 9⟩ = 54.546 ¢, ~3/2 = 701.942
error map: ⟨0.004, -0.008, 0.003]
  • CWE: ~[79 -63 9⟩ = 1\22, ~3/2 = 701.941
error map: ⟨0.000, -0.014, -0.003]

Optimal ET sequence: 22c, 132cc, 154c, 176, 330, 506, 836, 1342, 4532, 5874, 7216, 8558, 15774

Badness: 28.189

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

Optimal tunings:

  • WE: ~8388608/8103375 = 54.546 ¢, ~3/2 = 701.942
error map: ⟨-0.152, -0.165, 0.275, 0.384]
  • CWE: ~8388608/8103375 = 1\22, ~3/2 = 701.941
error map: ⟨0.000, 0.072, 0.604, 0.829]

Optimal ET sequence: 176, 330d, 506, 682

Badness: 28.987

2.3.5.11/7 subgroup

Subgroup: 2.3.5.11/7

Comma list: 9801/9800, 2.3.5.11/7 [-83 59 -2 -9⟩ (full 11-limit [-83 59 -2 9 -9⟩)

Mapping: [⟨22 0 -193 -160], ⟨0 1 7 5]]

mapping generators: ~3720087/3604480, ~3

Optimal tunings:

  • WE: ~3720087/3604480 = 54.546 ¢, ~3/2 = 701.943
error map: ⟨0.002, -0.010, 0.013, -0.053]
  • CWE: ~3720087/3604480 = 1\22, ~3/2 = 701.942
error map: ⟨0.000, -0.013, 0.010, -0.053]

Optimal ET sequence: 22*, 154c, 176, 330, 506, 1342, 1848, 2354, 4202, 6556, 8910c, 15466c

Badness: 0.999

Titanium

Titanium is named after the 22nd chemical 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, 4296875/3779136

Optimal tunings:

  • WE: ~48125/46656 = 54.545 ¢, ~4296875/3779136 = 224.101
error map: ⟨-0.002, -0.056, 0.151, -0.121, 0.051]
  • CWE: ~48125/46656 = 1\22, ~4296875/3779136 = 224.101
error map: ⟨0.000, -0.054, 0.153, -0.117, 0.056]

Optimal ET sequence: 198, 616, 814, 1012, 1826

Badness: 2.913

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, 338/297

Optimal tunings:

  • WE: ~65/63 = 54.545 ¢, ~338/297 = 224.101
error map: ⟨-0.003, -0.062, 0.142, -0.128, 0.047, 0.045]
  • CWE: ~65/63 = 1\22, ~338/297 = 224.101
error map: ⟨-0.000, -0.060, 0.146, -0.122, 0.055, 0.052]

Optimal ET sequence: 198, 616, 814, 1012, 1826, 2838

Badness: 1.373

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