Orwellismic temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
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: mapping generators: ~567/400, ~10/9
: mapping generators: ~567/400, ~10/9


[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~10/9 = 185.885{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 599.8218{{c}}, ~10/9 = 185.8303{{c}}
: [[error map]]: {{val| -0.356 -1.321 +1.141 +1.944 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~10/9 = 185.8618{{c}}
: error map: {{val| 0.000 -0.922 +1.613 +2.898 }}
<!-- * [[POTE]]: ~567/400 = 600.000{{c}}, ~10/9 = 185.885{{c}} -->


{{Optimal ET sequence|legend=1| 26, 58, 84, 142, 368cd }}
{{Optimal ET sequence|legend=1| 26, 58, 84, 142, 368cd }}


[[Badness]] (Smith): 0.095278
[[Badness]] (Sintel): 2.41


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 1 0 5 6 | 0 7 15 2 3 }}
Mapping: {{mapping| 2 1 0 5 6 | 0 7 15 2 3 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~10/9 = 185.962{{c}}
Optimal tunings:
* WE: ~99/70 = 599.5785{{c}}, ~10/9 = 185.8332{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 185.9140{{c}}
<!-- * POTE: ~99/70 = 600.000{{c}}, ~10/9 = 185.962{{c}} -->


{{Optimal ET sequence|legend=0| 26, 58, 142e, 200cdee }}
{{Optimal ET sequence|legend=0| 26, 58, 142e, 200cdee }}


Badness (Smith): 0.046373
Badness (Sintel): 1.53


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 1 0 5 6 4 | 0 7 15 2 3 11 }}
Mapping: {{mapping| 2 1 0 5 6 4 | 0 7 15 2 3 11 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~10/9 = 185.955{{c}}
Optimal tunings:
* WE: ~55/39 = 599.5432{{c}}, ~10/9 = 185.8136{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~10/9 = 185.9012{{c}}
<!-- * POTE: ~99/70 = 600.000{{c}}, ~10/9 = 185.955{{c}} -->


{{Optimal ET sequence|legend=0| 26, 58, 84, 142ef }}
{{Optimal ET sequence|legend=0| 26, 58, 84, 142ef }}


Badness (Smith): 0.025035
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Line 64: Line 75:
Mapping: {{mapping| 2 1 0 5 6 4 6 | 0 7 15 2 3 11 7 }}
Mapping: {{mapping| 2 1 0 5 6 4 6 | 0 7 15 2 3 11 7 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~10/9 = 185.916{{c}}
Optimal tunings:
* WE: ~17/12 = 599.7117{{c}}, ~10/9 = 185.8270{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/9 = 185.8834{{c}}
<!-- * POTE: ~17/12 = 600.000{{c}}, ~10/9 = 185.916{{c}} -->


{{Optimal ET sequence|legend=0| 26, 58, 84, 142ef }}
{{Optimal ET sequence|legend=0| 26, 58, 84 }}


Badness (Smith): 0.021642
Badness (Sintel): 1.10


=== 19-limit ===
=== 19-limit ===
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Mapping: {{mapping| 2 1 0 5 6 4 6 2 | 0 7 15 2 3 11 7 21 }}
Mapping: {{mapping| 2 1 0 5 6 4 6 2 | 0 7 15 2 3 11 7 21 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~10/9 = 185.858{{c}}
Optimal tunings:
* WE: ~17/12 = 599.7288{{c}}, ~10/9 = 185.7738{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/9 = 185.8279{{c}}
<!-- * POTE: ~17/12 = 600.000{{c}}, ~10/9 = 185.858{{c}} -->


{{Optimal ET sequence|legend=0| 26, 58h, 84, 142efh, 226deefgh }}
{{Optimal ET sequence|legend=0| 26, 58h, 84 }}


Badness (Smith): 0.018565
Badness (Sintel): 1.13


== Infraorwell ==
== Infraorwell ==
Line 88: Line 105:
[[Comma list]]: 1728/1715, 28672/28125
[[Comma list]]: 1728/1715, 28672/28125


{{Mapping|legend=1| 1 14 0 16 | 0 -16 3 -17 }}
{{Mapping|legend=1| 1 -2 3 -1 | 0 16 -3 17 }}
: mapping generators: ~2, ~12/7
: mapping generators: ~2, ~7/6


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/6 = 269.032{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.4649{{c}}, ~7/6 = 268.6878{{c}}
: [[error map]]: {{val| -1.535 +0.120 +3.018 +0.401 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 268.9844{{c}}
: error map: {{val| 0.000 +1.796 +6.733 +3.909 }}
<!-- * POTE: ~2 = 1200.000{{c}}, ~7/6 = 269.032{{c}} -->


{{Optimal ET sequence|legend=1| 9, 49, 58, 165cd, 223bcd }}
{{Optimal ET sequence|legend=1| 9, 40bd, 49, 58, 165cd, 223bccd, 281bcccdd }}


[[Badness]] (Smith): 0.117073
[[Badness]] (Sintel): 2.96


=== 11-limit ===
=== 11-limit ===
Line 102: Line 124:
Comma list: 176/175, 540/539, 1344/1331
Comma list: 176/175, 540/539, 1344/1331


Mapping: {{mapping| 1 14 0 16 12 | 0 -16 3 -17 -11 }}
Mapping: {{mapping| 1 -2 3 -1 1 | 0 16 -3 17 11 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 269.036{{c}}
Optimal tunings:
* WE: ~2 = 1198.2928{{c}}, ~7/6 = 268.6532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9825{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~7/6 = 269.036{{c}} -->


{{Optimal ET sequence|legend=0| 9, 49, 58, 165cdee }}
{{Optimal ET sequence|legend=0| 9, 40bde, 49, 58, 165cdee, 223bccdeee, 281bcccddeee }}


Badness (Smith): 0.040721
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
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Comma list: 144/143, 176/175, 196/195, 364/363
Comma list: 144/143, 176/175, 196/195, 364/363


Mapping: {{mapping| 1 14 0 16 12 20 | 0 -16 3 -17 -11 -21 }}
Mapping: {{mapping| 1 -2 3 -1 1 -1 | 0 16 -3 17 11 21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 269.021{{c}}
Optimal tunings:
* WE: ~2 = 1198.2216{{c}}, ~7/6 = 268.6220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9616{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~7/6 = 269.021{{c}} -->


{{Optimal ET sequence|legend=0| 9, 49f, 58 }}
{{Optimal ET sequence|legend=0| 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff }}


Badness (Smith): 0.023683
Badness (Sintel): 0.979


== Sentinel ==
== Sentinel ==

Revision as of 13:09, 21 October 2025

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.

This is a collection of rank-2 temperaments that temper out the orwellisma (monzo: [6 3 -1 -3⟩, ratio: 1728/1715).

Temperaments discussed elsewhere are:

Considered below are secant, infraorwell, quartonic, sentinel, diesic, philips, triskaidekic, and pentaorwell.

Secant

Subgroup: 2.3.5.7

Comma list: 1728/1715, 177147/175000

Mapping: [⟨2 1 0 5], ⟨0 7 15 2]]

mapping generators: ~567/400, ~10/9

Optimal tunings:

  • WE: ~567/400 = 599.8218 ¢, ~10/9 = 185.8303 ¢
error map: ⟨-0.356 -1.321 +1.141 +1.944]
  • CWE: ~567/400 = 600.0000 ¢, ~10/9 = 185.8618 ¢
error map: ⟨0.000 -0.922 +1.613 +2.898]

Optimal ET sequence: 26, 58, 84, 142, 368cd

Badness (Sintel): 2.41

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 1728/1715

Mapping: [⟨2 1 0 5 6], ⟨0 7 15 2 3]]

Optimal tunings:

  • WE: ~99/70 = 599.5785 ¢, ~10/9 = 185.8332 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~10/9 = 185.9140 ¢

Optimal ET sequence: 26, 58, 142e, 200cdee

Badness (Sintel): 1.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 351/350, 364/363, 441/440

Mapping: [⟨2 1 0 5 6 4], ⟨0 7 15 2 3 11]]

Optimal tunings:

  • WE: ~55/39 = 599.5432 ¢, ~10/9 = 185.8136 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~10/9 = 185.9012 ¢

Optimal ET sequence: 26, 58, 84, 142ef

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 144/143, 170/169, 221/220, 351/350, 441/440

Mapping: [⟨2 1 0 5 6 4 6], ⟨0 7 15 2 3 11 7]]

Optimal tunings:

  • WE: ~17/12 = 599.7117 ¢, ~10/9 = 185.8270 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~10/9 = 185.8834 ¢

Optimal ET sequence: 26, 58, 84

Badness (Sintel): 1.10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 144/143, 153/152, 170/169, 210/209, 221/220, 400/399

Mapping: [⟨2 1 0 5 6 4 6 2], ⟨0 7 15 2 3 11 7 21]]

Optimal tunings:

  • WE: ~17/12 = 599.7288 ¢, ~10/9 = 185.7738 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~10/9 = 185.8279 ¢

Optimal ET sequence: 26, 58h, 84

Badness (Sintel): 1.13

Infraorwell

Subgroup: 2.3.5.7

Comma list: 1728/1715, 28672/28125

Mapping: [⟨1 -2 3 -1], ⟨0 16 -3 17]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • WE: ~2 = 1198.4649 ¢, ~7/6 = 268.6878 ¢
error map: ⟨-1.535 +0.120 +3.018 +0.401]
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 268.9844 ¢
error map: ⟨0.000 +1.796 +6.733 +3.909]

Optimal ET sequence: 9, 40bd, 49, 58, 165cd, 223bccd, 281bcccdd

Badness (Sintel): 2.96

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 1344/1331

Mapping: [⟨1 -2 3 -1 1], ⟨0 16 -3 17 11]]

Optimal tunings:

  • WE: ~2 = 1198.2928 ¢, ~7/6 = 268.6532 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 268.9825 ¢

Optimal ET sequence: 9, 40bde, 49, 58, 165cdee, 223bccdeee, 281bcccddeee

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 364/363

Mapping: [⟨1 -2 3 -1 1 -1], ⟨0 16 -3 17 11 21]]

Optimal tunings:

  • WE: ~2 = 1198.2216 ¢, ~7/6 = 268.6220 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 268.9616 ¢

Optimal ET sequence: 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff

Badness (Sintel): 0.979

Sentinel

For the 5-limit version, see Miscellaneous 5-limit temperaments #Sentinel.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3645/3584

Mapping: [⟨1 3 -3 6], ⟨0 -4 15 -9]]

mapping generators: ~2, ~9/7

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 425.596 ¢

Optimal ET sequence: 14, 17, 31, 110, 141, 172b, 375bc

Badness (Smith): 0.093805

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 243/242, 385/384

Mapping: [⟨1 3 -3 6 7], ⟨0 -4 15 -9 -10]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 425.550 ¢

Optimal ET sequence: 14, 17, 31, 79, 110e, 141e

Badness (Smith): 0.039595

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 105/104, 144/143, 243/242

Mapping: [⟨1 3 -3 6 7 3], ⟨0 -4 15 -9 -10 2]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 425.501 ¢

Optimal ET sequence: 14, 17, 31, 79f, 110ef, 141ef

Badness (Smith): 0.037408

Diesic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Diesic.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 5103/5000

Mapping: [⟨1 2 3 3], ⟨0 -13 -21 -6]]

mapping generators: ~2, ~36/35

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~36/35 = 38.567 ¢

Optimal ET sequence: 31, 156c, 187c, 218bc, 249bc, 280bc

Badness (Smith): 0.107348

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 891/875

Mapping: [⟨1 2 3 3 4], ⟨0 -13 -21 -6 -17]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~36/35 = 38.565 ¢

Optimal ET sequence: 31, 156ce, 187ce, 218bcee, 249bcee

Badness (Smith): 0.044186

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 343/338, 441/440

Mapping: [⟨1 2 3 3 4 4], ⟨0 -13 -21 -6 -17 -9]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~36/35 = 38.444 ¢

Optimal ET sequence: 31, 94cff, 125cff

Badness (Smith): 0.038116

Phillips

For the 5-limit version, see Miscellaneous 5-limit temperaments #Phillips.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 6561/6272

Mapping: [⟨2 0 33 -7], ⟨0 1 -9 4]]

mapping generators: ~81/56, ~3

Optimal tuning (POTE): ~81/56 = 600.000 ¢, ~3/2 = 691.174 ¢

Optimal ET sequence: 14, 26, 66b, 92bc

Badness (Smith): 0.228602

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 729/704

Mapping: [⟨2 0 33 -7 -12], ⟨0 1 -9 4 6]]

Optimal tuning (POTE): ~63/44 = 600.000 ¢, ~3/2 = 691.040 ¢

Optimal ET sequence: 14, 26, 40, 66b

Badness (Smith): 0.095777

Triskaidekic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Triskaidekic.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 1875/1792

Mapping: [⟨13 0 30 16], ⟨0 1 0 1]]

mapping generators: ~15/14, ~3

Optimal tuning (POTE): ~15/14 = 92.308 ¢, ~3/2 = 694.143 ¢

Optimal ET sequence: 13d, 26

Badness (Smith): 0.219050

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 125/121, 385/384

Mapping: [⟨13 0 30 16 45], ⟨0 1 0 1 0]]

Optimal tuning (POTE): ~15/14 = 92.308 ¢, ~3/2 = 696.296 ¢

Optimal ET sequence: 13d, 26, 91cd

Badness (Smith): 0.098847

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 125/121, 1200/1183

Mapping: [⟨13 0 30 16 45 48], ⟨0 1 0 1 0 0]]

Optimal tuning (POTE): ~15/14 = 92.308 ¢, ~3/2 = 695.879 ¢

Optimal ET sequence: 13d, 26, 91cdf

Badness (Smith): 0.059216

Pentaorwell

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Counterpental.

Named by +merlan #flirora in 2021, pentaorwell tempers out 179200/177147 and is the 75 & 80 temperament.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 179200/177147

Mapping: [⟨5 0 -36 22], ⟨0 1 6 -1]]

Optimal tunings:

  • CTE: ~280/243 = 240.000 ¢, ~3/2 = 704.3709 ¢ (~64/63 = 15.6291 ¢)
  • POTE: ~280/243 = 240.000 ¢, ~3/2 = 704.9128 ¢ (~64/63 = 15.0872 ¢)

Optimal ET sequence: 5, 75, 80

Badness (Smith): 0.148665

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 1728/1715, 2200/2187

Mapping: [⟨5 0 -36 22 57], ⟨0 1 6 -1 -5]]

Optimal tunings:

  • CTE: ~55/48 = 240.000 ¢, ~3/2 = 704.8440 ¢ (~99/98 = 15.1560 ¢)
  • POTE: ~55/48 = 240.000 ¢, ~3/2 = 704.8788 ¢ (~99/98 = 15.1212 ¢)

Optimal ET sequence: 75e, 80, 315bcdddee

Badness (Smith): 0.071702

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1728/1715

Mapping: [⟨5 0 -36 22 57 82], ⟨0 1 6 -1 -5 -8]]

Optimal tunings:

  • CTE: ~55/48 = 240.000 ¢, ~3/2 = 704.7636 ¢ (~99/98 = 15.2364 ¢)
  • CWE: ~55/48 = 240.000 ¢, ~3/2 = 704.7112 ¢ (~99/98 = 15.2882 ¢)

Optimal ET sequence: 75e, 80, 155de

Badness (Smith): 0.0557

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