Orwellismic temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/)
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[[Comma list]]: 1728/1715, 3645/3584
[[Comma list]]: 1728/1715, 3645/3584


{{Mapping|legend=1| 1 3 -3 6 | 0 -4 15 -9 }}
{{Mapping|legend=1| 1 -1 12 -3 | 0 4 -15 9 }}
: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~14/9


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 425.596{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8626{{c}}, ~14/9 = 774.9609{{c}}
: [[error map]]: {{val| +0.863 -2.974 -0.376 +3.234 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.4021{{c}}
: error map: {{val| 0.000 -4.347 -2.345 +0.793 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 774.404{{c}} -->


{{Optimal ET sequence|legend=1| 14, 17, 31, 110, 141, 172b, 375bc }}
{{Optimal ET sequence|legend=1| 14, 17, 31, 110, 141, 172b }}


[[Badness]] (Smith): 0.093805
[[Badness]] (Sintel): 2.37


=== 11-limit ===
=== 11-limit ===
Line 172: Line 177:
Comma list: 99/98, 243/242, 385/384
Comma list: 99/98, 243/242, 385/384


Mapping: {{mapping| 1 3 -3 6 7 | 0 -4 15 -9 -10 }}
Mapping: {{mapping| 1 -1 12 -3 -3 | 0 4 -15 9 10 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 425.550{{c}}
Optimal tunings:
* WE: ~2 = 1201.0893{{c}}, ~14/9 = 775.1534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.4564{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~14/9 = 774.450{{c}} -->


{{Optimal ET sequence|legend=0| 14, 17, 31, 79, 110e, 141e }}
{{Optimal ET sequence|legend=0| 14, 17, 31, 79, 110e, 141e }}


Badness (Smith): 0.039595
Badness (Sintel): 1.31


=== 13-limit ===
=== 13-limit ===
Line 185: Line 193:
Comma list: 99/98, 105/104, 144/143, 243/242
Comma list: 99/98, 105/104, 144/143, 243/242


Mapping: {{mapping| 1 3 -3 6 7 3 | 0 -4 15 -9 -10 2 }}
Mapping: {{mapping| 1 -1 12 -3 -3 5 | 0 4 -15 9 10 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 425.501{{c}}
Optimal tunings:
* WE: ~2 = 1200.4045{{c}}, ~14/9 = 774.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.5000{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~14/9 = 774.499{{c}} -->


{{Optimal ET sequence|legend=0| 14, 17, 31, 79f, 110ef, 141ef }}
{{Optimal ET sequence|legend=0| 14, 17, 31 }}


Badness (Smith): 0.037408
Badness (Sintel): 1.55


== Diesic ==
== Diesic ==
Line 203: Line 214:
: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~36/35 = 38.567{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1293{{c}}, ~36/35 = 38.5709{{c}}
: [[error map]]: {{val| +0.129 -3.118 +4.086 +0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 38.5603{{c}}
: error map: {{val| 0.000 -3.239 +3.920 -0.188 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 38.567{{c}} -->


{{Optimal ET sequence|legend=1| 31, 156c, 187c, 218bc, 249bc, 280bc }}
{{Optimal ET sequence|legend=1| 1c, 30bc, 31, 156c }}


[[Badness]] (Smith): 0.107348
[[Badness]] (Sintel): 2.72


=== 11-limit ===
=== 11-limit ===
Line 216: Line 232:
Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~36/35 = 38.565{{c}}
Optimal tunings:
* WE: ~2 = 1200.4923{{c}}, ~36/35 = 38.5806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.5397{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~36/35 = 38.565{{c}} -->


{{Optimal ET sequence|legend=0| 31, 156ce, 187ce, 218bcee, 249bcee }}
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}


Badness (Smith): 0.044186
Badness (Sintel): 1.46


=== 13-limit ===
=== 13-limit ===
Line 229: Line 248:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~36/35 = 38.444{{c}}
Optimal tunings:
* WE: ~2 = 1199.3005{{c}}, ~36/35 = 38.4213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.4764{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~36/35 = 38.444{{c}} -->


{{Optimal ET sequence|legend=0| 31, 94cff, 125cff }}
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}


Badness (Smith): 0.038116
Badness (Sintel): 1.57


== Phillips ==
== Phillips ==
Line 245: Line 267:
: mapping generators: ~81/56, ~3
: mapping generators: ~81/56, ~3


[[Optimal tuning]] ([[POTE]]): ~81/56 = 600.000{{c}}, ~3/2 = 691.174{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~81/56 = 601.3944{{c}}, ~3/2 = 692.7807{{c}}
: [[error map]]: {{val| +2.789 -6.385 -0.424 +3.692 }}
* [[CWE]]: ~81/56 = 600.0000{{c}}, ~3/2 = 691.1141{{c}}
: error map: {{val| 0.000 -10.841 -6.340 -4.370 }}
<!-- * [[POTE]]: ~81/56 = 600.000{{c}}, ~3/2 = 691.174{{c}} -->


{{Optimal ET sequence|legend=1| 14, 26, 66b, 92bc }}
{{Optimal ET sequence|legend=1| 14, 26, 66b, 92bc }}


[[Badness]] (Smith): 0.228602
[[Badness]] (Sintel): 5.79


=== 11-limit ===
=== 11-limit ===
Line 258: Line 285:
Mapping: {{mapping| 2 0 33 -7 -12 | 0 1 -9 4 6 }}
Mapping: {{mapping| 2 0 33 -7 -12 | 0 1 -9 4 6 }}


Optimal tuning (POTE): ~63/44 = 600.000{{c}}, ~3/2 = 691.040{{c}}
Optimal tunings:
* WE: ~63/44 = 601.2349{{c}}, ~3/2 = 692.4623{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 691.0383{{c}}
<!-- * POTE: ~63/44 = 600.000{{c}}, ~3/2 = 691.040{{c}} -->


{{Optimal ET sequence|legend=0| 14, 26, 40, 66b }}
{{Optimal ET sequence|legend=0| 14, 26, 40, 66b }}


Badness (Smith): 0.095777
Badness (Sintel): 3.17


== Triskaidekic ==
== Triskaidekic ==

Revision as of 13:36, 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 -1 12 -3], ⟨0 4 -15 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.8626 ¢, ~14/9 = 774.9609 ¢
error map: ⟨+0.863 -2.974 -0.376 +3.234]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.4021 ¢
error map: ⟨0.000 -4.347 -2.345 +0.793]

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

Badness (Sintel): 2.37

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 12 -3 -3], ⟨0 4 -15 9 10]]

Optimal tunings:

  • WE: ~2 = 1201.0893 ¢, ~14/9 = 775.1534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.4564 ¢

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

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 12 -3 -3 5], ⟨0 4 -15 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1200.4045 ¢, ~14/9 = 774.7596 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.5000 ¢

Optimal ET sequence: 14, 17, 31

Badness (Sintel): 1.55

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 tunings:

  • WE: ~2 = 1200.1293 ¢, ~36/35 = 38.5709 ¢
error map: ⟨+0.129 -3.118 +4.086 +0.137]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 38.5603 ¢
error map: ⟨0.000 -3.239 +3.920 -0.188]

Optimal ET sequence: 1c, 30bc, 31, 156c

Badness (Sintel): 2.72

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 tunings:

  • WE: ~2 = 1200.4923 ¢, ~36/35 = 38.5806 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 38.5397 ¢

Optimal ET sequence: 1ce, 30bce, 31

Badness (Sintel): 1.46

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 tunings:

  • WE: ~2 = 1199.3005 ¢, ~36/35 = 38.4213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 38.4764 ¢

Optimal ET sequence: 1ce, 30bce, 31

Badness (Sintel): 1.57

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 tunings:

  • WE: ~81/56 = 601.3944 ¢, ~3/2 = 692.7807 ¢
error map: ⟨+2.789 -6.385 -0.424 +3.692]
  • CWE: ~81/56 = 600.0000 ¢, ~3/2 = 691.1141 ¢
error map: ⟨0.000 -10.841 -6.340 -4.370]

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

Badness (Sintel): 5.79

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 tunings:

  • WE: ~63/44 = 601.2349 ¢, ~3/2 = 692.4623 ¢
  • CWE: ~63/44 = 600.0000 ¢, ~3/2 = 691.0383 ¢

Optimal ET sequence: 14, 26, 40, 66b

Badness (Sintel): 3.17

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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