Orwellismic temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[orwellisma]] ({{monzo|legend=1| 6 3 -1 -3 }}, [[ratio]]: 1728/1715).
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[orwellisma]] ({{monzo|legend=1| 6 3 -1 -3 }}, [[ratio]]: 1728/1715).


Temperaments considered below are secant, infraorwell, quartonic, sentinel, diesic, philips, triskaidekic, and pentaorwell. Discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Beep]] (+21/20 or 27/25) → [[Bug family #Beep|Bug family]]
* [[Beep]] (+21/20 or 27/25) → [[Bug family #Beep|Bug family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
Line 11: Line 11:
* ''[[Secund]]'' (+405/392 or 525/512) → [[Greenwoodmic temperaments #Secund|Greenwoodmic temperaments]]
* ''[[Secund]]'' (+405/392 or 525/512) → [[Greenwoodmic temperaments #Secund|Greenwoodmic temperaments]]
* ''[[Quartonic]]'' (+4000/3969) → [[Quartonic family]]
* ''[[Quartonic]]'' (+4000/3969) → [[Quartonic family]]
* [[Buzzard]] (+5120/5103) → [[Vulture family #Buzzard|Vulture family]]
* [[Buzzard]] (+5120/5103) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* ''[[Trisensory]]'' (+78732/78125) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trisensory]]'' (+78732/78125) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trimabila]]'' (+268435456/263671875) → [[Mabila family #Trimabila|Mabila family]]
* ''[[Trimabila]]'' (+268435456/263671875) → [[Mabila family #Trimabila|Mabila family]]
Considered below are sentinel, secant, diesic, infraorwell, pentaorwell, triskaidekic, and philips.
== Sentinel ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sentinel]].''
Sentinel tempers out [[3645/3584]] and may be described as the {{nowrap| 14 & 17 }} temperament. It is related to [[squares]], but the mappings differ for the [[5/1|5th harmonic]]. Like squares, it splits the [[6/1|6th harmonic]] into four subminor sixths of 11/7~14/9 (or splits a perfect eleventh into four supermajor thirds of 9/7~14/11), and uses it for a generator; its [[ploidacot]] is beta-tetracot. However, the 5th harmonic is found by -15 generator steps instead of +16 as in squares.
As one might expect, [[31edo]] is a good tuning for this temperament, in which case it is identical to squares. Among the alternatives are [[48edo]] and [[79edo]], both in their [[patent val]]s.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1728/1715, 3645/3584
{{Mapping|legend=1| 1 -1 12 -3 | 0 4 -15 9 }}
: mapping generators: ~2, ~14/9
[[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 }}
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -1 12 -3 -3 | 0 4 -15 9 10 }}
Optimal tunings:
* WE: ~2 = 1201.0893{{c}}, ~11/7 = 775.1534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4564{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 12 -3 -3 5 | 0 4 -15 9 10 -2 }}
Optimal tunings:
* WE: ~2 = 1200.4045{{c}}, ~11/7 = 774.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.5000{{c}}
{{Optimal ET sequence|legend=0| 14, 17, 31 }}
Badness (Sintel): 1.55


== Secant ==
== Secant ==
Secant, the {{nowrap| 26 & 58 }} temperament, is generated by a slightly sharp ~10/9, seven of which plus a semi-octave period give the [[3/1|3rd harmonic]]. Its ploidacot is diploid alpha-heptacot. [[58edo]], [[84edo]] and [[142edo]] all make for excellent tunings.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 23: Line 81:
: 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 }}


{{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 ===
Line 36: Line 98:
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}}


{{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 ===
Line 49: Line 113:
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}}


{{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 62: Line 128:
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}}


{{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 ===
Line 75: Line 143:
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}}
 
{{Optimal ET sequence|legend=0| 26, 58h, 84 }}
 
Badness (Sintel): 1.13


{{Optimal ET sequence|legend=0| 26, 58h, 84, 142efh, 226deefgh }}
== Diesic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Diesic]].''


Badness (Smith): 0.018565
Diesic is generated by a diesis-sized interval of ~36/35, hence the name. It may be described as the {{nowrap| 31 & 32c }} temperament. It is related to [[slender]], both sharing the ploidacot of omega-13-cot, but the mappings differ for the [[5/1|5th harmonic]].  


== Infraorwell ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 28672/28125
[[Comma list]]: 1728/1715, 5103/5000


{{Mapping|legend=1| 1 14 0 16 | 0 -16 3 -17 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -13 -21 -6 }}
: mapping generators: ~2, ~12/7
: mapping generators: ~2, ~36/35


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/6 = 269.032{{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 }}


{{Optimal ET sequence|legend=1| 9, 49, 58, 165cd, 223bcd }}
{{Optimal ET sequence|legend=1| 1c, 30bc, 31, 156c }}


[[Badness]] (Smith): 0.117073
[[Badness]] (Sintel): 2.72


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 1344/1331
Comma list: 121/120, 441/440, 891/875


Mapping: {{mapping| 1 14 0 16 12 | 0 -16 3 -17 -11 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 269.036{{c}}
Optimal tunings:
* WE: ~2 = 1200.4923{{c}}, ~36/35 = 38.5806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.5397{{c}}


{{Optimal ET sequence|legend=0| 9, 49, 58, 165cdee }}
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}


Badness (Smith): 0.040721
Badness (Sintel): 1.46


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 196/195, 364/363
Comma list: 66/65, 121/120, 343/338, 441/440


Mapping: {{mapping| 1 14 0 16 12 20 | 0 -16 3 -17 -11 -21 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 269.021{{c}}
Optimal tunings:
* WE: ~2 = 1199.3005{{c}}, ~36/35 = 38.4213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.4764{{c}}


{{Optimal ET sequence|legend=0| 9, 49f, 58 }}
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}


Badness (Smith): 0.023683
Badness (Sintel): 1.57


== Sentinel ==
== Infraorwell ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sentinel]].''
Infraorwell may be described as the {{nowrap| 49 & 58 }} temperament. It is generated by a ~7/6, less sharp than the generator of [[orwell]] but still sharp of just, so that three generators make a ~8/5, but seven generators does not give the [[3/1|3rd harmonic]]. Instead, sixteen generators give the [[12/1|12th harmonic]]; the ploidacot for this temperament is therefore gamma-16-cot. [[58edo]] makes for a recommendable tuning, though [[49edo]] and [[67edo]] are among the possibilities.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 3645/3584
[[Comma list]]: 1728/1715, 28672/28125


{{Mapping|legend=1| 1 3 -3 6 | 0 -4 15 -9 }}
{{Mapping|legend=1| 1 -2 3 -1 | 0 16 -3 17 }}
: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~7/6


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 425.596{{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 }}


{{Optimal ET sequence|legend=1| 14, 17, 31, 110, 141, 172b, 375bc }}
{{Optimal ET sequence|legend=1| 9, 40bd, 49, 58, 165cd, 223bccd, 281bcccdd }}


[[Badness]] (Smith): 0.093805
[[Badness]] (Sintel): 2.96


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 243/242, 385/384
Comma list: 176/175, 540/539, 1344/1331


Mapping: {{mapping| 1 3 -3 6 7 | 0 -4 15 -9 -10 }}
Mapping: {{mapping| 1 -2 3 -1 1 | 0 16 -3 17 11 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 425.550{{c}}
Optimal tunings:
* WE: ~2 = 1198.2928{{c}}, ~7/6 = 268.6532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9825{{c}}


{{Optimal ET sequence|legend=0| 14, 17, 31, 79, 110e, 141e }}
{{Optimal ET sequence|legend=0| 9, 40bde, 49, 58, 165cdee, 223bccdeee, 281bcccddeee }}


Badness (Smith): 0.039595
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 105/104, 144/143, 243/242
Comma list: 144/143, 176/175, 196/195, 364/363


Mapping: {{mapping| 1 3 -3 6 7 3 | 0 -4 15 -9 -10 2 }}
Mapping: {{mapping| 1 -2 3 -1 1 -1 | 0 16 -3 17 11 21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 425.501{{c}}
Optimal tunings:
* WE: ~2 = 1198.2216{{c}}, ~7/6 = 268.6220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9616{{c}}


{{Optimal ET sequence|legend=0| 14, 17, 31, 79f, 110ef, 141ef }}
{{Optimal ET sequence|legend=0| 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff }}


Badness (Smith): 0.037408
Badness (Sintel): 0.979


== Diesic ==
== Pentaorwell ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Diesic]].''
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Counterpental]].''
 
Named by [[User:Flirora|+merlan #flirora]] in 2021, pentaorwell tempers out 179200/177147 and is the {{nowrap| 75 & 80 }} temperament. Its ploidacot is pentaploid monocot.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 5103/5000
[[Comma list]]: 1728/1715, 179200/177147


{{Mapping|legend=1| 1 2 3 3 | 0 -13 -21 -6 }}
{{Mapping|legend=1| 5 0 -36 22 | 0 1 6 -1 }}
: mapping generators: ~2, ~36/35


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~36/35 = 38.567{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~280/243 = 239.7268{{c}}, ~3/2 = 704.1103{{c}} (~64/63 = 15.0700{{c}})
: [[error map]]: {{val| -1.366 +0.789 -0.013 +2.419 }}
* [[CWE]]: ~280/243 = 240.0000{{c}}, ~3/2 = 704.7723{{c}} (~64/63 = 15.2277{{c}})
: error map: {{val| 0.000 +2.817 +2.320 +6.402 }}


{{Optimal ET sequence|legend=1| 31, 156c, 187c, 218bc, 249bc, 280bc }}
{{Optimal ET sequence|legend=1| 5, 75, 80 }}


[[Badness]] (Smith): 0.107348
[[Badness]] (Sintel): 3.76


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 441/440, 891/875
Comma list: 896/891, 1728/1715, 2200/2187


Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}
Mapping: {{mapping| 5 0 -36 22 57 | 0 1 6 -1 -5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~36/35 = 38.565{{c}}
Optimal tunings:
* WE: ~55/48 = 239.7357{{c}}, ~3/2 = 704.1025{{c}} (~99/98 = 15.1045{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.8440{{c}} (~99/98 = 15.1560{{c}})


{{Optimal ET sequence|legend=0| 31, 156ce, 187ce, 218bcee, 249bcee }}
{{Optimal ET sequence|legend=0| 5, 75e, 80, 315bcdddee }}


Badness (Smith): 0.044186
Badness (Sintel): 2.37


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 121/120, 343/338, 441/440
Comma list: 325/324, 352/351, 364/363, 1728/1715


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}
Mapping: {{mapping| 5 0 -36 22 57 82 | 0 1 6 -1 -5 -8 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~36/35 = 38.444{{c}}
Optimal tunings:
 
* WE: ~55/48 = 239.7711{{c}}, ~3/2 = 704.0302{{c}} (~99/98 = 15.2830{{c}})
{{Optimal ET sequence|legend=0| 31, 94cff, 125cff }}
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.7112{{c}} (~99/98 = 15.2888{{c}})
 
Badness (Smith): 0.038116
 
== Phillips ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Phillips]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 6561/6272


{{Mapping|legend=1| 2 0 33 -7 | 0 1 -9 4 }}
{{Optimal ET sequence|legend=0| 5f, 75e, 80, 155de }}
: mapping generators: ~81/56, ~3


[[Optimal tuning]] ([[POTE]]): ~81/56 = 600.000{{c}}, ~3/2 = 691.174{{c}}
Badness (Sintel): 2.30
 
{{Optimal ET sequence|legend=1| 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: {{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 ET sequence|legend=0| 14, 26, 40, 66b }}
 
Badness (Smith): 0.095777


== Triskaidekic ==
== Triskaidekic ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Triskaidekic]].''
: ''For the 5-limit version, see [[13th-octave temperaments #Triskaidekic]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 244: Line 314:
: mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


[[Optimal tuning]] ([[POTE]]): ~15/14 = 92.308{{c}}, ~3/2 = 694.143{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 92.5254{{c}}, ~3/2 = 695.7801{{c}}
: [[error map]]: {{val| +2.831 -3.344 -10.551 +10.192 }}
* [[CWE]]: ~15/14 = 92.3077{{c}}, ~3/2 = 695.8319{{c}}
: error map: {{val| 0.000 -6.123 -17.083 +3.929 }}


{{Optimal ET sequence|legend=1| 13d, 26 }}
{{Optimal ET sequence|legend=1| 13d, 26 }}


[[Badness]] (Smith): 0.219050
[[Badness]] (Sintel): 5.54


=== 11-limit ===
=== 11-limit ===
Line 257: Line 331:
Mapping: {{mapping| 13 0 30 16 45 | 0 1 0 1 0 }}
Mapping: {{mapping| 13 0 30 16 45 | 0 1 0 1 0 }}


Optimal tuning (POTE): ~15/14 = 92.308{{c}}, ~3/2 = 696.296{{c}}
Optimal tunings:
* WE: ~15/14 = 92.4364{{c}}, ~3/2 = 697.2675{{c}}
* CWE: ~15/14 = 92.3077{{c}}, ~3/2 = 697.0548{{c}}


{{Optimal ET sequence|legend=0| 13d, 26, 91cd }}
{{Optimal ET sequence|legend=0| 13d, 26 }}


Badness (Smith): 0.098847
Badness (Sintel): 3.27


=== 13-limit ===
=== 13-limit ===
Line 270: Line 346:
Mapping: {{mapping| 13 0 30 16 45 48 | 0 1 0 1 0 0 }}
Mapping: {{mapping| 13 0 30 16 45 48 | 0 1 0 1 0 0 }}


Optimal tuning (POTE): ~15/14 = 92.308{{c}}, ~3/2 = 695.879{{c}}
Optimal tunings:
* WE: ~15/14 = 92.4535{{c}}, ~3/2 = 696.9785{{c}}
* CWE: ~15/14 = 92.3077{{c}}, ~3/2 = 696.5711{{c}}


{{Optimal ET sequence|legend=0| 13d, 26, 91cdf }}
{{Optimal ET sequence|legend=0| 13d, 26 }}


Badness (Smith): 0.059216
Badness (Sintel): 2.45


== Pentaorwell ==
== Phillips ==
Named by [[User:Flirora|+merlan #flirora]] in 2021, pentaorwell tempers out 179200/177147 and is the {{nowrap| 75 & 80 }} temperament.  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Phillips]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 179200/177147
[[Comma list]]: 1728/1715, 6561/6272


{{Mapping|legend=1| 5 0 -36 22 | 0 1 6 -1 }}
{{Mapping|legend=1| 2 0 33 -7 | 0 1 -9 4 }}
: mapping generators: ~81/56, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~280/243 = 240.000{{c}}, ~3/2 = 704.3709{{c}} (~64/63 = 15.6291{{c}})
* [[WE]]: ~81/56 = 601.3944{{c}}, ~3/2 = 692.7807{{c}}
* [[POTE]]: ~280/243 = 240.000{{c}}, ~3/2 = 704.9128{{c}} (~64/63 = 15.0872{{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 }}


{{Optimal ET sequence|legend=1| 5, 75, 80 }}
{{Optimal ET sequence|legend=1| 14, 26, 66b, 92bc }}


[[Badness]] (Smith): 0.148665
[[Badness]] (Sintel): 5.79


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 896/891, 1728/1715, 2200/2187
Comma list: 99/98, 385/384, 729/704


Mapping: {{mapping| 5 0 -36 22 57 | 0 1 6 -1 -5 }}
Mapping: {{mapping| 2 0 33 -7 -12 | 0 1 -9 4 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/48 = 240.000{{c}}, ~3/2 = 704.8440{{c}} (~99/98 = 15.1560{{c}})
* WE: ~63/44 = 601.2349{{c}}, ~3/2 = 692.4623{{c}}
* POTE: ~55/48 = 240.000{{c}}, ~3/2 = 704.8788{{c}} (~99/98 = 15.1212{{c}})
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 691.0383{{c}}


{{Optimal ET sequence|legend=0| 75e, 80, 315bcdddee }}
{{Optimal ET sequence|legend=0| 14, 26, 40, 66b }}
 
Badness (Smith): 0.071702
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 364/363, 1728/1715
 
Mapping: {{mapping| 5 0 -36 22 57 82 | 0 1 6 -1 -5 -8 }}
 
Optimal tunings:
* CTE: ~55/48 = 240.000{{c}}, ~3/2 = 704.7636{{c}} (~99/98 = 15.2364{{c}})
* CWE: ~55/48 = 240.000{{c}}, ~3/2 = 704.7112{{c}} (~99/98 = 15.2882{{c}})
 
{{Optimal ET sequence|legend=0| 75e, 80, 155de }}


Badness (Smith): 0.0557
Badness (Sintel): 3.17


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
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[[Category:Orwellismic temperaments| ]] <!-- main article -->
[[Category:Orwellismic| ]] <!-- key article -->
[[Category:Orwellismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 22:34, 21 January 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.

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 sentinel, secant, diesic, infraorwell, pentaorwell, triskaidekic, and philips.

Sentinel

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

Sentinel tempers out 3645/3584 and may be described as the 14 & 17 temperament. It is related to squares, but the mappings differ for the 5th harmonic. Like squares, it splits the 6th harmonic into four subminor sixths of 11/7~14/9 (or splits a perfect eleventh into four supermajor thirds of 9/7~14/11), and uses it for a generator; its ploidacot is beta-tetracot. However, the 5th harmonic is found by -15 generator steps instead of +16 as in squares.

As one might expect, 31edo is a good tuning for this temperament, in which case it is identical to squares. Among the alternatives are 48edo and 79edo, both in their patent vals.

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 ¢, ~11/7 = 775.1534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 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 ¢, ~11/7 = 774.7596 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.5000 ¢

Optimal ET sequence: 14, 17, 31

Badness (Sintel): 1.55

Secant

Secant, the 26 & 58 temperament, is generated by a slightly sharp ~10/9, seven of which plus a semi-octave period give the 3rd harmonic. Its ploidacot is diploid alpha-heptacot. 58edo, 84edo and 142edo all make for excellent tunings.

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

Diesic

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

Diesic is generated by a diesis-sized interval of ~36/35, hence the name. It may be described as the 31 & 32c temperament. It is related to slender, both sharing the ploidacot of omega-13-cot, but the mappings differ for the 5th harmonic.

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

Infraorwell

Infraorwell may be described as the 49 & 58 temperament. It is generated by a ~7/6, less sharp than the generator of orwell but still sharp of just, so that three generators make a ~8/5, but seven generators does not give the 3rd harmonic. Instead, sixteen generators give the 12th harmonic; the ploidacot for this temperament is therefore gamma-16-cot. 58edo makes for a recommendable tuning, though 49edo and 67edo are among the possibilities.

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

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. Its ploidacot is pentaploid monocot.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 179200/177147

Mapping: [⟨5 0 -36 22], ⟨0 1 6 -1]]

Optimal tunings:

  • WE: ~280/243 = 239.7268 ¢, ~3/2 = 704.1103 ¢ (~64/63 = 15.0700 ¢)
error map: ⟨-1.366 +0.789 -0.013 +2.419]
  • CWE: ~280/243 = 240.0000 ¢, ~3/2 = 704.7723 ¢ (~64/63 = 15.2277 ¢)
error map: ⟨0.000 +2.817 +2.320 +6.402]

Optimal ET sequence: 5, 75, 80

Badness (Sintel): 3.76

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:

  • WE: ~55/48 = 239.7357 ¢, ~3/2 = 704.1025 ¢ (~99/98 = 15.1045 ¢)
  • CWE: ~55/48 = 240.0000 ¢, ~3/2 = 704.8440 ¢ (~99/98 = 15.1560 ¢)

Optimal ET sequence: 5, 75e, 80, 315bcdddee

Badness (Sintel): 2.37

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:

  • WE: ~55/48 = 239.7711 ¢, ~3/2 = 704.0302 ¢ (~99/98 = 15.2830 ¢)
  • CWE: ~55/48 = 240.0000 ¢, ~3/2 = 704.7112 ¢ (~99/98 = 15.2888 ¢)

Optimal ET sequence: 5f, 75e, 80, 155de

Badness (Sintel): 2.30

Triskaidekic

For the 5-limit version, see 13th-octave 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 tunings:

  • WE: ~15/14 = 92.5254 ¢, ~3/2 = 695.7801 ¢
error map: ⟨+2.831 -3.344 -10.551 +10.192]
  • CWE: ~15/14 = 92.3077 ¢, ~3/2 = 695.8319 ¢
error map: ⟨0.000 -6.123 -17.083 +3.929]

Optimal ET sequence: 13d, 26

Badness (Sintel): 5.54

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 tunings:

  • WE: ~15/14 = 92.4364 ¢, ~3/2 = 697.2675 ¢
  • CWE: ~15/14 = 92.3077 ¢, ~3/2 = 697.0548 ¢

Optimal ET sequence: 13d, 26

Badness (Sintel): 3.27

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 tunings:

  • WE: ~15/14 = 92.4535 ¢, ~3/2 = 696.9785 ¢
  • CWE: ~15/14 = 92.3077 ¢, ~3/2 = 696.5711 ¢

Optimal ET sequence: 13d, 26

Badness (Sintel): 2.45

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

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