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Orwellismic temperaments: Difference between revisions

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* ''[[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 secant, infraorwell, quartonic, sentinel, diesic, philips, triskaidekic, and pentaorwell.  
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 95: Line 151:
Badness (Sintel): 1.13
Badness (Sintel): 1.13


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.4649{{c}}, ~7/6 = 268.6878{{c}}
* [[WE]]: ~2 = 1200.1293{{c}}, ~36/35 = 38.5709{{c}}
: [[error map]]: {{val| -1.535 +0.120 +3.018 +0.401 }}
: [[error map]]: {{val| +0.129 -3.118 +4.086 +0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 268.9844{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 38.5603{{c}}
: error map: {{val| 0.000 +1.796 +6.733 +3.909 }}
: error map: {{val| 0.000 -3.239 +3.920 -0.188 }}


{{Optimal ET sequence|legend=1| 9, 40bd, 49, 58, 165cd, 223bccd, 281bcccdd }}
{{Optimal ET sequence|legend=1| 1c, 30bc, 31, 156c }}


[[Badness]] (Sintel): 2.96
[[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 -2 3 -1 1 | 0 16 -3 17 11 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.2928{{c}}, ~7/6 = 268.6532{{c}}
* WE: ~2 = 1200.4923{{c}}, ~36/35 = 38.5806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9825{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.5397{{c}}


{{Optimal ET sequence|legend=0| 9, 40bde, 49, 58, 165cdee, 223bccdeee, 281bcccddeee }}
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}


Badness (Sintel): 1.35
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 -2 3 -1 1 -1 | 0 16 -3 17 11 21 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.2216{{c}}, ~7/6 = 268.6220{{c}}
* WE: ~2 = 1199.3005{{c}}, ~36/35 = 38.4213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.4764{{c}}


{{Optimal ET sequence|legend=0| 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff }}
{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}


Badness (Sintel): 0.979
Badness (Sintel): 1.57


== Sentinel ==
== Infraorwell ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit 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 -1 12 -3 | 0 4 -15 9 }}
{{Mapping|legend=1| 1 -2 3 -1 | 0 16 -3 17 }}
: mapping generators: ~2, ~14/9
: mapping generators: ~2, ~7/6


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.8626{{c}}, ~14/9 = 774.9609{{c}}
* [[WE]]: ~2 = 1198.4649{{c}}, ~7/6 = 268.6878{{c}}
: [[error map]]: {{val| +0.863 -2.974 -0.376 +3.234 }}
: [[error map]]: {{val| -1.535 +0.120 +3.018 +0.401 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.4021{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 268.9844{{c}}
: error map: {{val| 0.000 -4.347 -2.345 +0.793 }}
: error map: {{val| 0.000 +1.796 +6.733 +3.909 }}


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


[[Badness]] (Sintel): 2.37
[[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 -1 12 -3 -3 | 0 4 -15 9 10 }}
Mapping: {{mapping| 1 -2 3 -1 1 | 0 16 -3 17 11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0893{{c}}, ~14/9 = 775.1534{{c}}
* WE: ~2 = 1198.2928{{c}}, ~7/6 = 268.6532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.4564{{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 (Sintel): 1.31
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 -1 12 -3 -3 5 | 0 4 -15 9 10 -2 }}
Mapping: {{mapping| 1 -2 3 -1 1 -1 | 0 16 -3 17 11 21 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4045{{c}}, ~14/9 = 774.7596{{c}}
* WE: ~2 = 1198.2216{{c}}, ~7/6 = 268.6220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.5000{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 268.9616{{c}}


{{Optimal ET sequence|legend=0| 14, 17, 31 }}
{{Optimal ET sequence|legend=0| 9, 40bdef, 49f, 58, 223bccdeeefff, 281bcccddeeeffff }}
 
Badness (Sintel): 0.979


Badness (Sintel): 1.55
== Pentaorwell ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Counterpental]].''


== Diesic ==
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.  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Diesic]].''


[[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]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1293{{c}}, ~36/35 = 38.5709{{c}}
* [[WE]]: ~280/243 = 239.7268{{c}}, ~3/2 = 704.1103{{c}} (~64/63 = 15.0700{{c}})
: [[error map]]: {{val| +0.129 -3.118 +4.086 +0.137 }}
: [[error map]]: {{val| -1.366 +0.789 -0.013 +2.419 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 38.5603{{c}}
* [[CWE]]: ~280/243 = 240.0000{{c}}, ~3/2 = 704.7723{{c}} (~64/63 = 15.2277{{c}})
: error map: {{val| 0.000 -3.239 +3.920 -0.188 }}
: error map: {{val| 0.000 +2.817 +2.320 +6.402 }}


{{Optimal ET sequence|legend=1| 1c, 30bc, 31, 156c }}
{{Optimal ET sequence|legend=1| 5, 75, 80 }}


[[Badness]] (Sintel): 2.72
[[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 tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4923{{c}}, ~36/35 = 38.5806{{c}}
* WE: ~55/48 = 239.7357{{c}}, ~3/2 = 704.1025{{c}} (~99/98 = 15.1045{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.5397{{c}}
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.8440{{c}} (~99/98 = 15.1560{{c}})


{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}
{{Optimal ET sequence|legend=0| 5, 75e, 80, 315bcdddee }}


Badness (Sintel): 1.46
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 tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3005{{c}}, ~36/35 = 38.4213{{c}}
* WE: ~55/48 = 239.7711{{c}}, ~3/2 = 704.0302{{c}} (~99/98 = 15.2830{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 38.4764{{c}}
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.7112{{c}} (~99/98 = 15.2888{{c}})


{{Optimal ET sequence|legend=0| 1ce, 30bce, 31 }}
{{Optimal ET sequence|legend=0| 5f, 75e, 80, 155de }}


Badness (Sintel): 1.57
Badness (Sintel): 2.30
 
== Phillips ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Phillips]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 6561/6272
 
{{Mapping|legend=1| 2 0 33 -7 | 0 1 -9 4 }}
: mapping generators: ~81/56, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 0 33 -7 -12 | 0 1 -9 4 6 }}
 
Optimal tunings:
* WE: ~63/44 = 601.2349{{c}}, ~3/2 = 692.4623{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 691.0383{{c}}
 
{{Optimal ET sequence|legend=0| 14, 26, 40, 66b }}
 
Badness (Sintel): 3.17


== Triskaidekic ==
== Triskaidekic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Triskaidekic]].''
: ''For the 5-limit version, see [[13th-octave temperaments #Triskaidekic]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 328: Line 354:
Badness (Sintel): 2.45
Badness (Sintel): 2.45


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


[[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:  
* [[WE]]: ~280/243 = 239.7268{{c}}, ~3/2 = 704.1103{{c}} (~64/63 = 15.0700{{c}})
* [[WE]]: ~81/56 = 601.3944{{c}}, ~3/2 = 692.7807{{c}}
: [[error map]]: {{val| -1.366 +0.789 -0.013 +2.419 }}
: [[error map]]: {{val| +2.789 -6.385 -0.424 +3.692 }}
* [[CWE]]: ~280/243 = 240.0000{{c}}, ~3/2 = 704.7723{{c}} (~64/63 = 15.2277{{c}})
* [[CWE]]: ~81/56 = 600.0000{{c}}, ~3/2 = 691.1141{{c}}
: error map: {{val| 0.000 +2.817 +2.320 +6.402 }}
: 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]] (Sintel): 3.76
[[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:  
* WE: ~55/48 = 239.7357{{c}}, ~3/2 = 704.1025{{c}} (~99/98 = 15.1045{{c}})
* WE: ~63/44 = 601.2349{{c}}, ~3/2 = 692.4623{{c}}
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.8440{{c}} (~99/98 = 15.1560{{c}})
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 691.0383{{c}}


{{Optimal ET sequence|legend=0| 5, 75e, 80, 315bcdddee }}
{{Optimal ET sequence|legend=0| 14, 26, 40, 66b }}


Badness (Sintel): 2.37
Badness (Sintel): 3.17
 
=== 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:
* WE: ~55/48 = 239.7711{{c}}, ~3/2 = 704.0302{{c}} (~99/98 = 15.2830{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~3/2 = 704.7112{{c}} (~99/98 = 15.2888{{c}})
 
{{Optimal ET sequence|legend=0| 5f, 75e, 80, 155de }}
 
Badness (Sintel): 2.30


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Orwellismic temperaments| ]] <!-- main article -->
[[Category:Orwellismic temperaments| ]] <!-- main article -->
[[Category:Orwellismic| ]] <!-- key article -->
[[Category:Orwellismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Orwellismic_temperaments"