Orwellismic temperaments: Difference between revisions

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{{Technical data page}}
These temper out {{monzo| 6 3 -1 -3 }} = 1728/1715, the orwellisma.
These temper out {{monzo| 6 3 -1 -3 }} = 1728/1715, the orwellisma.


Temperaments discussed elsewhere include:
Temperaments considered below are secant, infraorwell, quartonic, sentinel, diesic, philips, triskaidekic, and pentorwell. Discussed elsewhere are:  
* ''[[beep]]'', {21/20, 27/25} → [[Bug family #Beep]]
* ''[[Beep]]'' (+21/20 or 27/25) → [[Bug family #Beep|Bug family]]
* ''[[doublewide]]'', {50/49, 875/864} → [[Jubilismic clan #Doublewide]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[superpyth]], {64/63, 245/243} → [[Archytas clan #Superpyth]]
* [[Superpyth]] (+64/63) → [[Archytas clan #Superpyth|Archytas clan]]
* ''[[mothra]]'', {81/80, 1029/1024} → [[Meantone family #Mothra]]
* ''[[Mothra]]'' (+81/80) → [[Meantone family #Mothra|Meantone family]]
* [[myna]], {126/125, 1728/1715} → [[Starling temperaments #Myna]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[orwell]], {225/224, 1728/1715} → [[Semicomma family #Orwell]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[secund]]'', {405/392, 525/512} → [[Greenwoodmic temperaments #Secund]]
* ''[[Secund]]'' (+405/392 or 525/512) → [[Greenwoodmic temperaments #Secund|Greenwoodmic temperaments]]
* ''[[buzzard]]'', {1728/1715, 5120/5103} → [[Hemifamity temperaments #Buzzard]]
* ''[[Quartonic]]'' (+4000/3969) → [[Quartonic family]]
* ''[[Buzzard]]'' (+5120/5103) → [[Vulture family #Buzzard|Vulture family]]
* ''[[Trisensory]]'' (+78732/78125) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trimabila]]'' (+268435456/263671875) → [[Mabila family #Trimabila|Mabila family]]


== Secant ==
== Secant ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 177147/175000
[[Comma list]]: 1728/1715, 177147/175000


[[Mapping]]: [{{val| 2 1 0 5 }}, {{val| 0 7 15 2 }}]
{{Mapping|legend=1| 2 1 0 5 | 0 7 15 2 }}


{{Multival|legend=1| 14 30 4 15 -33 -75 }}
: Mapping generators: ~567/400, ~10/9


[[POTE generator]]: ~10/9 = 185.885
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~10/9 = 185.885


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


[[Badness]]: 0.095278
[[Badness]]: 0.095278
Line 31: Line 35:
Comma list: 441/440, 1344/1331, 1728/1715
Comma list: 441/440, 1344/1331, 1728/1715


Mapping: [{{val| 2 1 0 5 6 }}, {{val| 0 7 15 2 3 }}]
Mapping: {{mapping| 2 1 0 5 6 | 0 7 15 2 3 }}


POTE genertor: ~10/9 = 185.962
Optimal tuning (POTE): ~99/70 = 1\2, ~10/9 = 185.962


Vals: {{val list| 26, 58, 142e, 200cdee }}
Optimal ET sequence: {{Optimal ET sequence| 26, 58, 142e, 200cdee }}


Badness: 0.046373
Badness: 0.046373
Line 44: Line 48:
Comma list: 144/143, 351/350, 364/363, 441/440
Comma list: 144/143, 351/350, 364/363, 441/440


Mapping: [{{val| 2 1 0 5 6 4 }}, {{val| 0 7 15 2 3 11 }}]
Mapping: {{mapping| 2 1 0 5 6 4 | 0 7 15 2 3 11 }}


POTE generator: ~10/9 = 185.955
Optimal tuning (POTE): ~99/70 = 1\2, ~10/9 = 185.955


Vals: {{val list| 26, 58, 84, 142ef }}
Optimal ET sequence: {{Optimal ET sequence| 26, 58, 84, 142ef }}


Badness: 0.025035
Badness: 0.025035


== Infraorwell ==
=== 17-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 1728/1715, 28672/28125
Comma list: 144/143, 170/169, 221/220, 351/350, 441/440


[[Mapping]]: [{{val| 1 14 0 16 }}, {{val| 0 -16 3 -17 }}]
Mapping: {{mapping| 2 1 0 5 6 4 6 | 0 7 15 2 3 11 7 }}


{{Multival|legend=1| 16 -3 17 -42 -18 48 }}
Optimal tuning (POTE): ~17/12 = 1\2, ~10/9 = 185.916


[[POTE generator]]: ~7/6 = 269.032
Optimal ET sequence: {{Optimal ET sequence| 26, 58, 84, 142ef }}


{{Val list|legend=1| 9, 49, 58, 165cd, 223bcd }}
Badness: 0.021642


[[Badness]]: 0.117073
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 144/143, 153/152, 170/169, 210/209, 221/220, 400/399
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 1344/1331
Mapping: {{mapping| 2 1 0 5 6 4 6 2 | 0 7 15 2 3 11 7 21 }}


Mapping: [{{val| 1 14 0 16 12 }}, {{val| 0 -16 3 -17 -11 }}]
Optimal tuning (POTE): ~17/12 = 1\2, ~10/9 = 185.858


POTE generator: ~7/6 = 269.036
Optimal ET sequence: {{Optimal ET sequence| 26, 58h, 84, 142efh, 226deefgh }}


Vals: {{val list| 9, 49, 58, 165cdee }}
Badness: 0.018565


Badness: 0.040721
== Infraorwell ==
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 1728/1715, 28672/28125
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 196/195, 364/363
{{Mapping|legend=1| 1 14 0 16 | 0 -16 3 -17 }}


Mapping: [{{val| 1 14 0 16 12 20 }}, {{val| 0 -16 3 -17 -11 -21 }}]
: Mapping generators: ~2, ~12/7


POTE generator: ~7/6 = 269.021
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 269.032


Vals: {{val list| 9, 49f, 58 }}
{{Optimal ET sequence|legend=1| 9, 49, 58, 165cd, 223bcd }}


Badness: 0.023683
[[Badness]]: 0.117073
 
== Quartonic ==
Quartonic tempers out saleyo comma, {{monzo|3 -18 11}} = 390625000/387420489 in the 5-limit. This temperament can be described as 26&27 temperament, which tempers out the orwellisma and the octagar comma, 4000/3969. The name "quartonic" means quarter-tone, which is the generator of this temperament (representing ~[[36/35]] and ~[[40/39]]). Alternative extension [[Turkish maqam music temperaments #Yarman temperament|yarman]] (80&159) slices the quartonic generator in three.
 
Subgroup: 2.3.5
 
[[Comma list]]: 390625000/387420489
 
[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -11 -18 }}]
 
[[POTE generator]]: ~250/243 = 45.233
 
{{Val list|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}
 
[[Badness]]: 0.117250
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 1728/1715, 4000/3969
 
[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -11 -18 -5 }}]
 
{{Multival|legend=1| 11 18 5 3 -23 -39 }}
 
[[POTE generator]]: ~36/35 = 45.139
 
{{Val list|legend=1| 26, 27, 53, 80, 133d, 186d, 319d }}
 
[[Badness]]: 0.042632


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


Comma list: 176/175, 540/539, 2200/2187
Comma list: 176/175, 540/539, 1344/1331


Mapping: [{{val| 1 2 3 3 5 }}, {{val| 0 -11 -18 -5 -41 }}]
Mapping: {{mapping| 1 14 0 16 12 | 0 -16 3 -17 -11 }}


POTE generator: ~36/35 = 45.041
Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 269.036


Vals: {{val list| 26e, 27e, 53, 80, 213d, 293bcd }}
Optimal ET sequence: {{Optimal ET sequence| 9, 49, 58, 165cdee }}


Badness: 0.034031
Badness: 0.040721


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


Comma list: 169/168, 176/175, 325/324, 540/539
Comma list: 144/143, 176/175, 196/195, 364/363


Mapping: [{{val| 1 2 3 3 5 4 }}, {{val| 0 -11 -18 -5 -41 -8 }}]
Mapping: {{mapping| 1 14 0 16 12 20 | 0 -16 3 -17 -11 -21 }}


POTE generator: ~36/35 = 45.080
Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 269.021


Vals: {{val list| 26e, 27e, 53, 80, 133d, 213d}}
Optimal ET sequence: {{Optimal ET sequence| 9, 49f, 58 }}


Badness: 0.023875
Badness: 0.023683
 
=== Quarto ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 864/847
 
Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -11 -18 -5 -14 }}]
 
POTE generator: ~36/35 = 45.132
 
Vals: {{val list| 26, 27e, 53e, 80e }}
 
Badness: 0.041786
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 100/99, 144/143, 245/242
 
Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -11 -18 -5 -14 -8 }}]
 
POTE generator: ~36/35 = 45.205
 
Vals: {{val list| 26, 27e, 53e }}
 
Badness: 0.027692
 
=== Quartz ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 385/384, 4000/3969
 
Mapping: [{{val| 1 2 3 3 3 }}, {{val| 0 -11 -18 -5 12 }}]
 
POTE generator: ~36/35 = 45.385
 
Vals: {{val list| 26, 53 }}
 
Badness: 0.053285
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 169/168, 275/273, 385/384
 
Mapping: [{{val| 1 2 3 3 3 4 }}, {{val| 0 -11 -18 -5 12 -8 }}]
 
POTE generator: ~36/35 = 45.387
 
Vals: {{val list| 26, 53, 132e, 185cdef, 238cdef}}
 
Badness: 0.028818
 
=== Biquartonic ===
Subgroup: 2.3.5.7.11
 
Comma list: 1728/1715, 2420/2401, 2560/2541
 
Mapping: [{{val|2 4 6 6 7}}, {{val|0 -11 -18 -5 -1}}]
 
POTE generator: ~36/35 = 45.129
 
Vals: {{Val list| 26, 54c, 80, 186de, 266dde }}
 
Badness: 0.060737
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 640/637
 
Mapping: [{{val|2 4 6 6 7 8}}, {{val|0 -11 -18 -5 -1 -8}}]
 
POTE generator: ~36/35 = 45.153
 
Vals: {{Val list| 26, 54c, 80, 106, 186de }}
 
Badness: 0.039891


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 3 -3 6 }}, {{val| 0 -4 15 -9 }}]
{{Mapping|legend=1| 1 3 -3 6 | 0 -4 15 -9 }}


{{Multival|legend=1| 4 -15 9 -33 3 63 }}
: Mapping generators: ~2, ~9/7


[[POTE generator]]: ~9/7 = 425.596
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 425.596


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


[[Badness]]: 0.093805
[[Badness]]: 0.093805
Line 249: Line 145:
Comma list: 99/98, 243/242, 385/384
Comma list: 99/98, 243/242, 385/384


Mapping: [{{val| 1 3 -3 6 7 }}, {{val| 0 -4 15 -9 -10 }}]
Mapping: {{mapping| 1 3 -3 6 7 | 0 -4 15 -9 -10 }}


POTE generator: ~9/7 = 425.550
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550


Vals: {{val list| 14, 17, 31, 79, 110e, 141e }}
Optimal ET sequence: {{Optimal ET sequence| 14, 17, 31, 79, 110e, 141e }}


Badness: 0.039595
Badness: 0.039595
Line 262: Line 158:
Comma list: 99/98, 105/104, 144/143, 243/242
Comma list: 99/98, 105/104, 144/143, 243/242


Mapping: [{{val| 1 3 -3 6 7 3 }}, {{val| 0 -4 15 -9 -10 2 }}]
Mapping: {{mapping| 1 3 -3 6 7 3 | 0 -4 15 -9 -10 2 }}


POTE generator: ~9/7 = 425.501
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.501


Vals: {{val list| 14, 17, 31, 79f, 110ef, 141ef }}
Optimal ET sequence: {{Optimal ET sequence| 14, 17, 31, 79f, 110ef, 141ef }}


Badness: 0.037408
Badness: 0.037408
Line 273: Line 169:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Diesic]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Diesic]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -13 -21 -6 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -13 -21 -6 }}


{{Multival|legend=1| 13 21 6 3 -27 -45 }}
: Mapping generators: ~2, ~36/35


[[POTE generator]]: ~36/35 = 38.567
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~36/35 = 38.567


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


[[Badness]]: 0.107348
[[Badness]]: 0.107348
Line 292: Line 188:
Comma list: 121/120, 441/440, 891/875
Comma list: 121/120, 441/440, 891/875


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -13 -21 -6 -17 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -13 -21 -6 -17 }}


POTE generator: ~36/35 = 38.565
Optimal tuning (POTE): ~2 = 1\1, ~36/35 = 38.565


Vals: {{val list| 31, 156ce, 187ce, 218bcee, 249bcee }}
Optimal ET sequence: {{Optimal ET sequence| 31, 156ce, 187ce, 218bcee, 249bcee }}


Badness: 0.044186
Badness: 0.044186
Line 305: Line 201:
Comma list: 66/65, 121/120, 343/338, 441/440
Comma list: 66/65, 121/120, 343/338, 441/440


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -13 -21 -6 -17 -9 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -13 -21 -6 -17 -9 }}


POTE generator: ~36/35 = 38.444
Optimal tuning (POTE): ~2 = 1\1, ~36/35 = 38.444


Vals: {{val list| 31, 94cff, 125cff }}
Optimal ET sequence: {{Optimal ET sequence| 31, 94cff, 125cff }}


Badness: 0.038116
Badness: 0.038116
Line 316: Line 212:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Phillips]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Phillips]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{Val|2 0 33 -7}}, {{Val|0 1 -9 4}}]
{{Mapping|legend=1| 2 0 33 -7 | 0 1 -9 4 }}


{{Multival|legend=1|2 -18 8 -33 7 69}}
: Mapping generators: ~81/56, ~3


[[POTE generator]]: ~3/2 = 691.174
[[Optimal tuning]] ([[POTE]]): ~81/56 = 1\2, ~3/2 = 691.174


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


[[Badness]]: 0.228602
[[Badness]]: 0.228602
Line 335: Line 231:
Comma list: 99/98, 385/384, 729/704
Comma list: 99/98, 385/384, 729/704


Mapping: [{{Val|2 0 33 -7 -12}}, {{Val|0 1 -9 4 6}}]
Mapping: {{mapping| 2 0 33 -7 -12 | 0 1 -9 4 6 }}


POTE generator: ~3/2 = 691.040
Optimal tuning (POTE): ~63/44 = 1\2, ~3/2 = 691.040


Vals: {{Val list| 14, 26, 40, 66b }}
Optimal ET sequence: {{Optimal ET sequence| 14, 26, 40, 66b }}


Badness: 0.095777
Badness: 0.095777
Line 346: Line 242:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Triskaidekic]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Triskaidekic]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 1875/1792
[[Comma list]]: 1728/1715, 1875/1792


[[Mapping]]: [{{Val|13 0 30 16}}, {{Val|0 1 0 1}}]
{{Mapping|legend=1| 13 0 30 16 | 0 1 0 1 }}


{{Multival|legend=1|13 0 13 -30 -16 30}}
: Mapping generators: ~15/14, ~3


[[POTE generator]]: ~3/2 = 694.143
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\13, ~3/2 = 694.143


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


[[Badness]]: 0.219050
[[Badness]]: 0.219050
Line 365: Line 261:
Comma list: 99/98, 125/121, 385/384
Comma list: 99/98, 125/121, 385/384


Mapping: [{{Val|13 0 30 16 45}}, {{Val|0 1 0 1 0}}]
Mapping: {{mapping| 13 0 30 16 45 | 0 1 0 1 0 }}


POTE generator: ~3/2 = 696.296
Optimal tuning (POTE): ~15/14 = 1\13, ~3/2 = 696.296


Vals: {{Val list| 13d, 26, 91cd }}
Optimal ET sequence: {{Optimal ET sequence| 13d, 26, 91cd }}


Badness: 0.098847
Badness: 0.098847
Line 378: Line 274:
Comma list: 78/77, 99/98, 125/121, 1200/1183
Comma list: 78/77, 99/98, 125/121, 1200/1183


Mapping: [{{Val|13 0 30 16 45 48}}, {{Val|0 1 0 1 0 0}}]
Mapping: {{mapping| 13 0 30 16 45 48 | 0 1 0 1 0 0 }}


POTE generator: ~3/2 = 695.879
Optimal tuning (POTE): ~15/14 = 1\13, ~3/2 = 695.879


Vals: {{Val list| 13d, 26, 91cdf }}
Optimal ET sequence: {{Optimal ET sequence| 13d, 26, 91cdf }}


Badness: 0.059216
Badness: 0.059216


== Pentorwell ==
== Pentorwell ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 5 8 12 14 }}, {{val| 0 -1 -6 1 }}]
{{Mapping|legend=1| 5 0 -36 22 | 0 1 6 -1 }}


[[POTE generator]]: ~7/4 = 975.0872
[[Optimal tuning]]s:
* [[CTE]]: ~280/243 = 1\5, ~3/2 = 704.3709 (~64/63 = 15.6291)
* [[POTE]]: ~280/243 = 1\5, ~3/2 = 704.9128 (~64/63 = 15.0872)


{{Val list|legend=1| 5, 75, 80 }}
{{Optimal ET sequence|legend=1| 5, 75, 80 }}


[[Badness]]: 0.148665
[[Badness]]: 0.148665
Line 404: Line 302:
Comma list: 896/891, 1728/1715, 2200/2187
Comma list: 896/891, 1728/1715, 2200/2187


Mapping: [{{val| 5 8 12 14 17 }}, {{val| 0 -1 -6 1 5 }}]
Mapping: {{mapping| 5 0 -36 22 57 | 0 1 6 -1 -5 }}


POTE generator: ~7/4 = 975.1212
Optimal tunings:
* CTE: ~55/48 = 1\5, ~3/2 = 704.8440 (~99/98 = 15.1560)
* POTE: ~55/48 = 1\5, ~3/2 = 704.8788 (~99/98 = 15.1212)


Vals: {{val list| 5, 75e, 80 }}
Optimal ET sequence: {{Optimal ET sequence| 75e, 80, 315bcdddee }}


Badness: 0.071702
Badness: 0.071702


[[Category:Regular temperament theory]]
=== 13-limit ===
[[Category:Temperament collection]]
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 = 1\5, ~3/2 = 704.7636 (~99/98 = 15.2364)
* CWE: ~55/48 = 1\5, ~3/2 = 704.7112 (~99/98 = 15.2882)
 
Optimal ET sequence: {{Optimal ET sequence| 75e, 80, 155de }}
 
Badness: 0.0557
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Orwellismic temperaments| ]] <!-- main article -->
[[Category:Orwellismic temperaments| ]] <!-- main article -->
[[Category:Orwellismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:29, 24 June 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.

These temper out [6 3 -1 -3⟩ = 1728/1715, the orwellisma.

Temperaments considered below are secant, infraorwell, quartonic, sentinel, diesic, philips, triskaidekic, and pentorwell. Discussed elsewhere are:

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 tuning (POTE): ~567/400 = 1\2, ~10/9 = 185.885

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

Badness: 0.095278

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 tuning (POTE): ~99/70 = 1\2, ~10/9 = 185.962

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

Badness: 0.046373

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 tuning (POTE): ~99/70 = 1\2, ~10/9 = 185.955

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

Badness: 0.025035

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 tuning (POTE): ~17/12 = 1\2, ~10/9 = 185.916

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

Badness: 0.021642

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 tuning (POTE): ~17/12 = 1\2, ~10/9 = 185.858

Optimal ET sequence: 26, 58h, 84, 142efh, 226deefgh

Badness: 0.018565

Infraorwell

Subgroup: 2.3.5.7

Comma list: 1728/1715, 28672/28125

Mapping: [⟨1 14 0 16], ⟨0 -16 3 -17]]

Mapping generators: ~2, ~12/7

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 269.032

Optimal ET sequence: 9, 49, 58, 165cd, 223bcd

Badness: 0.117073

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 14 0 16 12], ⟨0 -16 3 -17 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 269.036

Optimal ET sequence: 9, 49, 58, 165cdee

Badness: 0.040721

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 14 0 16 12 20], ⟨0 -16 3 -17 -11 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 269.021

Optimal ET sequence: 9, 49f, 58

Badness: 0.023683

Sentinel

For the 5-limit version of this temperament, see High badness 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 = 1\1, ~9/7 = 425.596

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

Badness: 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 = 1\1, ~9/7 = 425.550

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

Badness: 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 = 1\1, ~9/7 = 425.501

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

Badness: 0.037408

Diesic

For the 5-limit version of this temperament, see High badness 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 = 1\1, ~36/35 = 38.567

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

Badness: 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 = 1\1, ~36/35 = 38.565

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

Badness: 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 = 1\1, ~36/35 = 38.444

Optimal ET sequence: 31, 94cff, 125cff

Badness: 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: [⟨2 0 33 -7], ⟨0 1 -9 4]]

Mapping generators: ~81/56, ~3

Optimal tuning (POTE): ~81/56 = 1\2, ~3/2 = 691.174

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

Badness: 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 = 1\2, ~3/2 = 691.040

Optimal ET sequence: 14, 26, 40, 66b

Badness: 0.095777

Triskaidekic

For the 5-limit version of this temperament, see High badness 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 = 1\13, ~3/2 = 694.143

Optimal ET sequence: 13d, 26

Badness: 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 = 1\13, ~3/2 = 696.296

Optimal ET sequence: 13d, 26, 91cd

Badness: 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 = 1\13, ~3/2 = 695.879

Optimal ET sequence: 13d, 26, 91cdf

Badness: 0.059216

Pentorwell

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 = 1\5, ~3/2 = 704.3709 (~64/63 = 15.6291)
  • POTE: ~280/243 = 1\5, ~3/2 = 704.9128 (~64/63 = 15.0872)

Optimal ET sequence: 5, 75, 80

Badness: 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 = 1\5, ~3/2 = 704.8440 (~99/98 = 15.1560)
  • POTE: ~55/48 = 1\5, ~3/2 = 704.8788 (~99/98 = 15.1212)

Optimal ET sequence: 75e, 80, 315bcdddee

Badness: 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 = 1\5, ~3/2 = 704.7636 (~99/98 = 15.2364)
  • CWE: ~55/48 = 1\5, ~3/2 = 704.7112 (~99/98 = 15.2882)

Optimal ET sequence: 75e, 80, 155de

Badness: 0.0557

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