Semicomma family: Difference between revisions

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{{Mapping|legend=1| 1 0 3 | 0 7 -3 }}
{{Mapping|legend=1| 1 0 3 | 0 7 -3 }}
: mapping generators: ~2, ~75/64
: mapping generators: ~2, ~75/64


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.670
* [[WE]]: ~2 = 1200.2902{{c}}, ~75/64 = 271.6929{{c}}
: [[error map]]: {{val| 0.000 -0.264 -1.324 }}
: [[error map]]: {{val| +0.290 -0.104 -0.522 }}
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.627
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}}
: error map: {{val| 0.000 -0.564 -1.195 }}
: error map: {{val| 0.000 -0.479 -1.232 }}


[[Tuning ranges]]:
[[Tuning ranges]]:
* 5-odd-limit [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
* [[5-odd-limit]] [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)
* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)


{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c }}
{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c, 1586bccc }}


[[Badness]] (Smith): 0.040807
[[Badness]] (Sintel): 0.957


=== Overview to extensions ===
=== Overview to extensions ===
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{{Main| Orwell }}
{{Main| Orwell }}


So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the {{nowrap| 22 & 31 }} temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.
So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the {{nowrap| 22 & 31 }} temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of its slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]] (the nuwell comma), [[1728/1715]] (the orwellisma), [[225/224]] (the marvel comma or septimal kleisma), and [[6144/6125]] (the porwell comma).


The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.
The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.
Line 49: Line 48:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/6 = 271.513
* [[WE]]: ~2 = 1200.0192{{c}}, ~7/6 = 271.5130{{c}}
: [[error map]]: {{val| 0.000 -1.364 -0.853 +3.278 }}
: [[error map]]: {{val| +0.019 -1.364 -0.795 +3.297 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 271.509
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 271.5097{{c}}
: error map: {{val| 0.000 -1.394 -0.840 +3.243 }}
: error map: {{val| 0.000 -1.387 -0.843 +3.252 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges]]:
[[Tuning ranges]]:
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[[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.
[[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.


{{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84, 137, 221d, 358d }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84, 137, 221d }}


[[Badness]] (Smith): 0.020735
[[Badness]] (Sintel): 0.525


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 271.560
* WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 271.426
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}}


Minimax tuning:
Minimax tuning:
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{{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }}
{{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }}


Badness (Smith): 0.015231
Badness (Sintel): 0.504


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 271.556
* WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 271.546
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}}


Tuning ranges:
Tuning ranges:
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{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}
{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}


Badness (Smith): 0.019718
Badness (Sintel): 0.815


==== Blair ====
==== Blair ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 271.747
* WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 271.301
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.3846{{c}}


{{Optimal ET sequence|legend=0| 9, 22, 31f }}
{{Optimal ET sequence|legend=0| 9, 22, 31f }}


Badness (Smith): 0.023086
Badness (Sintel): 0.954


==== Winston ====
==== Winston ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 271.163
* WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 271.088
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}}


Tuning ranges:
Tuning ranges:
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{{Optimal ET sequence|legend=0| 9, 22f, 31 }}
{{Optimal ET sequence|legend=0| 9, 22f, 31 }}


Badness (Smith): 0.019931
Badness (Sintel): 0.824


==== Doublethink ====
==== Doublethink ====
Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]]. Its ploidacot is alpha-tetradecacot.  
Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]]. Its ploidacot is alpha-14-cot.  


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 135.811
* WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 135.723
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}}


Tuning ranges:
Tuning ranges:
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* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]


{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 62, 115ef }}
{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 115ef }}


Badness (Smith): 0.027120
Badness (Sintel): 1.12


=== Newspeak ===
=== Newspeak ===
In newspeak, the simplicity of obtaining ~[[11/8]] by stacking the generator ~[[7/6]] twice (as in basic 11-limit orwell) is sacrificed to gain accuracy for larger equal temperaments (such as [[84edo]] and [[115edo]]), at the cost of much higher complexity: it is reached only after stacking the generator 33 times and octave-reducing. Newspeak intersects with undecimal orwell at [[31edo]].
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 180: Line 181:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 271.316
* WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 271.288
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}}


Tuning ranges:
Tuning ranges:
Line 189: Line 190:
{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}
{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}


Badness (Smith): 0.031438
Badness (Sintel): 1.04


=== Borwell ===
=== Borwell ===
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Comma list: 225/224, 243/242, 1728/1715
Comma list: 225/224, 243/242, 1728/1715


Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}
Mapping: {{mapping| 1 -7 6 -7 -18 | 0 14 -6 16 35 }}
 
: mapping generators: ~2, ~55/36
: mapping generators: ~2, ~72/55


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~55/36 = 735.754
* WE: ~2 = 1200.0194{{c}}, ~55/36 = 735.7641{{c}}
* POTE: ~2 = 1200.000, ~55/36 = 735.752
* CWE: ~2 = 1200.000{{c}}, ~55/36 = 735.7527{{c}}


{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}
{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}


Badness (Smith): 0.038377
Badness (Sintel): 1.27


== Sabric ==
== Sabric ==
The sabric temperament ({{nowrap| 53 & 190 }}) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).
The sabric temperament tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 53 & 190 }} temperament. It was named by [[Xenllium]] in 2021 for its relation to the Sabra2 tuning (generator: 271.607278 cents).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.622
* [[WE]]: ~2 = 1200.3056{{c}}, ~75/64 = 271.6760{{c}}
: [[error map]]: {{val| 0.000 -0.599 -1.180 +0.131 }}
: [[error map]]: {{val| +0.306 -0.223 -0.425 +0.049 }}
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.607
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6110{{c}}
: error map: {{val| 0.000 -0.707 -1.134 -0.808 }}
: error map: {{val| 0.000 -0.678 -1.147 -0.558 }}


{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}
{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}


[[Badness]] (Smith): 0.088355
[[Badness]] (Sintel): 2.24


== Triwell ==
== Triwell ==
The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the 5th harmonic.
Triwell tempers out the gamelisma, [[1029/1024]], and the triwellisma, [[235298/234375]]. It may be described as the {{nowrap| 31 & 159 }} temperament. It slices orwell's generator plus two octaves into three generators, and seven generators octave reduced make a ~8/7, which is the generator of [[slendric]]. Its ploidacot is 15-sheared-21-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 1029/1024, 235298/234375
[[Comma list]]: 1029/1024, 235298/234375


{{Mapping|legend=1| 1 7 0 1 | 0 -21 9 7 }}
{{Mapping|legend=1| 1 -14 9 8 | 0 21 -9 -7 }}
 
: mapping generators: ~2, ~375/224
: mapping generators: ~2, ~448/375


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~448/375 = 309.456
* [[WE]]: ~2 = 1200.4763{{c}}, ~375/224 = 890.8812{{c}}
: [[error map]]: {{val| 0.000 -0.522 -1.213 -2.637 }}
: [[error map]]: {{val| +0.476 -0.118 +0.042 -1.184 }}
* [[POTE]]: ~2 = 1200.000, ~448/375 = 309.472
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/224 = 890.5312{{c}}
: error map: {{val| 0.000 -0.872 -1.063 -2.520 }}
: error map: {{val| 0.000 -0.799 -1.095 -2.545 }}


{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}


[[Badness]] (Smith): 0.080575
[[Badness]] (Sintel): 2.04


=== 11-limit ===
=== 11-limit ===
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Comma list: 385/384, 441/440, 456533/455625
Comma list: 385/384, 441/440, 456533/455625


Mapping: {{mapping| 1 7 0 1 13 | 0 -21 9 7 -37 }}
Mapping: {{mapping| 1 -14 9 8 -24 | 0 21 -9 -7 37 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~448/375 = 309.444
* WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}}
* POTE: ~2 = 1200.000, ~448/375 = 309.471
* CWE: ~2 = 1200.0000{{c}}, ~375/224 = 890.5344{{c}}


{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}


Badness (Smith): 0.029807
Badness (Sintel): 0.985


== Quadrawell ==
== Quadrawell ==
The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the 5th harmonic.
Quadrawell tempers out [[2401/2400]] and may be described as the {{nowrap| 31 & 212 }} temperament. It has a [[7/4]] generator of about 968 cents, four of which minus three octaves give the original generator of orwell. It can also be viewed as [[2.5.7|2.5.7-subgroup]] [[mothra]] with a different mapping of prime [[3/1|3]]. Its ploidacot is 22-sheared-28-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 2401/2400, 2109375/2097152
[[Comma list]]: 2401/2400, 2109375/2097152


{{Mapping|legend=1| 1 7 0 3 | 0 -28 12 -1 }}
{{Mapping|legend=1| 1 -21 12 2 | 0 28 -12 1 }}
 
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~8/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.082
* [[WE]]: ~2 = 1200.3006{{c}}, ~7/4 = 968.1489{{c}}
: [[error map]]: {{val| 0.000 -0.255 -1.328 -0.908 }}
: [[error map]]: {{val| +0.301 -0.098 -0.493 -0.076 }}
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.094
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 967.9090{{c}}
: error map: {{val| 0.000 -0.574 -1.191 -0.919 }}
: error map: {{val| 0.000 -0.503 -1.222 -0.917 }}


{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}
{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}


[[Badness]] (Smith): 0.075754
[[Badness]] (Sintel): 1.92


=== 11-limit ===
=== 11-limit ===
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Comma list: 385/384, 1375/1372, 14641/14580
Comma list: 385/384, 1375/1372, 14641/14580


Mapping: {{mapping| 1 7 0 3 11 | 0 -28 12 -1 -39 }}
Mapping: {{mapping| 1 -21 12 2 -28 | 0 28 -12 1 39 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~8/7 = 232.065
* WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}}
* POTE: ~2 = 1200.000, ~8/7 = 232.083
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 967.9206{{c}}


{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}
{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}


Badness (Smith): 0.036493
Badness (Sintel): 1.21


== Rainwell ==
== Rainwell ==
The ''rainwell'' temperament ({{nowrap| 31 & 265 }}) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.
The rainwell temperament tempers out the mirkwai comma, [[16875/16807]], and the rainy comma, [[2100875/2097152]]. It may be described as the {{nowrap| 31 & 265 }} temperament. Its ploidacot is 22-sheared-35-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 306: Line 304:
[[Comma list]]: 16875/16807, 2100875/2097152
[[Comma list]]: 16875/16807, 2100875/2097152


{{Mapping|legend=1| 1 14 -3 6 | 0 -35 15 -9 }}
{{Mapping|legend=1| 1 -21 12 -3 | 0 35 -15 9 }}


: mapping generators: ~2, ~2625/2048
: mapping generators: ~2, ~2625/2048


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~2625/2048 = 425.666
* [[WE]]: ~2 = 1200.2032{{c}}, ~2401/1536 = 774.4577{{c}}
: [[error map]]: {{val| 0.000 -0.278 -1.318 0.177 }}
: [[error map]]: {{val| +0.203 -0.204 -0.740 +0.683 }}
* [[POTE]]: ~2 = 1200.000, ~2625/2048 = 425.673
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2401/1536 = 774.3282{{c}}
: error map: {{val| 0.000 -0.526 -1.212 0.113 }}
: error map: {{val| 0.000 -0.469 -1.236 +0.128 }}


{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}


[[Badness]] (Smith): 0.143488
[[Badness]] (Sintel): 3.63


=== 11-limit ===
=== 11-limit ===
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Comma list: 540/539, 1375/1372, 2100875/2097152
Comma list: 540/539, 1375/1372, 2100875/2097152


Mapping: {{mapping| 1 14 -3 6 29 | 0 -35 15 -9 -72 }}
Mapping: {{mapping| 1 -21 12 -3 -43 | 0 35 -15 9 72 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~2625/2048 = 425.671
* WE: ~2 = 1200.1915{{c}}, ~2205/1408 = 774.4451{{c}}
* POTE: ~2 = 1200.000, ~2625/2048 = 425.679
* CWE: ~2 = 1200.0000{{c}}, ~2205/1408 = 774.3233{{c}}


{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}
{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}


Badness (Smith): 0.052774
Badness (Sintel): 1.74


== Quinwell ==
== Quinwell ==
The quinwell temperament ({{nowrap| 22 & 243 }}) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.
The quinwell temperament tempers out the wizma, [[420175/419904]], and may be described as the {{nowrap| 22 & 243 }} temperament. It slices orwell's generator into five quartertones. Its ploidacot is alpha-35-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 343: Line 341:


{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}
{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}
: mapping generators: ~2, ~405/392
: mapping generators: ~2, ~405/392


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~405/392 = 54.335
* [[WE]]: ~2 = 1200.2860{{c}}, ~405/392 = 54.3373{{c}}
: [[error map]]: {{val| 0.000 -0.233 -1.338 -0.061 }}
: [[error map]]: {{val| +0.286 -0.151 -0.515 +0.084 }}
* [[POTE]]: ~2 = 1200.000, ~405/392 = 54.324
* [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}}
: error map: {{val| 0.000 -0.604 -1.178 -0.718 }}
: error map: {{val| 0.000 -0.501 -1.223 -0.536 }}


{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}
{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}


[[Badness]] (Smith): 0.168897
[[Badness]] (Sintel): 4.27


=== 11-limit ===
=== 11-limit ===
Line 364: Line 361:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~33/32 = 54.338
* WE: ~2 = 1200.0642{{c}}, ~33/32 = 54.3395{{c}}
* POTE: ~2 = 1200.000, ~33/32 = 54.334
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 54.3369{{c}}


{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}
{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}


Badness (Smith): 0.097202
Badness (Sintel): 3.21


=== Quinbetter ===
=== Quinbetter ===
Line 379: Line 376:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~405/392 = 54.332
* WE: ~2 = 1200.0642{{c}}, ~405/392 = 54.3373{{c}}
* POTE: ~2 = 1200.000, ~405/392 = 54.316
* CWE: ~2 = 1200.0000{{c}}, ~405/392 = 54.3192{{c}}


{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}
{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}


Badness (Smith): 0.078657
Badness (Sintel): 2.60


[[Category:Temperament families]]
[[Category:Temperament families]]

Latest revision as of 17:12, 1 March 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.

The 5-limit parent comma for the semicomma family of temperaments is the semicomma (monzo: [-21 3 7⟩, ratio: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

Orson

Orson, first discovered by Erv Wilson[citation needed], is the 5-limit temperament tempering out the semicomma. It has a generator of ~75/64, seven of which give the perfect twelfth; its ploidacot is alpha-heptacot. The generator is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example 53edo or 84edo. These give tunings to the generator which are sharp of 7/6 by less than five cents, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

Subgroup: 2.3.5

Comma list: 2109375/2097152

Mapping: [⟨1 0 3], ⟨0 7 -3]]

mapping generators: ~2, ~75/64

Optimal tunings:

  • WE: ~2 = 1200.2902 ¢, ~75/64 = 271.6929 ¢
error map: ⟨+0.290 -0.104 -0.522]
  • CWE: ~2 = 1200.0000 ¢, ~75/64 = 271.6394 ¢
error map: ⟨0.000 -0.479 -1.232]

Tuning ranges:

Optimal ET sequence: 22, 31, 53, 190, 243, 296, 645c, 1586bccc

Badness (Sintel): 0.957

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add

  • 1029/1024, leading to the 31 & 159 temperament (triwell), or
  • 2401/2400, giving the 31 & 243 temperament (quadrawell), or
  • 4375/4374, giving the 53 & 243 temperament (sabric).

Orwell

Main article: Orwell

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with 22, 31, 53 and 84 equal, and may be described as the 22 & 31 temperament. It is a good system in the 7-limit and naturally extends into the 11-limit. 84edo, with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit POTE tuning, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. 53edo might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of its slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401 (the nuwell comma), 1728/1715 (the orwellisma), 225/224 (the marvel comma or septimal kleisma), and 6144/6125 (the porwell comma).

The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.

Orwell has mos scales of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.

Subgroup: 2.3.5.7

Comma list: 225/224, 1728/1715

Mapping: [⟨1 0 3 1], ⟨0 7 -3 8]]

Optimal tunings:

  • WE: ~2 = 1200.0192 ¢, ~7/6 = 271.5130 ¢
error map: ⟨+0.019 -1.364 -0.795 +3.297]
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 271.5097 ¢
error map: ⟨0.000 -1.387 -0.843 +3.252]

Minimax tuning:

[[1 0 0 0⟩, [14/11 0 -7/11 7/11⟩, [27/11 0 3/11 -3/11⟩, [27/11 0 -8/11 8/11⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[1 0 0 0⟩, [21/17 14/17 -7/17 0⟩, [42/17 -6/17 3/17 0⟩, [41/17 16/17 -8/17 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

  • 7-odd-limit diamond monotone: ~7/6 = [266.667, 272.727] (2\9 to 5\22)
  • 9-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
  • 7-odd-limit diamond tradeoff: ~7/6 = [266.871, 271.708]
  • 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

Algebraic generator: Sabra3, the real root of 12x3 - 7x - 48.

Optimal ET sequence: 9, 22, 31, 53, 84, 137, 221d

Badness (Sintel): 0.525

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 176/175

Mapping: [⟨1 0 3 1 3], ⟨0 7 -3 8 2]]

Optimal tunings:

  • WE: ~2 = 1200.5989 ¢, ~7/6 = 271.5616 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 271.4552 ¢

Minimax tuning:

  • 11-odd-limit: ~7/6 = [2/11 0 -1/11 1/11⟩
[[1 0 0 0 0⟩, [14/11 0 -7/11 7/11 0⟩, [27/11 0 3/11 -3/11 0⟩, [27/11 0 -8/11 8/11 0⟩, [37/11 0 -2/11 2/11 0⟩]
Unchanged-interval (eigenmonzo) basis: 2.7/5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
  • 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

Optimal ET sequence: 9, 22, 31, 53, 84e

Badness (Sintel): 0.504

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 176/175, 275/273

Mapping: [⟨1 0 3 1 3 8], ⟨0 7 -3 8 2 -19]]

Optimal tunings:

  • WE: ~2 = 1200.3621 ¢, ~7/6 = 271.6283 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 271.5477 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
  • 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

Optimal ET sequence: 22, 31, 53, 84e

Badness (Sintel): 0.815

Blair

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 99/98

Mapping: [⟨1 0 3 1 3 3], ⟨0 7 -3 8 2 3]]

Optimal tunings:

  • WE: ~2 = 1201.8031 ¢, ~7/6 = 271.7083 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 271.3846 ¢

Optimal ET sequence: 9, 22, 31f

Badness (Sintel): 0.954

Winston

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 121/120

Mapping: [⟨1 0 3 1 3 1], ⟨0 7 -3 8 2 12]]

Optimal tunings:

  • WE: ~2 = 1200.2846 ¢, ~7/6 = 271.1524 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 271.1032 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
  • 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]

Optimal ET sequence: 9, 22f, 31

Badness (Sintel): 0.824

Doublethink

Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two 13/12~14/13's by tempering out their difference, 169/168. Its ploidacot is alpha-14-cot.

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 176/175

Mapping: [⟨1 0 3 1 3 2], ⟨0 14 -6 16 4 15]]

Optimal tunings:

  • WE: ~2 = 1200.6876 ¢, ~13/12 = 135.8006 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 135.7410 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
  • 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]

Optimal ET sequence: 9, 35bd, 44, 53, 115ef

Badness (Sintel): 1.12

Newspeak

In newspeak, the simplicity of obtaining ~11/8 by stacking the generator ~7/6 twice (as in basic 11-limit orwell) is sacrificed to gain accuracy for larger equal temperaments (such as 84edo and 115edo), at the cost of much higher complexity: it is reached only after stacking the generator 33 times and octave-reducing. Newspeak intersects with undecimal orwell at 31edo.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1728/1715

Mapping: [⟨1 0 3 1 -4], ⟨0 7 -3 8 33]]

Optimal tunings:

  • WE: ~2 = 1200.2072 ¢, ~7/6 = 271.3353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 271.2952 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
  • 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

Optimal ET sequence: 22e, 31, 84, 115

Badness (Sintel): 1.04

Borwell

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 1728/1715

Mapping: [⟨1 -7 6 -7 -18], ⟨0 14 -6 16 35]]

mapping generators: ~2, ~55/36

Optimal tunings:

  • WE: ~2 = 1200.0194 ¢, ~55/36 = 735.7641 ¢
  • CWE: ~2 = 1200.000 ¢, ~55/36 = 735.7527 ¢

Optimal ET sequence: 31, 75e, 106, 137

Badness (Sintel): 1.27

Sabric

The sabric temperament tempers out the ragisma, 4375/4374, and may be described as the 53 & 190 temperament. It was named by Xenllium in 2021 for its relation to the Sabra2 tuning (generator: 271.607278 cents).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2109375/2097152

Mapping: [⟨1 0 3 -11], ⟨0 7 -3 61]]

Optimal tunings:

  • WE: ~2 = 1200.3056 ¢, ~75/64 = 271.6760 ¢
error map: ⟨+0.306 -0.223 -0.425 +0.049]
  • CWE: ~2 = 1200.0000 ¢, ~75/64 = 271.6110 ¢
error map: ⟨0.000 -0.678 -1.147 -0.558]

Optimal ET sequence: 53, 137d, 190, 243, 1511bccd

Badness (Sintel): 2.24

Triwell

Triwell tempers out the gamelisma, 1029/1024, and the triwellisma, 235298/234375. It may be described as the 31 & 159 temperament. It slices orwell's generator plus two octaves into three generators, and seven generators octave reduced make a ~8/7, which is the generator of slendric. Its ploidacot is 15-sheared-21-cot.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 235298/234375

Mapping: [⟨1 -14 9 8], ⟨0 21 -9 -7]]

mapping generators: ~2, ~375/224

Optimal tunings:

  • WE: ~2 = 1200.4763 ¢, ~375/224 = 890.8812 ¢
error map: ⟨+0.476 -0.118 +0.042 -1.184]
  • CWE: ~2 = 1200.0000 ¢, ~375/224 = 890.5312 ¢
error map: ⟨0.000 -0.799 -1.095 -2.545]

Optimal ET sequence: 31, 97, 128, 159, 190

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 456533/455625

Mapping: [⟨1 -14 9 8 -24], ⟨0 21 -9 -7 37]]

Optimal tunings:

  • WE: ~2 = 1200.4804 ¢, ~375/224 = 890.8854 ¢
  • CWE: ~2 = 1200.0000 ¢, ~375/224 = 890.5344 ¢

Optimal ET sequence: 31, 97, 128, 159, 190

Badness (Sintel): 0.985

Quadrawell

Quadrawell tempers out 2401/2400 and may be described as the 31 & 212 temperament. It has a 7/4 generator of about 968 cents, four of which minus three octaves give the original generator of orwell. It can also be viewed as 2.5.7-subgroup mothra with a different mapping of prime 3. Its ploidacot is 22-sheared-28-cot.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 2109375/2097152

Mapping: [⟨1 -21 12 2], ⟨0 28 -12 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1200.3006 ¢, ~7/4 = 968.1489 ¢
error map: ⟨+0.301 -0.098 -0.493 -0.076]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 967.9090 ¢
error map: ⟨0.000 -0.503 -1.222 -0.917]

Optimal ET sequence: 31, 119, 150, 181, 212, 243, 698cd, 941cd

Badness (Sintel): 1.92

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 14641/14580

Mapping: [⟨1 -21 12 2 -28], ⟨0 28 -12 1 39]]

Optimal tunings:

  • WE: ~2 = 1200.3622 ¢, ~7/4 = 968.2089 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 967.9206 ¢

Optimal ET sequence: 31, 119, 150, 181, 212, 455ee, 667cdee

Badness (Sintel): 1.21

Rainwell

The rainwell temperament tempers out the mirkwai comma, 16875/16807, and the rainy comma, 2100875/2097152. It may be described as the 31 & 265 temperament. Its ploidacot is 22-sheared-35-cot.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 2100875/2097152

Mapping: [⟨1 -21 12 -3], ⟨0 35 -15 9]]

mapping generators: ~2, ~2625/2048

Optimal tunings:

  • WE: ~2 = 1200.2032 ¢, ~2401/1536 = 774.4577 ¢
error map: ⟨+0.203 -0.204 -0.740 +0.683]
  • CWE: ~2 = 1200.0000 ¢, ~2401/1536 = 774.3282 ¢
error map: ⟨0.000 -0.469 -1.236 +0.128]

Optimal ET sequence: 31, 172, 203, 234, 265, 296

Badness (Sintel): 3.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 2100875/2097152

Mapping: [⟨1 -21 12 -3 -43], ⟨0 35 -15 9 72]]

Optimal tunings:

  • WE: ~2 = 1200.1915 ¢, ~2205/1408 = 774.4451 ¢
  • CWE: ~2 = 1200.0000 ¢, ~2205/1408 = 774.3233 ¢

Optimal ET sequence: 31, 234, 265, 296, 919bc

Badness (Sintel): 1.74

Quinwell

The quinwell temperament tempers out the wizma, 420175/419904, and may be described as the 22 & 243 temperament. It slices orwell's generator into five quartertones. Its ploidacot is alpha-35-cot.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 2109375/2097152

Mapping: [⟨1 0 3 0], ⟨0 35 -15 62]]

mapping generators: ~2, ~405/392

Optimal tunings:

  • WE: ~2 = 1200.2860 ¢, ~405/392 = 54.3373 ¢
error map: ⟨+0.286 -0.151 -0.515 +0.084]
  • CWE: ~2 = 1200.0000 ¢, ~405/392 = 54.3273 ¢
error map: ⟨0.000 -0.501 -1.223 -0.536]

Optimal ET sequence: 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd

Badness (Sintel): 4.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 2109375/2097152

Mapping: [⟨1 0 3 0 5], ⟨0 35 -15 62 -34]]

Optimal tunings:

  • WE: ~2 = 1200.0642 ¢, ~33/32 = 54.3395 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 54.3369 ¢

Optimal ET sequence: 22, 221, 243, 265

Badness (Sintel): 3.21

Quinbetter

Subgroup: 2.3.5.7.11

Comma list: 385/384, 24057/24010, 43923/43750

Mapping: [⟨1 0 3 0 4], ⟨0 35 -15 62 -12]]

Optimal tunings:

  • WE: ~2 = 1200.0642 ¢, ~405/392 = 54.3373 ¢
  • CWE: ~2 = 1200.0000 ¢, ~405/392 = 54.3192 ¢

Optimal ET sequence: 22, …, 199d, 221e, 243e, 707bcdeee

Badness (Sintel): 2.60

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