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Semicomma family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/2)
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}}
: error map: {{val| 0.000 -0.479 -1.232 }}
: error map: {{val| 0.000 -0.479 -1.232 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~75/64 = 271.670{{c}}
: [[error map]]: {{val| 0.000 -0.264 -1.324 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~75/64 = 271.627{{c}}
: error map: {{val| 0.000 -0.564 -1.195 }} -->


[[Tuning ranges]]:
[[Tuning ranges]]:
Line 56: Line 52:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 271.5097{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 271.5097{{c}}
: error map: {{val| 0.000 -1.387 -0.843 +3.252 }}
: error map: {{val| 0.000 -1.387 -0.843 +3.252 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~7/6 = 271.513{{c}}
: [[error map]]: {{val| 0.000 -1.364 -0.853 +3.278 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~7/6 = 271.509{{c}}
: error map: {{val| 0.000 -1.394 -0.840 +3.243 }} -->


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 91: Line 83:
* WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}}
* WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/6 = 271.560{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/6 = 271.426{{c}} -->


Minimax tuning:
Minimax tuning:
Line 117: Line 107:
* WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}}
* WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/6 = 271.556{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/6 = 271.546{{c}} -->


Tuning ranges:
Tuning ranges:
Line 127: Line 115:


Badness (Sintel): 0.815
Badness (Sintel): 0.815
==== 2.3.5.7.11.13.19 ====
Subgroup: 2.3.5.7.11.13.19
Comma list: 99/98, 121/120, 176/175, 275/273, [[400/399]]
Mapping: {{mapping| 1 0 3 1 3 8 9| 0 7 -3 8 2 -19 -21}}
Optimal tunings:
* WE: ~2 = 1200.320{{c}}, ~7/6 = 271.640{{c}}
* CWE: ~2 = 1200.000{{c}}, ~7/6 = 271.566{{c}}
Badness (Sintel): 0.881


==== Blair ====
==== Blair ====
Line 138: Line 140:
* WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}}
* WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.3846{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.3846{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/6 = 271.747{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/6 = 271.301{{c}} -->


{{Optimal ET sequence|legend=0| 9, 22, 31f }}
{{Optimal ET sequence|legend=0| 9, 22, 31f }}
Line 155: Line 155:
* WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}}
* WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/6 = 271.163{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/6 = 271.088{{c}} -->


Tuning ranges:
Tuning ranges:
Line 167: Line 165:


==== 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
Line 178: Line 176:
* WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}}
* WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~13/12 = 135.811{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/12 = 135.723{{c}} -->


Tuning ranges:
Tuning ranges:
Line 190: Line 186:


=== 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 basic 11-limit Orwell at [[31edo]].
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 201: Line 197:
* WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}}
* WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/6 = 271.316{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/6 = 271.288{{c}} -->


Tuning ranges:
Tuning ranges:
Line 218: Line 212:


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


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


{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}
{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}
Line 231: Line 223:


== 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
Line 244: Line 236:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6110{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6110{{c}}
: error map: {{val| 0.000 -0.678 -1.147 -0.558 }}
: error map: {{val| 0.000 -0.678 -1.147 -0.558 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~75/64 = 271.622{{c}}
: [[error map]]: {{val| 0.000 -0.599 -1.180 +0.131 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~75/64 = 271.607{{c}}
: error map: {{val| 0.000 -0.707 -1.134 -0.808 }} -->


{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}
{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}
Line 254: Line 242:


== 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
Line 268: Line 256:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/224 = 890.5312{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/224 = 890.5312{{c}}
: error map: {{val| 0.000 -0.799 -1.095 -2.545 }}
: error map: {{val| 0.000 -0.799 -1.095 -2.545 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~375/224 = 890.544{{c}}
: [[error map]]: {{val| 0.000 -0.522 -1.213 -2.637 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~375/224 = 890.528{{c}}
: error map: {{val| 0.000 -0.872 -1.063 -2.520 }} -->


{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}
Line 287: Line 271:
* WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}}
* WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~375/224 = 890.5344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~375/224 = 890.5344{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~375/224 = 890.556{{c}}
* POTE: ~2 = 1200.000{{c}}, ~375/224 = 890.529{{c}} -->


{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}
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== 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
Line 309: Line 291:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 967.9090{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 967.9090{{c}}
: error map: {{val| 0.000 -0.503 -1.222 -0.917 }}
: error map: {{val| 0.000 -0.503 -1.222 -0.917 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~7/4 = 967.918{{c}}
: [[error map]]: {{val| 0.000 -0.255 -1.328 -0.908 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~7/4 = 967.906{{c}}
: error map: {{val| 0.000 -0.574 -1.191 -0.919 }} -->


{{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 }}
Line 328: Line 306:
* WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}}
* WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 967.9206{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 967.9206{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/4 = 967.935{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/4 = 967.917{{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 }}
Line 336: Line 312:


== 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 351: Line 327:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2401/1536 = 774.3282{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2401/1536 = 774.3282{{c}}
: error map: {{val| 0.000 -0.469 -1.236 +0.128 }}
: error map: {{val| 0.000 -0.469 -1.236 +0.128 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~2401/1536 = 774.334{{c}}
: [[error map]]: {{val| 0.000 -0.278 -1.318 0.177 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~2401/1536 = 774.327{{c}}
: error map: {{val| 0.000 -0.526 -1.212 0.113 }} -->


{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
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* WE: ~2 = 1200.1915{{c}}, ~2205/1408 = 774.4451{{c}}
* WE: ~2 = 1200.1915{{c}}, ~2205/1408 = 774.4451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~2205/1408 = 774.3233{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~2205/1408 = 774.3233{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~2205/1408 = 774.329{{c}}
* POTE: ~2 = 1200.000{{c}}, ~2205/1408 = 774.321{{c}} -->


{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}
{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}
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== 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 392: Line 362:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}}
: error map: {{val| 0.000 -0.501 -1.223 -0.536 }}
: error map: {{val| 0.000 -0.501 -1.223 -0.536 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~405/392 = 54.335{{c}}
: [[error map]]: {{val| 0.000 -0.233 -1.338 -0.061 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~405/392 = 54.324{{c}}
: error map: {{val| 0.000 -0.604 -1.178 -0.718 }} -->


{{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 }}
Line 411: Line 377:
* WE: ~2 = 1200.0642{{c}}, ~33/32 = 54.3395{{c}}
* WE: ~2 = 1200.0642{{c}}, ~33/32 = 54.3395{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 54.3369{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 54.3369{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~33/32 = 54.338{{c}}
* POTE: ~2 = 1200.000{{c}}, ~33/32 = 54.334{{c}} -->


{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}
{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}
Line 428: Line 392:
* WE: ~2 = 1200.0642{{c}}, ~405/392 = 54.3373{{c}}
* WE: ~2 = 1200.0642{{c}}, ~405/392 = 54.3373{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~405/392 = 54.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~405/392 = 54.3192{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~405/392 = 54.332{{c}}
* POTE: ~2 = 1200.000{{c}}, ~405/392 = 54.316{{c}} -->


{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}
{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}
Line 436: Line 398:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Orson]]
[[Category:Orson]]
[[Category:Orwell]]
[[Category:Orwell]]
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