Semicomma family: Difference between revisions
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.2902{{c}}, ~75/64 = 271.6929{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.290 -0.104 -0.522 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}} | ||
: error map: {{val| 0.000 -0. | : error map: {{val| 0.000 -0.479 -1.232 }} | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 22: | Line 22: | ||
* 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]] ( | [[Badness]] (Sintel): 0.957 | ||
=== Overview to extensions === | === Overview to extensions === | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.0192{{c}}, ~7/6 = 271.5130{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.019 -1.364 -0.795 +3.297 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 271.5097{{c}} | ||
: error map: {{val| 0.000 -1. | : error map: {{val| 0.000 -1.387 -0.843 +3.252 }} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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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 | {{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84, 137, 221d }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 0.525 | ||
=== 11-limit === | === 11-limit === | ||
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Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 95: | Line 95: | ||
{{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }} | {{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }} | ||
Badness ( | Badness (Sintel): 0.504 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 105: | Line 105: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}} | ||
Tuning ranges: | Tuning ranges: | ||
| Line 114: | Line 114: | ||
{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }} | {{Optimal ET sequence|legend=0| 22, 31, 53, 84e }} | ||
Badness ( | Badness (Sintel): 0.815 | ||
==== Blair ==== | ==== Blair ==== | ||
| Line 124: | Line 124: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}} | ||
* | * 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 ( | Badness (Sintel): 0.954 | ||
==== Winston ==== | ==== Winston ==== | ||
| Line 139: | Line 139: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}} | ||
Tuning ranges: | Tuning ranges: | ||
| Line 148: | Line 148: | ||
{{Optimal ET sequence|legend=0| 9, 22f, 31 }} | {{Optimal ET sequence|legend=0| 9, 22f, 31 }} | ||
Badness ( | 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- | 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 160: | Line 160: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}} | ||
Tuning ranges: | Tuning ranges: | ||
| Line 167: | Line 167: | ||
* 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 | {{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 115ef }} | ||
Badness ( | Badness (Sintel): 1.12 | ||
=== Newspeak === | === Newspeak === | ||
In | 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 | ||
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Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}} | ||
Tuning ranges: | Tuning ranges: | ||
| Line 190: | Line 190: | ||
{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }} | {{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }} | ||
Badness ( | 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 | Mapping: {{mapping| 1 -7 6 -7 -18 | 0 14 -6 16 35 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~55/36 | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0194{{c}}, ~55/36 = 735.7641{{c}} | ||
* | * 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 ( | Badness (Sintel): 1.27 | ||
== Sabric == | == Sabric == | ||
The sabric temperament | 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: | ||
* [[ | * [[WE]]: ~2 = 1200.3056{{c}}, ~75/64 = 271.6760{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.306 -0.223 -0.425 +0.049 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6110{{c}} | ||
: error map: {{val| 0.000 -0. | : 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]] ( | [[Badness]] (Sintel): 2.24 | ||
== Triwell == | == Triwell == | ||
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 234: | Line 234: | ||
[[Comma list]]: 1029/1024, 235298/234375 | [[Comma list]]: 1029/1024, 235298/234375 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -14 9 8 | 0 21 -9 -7 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~375/224 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.4763{{c}}, ~375/224 = 890.8812{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.476 -0.118 +0.042 -1.184 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~375/224 = 890.5312{{c}} | ||
: error map: {{val| 0.000 -0. | : 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]] ( | [[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 | Mapping: {{mapping| 1 -14 9 8 -24 | 0 21 -9 -7 37 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}} | ||
* | * 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 ( | Badness (Sintel): 0.985 | ||
== Quadrawell == | == Quadrawell == | ||
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 | {{Mapping|legend=1| 1 -21 12 2 | 0 28 -12 1 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~7/4 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.3006{{c}}, ~7/4 = 968.1489{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.301 -0.098 -0.493 -0.076 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 967.9090{{c}} | ||
: error map: {{val| 0.000 -0. | : 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]] ( | [[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 | Mapping: {{mapping| 1 -21 12 2 -28 | 0 28 -12 1 39 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}} | ||
* | * 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 ( | Badness (Sintel): 1.21 | ||
== Rainwell == | == Rainwell == | ||
The | 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 304: | Line 304: | ||
[[Comma list]]: 16875/16807, 2100875/2097152 | [[Comma list]]: 16875/16807, 2100875/2097152 | ||
{{Mapping|legend=1| 1 | {{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: | ||
* [[ | * [[WE]]: ~2 = 1200.2032{{c}}, ~2401/1536 = 774.4577{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.203 -0.204 -0.740 +0.683 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~2401/1536 = 774.3282{{c}} | ||
: error map: {{val| 0.000 -0. | : 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]] ( | [[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 | Mapping: {{mapping| 1 -21 12 -3 -43 | 0 35 -15 9 72 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.1915{{c}}, ~2205/1408 = 774.4451{{c}} | ||
* | * 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 ( | Badness (Sintel): 1.74 | ||
== Quinwell == | == Quinwell == | ||
The quinwell temperament | 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 | ||
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{{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: | ||
* [[ | * [[WE]]: ~2 = 1200.2860{{c}}, ~405/392 = 54.3373{{c}} | ||
: [[error map]]: {{val| 0. | : [[error map]]: {{val| +0.286 -0.151 -0.515 +0.084 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}} | ||
: error map: {{val| 0.000 -0. | : 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]] ( | [[Badness]] (Sintel): 4.27 | ||
=== 11-limit === | === 11-limit === | ||
| Line 362: | Line 361: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0642{{c}}, ~33/32 = 54.3395{{c}} | ||
* | * 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 ( | Badness (Sintel): 3.21 | ||
=== Quinbetter === | === Quinbetter === | ||
| Line 377: | Line 376: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0642{{c}}, ~405/392 = 54.3373{{c}} | ||
* | * 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 ( | Badness (Sintel): 2.60 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Semicomma family| ]] <!-- main article --> | [[Category:Semicomma family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Orson]] | [[Category:Orson]] | ||
[[Category:Orwell]] | [[Category:Orwell]] | ||