Semicomma family: Difference between revisions

* [[WE]]: ~2 = 1200.2902{{c}}, ~75/64 = 271.6929{{c}}

: [[error map]]: {{val| +0.290 -0.104 -0.522 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}}

: error map: {{val| 0.000 -0.479 -1.232 }}

* [[5-odd-limit]] [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)

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

[[Badness]] (Sintel): 0.957

* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or

* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or

* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).

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).

* [[WE]]: ~2 = 1200.0192{{c}}, ~7/6 = 271.5130{{c}}

: [[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 -1.387 -0.843 +3.252 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

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

[[Badness]] (Sintel): 0.525

* WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}}

: Unchanged-interval (eigenmonzo) basis: 2.7/5

Badness (Sintel): 0.504

* WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}}

Badness (Sintel): 0.815

* WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.3846{{c}}

Badness (Sintel): 0.954

* WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}}

Badness (Sintel): 0.824

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.  

* WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}}

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

Badness (Sintel): 1.12

* WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}}

Badness (Sintel): 1.04

Mapping: {{mapping| 1 -7 6 -7 -18 | 0 14 -6 16 35 }}

: mapping generators: ~2, ~55/36

* WE: ~2 = 1200.0194{{c}}, ~55/36 = 735.7641{{c}}

* CWE: ~2 = 1200.000{{c}}, ~55/36 = 735.7527{{c}}

Badness (Sintel): 1.27

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).

* [[WE]]: ~2 = 1200.3056{{c}}, ~75/64 = 271.6760{{c}}

: [[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 -0.678 -1.147 -0.558 }}

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

[[Badness]] (Sintel): 2.24

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.  

{{Mapping|legend=1| 1 -14 9 8 | 0 21 -9 -7 }}

* [[WE]]: ~2 = 1200.4763{{c}}, ~375/224 = 890.8812{{c}}

: [[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 -0.799 -1.095 -2.545 }}

[[Badness]] (Sintel): 2.04

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

Optimal tunings:

* WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~375/224 = 890.5344{{c}}

Badness (Sintel): 0.985

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.  

{{Mapping|legend=1| 1 -21 12 2 | 0 28 -12 1 }}

* [[WE]]: ~2 = 1200.3006{{c}}, ~7/4 = 968.1489{{c}}

: [[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 -0.503 -1.222 -0.917 }}

[[Badness]] (Sintel): 1.92

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

Optimal tunings:

* WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 967.9206{{c}}

Badness (Sintel): 1.21

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.  

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

[[Optimal tuning]]s:

: [[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 -0.469 -1.236 +0.128 }}

[[Badness]] (Sintel): 3.63

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

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~2205/1408 = 774.3233{{c}}

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

Badness (Sintel): 1.74

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.  

* [[WE]]: ~2 = 1200.2860{{c}}, ~405/392 = 54.3373{{c}}

: [[error map]]: {{val| +0.286 -0.151 -0.515 +0.084 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}}

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

[[Badness]] (Sintel): 4.27

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 54.3369{{c}}

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

Badness (Sintel): 3.21

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~405/392 = 54.3192{{c}}

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

Badness (Sintel): 2.60