Semicomma family: Difference between revisions

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 1 3 | 0 7 -3 8 2 }}

Optimal tunings:

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}

Optimal tunings:

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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]]

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 ====

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}

Optimal tunings:

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}

Optimal tunings:

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==== 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

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}

Optimal tunings:

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=== 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

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 1 -4 | 0 7 -3 8 33 }}

Optimal tunings:

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

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

: mapping generators: ~2, ~72/55

: mapping generators: ~2, ~55/36

Optimal tunings:

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== 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

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== 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

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

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

Optimal tunings:

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== 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

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

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

Optimal tunings:

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== 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

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

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

Optimal tunings:

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== 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

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 0 5 | 0 35 -15 62 -34 }}

Optimal tunings:

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

~~Mapping:~~ {{~~mapping~~| 1 0 3 0 4 | 0 35 -15 62 -12 }}

Optimal tunings:

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[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Semicomma family| ]]

[[Category:Rank 2]]

[[Category:Orson]]

[[Category:Orwell]]

@@ Line 78: / Line 78: @@
 Comma list: 99/98, 121/120, 176/175
-Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}
+{{Mapping|legend=0| 1 0 3 1 3 | 0 7 -3 8 2 }}
 Optimal tunings:
@@ Line 102: / Line 102: @@
 Comma list: 99/98, 121/120, 176/175, 275/273
-Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}
+{{Mapping|legend=0| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}
 Optimal tunings:
@@ Line 115: / Line 115: @@
 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|legend=0| 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 ====
@@ Line 121: / Line 135: @@
 Comma list: 65/64, 78/77, 91/90, 99/98
-Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}
+{{Mapping|legend=0| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}
 Optimal tunings:
@@ Line 136: / Line 150: @@
 Comma list: 66/65, 99/98, 105/104, 121/120
-Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}
+{{Mapping|legend=0| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}
 Optimal tunings:
@@ Line 151: / Line 165: @@
 ==== 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
@@ Line 157: / Line 171: @@
 Comma list: 99/98, 121/120, 169/168, 176/175
-Mapping: {{mapping| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}
+{{Mapping|legend=0| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}
 Optimal tunings:
@@ Line 172: / Line 186: @@
 === 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
@@ Line 178: / Line 192: @@
 Comma list: 225/224, 441/440, 1728/1715
-Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}
+{{Mapping|legend=0| 1 0 3 1 -4 | 0 7 -3 8 33 }}
 Optimal tunings:
@@ Line 197: / Line 211: @@
 Comma list: 225/224, 243/242, 1728/1715
-Mapping: {{mapping| 1 -7 6 -7 -18 | 0 14 -6 16 35 }}
+{{Mapping|legend=0| 1 -7 6 -7 -18 | 0 14 -6 16 35 }}
-: mapping generators: ~2, ~72/55
+: mapping generators: ~2, ~55/36
 Optimal tunings:
@@ Line 209: / Line 223: @@
 == 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
@@ Line 228: / Line 242: @@
 == 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
@@ Line 252: / Line 266: @@
 Comma list: 385/384, 441/440, 456533/455625
-Mapping: {{mapping| 1 -14 9 8 -24 | 0 21 -9 -7 37 }}
+{{Mapping|legend=0| 1 -14 9 8 -24 | 0 21 -9 -7 37 }}
 Optimal tunings:
@@ Line 263: / Line 277: @@
 == 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
@@ Line 287: / Line 301: @@
 Comma list: 385/384, 1375/1372, 14641/14580
-Mapping: {{mapping| 1 -21 12 2 -28 | 0 28 -12 1 39 }}
+{{Mapping|legend=0| 1 -21 12 2 -28 | 0 28 -12 1 39 }}
 Optimal tunings:
@@ Line 298: / Line 312: @@
 == 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
@@ Line 323: / Line 337: @@
 Comma list: 540/539, 1375/1372, 2100875/2097152
-Mapping: {{mapping| 1 -21 12 -3 -43 | 0 35 -15 9 72 }}
+{{Mapping|legend=0| 1 -21 12 -3 -43 | 0 35 -15 9 72 }}
 Optimal tunings:
@@ Line 334: / Line 348: @@
 == 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
@@ Line 358: / Line 372: @@
 Comma list: 540/539, 4375/4356, 2109375/2097152
-Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}
+{{Mapping|legend=0| 1 0 3 0 5 | 0 35 -15 62 -34 }}
 Optimal tunings:
@@ Line 373: / Line 387: @@
 Comma list: 385/384, 24057/24010, 43923/43750
-Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}
+{{Mapping|legend=0| 1 0 3 0 4 | 0 35 -15 62 -12 }}
 Optimal tunings:
@@ Line 384: / Line 398: @@
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Semicomma family| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Orson]]
 [[Category:Orwell]]