Semicomma family

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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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