Semicomma family

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The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = [-21 3 7⟩. 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, is the 5-limit temperament tempering out the semicomma. It has a generator of 75/64, which 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 tuning (POTE): ~2 = 1\1, ~75/64 = 271.627

Tuning ranges:

  • 5-odd-limit diamond monotone: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
  • 5-odd-limit diamond tradeoff: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)
  • 5-odd-limit diamond monotone and tradeoff: ~75/64 = [271.229, 271.708]

Optimal ET sequence: 22, 31, 53, 190, 243, 296, 645c

Badness: 0.040807

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) with wedgie ⟨⟨ 21 -9 -7 -63 -70 9 ]], or
  • 2401/2400, giving the 31&243 temperament (quadrawell) with wedgie ⟨⟨ 28 -12 1 -84 -77 36 ]], or
  • 4375/4374, giving the 53&243 temperament (sabric) with wedgie ⟨⟨ 7 -3 61 -21 77 150 ]].

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 it 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 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]]

Wedgie: ⟨⟨ 7 -3 8 -21 -7 27 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.509

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⟩]
eigenmonzo (unchanged-interval) basis: 2.7/5
  • 9-odd-limit: ~7/6 = [3/17 2/17 -1/17⟩
[[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⟩]
eigenmonzo (unchanged-interval) 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]
  • 7-odd-limit diamond monotone and tradeoff: ~7/6 = [266.871, 271.708]
  • 9-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.514]

Algebraic generator: Sabra3, the real root of 12x3 - 7x - 48.

Optimal ET sequence: 9, 22, 31, 53, 84, 137, 221d, 358d

Badness: 0.020735

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]]

Wedgie: ⟨⟨ 7 -3 8 2 -21 -7 -21 27 15 -22 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.426

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⟩]
Eigenmonzo (unchanged-interval) 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]
  • 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]

Optimal ET sequence: 9, 22, 31, 53, 84e

Badness: 0.015231

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 tuning (POTE): ~2 = 1\1, ~7/6 = 271.546

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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]

Optimal ET sequence: 22, 31, 53, 84e, 137e

Badness: 0.019718

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 tuning (POTE): ~2 = 1\1, ~7/6 = 271.301

Optimal ET sequence: 9, 22, 31f

Badness: 0.023086

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 tuning (POTE): ~2 = 1\1, ~7/6 = 271.088

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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]

Optimal ET sequence: 22f, 31

Badness: 0.019931

Doublethink

Doublethink is a weak extension of orwell to the 13-limit because it splits the generator of ~7/6 into two ~13/12~14/13's (by tempering their difference, 169/168).

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 tuning (POTE): ~2 = 1\1, ~13/12 = 135.723

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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~13/12 = [135.484, 136.364]

Optimal ET sequence: 9, 35bd, 44, 53, 62, 115ef, 168eef

Badness: 0.027120

Newspeak

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 tuning (POTE): ~2 = 1\1, ~7/6 = 271.288

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]
  • 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]

Optimal ET sequence: 31, 84, 115, 376b, 491bd, 606bde

Badness: 0.031438

Borwell

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 7 0 9 17], ⟨0 -14 6 -16 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.752

Optimal ET sequence: 31, 106, 137, 442bd

Badness: 0.038377

Sabric

The sabric temperament (53 & 190) tempers out the ragisma (4375/4374). It is so named because it is closely related 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]]

Wedgie: ⟨⟨ 7 -3 61 -21 77 150 ]]

Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 271.607

Optimal ET sequence: 53, 137d, 190, 243

Badness: 0.088355

Triwell

The triwell temperament (31 & 159) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 235298/234375

Mapping: [⟨1 7 0 1], ⟨0 -21 9 7]]

Wedgie: ⟨⟨ 21 -9 -7 -63 -70 9 ]]

Optimal tuning (POTE): ~2 = 1\1, ~448/375 = 309.472

Optimal ET sequence: 31, 97, 128, 159, 190

Badness: 0.080575

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 7 0 1 13], ⟨0 -21 9 7 -37]]

Optimal tuning (POTE): ~2 = 1\1, ~448/375 = 309.471

Optimal ET sequence: 31, 97, 128, 159, 190

Badness: 0.029807

Quadrawell

The quadrawell temperament (31&212) has an 8/7 generator of about 232 cents, twelve of which give the fifth harmonic.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 2109375/2097152

Mapping: [⟨1 7 0 3], ⟨0 -28 12 -1]]

Wedgie: ⟨⟨ 28 -12 1 -84 -77 36 ]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.094

Optimal ET sequence: 31, 119, 150, 181, 212, 243, 698cd, 941cd

Badness: 0.075754

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 7 0 3 11], ⟨0 -28 12 -1 -39]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.083

Optimal ET sequence: 31, 119, 150, 181, 212, 455ee, 667cdee

Badness: 0.036493

Rainwell

The rainwell temperament (31 & 265) tempers out the mirkwai comma, 16875/16807 and the rainy comma, 2100875/2097152.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 2100875/2097152

Mapping: [⟨1 14 -3 6], ⟨0 -35 15 -9]]

Wedgie: ⟨⟨ 35 -15 9 -105 -84 63 ]]

Optimal tuning (POTE): ~2 = 1\1, ~2625/2048 = 425.673

Optimal ET sequence: 31, 172, 203, 234, 265, 296

Badness: 0.143488

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 14 -3 6 29], ⟨0 -35 15 -9 -72]]

Optimal tuning (POTE): ~2 = 1\1, ~2625/2048 = 425.679

Optimal ET sequence: 31, 172e, 203e, 234, 265, 296, 919bc, 1215bcc, 1511bcc

Badness: 0.052774

Quinwell

The quinwell temperament (22 & 243) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 2109375/2097152

Mapping: [⟨1 0 3 0], ⟨0 35 -15 62]]

Wedgie: ⟨⟨ 35 -15 62 -105 0 186 ]]

Optimal tuning (POTE): ~2 = 1\1, ~405/392 = 54.324

Optimal ET sequence: 22, 221, 243, 751c, 994cd, 1237bccd, 1480bccd

Badness: 0.168897

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 tuning (POTE): ~2 = 1\1, ~33/32 = 54.334

Optimal ET sequence: 22, 221, 243, 265, 773ce, 1038ccee, 1303ccee

Badness: 0.097202

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 tuning (POTE): ~2 = 1\1, ~405/392 = 54.316

Optimal ET sequence: 22, 199d, 221e, 243e

Badness: 0.078657

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