Magic family

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The magic family of temperaments tempers out 3125/3072, the small diesis or magic comma. The septimal version of magic is locally optimal, for some searches, in the 9-odd-limit. Magic has a slightly higher complexity than meantone but it is closer to just intonation. It is the simplest rank-2 temperament that tunes every 9-odd-limit interval better than is possible in 12edo. The most prominent deficiency is that it lacks proper or nearly-proper mos scales in the 5- to 10-note region. Properties may depend on tuning and extension.

Magic

Main article: Magic

The generator of magic is a major third, and to get to the interval class of fifths requires five of these. In fact, (5/4)5 = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the optimal patent val generator, also makes a lot of sense, and using 19edo or 22edo is always possible.

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.499
error map: ⟨0.000 +0.542 -5.814]
  • POTE: ~2 = 1200.000, ~5/4 = 380.058
error map: ⟨0.000 -1.663 -6.255]

Minimax tuning:

eigenmonzo (unchanged-interval) basis: 2.3

Tuning ranges:

Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.

Optimal ET sequence: 3, 13b, 16, 19, 41, 60, 221cc, 281cc

Badness (Smith): 0.039163

Overview to extensions

Apart from magic, we also consider other extensions. The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives septimal magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator, as well as low-accuracy extensions including darkstone and brightstone.

Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are

Septimal magic

Main article: Magic

Septimal magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. 41edo is a good magic tuning, and 19- or 22-note mosses are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.

This temperament, with its accurate fifths, works well with 9-odd-limit harmony. It is more accurate than meantone and simpler than garibaldi. It is a little tricky to work with because its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.

225/224 is the marvel comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the sensamagic comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.

By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, ⟨⟨ 5 1 12 -8 … ]]. For this, 104edo provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1], ⟨0 5 1 12]]

mapping generators: ~2, ~5/4

Wedgie: ⟨⟨ 5 1 12 -10 5 25 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.651
error map: ⟨0.000 +1.301 -5.662 -1.011]
  • POTE: ~2 = 1200.000, ~5/4 = 380.352
error map: ⟨0.000 -0.195 -5.962 -4.602]

Minimax tuning:

eigenmonzo (unchanged-interval) basis: 2.3

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 7- and 9-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)

Algebraic generator: Tirzbirat or Septimage, the real root of 5x5 + 4x - 20, 380.7604 cents.

Optimal ET sequence: 19, 41, 142cd, 183cd, 224ccd

Badness (Smith): 0.018918

11-limit

Tempering out 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/243

Mapping: [⟨1 0 2 -1 6], ⟨0 5 1 12 -8]]

Wedgie: ⟨⟨ 5 1 12 -8 -10 5 -30 25 -22 -64 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.720
  • POTE: ~2 = 1200.000, ~5/4 = 380.696

Minimax tuning:

  • 11-odd-limit: ~5/4 = [1/3 1/9 0 0 -1/18⟩
eigenmonzo (unchanged-interval) basis: 2.11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)

Optimal ET sequence: 19, 22, 41, 104

Badness (Smith): 0.020352

13-limit

A notable patent val tuning beyond the optimal patent val of 41edo is 19 + 41 = 60edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.435
  • POTE: ~2 = 1200.000, ~5/4 = 380.427

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]

Optimal ET sequence: 19, 22f, 41

Badness (Smith): 0.021509

Magical

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153

Mapping: [⟨1 0 2 -1 6 -2 6], ⟨0 5 1 12 -8 18 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.510
  • POTE: ~2 = 1200.000, ~5/4 = 380.604

Optimal ET sequence: 19, 22f, 41

Badness (Smith): 0.020633

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153

Mapping: [⟨1 0 2 -1 6 -2 6 9], ⟨0 5 1 12 -8 18 -6 -15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.451

Optimal ET sequence: 19, 41

Badness (Smith): 0.020881

Magica

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169

Mapping: [⟨1 0 2 -1 6 -2 6 -4], ⟨0 5 1 12 -8 18 -6 26]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.573

Optimal ET sequence: 22fh, 41

Badness (Smith): 0.019945

Magia

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7], ⟨0 5 1 12 -8 18 35]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.271

Optimal ET sequence: 19g, 41, 60

Badness (Smith): 0.026232

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7 -4], ⟨0 5 1 12 -8 18 35 26]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.360

Optimal ET sequence: 19gh, 41

Badness (Smith): 0.023709

Evening

Evening is a remarkable subgroup temperament of 19 & 41 with prime harmonics of 29 and 31.

Subgroup: 2.3.5.7.11.13.29.31

Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195

Sval mapping: [⟨1 0 2 -1 6 -2 2 4], ⟨0 5 1 12 -8 18 9 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.478
  • POTE: ~2 = 1200.000, ~5/4 = 380.416

Optimal ET sequence: 19, 22f, 41

Sorcery

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 100/99

Mapping: [⟨1 0 2 -1 6 4], ⟨0 5 1 12 -8 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.674
  • POTE: ~2 = 1200.000, ~5/4 = 380.477

Optimal ET sequence: 19, 22, 41f

Badness (Smith): 0.025829

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 52/51, 65/64, 78/77, 91/90, 100/99

Mapping: [⟨1 0 2 -1 6 4 6], ⟨0 5 1 12 -8 -1 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.784
  • POTE: ~2 = 1200.000, ~5/4 = 380.729

Optimal ET sequence: 19, 22

Badness (Smith): 0.023768

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 52/51, 65/64, 78/77, 91/90, 100/99, 133/132

Mapping: [⟨1 0 2 -1 6 4 6 9], ⟨0 5 1 12 -8 -1 -6 -15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.625
  • CWE: ~2 = 1200.000, ~5/4 = 380.621

Optimal ET sequence: 19, 22

Badness (Smith): 0.023232

Necromancy

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 275/273

Mapping: [⟨1 0 2 -1 6 11], ⟨0 5 1 12 -8 -23]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.788
  • POTE: ~2 = 1200.000, ~5/4 = 380.787

Optimal ET sequence: 19f, 22, 41, 63, 104

Badness (Smith): 0.025275

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 154/153, 225/224, 273/272

Mapping: [⟨1 0 2 -1 6 11 6], ⟨0 5 1 12 -8 -23 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.837
  • POTE: ~2 = 1200.000, ~5/4 = 380.827

Optimal ET sequence: 19f, 22, 41, 63

Badness (Smith): 0.022032

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 133/132, 154/153, 209/208, 225/224

Mapping: [⟨1 0 2 -1 6 11 6 9], ⟨0 5 1 12 -8 -23 -6 -15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.741
  • CWE: ~2 = 1200.000, ~5/4 = 380.735

Optimal ET sequence: 19f, 22, 41

Badness (Smith): 0.021101

Soothsaying

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 1352/1331

Mapping: [⟨2 0 4 -2 12 15], ⟨0 5 1 12 -8 -12]]

Optimal tunings:

  • CTE: ~55/39 = 600.000, ~5/4 = 380.539
  • POTE: ~55/39 = 600.000, ~5/4 = 380.508

Optimal ET sequence: 22, 60, 82

Badness (Smith): 0.055443

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5], ⟨0 5 1 12 -8 -12 5]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~5/4 = 380.553
  • POTE: ~17/12 = 600.000, ~5/4 = 380.508

Optimal ET sequence: 22, 60, 82

Badness (Smith): 0.035654

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5 18], ⟨0 5 1 12 -8 -12 5 -15]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~5/4 = 380.470
  • CWE: ~17/12 = 600.000, ~5/4 = 380.470

Optimal ET sequence: 22, 60, 82

Badness (Smith): 0.031291

Telepathy

Subgroup: 2.3.5.7.11

Comma list: 55/54, 99/98, 176/175

Mapping: [⟨1 0 2 -1 -1], ⟨0 5 1 12 14]]

Wedgie: ⟨⟨ 5 1 12 14 -10 5 5 25 29 -2 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.231
  • POTE: ~2 = 1200.000, ~5/4 = 381.019

Optimal ET sequence: 19e, 22, 41e, 63e

Badness (Smith): 0.027109

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.196
  • POTE: ~2 = 1200.000, ~5/4 = 380.520

Optimal ET sequence: 19e, 22, 41ef

Badness (Smith): 0.025522

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 55/54, 65/64, 85/84, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4 -1], ⟨0 5 1 12 14 -1 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.288
  • POTE: ~2 = 1200.000, ~5/4 = 380.619

Optimal ET sequence: 19eg, 22, 41efg

Badness (Smith): 0.020201

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 55/54, 57/56, 65/64, 76/75, 85/84, 99/98

Mapping: [⟨1 0 2 -1 -1 4 -1 2], ⟨0 5 1 12 14 -1 16 7]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.464
  • CWE: ~2 = 1200.000, ~5/4 = 380.735

Optimal ET sequence: 19egh, 22, 41efghh

Badness (Smith): 0.019004

Intuition

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 99/98, 105/104

Mapping: [⟨1 0 2 -1 -1 -2], ⟨0 5 1 12 14 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.816
  • POTE: ~2 = 1200.000, ~5/4 = 380.483

Optimal ET sequence: 19e, 22f

Badness (Smith): 0.026089

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 55/54, 66/65, 85/84, 99/98, 105/104

Mapping: [⟨1 0 2 -1 -1 -2 -1], ⟨0 5 1 12 14 18 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.952
  • POTE: ~2 = 1200.000, ~5/4 = 380.604

Optimal ET sequence: 19eg, 22f

Badness (Smith): 0.020274

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 55/54, 66/65, 77/76, 85/84, 99/98, 105/104

Mapping: [⟨1 0 2 -1 -1 -2 -1 -4], ⟨0 5 1 12 14 18 16 26]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.866
  • CWE: ~2 = 1200.000, ~5/4 = 380.628

Optimal ET sequence: 19egh, 22fh

Badness (Smith): 0.019518

Horcrux

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 379.796
  • POTE: ~2 = 1200.000, ~5/4 = 379.642

Optimal ET sequence: 3de, 16d, 19

Badness (Smith): 0.039282

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 245/243

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 11 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 379.885
  • POTE: ~2 = 1200.000, ~5/4 = 379.791

Optimal ET sequence: 3def, 16dff, 19

Badness (Smith): 0.031938

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 85/84, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.216
  • CWE: ~2 = 1200.000, ~5/4 = 380.148

Optimal ET sequence: 3defg, 16dffgg, 19g

Badness (Smith): 0.028074

Horcruxic

Subgroup: 2.3.5.7.11.13.17

Comma list: 35/34, 45/44, 52/51, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 13]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 379.524
  • CWE: ~2 = 1200.000, ~5/4 = 379.571

Optimal ET sequence: 3defg, 16dff, 19

Badness (Smith): 0.029556

Glamour

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 245/243

Mapping: [⟨1 0 2 -1 0 4], ⟨0 5 1 12 11 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 379.757
  • POTE: ~2 = 1200.000, ~5/4 = 379.116

Optimal ET sequence: 3de, 16d, 19

Badness (Smith): 0.033317

Witchcraft

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 441/440

Mapping: [⟨1 0 2 -1 -7], ⟨0 5 1 12 33]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.416
  • POTE: ~2 = 1200.000, ~5/4 = 380.232

Optimal ET sequence: 19e, 41, 60e, 101cd, 243ccdde

Badness (Smith): 0.030706

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243, 275/273

Mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.352
  • POTE: ~2 = 1200.000, ~5/4 = 380.189

Optimal ET sequence: 19e, 41, 60e, 101cd

Badness (Smith): 0.023547

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 245/243

Mapping: [⟨1 0 2 -1 -7 -2 -7], ⟨0 5 1 12 33 18 35]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.281
  • POTE: ~2 = 1200.000, ~5/4 = 380.114

Optimal ET sequence: 19eg, 41, 60e, 101cd

Badness (Smith): 0.020756

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 105/104, 133/132, 154/153, 170/169, 171/169, 196/195

Mapping: [⟨1 0 2 -1 -7 -2 -7 -4], ⟨0 5 1 12 33 18 35 26]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 380.339
  • CWE: ~2 = 1200.000, ~5/4 = 380.205

Optimal ET sequence: 19egh, 41, 60eh

Badness (Smith): 0.018625

Divination

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 245/243

Mapping: [⟨2 0 4 -2 5], ⟨0 5 1 12 3]]

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~5/4 = 380.732
  • POTE: ~99/70 = 600.000, ~5/4 = 380.223

Optimal ET sequence: 22, 38d, 60e, 142cdee, 202ccddeee

Badness (Smith): 0.035864

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4], ⟨0 5 1 12 3 18]]

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~5/4 = 380.417
  • POTE: ~99/70 = 600.000, ~5/4 = 379.920

Optimal ET sequence: 22f, 38df, 60e

Badness (Smith): 0.034551

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4 5], ⟨0 5 1 12 3 18 5]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~5/4 = 380.433
  • CWE: ~17/12 = 600.000, ~5/4 = 380.067

Optimal ET sequence: 22f, 38df, 60e

Badness (Smith): 0.023775

Hocus

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 245/242

Mapping: [⟨1 5 3 11 12], ⟨0 -10 -2 -24 -25]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/11 = 409.760
  • POTE: ~2 = 1200.000, ~14/11 = 409.910

Optimal ET sequence: 38d, 41, 120cd

Badness (Smith): 0.038519

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 243/242, 245/242

Mapping: [⟨1 5 3 11 12 16], ⟨0 -10 -2 -24 -25 -36]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/11 = 409.842
  • POTE: ~2 = 1200.000, ~14/11 = 410.004

Optimal ET sequence: 38df, 41, 79d, 120cd

Badness (Smith): 0.030280

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 196/195, 243/242, 245/242

Mapping: [⟨1 5 3 11 12 16 14], ⟨0 -10 -2 -24 -25 -36 -29]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/11 = 409.896
  • CWE: ~2 = 1200.000, ~14/11 = 409.994

Optimal ET sequence: 38df, 41, 79d

Badness (Smith): 0.025491

19-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 196/195, 243/242, 245/242

Mapping: [⟨1 5 3 11 12 16 14 8], ⟨0 -10 -2 -24 -25 -36 -29 -11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/11 = 409.884
  • CWE: ~2 = 1200.000, ~14/11 = 410.012

Optimal ET sequence: 38df, 41, 79dh

Badness (Smith): 0.020277

Muggles

Main article: Muggles

Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is 19edo, in which tuning it is the same thing as magic. Muggles works better for small scales than magic in the sense that 7- or 10-note mosses are reasonable choices, as while the flatter generator compromises the accuracy of the 5-limit intervals, it grants simpler access to some higher-limit ones, and makes the small steps larger and more melodically effective.

Subgroup: 2.3.5.7

Comma list: 126/125, 525/512

Mapping: [⟨1 0 2 5], ⟨0 5 1 -7]]

Wedgie: ⟨⟨ 5 1 -7 -10 -25 -19 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 378.744
error map: ⟨0.000 -8.235 -7.570 -20.034]
  • POTE: ~2 = 1200.000, ~5/4 = 378.479
error map: ⟨0.000 -9.558 -7.834 -18.181]

Tuning ranges:

Optimal ET sequence: 16, 19, 73bcd, 92bcdd, 111bcddd

Badness (Smith): 0.056206

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 126/125, 385/384

Mapping: [⟨1 0 2 5 0], ⟨0 5 1 -7 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 378.228
  • POTE: ~2 = 1200.000, ~5/4 = 377.724

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
  • 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]

Optimal ET sequence: 16, 19, 35, 54bd

Badness (Smith): 0.048038

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 126/125

Mapping: [⟨1 0 2 5 0 4], ⟨0 5 1 -7 11 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 378.176
  • POTE: ~2 = 1200.000, ~5/4 = 377.653

Optimal ET sequence: 16, 19, 35f, 54bdf

Badness (Smith): 0.030386

Muggloid

Subgroup: 2.3.5.7.11

Comma list: 33/32, 126/125, 176/175

Mapping: [⟨1 0 2 5 5], ⟨0 5 1 -7 -5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 377.724
  • POTE: ~2 = 1200.000, ~5/4 = 377.832

Optimal ET sequence: 3, 16, 19e, 35ee, 54bdeee

Badness (Smith): 0.046970

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 65/64, 105/104, 126/125

Mapping: [⟨1 0 2 5 5 4], ⟨0 5 1 -7 -5 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 377.652
  • POTE: ~2 = 1200.000, ~5/4 = 377.838

Optimal ET sequence: 3, 16, 19e, 35eef

Badness (Smith): 0.028732

Darkstone

Darkstone (16 & 19d) is a low-accuacy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles. In Encyclopedia of Microtonal Music Theory, Tonalsoft, this temperament is given a name witch.

Subgroup: 2.3.5.7

Comma list: 36/35, 1875/1792

Mapping: [⟨1 0 2 0], ⟨0 5 1 9]]

Wedgie: ⟨⟨ 5 1 9 -10 0 18 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 377.385
error map: ⟨0.000 -15.028 -8.928 +27.643]
  • CWE: ~2 = 1200.000, ~5/4 = 376.963
error map: ⟨0.000 -18.198 -9.562 +21.937]

Optimal ET sequence: 3d, …, 13b, 16

Badness (Smith): 0.084213

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0], ⟨0 5 1 9 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 377.388
  • CWE: ~2 = 1200.000, ~5/4 = 376.973

Optimal ET sequence: 3de, 13be, 16

Badness (Smith): 0.046775

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0 -1], ⟨0 5 1 9 11 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 376.914
  • CWE: ~2 = 1200.000, ~5/4 = 376.422

Optimal ET sequence: 3def, 13beff, 16

Badness (Smith): 0.038328

Brightstone

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Wedgie: ⟨⟨ 5 1 -10 -10 -30 -26 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.955
error map: ⟨0.000 +7.818 -4.359 +11.627]
  • CWE: ~2 = 1200.000, ~5/4 = 381.956
error map: ⟨0.000 +7.826 -4.358 +11.613]

Optimal ET sequence: 3, 19d, 22

Badness (Smith): 0.088072

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 605/588

Mapping: [⟨1 0 2 6 6], ⟨0 5 1 -10 -8]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.790
  • CWE: ~2 = 1200.000, ~5/4 = 381.712

Optimal ET sequence: 3, 19d, 22

Badness (Smith): 0.047379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 65/63, 100/99, 169/165

Mapping: [⟨1 0 2 6 6 4], ⟨0 5 1 -10 -8 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 381.732
  • CWE: ~2 = 1200.000, ~5/4 = 381.736

Optimal ET sequence: 3, 22

Badness (Smith): 0.039703

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 5 3 -3], ⟨0 -10 -2 17]]

mapping generators: ~2, ~63/50

Wedgie: ⟨⟨ 10 2 -17 -20 -55 -45 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~63/50 = 409.836
error map: ⟨0.000 -0.316 -5.986 -1.612]
  • POTE: ~2 = 1200.000, ~63/50 = 410.108
error map: ⟨0.000 -0.437 -6.010 -1.406]

Optimal ET sequence: 3, 38, 41, 161c

Badness (Smith): 0.107115

Trismegistus

See also: No-threes subgroup temperaments § Mabilic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 10 4 0], ⟨0 -15 -3 5]]

mapping generators: ~2, ~147/100

Wedgie: ⟨⟨ 15 3 -5 -30 -50 -20 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~147/100 = 673.187
error map: ⟨0.000 +0.240 -5.875 -2.891]
  • POTE: ~2 = 1200.000, ~147/100 = 673.290
error map: ⟨0.000 -0.932 -6.109 -2.500]

Optimal ET sequence: 16, 25, 41, 139c, 180c, 221cc, 262ccd

Badness (Smith): 0.098334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 625/616

Mapping: [⟨1 10 4 0 13], ⟨0 -15 -3 5 -17]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~22/15 = 673.241
  • POTE: ~2 = 1200.000, ~22/15 = 673.340

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Smith): 0.045623

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273, 625/616

Mapping: [⟨1 10 4 0 13 11], ⟨0 -15 -3 5 -17 -13]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~22/15 = 673.294
  • POTE: ~2 = 1200.000, ~22/15 = 673.359

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Smith): 0.033081

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 5 3 4], ⟨0 -20 -4 -7]]

mapping generators: ~2, ~28/25

Wedgie: ⟨⟨ 20 4 7 -40 -45 5 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~28/25 = 204.860
error map: ⟨0.000 +0.853 -5.752 -2.843]
  • POTE: ~2 = 1200.000, ~28/25 = 204.987
error map: ⟨0.000 -1.691 -6.261 -3.733]

Optimal ET sequence: 6, 29b, 35, 41, 158cd, 199ccd, 240ccd, 281ccd

Badness (Smith): 0.127422

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 625/616

Mapping: [⟨1 5 3 4 5], ⟨0 -20 -4 -7 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~28/25 = 204.881
  • POTE: ~2 = 1200.000, ~28/25 = 204.956

Optimal ET sequence: 6, 35, 41

Badness (Smith): 0.061572

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/242, 625/616

Mapping: [⟨1 5 3 4 5 9], ⟨0 -20 -4 -7 -9 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~28/25 = 204.956
  • POTE: ~2 = 1200.000, ~28/25 = 205.028

Optimal ET sequence: 6f, 35f, 41, 117c

Badness (Smith): 0.044047

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

mapping generators: ~2, ~48/35

Wedgie: ⟨⟨ 25 5 19 -50 -40 30 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~48/35 = 556.123
error map: ⟨0.000 +1.132 -5.696 -2.480]
  • CWE: ~2 = 1200.000, ~48/35 = 556.050
error map: ⟨0.000 -0.695 -6.062 -3.868]

Optimal ET sequence: 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd

Badness (Smith): 0.194548

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 625/616, 2401/2376

Mapping: [⟨1 -10 0 -6 3], ⟨0 25 5 19 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 556.122
  • CWE: ~2 = 1200.000, ~11/8 = 556.095

Optimal ET sequence: 13b, 28b, 41

Badness (Smith): 0.101724

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/616

Mapping: [⟨1 -10 0 -6 3 0], ⟨0 25 5 19 1 8]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 556.106
  • CWE: ~2 = 1200.000, ~11/8 = 556.117

Optimal ET sequence: 13b, 28b, 41

Badness (Smith): 0.067742

Warlock

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16807/16384

Mapping: [⟨5 0 10 14], ⟨0 5 1 0]]

mapping generators: ~8/7, ~5/4

Optimal tunings:

  • CTE: ~8/7 = 240.000, ~5/4 = 380.499 (~256/245 = 99.501)
error map: ⟨0.000 +0.542 -5.814 -8.826]
  • CWE: ~8/7 = 240.000, ~5/4 = 379.997 (~256/245 = 100.003)
error map: ⟨0.000 -1.972 -6.317 -8.826]

Optimal ET sequence: 25, 35, 60

Badness (Smith): 0.287190

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