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 optimal, for some searches, in the 9-odd-limit. It has slightly higher complexity than meantone and is also 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 monzo of the magic comma is [-10 -1 5⟩, and flipping that yields ⟨⟨ 5 1 -10 ]] for the wedgie. This tells us the generator is a major third, and that to get to the interval class of fifths will require 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 = 1\1, ~5/4 = 380.4994
  • POTE: ~2 = 1\1, ~5/4 = 380.058

Minimax tuning:

eigenmonzo (unchanged-interval) basis: 2.3

Tuning ranges:

  • 5-odd-limit diamond monotone: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
  • 5-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
  • 5-odd-limit diamond monotone and tradeoff: ~5/4 = [378.910, 386.314]

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, 22, 41

Badness: 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 = 1\1, ~5/4 = 380.6512
  • POTE: ~2 = 1\1, ~5/4 = 380.352

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)
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

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

Optimal ET sequence: 19, 22, 41, 104, 145c, 186c

Badness: 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 = 1\1, ~5/4 = 380.7200
  • POTE: ~2 = 1\1, ~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)
  • 11-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

Optimal ET sequence: 19, 22, 41, 104, 145c, 249cce

Badness: 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 = 1\1, ~5/4 = 380.4354
  • POTE: ~2 = 1\1, ~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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

Optimal ET sequence: 19, 22f, 41

Badness: 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 = 1\1, ~5/4 = 380.5103
  • POTE: ~2 = 1\1, ~5/4 = 380.604

Optimal ET sequence: 19, 22f, 41

Badness: 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 = 1\1, ~5/4 = 380.4507

Optimal ET sequence: 19, 22f, 41

Badness: 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 = 1\1, ~5/4 = 380.5733

Optimal ET sequence: 19h, 22fh, 41

Badness: 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 = 1\1, ~5/4 = 380.2706

Optimal ET sequence: 19g, 41, 101cde, 142cdefg

Badness: 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 = 1\1, ~5/4 = 380.3604

Optimal ET sequence: 19g, 41

Badness: 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:

  • POTE: ~2 = 1\1, ~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 = 1\1, ~5/4 = 380.6741
  • POTE: ~2 = 1\1, ~5/4 = 380.477

Optimal ET sequence: 19, 22, 41f

Badness: 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 = 1\1, ~5/4 = 380.7839
  • POTE: ~2 = 1\1, ~5/4 = 380.729

Optimal ET sequence: 19, 22, 41f

Badness: 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 = 1\1, ~5/4 = 380.6248

Optimal ET sequence: 19, 22, 41f

Badness: 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 = 1\1, ~5/4 = 380.7876
  • POTE: ~2 = 1\1, ~5/4 = 380.787

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

Badness: 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 = 1\1, ~5/4 = 380.8373
  • POTE: ~2 = 1\1, ~5/4 = 380.827

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

Badness: 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 = 1\1, ~5/4 = 380.7411

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

Badness: 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 = 1\2, ~5/4 = 380.5385
  • POTE: ~55/39 = 1\2, ~5/4 = 380.508

Optimal ET sequence: 22, 60, 82

Badness: 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 = 1\2, ~5/4 = 380.5533
  • POTE: ~17/12 = 1\2, ~5/4 = 380.508

Optimal ET sequence: 22, 60, 82

Badness: 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 = 1\2, ~5/4 = 380.4704

Optimal ET sequence: 22, 60, 82

Badness: 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 = 1\1, ~5/4 = 381.2311
  • POTE: ~2 = 1\1, ~5/4 = 381.019

Optimal ET sequence: 19e, 22, 63e, 85ee

Badness: 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 = 1\1, ~5/4 = 381.1957
  • POTE: ~2 = 1\1, ~5/4 = 380.520

Optimal ET sequence: 19e, 22, 63eff

Badness: 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 = 1\1, ~5/4 = 381.2884
  • POTE: ~2 = 1\1, ~5/4 = 380.619

Optimal ET sequence: 19eg, 22, 63effg

Badness: 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 = 1\1, ~5/4 = 381.4641

Optimal ET sequence: 19egh, 22

Badness: 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 = 1\1, ~5/4 = 380.8157
  • POTE: ~2 = 1\1, ~5/4 = 380.483

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

Badness: 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 = 1\1, ~5/4 = 380.9518
  • POTE: ~2 = 1\1, ~5/4 = 380.604

Optimal ET sequence: 19eg, 22f, 41eg, 63efg

Badness: 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 = 1\1, ~5/4 = 380.8658

Optimal ET sequence: 19egh, 22fh, 41eg, 63efg

Badness: 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 = 1\1, ~5/4 = 379.7959
  • POTE: ~2 = 1\1, ~5/4 = 379.642

Optimal ET sequence: 3de, …, 16d, 19, 41ee, 60ee

Badness: 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 = 1\1, ~5/4 = 379.8847
  • POTE: ~2 = 1\1, ~5/4 = 379.791

Optimal ET sequence: 3def, …, 19, 41ee, 60ee

Badness: 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 = 1\1, ~5/4 = 380.2159

Optimal ET sequence: 3defg, …, 19g, 41eeg

Badness: 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 = 1\1, ~5/4 = 379.5237

Optimal ET sequence: 3defg, …, 19

Badness: 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 = 1\1, ~5/4 = 379.7574
  • POTE: ~2 = 1\1, ~5/4 = 379.116

Optimal ET sequence: 3de, …, 16d, 19, 41eef, 60eeff

Badness: 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 = 1\1, ~5/4 = 380.4163
  • POTE: ~2 = 1\1, ~5/4 = 380.232

Optimal ET sequence: 19e, 22e, 41

Badness: 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 = 1\1, ~5/4 = 380.3523
  • POTE: ~2 = 1\1, ~5/4 = 380.189

Optimal ET sequence: 19e, 41, 142cdf

Badness: 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 = 1\1, ~5/4 = 380.2809
  • POTE: ~2 = 1\1, ~5/4 = 380.114

Optimal ET sequence: 19eg, 41, 101cd, 142cdfg

Badness: 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 = 1\1, ~5/4 = 380.3389

Optimal ET sequence: 19egh, 41, 101cdhh, 142cdfghh

Badness: 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 = 1\2, ~5/4 = 380.7317
  • POTE: ~99/70 = 1\2, ~5/4 = 380.223

Optimal ET sequence: 22, 60e, 82e, 104e

Badness: 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 = 1\2, ~5/4 = 380.4171
  • POTE: ~99/70 = 1\2, ~5/4 = 379.920

Optimal ET sequence: 22f, 60e, 82ef

Badness: 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 = 1\2, ~5/4 = 380.4329

Optimal ET sequence: 22f, 60e, 82ef

Badness: 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 = 1\1, ~14/11 = 409.7603
  • POTE: ~2 = 1\1, ~14/11 = 409.910

Optimal ET sequence: 38d, 41

Badness: 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 = 1\1, ~14/11 = 409.8421
  • POTE: ~2 = 1\1, ~14/11 = 410.004

Optimal ET sequence: 38df, 41

Badness: 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 = 1\1, ~14/11 = 409.8958

Optimal ET sequence: 38df, 41, 120cdg, 161cdg, 202ccddfgg

Badness: 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 = 1\1, ~14/11 = 409.8836

Optimal ET sequence: 38df, 41, 120cdgh, 161cdgh, 202ccddfgghh

Badness: 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 = 1\1, ~5/4 = 378.744
  • POTE: ~2 = 1\1, ~5/4 = 378.479

Tuning ranges:

  • 7-odd-limit diamond monotone: ~5/4 = [375.000, 378.947] (5\16 to 6\19)
  • 9-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
  • 7- and 9-odd-limit diamond tradeoff: ~5/4 = [375.882, 386.314]
  • 7-odd-limit diamond monotone and tradeoff: ~5/4 = [375.882, 378.947]
  • 9-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947

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

Badness: 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 = 1\1, ~5/4 = 378.228
  • POTE: ~2 = 1\1, ~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]
  • 11-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947

Optimal ET sequence: 16, 19, 54bd

Badness: 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 = 1\1, ~5/4 = 378.1761
  • POTE: ~2 = 1\1, ~5/4 = 377.653

Optimal ET sequence: 16, 19, 54bdf

Badness: 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 = 1\1, ~5/4 = 377.724
  • POTE: ~2 = 1\1, ~5/4 = 377.832

Optimal ET sequence: 3, 10bd, 13bd, 16, 19e, 35ee

Badness: 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 = 1\1, ~5/4 = 377.652
  • POTE: ~2 = 1\1, ~5/4 = 377.838

Optimal ET sequence: 3, 10bd, 13bd, 16, 19e, 35eef

Badness: 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 = 1\1, ~5/4 = 377.385

Optimal ET sequence: 3d, …, 13b, 16, 19d, 35d

Badness: 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 = 1\1, ~5/4 = 377.388

Optimal ET sequence: 3de, …, 13be, 16, 19d, 35d

Badness: 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 = 1\1, ~5/4 = 376.914

Optimal ET sequence: 3def, …, 16, 35d

Badness: 0.038328

Brightstone

See also: Archytas clan

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 = 1\1, ~5/4 = 381.955

Optimal ET sequence: 3, …, 19d, 22

Badness: 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 = 1\1, ~5/4 = 381.790

Optimal ET sequence: 3, …, 19d, 22

Badness: 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 = 1\1, ~5/4 = 381.732

Optimal ET sequence: 3, …, 19d, 22

Badness: 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 = 1\1, ~63/50 = 409.836
  • POTE: ~2 = 1\1, ~63/50 = 410.108

Optimal ET sequence: 3, …, 35d, 38, 41, 202cc, 243cc, 284cc

Badness: 0.107115

Trismegistus

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 = 1\1, ~147/100 = 673.187
  • POTE: ~2 = 1\1, ~147/100 = 673.290

Optimal ET sequence: 16, 25, 41

Badness: 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 = 1\1, ~22/15 = 673.241
  • POTE: ~2 = 1\1, ~22/15 = 673.340

Optimal ET sequence: 16, 25e, 41, 221cc, 262ccde

Badness: 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 = 1\1, ~22/15 = 673.294
  • POTE: ~2 = 1\1, ~22/15 = 673.359

Optimal ET sequence: 16, 25e, 41, 139cf, 180cf, 221ccf

Badness: 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 = 1\1, ~28/25 = 204.860
  • POTE: ~2 = 1\1, ~28/25 = 204.987

Optimal ET sequence: 6, …, 29b, 35, 41

Badness: 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 = 1\1, ~28/25 = 204.881
  • POTE: ~2 = 1\1, ~28/25 = 204.956

Optimal ET sequence: 6, …, 29b, 35, 41

Badness: 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 = 1\1, ~28/25 = 204.956
  • POTE: ~2 = 1\1, ~28/25 = 205.028

Optimal ET sequence: 35f, 41, 199ccdef, 240ccdef, 281ccdeff

Badness: 0.044047

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

Wedgie: ⟨⟨ 25 5 19 -50 -40 30 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~48/35 = 556.123

Optimal ET sequence: 13b, 28b, 41

Badness: 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 = 1\1, ~11/8 = 556.122

Optimal ET sequence: 13b, 28b, 41

Badness: 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 = 1\1, ~11/8 = 556.106

Optimal ET sequence: 13b, 28b, 41

Badness: 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 = 1\5, ~5/4 = 380.499 (~256/245 = 99.501)
  • POTE: ~8/7 = 1\5, ~5/4 = 379.7131 (~256/245 = 100.2869)

Optimal ET sequence: 25, 35, 60

Badness: 0.287190

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