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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.4994

Minimax tuning:

Eigenmonzos (unchanged-intervals): 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]
  • 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.

Weak extensions considered below are hocum, trismegistus, quadrimage, 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 MOSes 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 in 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.6512

Minimax tuning:

Eigenmonzos (unchanged-intervals): 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]
  • 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 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]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7200

Minimax tuning:

Eigenmonzos (unchanged-intervals): 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]
  • 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

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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.4354

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

17-limit

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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.5103

Optimal ET sequence: 19, 22f, 41

Badness: 0.020633

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

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

POTE generator: ~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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.6741

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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7839

Optimal ET sequence: 19, 22, 41f

Badness: 0.023768

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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7876

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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.8373

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

Badness: 0.022032

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]]

POTE generator: ~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]]

POTE generator: ~5/4 = 380.508

Optimal ET sequence: 22, 60, 82

Badness: 0.035654

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]]

POTE generator: ~5/4 = 381.019

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

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]]

POTE generator: ~5/4 = 380.520

Optimal ET sequence: 19e, 22, 41ef

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]]

POTE generator: ~5/4 = 380.619

Optimal ET sequence: 19eg, 22, 41efg

Badness: 0.020201

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]]

POTE generator: ~5/4 = 380.483

Optimal ET sequence: 19e, 22f, 41e

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]]

POTE generator: ~5/4 = 380.604

Optimal ET sequence: 19eg, 22f, 41eg

Badness: 0.020274

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]]

POTE generator: ~5/4 = 379.642

Optimal ET sequence: 3de, 19, 41ee, 60ee

Badness: 0.039282

Charisma

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]]

POTE generator: ~5/4 = 379.791

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

Badness: 0.031938

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]]

POTE generator: ~5/4 = 379.116

Optimal ET sequence: 3de, 19

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]]

POTE generator: ~5/4 = 380.232

Optimal ET sequence: 41, 60e, 101cd, 243cde

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]]

POTE generator: ~5/4 = 380.189

Optimal ET sequence: 41, 60e, 101cd

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]]

POTE generator: ~5/4 = 380.114

Optimal ET sequence: 41, 60e, 101cd, 161cde

Badness: 0.020756

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]]

POTE generator: ~5/4 = 380.233

Optimal ET sequence: 22, 38d, 60e, 142cde

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]]

POTE generator: ~5/4 = 379.920

Optimal ET sequence: 22f, 60e

Badness: 0.034551

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]]

POTE generator: ~14/11 = 409.910

Optimal ET sequence: 38d, 41, 120cd, 161cd, 202cd

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]]

POTE generator: ~14/11 = 410.004

Optimal ET sequence: 41, 79d, 120cd

Badness: 0.030280

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's the same thing as magic. Muggles works better for small scales than magic in the sense that 7- or 10-note MOS 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]]

POTE generator: ~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, 92bcd

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]]

POTE generator: ~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, 35, 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]]

POTE generator: ~5/4 = 377.653

Optimal ET sequence: 16, 19, 35f, 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]]

POTE generator: ~5/4 = 377.832

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

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]]

POTE generator: ~5/4 = 377.838

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

Badness: 0.028732

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 5 3 -3], ⟨0 -10 -2 17]]

Wedgie: ⟨⟨10 2 -17 -20 -55 -45]]

POTE generator: ~63/50 = 410.108

Optimal ET sequence: 38, 41, 161c, 202c, 243c, 284c

Badness: 0.107115

Trismegistus

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 10 4 0], ⟨0 -15 -3 5]]

Wedgie: ⟨⟨15 3 -5 -30 -50 -20]]

POTE generator: ~147/100 = 673.290

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

Badness: 0.098334

11-limit

Subgroup: 2.3.5.7.11

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

POTE generator: ~22/15 = 673.340

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

Optimal ET sequence: 16, 25e, 41, 98c, 139c, 180c

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]]

POTE generator: ~22/15 = 673.359

Optimal ET sequence: 16, 25e, 41, 98c, 139cf

Badness: 0.033081

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 5 3 4], ⟨0 -20 -4 -7]]

Wedgie: ⟨⟨20 4 7 -40 -45 5]]

POTE generator: ~28/25 = 204.987

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

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]]

POTE generator: ~28/25 = 204.956

Optimal ET sequence: 6, 35, 41, 199cde, 240cde, 281cde

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]]

POTE generator: ~28/25 = 205.028

Optimal ET sequence: 41, 117c, 158cd, 199cdef

Badness: 0.044047

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

POTE generator: ~5/4 = 379.7131

Optimal ET sequence: 25, 35, 60

Badness: 0.287190

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