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.

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 include astrology, spell, hocum, trismegistus, quadrimage, and warlock.

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

POTE generator: ~5/4 = 380.058

Minimax tuning:

[[1 0 0, [0 1 0, [2 1/5 0]
Eigenmonzos (unchanged intervals): 2, 3

Tuning ranges:

  • Diamond monotone range: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
  • Diamond tradeoff range: ~5/4 = [378.910, 386.314]
  • 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.

Vals19, 41, 60, 221c, 281c

Badness: 0.039163

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]]

POTE generator: ~5/4 = 380.352

Minimax tuning:

[[1 0 0 0, [0 1 0 0, [2 1/5 0 0, [-1 12/5 0 0]
Eigenmonzos (unchanged intervals): 2, 3

Tuning ranges:

  • Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • Diamond tradeoff range: ~5/4 = [378.910, 386.314]
  • 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.

Vals19, 41, 142cd, 183cd, 224cd

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]]

POTE generator: ~5/4 = 380.696

Tuning ranges:

  • Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • Diamond tradeoff range: ~5/4 = [378.910, 386.314]
  • Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

Vals: 19, 22, 41, 104, 145c

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]]

POTE generator: ~5/4 = 380.427

Tuning ranges:

  • Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • Diamond tradeoff range: ~5/4 = [378.617, 386.314]
  • Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

Vals: 19, 22f, 41, 265cdef

Badness: 0.021509

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]]

POTE generator: ~5/4 = 380.477

Tuning ranges:

  • Diamond monotone range: ~5/4 = 378.947 (6\19)
  • Diamond tradeoff range: ~5/4 = [359.472, 386.314]
  • Diamond monotone and tradeoff: ~5/4 = 378.947

Vals: 19, 22, 41f

Badness: 0.025829

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]]

POTE generator: ~5/4 = 380.787

Tuning ranges:

  • Diamond monotone range: ~5/4 = [380.488, 380.952] (13\41 to 20\63)
  • Diamond tradeoff range: ~5/4 = [378.910, 386.314]
  • Diamond monotone and tradeoff: ~5/4 = [380.488, 380.952]

Vals: 19f, 22, 41, 63, 104

Badness: 0.025275

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

Vals: 22, 60, 82

Badness: 0.055443

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

Vals: 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

Vals: 19e, 22, 41ef

Badness: 0.025522

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

Vals: 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

Vals: 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 -2]]

POTE generator: ~5/4 = 379.116

Vals: 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

Vals: 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

Vals: 41, 60e, 101cd

Badness: 0.023547

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

Vals: 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

Vals: 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

Vals: 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

Vals: 41, 79d, 120cd

Badness: 0.030280

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

Vals16, 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

Vals: 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 = 377.724

Vals: 16, 19, 35f, 54bdf

Badness: 0.030386

Astrology

Subgroup: 2.3.5.7

Comma list: 50/49, 3125/3072

Mapping: [2 0 4 5], 0 5 1 1]]

Wedgie⟨⟨10 2 2 -20 -25 -1]]

POTE generator: ~5/4 = 380.578

Vals6, 16, 22, 60d, 82d

Badness: 0.082673

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 176/175

Mapping: [2 0 4 5 5], 0 5 1 1 3]]

POTE generator: ~5/4 = 380.530

Vals: 6, 16, 22, 60de, 82de

Badness: 0.039151

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/64, 78/77, 121/120

Mapping: [2 0 4 5 5 8], 0 5 1 1 3 -1]]

POTE generator: ~5/4 = 379.787

Vals: 6, 16, 22, 38f

Badness: 0.034376

Music

Horoscope

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 66/65, 105/104, 121/120

Mapping: [2 0 4 5 5 3], 0 5 1 1 3 7]]

POTE generator: ~5/4 = 379.837

Vals: 16, 22f, 38

Badness: 0.035284

Spell

See also: Hemimean clan

Subgroup: 2.3.5.7

Comma list: 49/48, 3125/3072

Mapping: [1 0 2 2], 0 10 2 5]]

Wedgie⟨⟨10 2 5 -20 -20 6]]

POTE generator: ~28/25 = 189.927

Vals6, 19, 82dd

Badness: 0.080958

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 125/121

Mapping: [1 0 2 2 3], 0 10 2 5 3]]

POTE generator: ~11/10 = 190.285

Vals: 6, 19, 44de, 63dee, 82ddee

Badness: 0.059791

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: [1 0 2 2 3 4], 0 10 2 5 3 -2]]

POTE generator: ~11/10 = 189.928

Vals: 6, 19, 82ddeeff

Badness: 0.045591

Cantrip

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: [1 0 2 2 3 1], 0 10 2 5 3 17]]

POTE generator: ~11/10 = 190.360

Vals: 19, 44de, 63dee, 82ddee

Badness: 0.041603

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

Vals38, 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

Vals16, 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]]

Vals: 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

Vals: 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

Vals6, 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

Vals: 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

Vals: 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

Vals25, 35, 60

Badness: 0.287