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A magic temperament is optimal, for some searches, in the 9-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-limit interval better than is possible in 12edo. Properties may depend on tuning and extension.

The most prominent deficiency of magic temperaments is that they lack proper or nearly-proper MOS scales in the 5 to 10 note "diatonic" region.

Five limit magic

The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo 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.

Comma: 3125/3072

5-limit minimax
[<1 0 0|, <0 1 0|, <2 1/5 0|]
Eigenmonzos: 2, 3

valid range: [360.000, 400.000] (10 to 3)
nice range: [378.910, 386.314]
strict range: [378.910, 386.314]

POTE generator: ~5/4 = 380.058

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

Map: [<1 0 2|, <0 5 1|]
Generators: 2, 5/4
Edos: 6, 16, 19, 22, 41, 60, 221c, 281c

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.

Magic

(See also Magic.)
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 MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.

Magic, with its accurate fifths, works well with 9-limit harmony. It's more accurate than meantone and simpler than garibaldi. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is <<5 1 12 -10 5 25||.

225/224 is the marvel comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.

243/242 leads to another essentially tempered 9-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. (This "octarod comma" is shared with sensi, godzilla, superpyth, octacot, rodan, shrutar, hedgehog, clyde, and bohpier. See temperament finder.)

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 three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.

Commas: 225/224, 245/243

7 and 9 limit minimax
[|1 0 0 0>, |0 1 0 0>, |2 1/5 0 0>, |-1 12/5 0 0>]
Eigenmonzos: 2, 3

valid range: [378.947, 381.818] (19 to 22)
nice range: [378.910, 386.314]
strict range: [378.947, 381.818]

POTE generator: 380.352

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

Map: [<1 0 2 -1|, <0 5 1 12|]
Generators: 2, 5/4

EDOs: 41, 142cd, 183cd, 224cd

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)/(26/25)=100/99.)

See also Chords of magic

Commas: 225/224, 245/243, 100/99

valid range: [378.947, 381.818] (19 to 22)
nice range: [378.910, 386.314]
strict range: [378.947, 381.818]

POTE generator: 380.696

Map: [<1 0 2 -1 6|, <0 5 1 12 -8|]
EDOs: 19, 22, 41, 104, 145c
Badness: 0.0204

13-limit

Commas: 100/99, 105/104, 144/143, 196/195

valid range: [378.947, 381.818] (19 to 22f)
nice range: [378.617, 386.314]
strict range: [378.947, 381.818]

POTE generator: ~5/4 = 380.427

Map: [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]
EDOS: 19, 41, 265cdef
Badness: 0.0215

Sorcery

Commas: 65/64, 78/77, 91/90, 100/99

valid range: 378.947 (19)
nice range: [359.472, 386.314]
strict range: 378.947

POTE generator: ~5/4 = 380.477

Map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|]
EDOs: 19, 22, 31f, 41f
Badness: 0.0258

Necromancy

Commas: 100/99, 225/224, 245/243, 275/273

valid range: [380.488, 380.952] (41 to 63)
nice range: [378.910, 386.314]
strict range: [380.488, 380.952]

POTE generator: ~5/4 = 380.787

Map: [<1 0 2 -1 6 11|, <0 5 1 12 -8 -23|]
EDOs: 19, 22, 41, 63, 104
Badness: 0.0253

Telepathy

Commas: 55/54, 99/98, 176/175

POTE generator: ~5/4 = 381.019

Map: [<1 0 2 -1 -1|, <0 5 1 12 14|]
EDOs: 19e, 22, 41e, 63e
Badness: 0.0271

13-limit telepathy

Commas: 55/54, 65/64, 91/90, 99/98

POTE generator: ~5/4 = 380.520

Map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]
EDOs: 19e, 22, 41ef
Badness: 0.0255

Horcrux

Commas: 45/44, 56/55, 245/243

POTE generator: ~5/4 = 379.642

Map: [<1 0 2 -1 0|, <0 5 1 12 11|]
EDOs: 19, 60e
Badness: 0.0393

Divination

Commas: 121/120, 225/224, 245/243

POTE generator: ~5/4 = 380.233

Map: [<2 0 4 -2 5|, <0 5 1 12 3|]
EDOs: 22, 38d, 60e, 142cde
Badness: 0.0359

13-limit

Commas: 105/104, 121/120, 196/195, 245/243

POTE generator: ~5/4 = 379.920

Map: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|]
EDOs: 22f, 60e
Badness: 0.0346

Soothsaying

Commas: 100/99, 225/224, 245/243, 1352/1331

POTE generator: ~5/4 = 380.508

Map: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|]
EDOs: 22, 60, 82
Badness: 0.0554

Witchcraft

Commas: 225/224, 245/243, 441/440

POTE generator: ~5/4 = 380.232

Map: [<1 0 2 -1 -7|, <0 5 1 12 33|]
EDOs: 41, 60e, 101cd, 243cde
Badness: 0.0307

13-limit

Commas: 105/104, 196/195, 245/243, 275/273

POTE generator: ~5/4 = 380.189

Map: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]
EDOs: 41, 60e, 101cd
Badness: 0.0235

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. The muggles wedgie is <<5 1 -7 -10 -25 -19||.

Commas: 126/125, 525/512

POTE generator: ~5/4 = 378.479

Map: [<1 0 2 5|, <0 5 1 -7|]
EDOs: 19, 73bcd, 92bcd
Badness: 0.0562

11-limit

Commas: 45/44, 126/125, 385/384

POTE generator: ~5/4 = 377.724

Map: [<1 0 2 5 0|, <0 5 1 -7 11|]
EDOs: 16, 19, 35, 54bd
Badness: 0.0480

13-limit

Commas: 45/44, 65/64, 78/77, 126/125

POTE generator: ~5/4 = 377.724

Map: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|]
EDOs: 16, 19, 35f, 54bdf
Badness: 0.0309

Astrology

Commas: 50/49, 3125/3072

POTE generator: ~5/4 = 380.578

Map: [<2 0 4 5|, <0 5 1 1|]
Wedgie: <<10 2 2 -20 -25 -1||
EDOs: 6, 16, 22, 60d, 82d
Badness: 0.0827

11-limit

Commas: 50/49, 121/120, 176/175

POTE generator: ~5/4 = 380.530

Map: [<2 0 4 5 5|, <0 5 1 1 3|]
EDOs: 6, 16, 22, 60de, 82de
Badness: 0.0392

13-limit

Commas: 50/49, 65/64, 78/77, 121/120

POTE generator: ~5/4 = 379.787

Map: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|]
EDOs: 6, 16, 22, 38f
Badness: 0.0344

Astrology Percussion Quintet No 1 by Joel Taylor

Horoscope

Commas: 50/49, 66/65, 105/104, 121/120

POTE generator: ~5/4 = 379.837

Map: [<2 0 4 5 5 3|, <0 5 1 1 3 7|]
EDOs: 16, 22f, 38
Badness: 0.0353

Spell

Commas: 49/48, 3125/3072

POTE generator: ~28/25 = 189.927

Map: [<1 0 2 2|, <0 10 2 5|]
Wedgie: <<10 2 5 -20 -20 6||
EDOs: 6, 19, 82d
Badness: 0.0810

11-limit

Commas: 49/48, 56/55, 125/121

POTE generator: ~11/10 = 190.285

Map: [<1 0 2 2 3|, <0 10 2 5 3|]
EDOs: 6, 19, 44de, 63de, 82de
Badness: 0.0598

13-limit

Commas: 49/48, 56/55, 78/77, 125/121

POTE generator: ~11/10 = 189.928

Map: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|]
EDOs: 6, 19, 82def
Badness: 0.0456

Cantrip

Commas: 49/48, 56/55, 91/90, 125/121

POTE generator: ~11/10 = 190.360

Map: [<1 0 2 2 3 1|, <0 10 2 5 3 17|]
EDOs: 19, 44de, 63de, 82de
Badness: 0.0416

Hocum

Commas: 3125/3072, 4000/3969

POTE generator: ~63/50 = 400.108

Map: [<1 5 3 -3|, <0 -10 -2 17|]
Wedgie: <<10 2 -17 -20 -55 -45||
EDOs: 38, 41, 161c, 202c, 243c, 284c
Badness: 0.1071

Hocus

Commas: 225/224, 243/242, 245/242

POTE generator: ~14/11 = 409.910

Map: [<1 5 3 11 12|, <0 -10 -2 -24 -25|]
EDOs: 38d, 41, 120cd, 161cd, 202cd
Badness: 0.0385

13-limit

Commas: 105/104, 196/195, 243/242, 245/242

POTE generator: ~14/11 = 410.004

Map: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|]
EDOs: 41, 79d, 120cd
Badness: 0.0303

Trismegistus

Commas: 3125/3072, 1029/1024

POTE generator: ~147/100 = 673.290

Map: [<1 10 4 0|, <0 -15 -3 5|]
Wedgie: <<15 3 -5 -30 -50 -20||
EDOs: 16, 25, 41, 139c, 180c, 221c, 262c
Badness: 0.0983

11-limit

Commas: 385/384, 441/440, 625/616

POTE generator: ~22/15 = 673.340

Map: [<1 10 4 0 13|, <0 -15 -3 5 -17|]
EDOs: 16, 25e, 41, 98c, 139c, 180c
Badness: 0.0456

13-limit

Commas: 105/104, 144/143, 275/273, 625/616

POTE generator: ~22/15 = 673.359

Map: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|]
EDOs: 16, 25e, 41, 98c, 139cf
Badness: 0.0331

Quadrimage

Commas: 2401/2400, 3125/3072

POTE generator: ~28/25 = 204.987

Map: [<1 5 3 4|, <0 -20 -4 -7|]
Wedgie: <<20 4 7 -40 -45 5||
EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd
Badness: 0.1274

11-limit

Commas: 245/242, 385/384, 625/616

POTE generator: ~28/25 = 204.956

Map: [<1 5 3 4 5|, <0 -20 -4 -7 -9|]
EDOs: 6, 35, 41, 199cde, 240cde, 281cde
Badness: 0.0616

13-limit

Commas: 105/104, 144/143, 245/242, 625/616

POTE generator: ~28/25 = 205.028

Map: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|]
EDOs: 41, 117c, 158cd, 199cdef
Badness: 0.0440