Magic family
The magic family of temperaments tempers out 3125/3072, the small diesis or magic comma. A magic temperament 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-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.
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
- [[1 0 0⟩, [0 1 0⟩, [2 1/5 0⟩]
- Eigenmonzos: 2, 3
- valid range: [360.000, 400.000] (3\10 to 1\3)
- nice range: [378.910, 386.314]
- strict range: [378.910, 386.314]
Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.
Seven-limit 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 magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.
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-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
- 7- and 9-odd-limit
- [[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] (6\19 to 7\22)
- nice range: [378.910, 386.314]
- strict range: [378.947, 381.818]
Algebraic generator: Tirzbirat or Septimage, the real root of 5x5 + 4x - 20, 380.7604 cents.
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: 225/224, 245/243, 100/99
Mapping: [⟨1 0 2 -1 6], ⟨0 5 1 12 -8]]
POTE generator: ~5/4 = 380.696
Tuning ranges:
- valid range: [378.947, 381.818] (6\19 to 7\22)
- nice range: [378.910, 386.314]
- strict range: [378.947, 381.818]
Badness: 0.0204
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:
- valid range: [378.947, 381.818] (6\19 to 7\22)
- nice range: [378.617, 386.314]
- strict range: [378.947, 381.818]
Badness: 0.0215
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:
- valid range: 378.947 (6\19)
- nice range: [359.472, 386.314]
- strict range: 378.947
Badness: 0.0258
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:
- valid range: [380.488, 380.952] (13\41 to 20\63)
- nice range: [378.910, 386.314]
- strict range: [380.488, 380.952]
Badness: 0.0253
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
Badness: 0.0271
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
Badness: 0.0255
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
Badness: 0.0393
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
Badness: 0.0359
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
Badness: 0.0346
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
Badness: 0.0554
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
Badness: 0.0307
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
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.
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
Badness: 0.0562
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
Badness: 0.0480
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
Badness: 0.0309
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
Badness: 0.0827
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
Badness: 0.0392
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
Badness: 0.0344
- 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
Badness: 0.0353
Spell
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
Badness: 0.0810
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
Badness: 0.0598
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
Badness: 0.0456
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
Badness: 0.0416
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 = 400.108
Badness: 0.1071
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
Badness: 0.0385
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
Badness: 0.0303
Trismegistus
Subgroup: 2.3.5.7
Comma list: 3125/3072, 1029/1024
Mapping: [<1 10 4 0|, <0 -15 -3 5|]
Wedgie: ⟨⟨ 15 3 -5 -30 -50 -20 ]]
POTE generator: ~147/100 = 673.290
Badness: 0.0983
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|]
Badness: 0.0456
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
Badness: 0.0331
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
Badness: 0.1274
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
Badness: 0.0616
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
Badness: 0.0440