Magic family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The magic family of temperaments tempers out 3125/3072, the small diesis or magic comma. The septimal version of magic is locally optimal, for some searches, in the 9-odd-limit. Magic has a slightly higher complexity than meantone but it is 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 generator of magic is a major third, and to get to the interval class of fifths requires 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 = 1200.000 ¢, ~5/4 = 380.499 ¢

error map: ⟨0.000 +0.542 -5.814]

• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.058 ¢

error map: ⟨0.000 -1.663 -6.255]

Minimax tuning:

• 5-odd-limit: ~5/4 = [0 1/5 0⟩

unchanged-interval (eigenmonzo) 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)

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, 41, 60, 221cc, 281cc

Badness (Smith): 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

• Astrology → Jubilismic clan
• Spell → Hemimean clan

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. 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

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 380.651 ¢

error map: ⟨0.000 +1.301 -5.662 -1.011]

• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.352 ¢

error map: ⟨0.000 -0.195 -5.962 -4.602]

Minimax tuning:

• 7- and 9-odd-limit: ~5/4 = [0 1/5 0 0⟩

unchanged-interval (eigenmonzo) 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)

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

Optimal ET sequence: 19, 41, 142cd, 183cd, 224ccd

Badness (Smith): 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]]

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 380.720 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.696 ¢

Minimax tuning:

• 11-odd-limit: ~5/4 = [1/3 1/9 0 0 -1/18⟩

unchanged-interval (eigenmonzo) 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)

Optimal ET sequence: 19, 22, 41, 104

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.435 ¢
• POTE: ~2 = 1200.000 ¢, ~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]

Optimal ET sequence: 19, 22f, 41

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.510 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.604 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.451 ¢

Optimal ET sequence: 19, 41

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.573 ¢

Optimal ET sequence: 22fh, 41

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.271 ¢

Optimal ET sequence: 19g, 41, 60

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.360 ¢

Optimal ET sequence: 19gh, 41

Badness (Smith): 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:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 380.478 ¢
• POTE: ~2 = 1200.000 ¢, ~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 = 1200.000 ¢, ~5/4 = 380.674 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.477 ¢

Optimal ET sequence: 19, 22, 41f

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.784 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.729 ¢

Optimal ET sequence: 19, 22

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.625 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 380.621 ¢

Optimal ET sequence: 19, 22

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.788 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.787 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.837 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.827 ¢

Optimal ET sequence: 19f, 22, 41, 63

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.741 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 380.735 ¢

Optimal ET sequence: 19f, 22, 41

Badness (Smith): 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 = 600.000 ¢, ~5/4 = 380.539 ¢
• POTE: ~55/39 = 600.000 ¢, ~5/4 = 380.508 ¢

Optimal ET sequence: 22, 60, 82

Badness (Smith): 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 = 600.000 ¢, ~5/4 = 380.553 ¢
• POTE: ~17/12 = 600.000 ¢, ~5/4 = 380.508 ¢

Optimal ET sequence: 22, 60, 82

Badness (Smith): 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 = 600.000 ¢, ~5/4 = 380.470 ¢
• CWE: ~17/12 = 600.000 ¢, ~5/4 = 380.470 ¢

Optimal ET sequence: 22, 60, 82

Badness (Smith): 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]]

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 381.231 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 381.019 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 381.196 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.520 ¢

Optimal ET sequence: 19e, 22, 41ef

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 381.288 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.619 ¢

Optimal ET sequence: 19eg, 22, 41efg

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 381.464 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 380.735 ¢

Optimal ET sequence: 19egh, 22, 41efghh

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.816 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.483 ¢

Optimal ET sequence: 19e, 22f

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.952 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.604 ¢

Optimal ET sequence: 19eg, 22f

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.866 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 380.628 ¢

Optimal ET sequence: 19egh, 22fh

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 379.796 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 379.642 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 379.885 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 379.791 ¢

Optimal ET sequence: 3def, 16dff, 19

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.216 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 380.148 ¢

Optimal ET sequence: 3defg, 16dffgg, 19g

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 379.524 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 379.571 ¢

Optimal ET sequence: 3defg, 16dff, 19

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 379.757 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 379.116 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.416 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.232 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd, 243ccdde

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.352 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.189 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.281 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 380.114 ¢

Optimal ET sequence: 19eg, 41, 60e, 101cd

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 380.339 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 380.205 ¢

Optimal ET sequence: 19egh, 41, 60eh

Badness (Smith): 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 = 600.000 ¢, ~5/4 = 380.732 ¢
• POTE: ~99/70 = 600.000 ¢, ~5/4 = 380.223 ¢

Optimal ET sequence: 22, 38d, 60e, 142cdee, 202ccddeee

Badness (Smith): 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 = 600.000 ¢, ~5/4 = 380.417 ¢
• POTE: ~99/70 = 600.000 ¢, ~5/4 = 379.920 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Smith): 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 = 600.000 ¢, ~5/4 = 380.433 ¢
• CWE: ~17/12 = 600.000 ¢, ~5/4 = 380.067 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 409.760 ¢
• POTE: ~2 = 1200.000 ¢, ~14/11 = 409.910 ¢

Optimal ET sequence: 38d, 41, 120cd

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 409.842 ¢
• POTE: ~2 = 1200.000 ¢, ~14/11 = 410.004 ¢

Optimal ET sequence: 38df, 41, 79d, 120cd

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 409.896 ¢
• CWE: ~2 = 1200.000 ¢, ~14/11 = 409.994 ¢

Optimal ET sequence: 38df, 41, 79d

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 409.884 ¢
• CWE: ~2 = 1200.000 ¢, ~14/11 = 410.012 ¢

Optimal ET sequence: 38df, 41, 79dh

Badness (Smith): 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]]

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 378.744 ¢

error map: ⟨0.000 -8.235 -7.570 -20.034]

• POTE: ~2 = 1200.000 ¢, ~5/4 = 378.479 ¢

error map: ⟨0.000 -9.558 -7.834 -18.181]

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]

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

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 378.228 ¢
• POTE: ~2 = 1200.000 ¢, ~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]

Optimal ET sequence: 16, 19, 35, 54bd

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 378.176 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 377.653 ¢

Optimal ET sequence: 16, 19, 35f, 54bdf

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 377.724 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 377.832 ¢

Optimal ET sequence: 3, 16, 19e, 35ee, 54bdeee

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 377.652 ¢
• POTE: ~2 = 1200.000 ¢, ~5/4 = 377.838 ¢

Optimal ET sequence: 3, 16, 19e, 35eef

Badness (Smith): 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]]

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 377.385 ¢

error map: ⟨0.000 -15.028 -8.928 +27.643]

• CWE: ~2 = 1200.000 ¢, ~5/4 = 376.963 ¢

error map: ⟨0.000 -18.198 -9.562 +21.937]

Optimal ET sequence: 3d, …, 13b, 16

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 377.388 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 376.973 ¢

Optimal ET sequence: 3de, 13be, 16

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 376.914 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 376.422 ¢

Optimal ET sequence: 3def, 13beff, 16

Badness (Smith): 0.038328

Brightstone

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~5/4 = 381.955 ¢

error map: ⟨0.000 +7.818 -4.359 +11.627]

• CWE: ~2 = 1200.000 ¢, ~5/4 = 381.956 ¢

error map: ⟨0.000 +7.826 -4.358 +11.613]

Optimal ET sequence: 3, 19d, 22

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 381.790 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 381.712 ¢

Optimal ET sequence: 3, 19d, 22

Badness (Smith): 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 = 1200.000 ¢, ~5/4 = 381.732 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 381.736 ¢

Optimal ET sequence: 3, 22

Badness (Smith): 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

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~63/50 = 409.836 ¢

error map: ⟨0.000 -0.316 -5.986 -1.612]

• POTE: ~2 = 1200.000 ¢, ~63/50 = 410.108 ¢

error map: ⟨0.000 -0.437 -6.010 -1.406]

Optimal ET sequence: 3, 38, 41, 161c

Badness (Smith): 0.107115

Trismegistus

See also: No-threes subgroup temperaments § Mabilic

Main article: Mabilic and 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

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~147/100 = 673.187 ¢

error map: ⟨0.000 +0.240 -5.875 -2.891]

• POTE: ~2 = 1200.000 ¢, ~147/100 = 673.290 ¢

error map: ⟨0.000 -0.932 -6.109 -2.500]

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

Badness (Smith): 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 = 1200.000 ¢, ~22/15 = 673.241 ¢
• POTE: ~2 = 1200.000 ¢, ~22/15 = 673.340 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Smith): 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 = 1200.000 ¢, ~22/15 = 673.294 ¢
• POTE: ~2 = 1200.000 ¢, ~22/15 = 673.359 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Smith): 0.033081

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 105/104, 144/143, 441/440, 210/209, 625/616

Subgroup-val mapping: [⟨1 10 4 0 13 11 2], ⟨0 -15 -3 5 -17 -13 4]]

Optimal tunings:

• CWE: ~2 = 1200.000 ¢, ~22/15 = 673.363 ¢

Optimal ET sequence: 16, 25e, 41, 98c

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

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~28/25 = 204.860 ¢

error map: ⟨0.000 +0.853 -5.752 -2.843]

• POTE: ~2 = 1200.000 ¢, ~28/25 = 204.987 ¢

error map: ⟨0.000 -1.691 -6.261 -3.733]

Optimal ET sequence: 6, 29b, 35, 41, 158cd, 199ccd, 240ccd, 281ccd

Badness (Smith): 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 = 1200.000 ¢, ~28/25 = 204.881 ¢
• POTE: ~2 = 1200.000 ¢, ~28/25 = 204.956 ¢

Optimal ET sequence: 6, 35, 41

Badness (Smith): 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 = 1200.000 ¢, ~28/25 = 204.956 ¢
• POTE: ~2 = 1200.000 ¢, ~28/25 = 205.028 ¢

Optimal ET sequence: 6f, 35f, 41, 117c

Badness (Smith): 0.044047

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

mapping generators: ~2, ~48/35

Optimal tunings:

• CTE: ~2 = 1200.000 ¢, ~48/35 = 556.123 ¢

error map: ⟨0.000 +1.132 -5.696 -2.480]

• CWE: ~2 = 1200.000 ¢, ~48/35 = 556.050 ¢

error map: ⟨0.000 -0.695 -6.062 -3.868]

Optimal ET sequence: 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd

Badness (Smith): 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 = 1200.000 ¢, ~11/8 = 556.122 ¢
• CWE: ~2 = 1200.000 ¢, ~11/8 = 556.095 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Smith): 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 = 1200.000 ¢, ~11/8 = 556.106 ¢
• CWE: ~2 = 1200.000 ¢, ~11/8 = 556.117 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Smith): 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 = 240.000 ¢, ~5/4 = 380.499 ¢ (~256/245 = 99.501 ¢)

error map: ⟨0.000 +0.542 -5.814 -8.826]

• CWE: ~8/7 = 240.000 ¢, ~5/4 = 379.997 ¢ (~256/245 = 100.003 ¢)

error map: ⟨0.000 -1.972 -6.317 -8.826]

Optimal ET sequence: 25, 35, 60

Badness (Smith): 0.287190