Magic family: Difference between revisions

Latest revision as of 01:34, 12 May 2026

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:

WE: ~2 = 1201.2449 ¢, ~5/4 = 380.4527 ¢

error map: ⟨+1.245 +0.309 -3.371]

CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2194 ¢

error map: ⟨0.000 -0.858 -6.094]

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 (Sintel): 0.919

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 and 105/104 to the list of commas, magic can be extended to the 11-limit and 13-limit. 11-limit magic 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. 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. For the 13-limit, 41edo makes for a recommendable 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:

WE: ~2 = 1201.0786 ¢, ~5/4 = 380.6939 ¢

error map: ⟨+1.079 +1.514 -3.463 -1.578]

CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4576 ¢

error map: ⟨0.000 +0.333 -5.856 -3.335]

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 (Sintel): 0.479

11-limit

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:

WE: ~2 = 1200.1372 ¢, ~5/4 = 380.7399 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7008 ¢

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 (Sintel): 0.673

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:

WE: ~2 = 1200.0331 ¢, ~5/4 = 380.4377 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4284 ¢

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 (Sintel): 0.889

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:

WE: ~2 = 1199.3584 ¢, ~5/4 = 380.4006 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5896 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 1.05

Magicus

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:

WE: ~2 = 1199.7173 ¢, ~5/4 = 380.3808 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4680 ¢

Optimal ET sequence: 19, 41

Badness (Sintel): 1.27

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:

WE: ~2 = 1199.3670 ¢, ~5/4 = 380.4681 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6541 ¢

Optimal ET sequence: 22fh, 41

Badness (Sintel): 1.21

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:

WE: ~2 = 1200.1727 ¢, ~5/4 = 380.2982 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2483 ¢

Optimal ET sequence: 19g, 41, 60

Badness (Sintel): 1.34

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:

WE: ~2 = 1200.2179 ¢, ~5/4 = 380.3942 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.3314 ¢

Optimal ET sequence: 19gh, 41

Badness (Sintel): 1.44

Evening

Evening is a remarkable subgroup temperament of 19 & 22f 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

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

Optimal tunings:

WE: ~2 = 1200.2802 ¢, ~5/4 = 380.5053 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4258 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.807

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:

WE: ~2 = 1201.2397 ¢, ~5/4 = 380.8698 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5080 ¢

Optimal ET sequence: 19, 22, 41f

Badness (Sintel): 1.07

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:

WE: ~2 = 1199.9675 ¢, ~5/4 = 380.7770 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7874 ¢

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

Badness (Sintel): 1.04

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:

WE: ~2 = 1199.6176 ¢, ~5/4 = 380.7053 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.8280 ¢

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

Badness (Sintel): 1.12

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:

WE: ~55/39 = 600.2918 ¢, ~5/4 = 380.6928 ¢
CWE: ~55/39 = 600.0000 ¢, ~5/4 = 380.5121 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 2.29

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:

WE: ~17/12 = 600.2918 ¢, ~5/4 = 380.6927 ¢
CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.5135 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.82

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:

WE: ~17/12 = 600.3301 ¢, ~5/4 = 380.6797 ¢
CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.4704 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.90

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:

WE: ~2 = 1200.7724 ¢, ~5/4 = 381.2641 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.0913 ¢

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

Badness (Sintel): 0.896

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:

WE: ~2 = 1202.5634 ¢, ~5/4 = 381.3348 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6886 ¢

Optimal ET sequence: 19e, 22, 41ef

Badness (Sintel): 1.05

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:

WE: ~2 = 1201.3172 ¢, ~5/4 = 380.9004 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5942 ¢

Optimal ET sequence: 19e, 22f

Badness (Sintel): 1.08

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:

WE: ~2 = 1200.4670 ¢, ~5/4 = 379.7895 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.6889 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.30

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:

WE: ~2 = 1200.2953 ¢, ~5/4 = 379.8842 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.8165 ¢

Optimal ET sequence: 3def, 16dff, 19

Badness (Sintel): 1.32

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:

WE: ~2 = 1200.2484 ¢, ~5/4 = 380.2053 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1482 ¢

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

Badness (Sintel): 1.43

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:

WE: ~2 = 1199.5457 ¢, ~5/4 = 379.4681 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.5713 ¢

Optimal ET sequence: 3defg, 16dff, 19

Badness (Sintel): 1.51

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:

WE: ~2 = 1202.2187 ¢, ~5/4 = 379.8171 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.2709 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.38

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:

WE: ~2 = 1201.2634 ¢, ~5/4 = 380.6321 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2849 ¢

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

Badness (Sintel): 1.02

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:

WE: ~2 = 1201.0424 ¢, ~5/4 = 380.5193 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2349 ¢

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

Badness (Sintel): 0.973

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:

WE: ~2 = 1201.1638 ¢, ~5/4 = 380.4827 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1599 ¢

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

Badness (Sintel): 1.06

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:

WE: ~99/70 = 600.8306 ¢, ~5/4 = 380.7598 ¢
CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.3800 ¢

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

Badness (Sintel): 1.19

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:

WE: ~99/70 = 600.9624 ¢, ~5/4 = 380.5297 ¢
CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.0614 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.43

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:

WE: ~17/12 = 600.8921 ¢, ~5/4 = 380.5094 ¢
CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.0672 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.21

Hocus

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13], ⟨0 10 2 24 25]]

Optimal tunings:

WE: ~2 = 1201.0749 ¢, ~11/7 = 790.7980 ¢
CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.1429 ¢

Optimal ET sequence: 38d, 41, 120cd

Badness (Sintel): 1.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20], ⟨0 10 2 24 25 36]]

Optimal tunings:

WE: ~2 = 1201.2830 ¢, ~11/7 = 790.8409 ¢
CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0516 ¢

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

Badness (Sintel): 1.25

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 1 -13 -13 -20 -15], ⟨0 10 2 24 25 36 29]]

Optimal tunings:

WE: ~2 = 1201.1557 ¢, ~11/7 = 790.7157 ¢
CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0057 ¢

Optimal ET sequence: 38df, 41, 79d

Badness (Sintel): 1.30

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 1 -13 -13 -20 -3], ⟨0 10 2 24 25 36 29 11]]

Optimal tunings:

WE: ~2 = 1201.3558 ¢, ~11/7 = 790.8266 ¢
CWE: ~2 = 1200.0000 ¢, ~11/7 = 789.9880 ¢

Optimal ET sequence: 38df, 41, 79dh

Badness (Sintel): 1.23

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:

WE: ~2 = 1203.9554 ¢, ~5/4 = 379.7269 ¢

error map: ⟨+3.955 -3.321 +1.324 -7.137]

CWE: ~2 = 1200.0000 ¢, ~5/4 = 378.5328 ¢

error map: ⟨0.000 -9.291 -7.781 -18.555]

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 (Sintel): 1.42

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:

WE: ~2 = 1203.0804 ¢, ~5/4 = 378.6936 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8174 ¢

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 (Sintel): 1.59

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:

WE: ~2 = 1203.4291 ¢, ~5/4 = 378.7321 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.7336 ¢

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

Badness (Sintel): 1.26

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:

WE: ~2 = 1205.6044 ¢, ~5/4 = 379.5966 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8142 ¢

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

Badness (Sintel): 1.55

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:

WE: ~2 = 1205.4897 ¢, ~5/4 = 379.5667 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8118 ¢

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

Badness (Sintel): 1.19

Brightstone

Brightstone tempers out 64/63 and may be described as 22 & 25. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Optimal tunings:

WE: ~2 = 1198.1701 ¢, ~5/4 = 381.3741 ¢

error map: ⟨-1.830 +4.915 -8.599 +6.454]

CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.9562 ¢

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

Optimal ET sequence: 3, 19d, 22

Badness (Sintel): 2.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 625/616

Mapping: [⟨1 0 2 6 -1], ⟨0 5 1 -10 14]]

Optimal tunings:

WE: ~2 = 1198.5372 ¢, ~5/4 = 381.7556 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 382.1943 ¢

Optimal ET sequence: 22, 69b

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 55/54, 64/63, 625/616

Mapping: [⟨1 0 2 6 6 4], ⟨0 5 1 -10 14 -4]]

Optimal tunings:

WE: ~2 = 1197.2300 ¢, ~5/4 = 381.6164 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 382.4690 ¢

Optimal ET sequence: 22f, 47bff

Badness (Sintel): 1.99

Darkstone

Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.

This temperament is known as witch in Tonalsoft Encyclopedia.

Subgroup: 2.3.5.7

Comma list: 36/35, 1875/1792

Mapping: [⟨1 0 2 0], ⟨0 5 1 9]]

Optimal tunings:

WE: ~2 = 1201.7458 ¢, ~5/4 = 377.2996 ¢

error map: ⟨+1.746 -15.457 -5.523 +26.870]

CWE: ~2 = 1200.000 ¢, ~5/4 = 376.9630 ¢

error map: ⟨0.000 -17.140 -9.351 +23.841]

Optimal ET sequence: 3d, …, 13b, 16

Badness (Sintel): 2.13

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:

WE: ~2 = 1201.7428 ¢, ~5/4 = 377.3134 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 376.9735 ¢

Optimal ET sequence: 3de, 13be, 16

Badness (Sintel): 1.55

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:

WE: ~2 = 1201.7428 ¢, ~5/4 = 377.3134 ¢
CWE: ~2 = 1200.0000 ¢, ~5/4 = 376.4221 ¢

Optimal ET sequence: 3def, 13beff, 16

Badness (Sintel): 1.58

Scales: User:BudjarnLambeth/Volcanic glass

Music: Rain in the crystal mirror caves - Budjarn Lambeth (2026)

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 -5 1 14], ⟨0 10 2 -17]]

mapping generators: ~2, ~63/40

Optimal tunings:

WE: ~2 = 1200.8375 ¢, ~63/40 = 790.7032 ¢

error map: ⟨+0.838 +0.890 -4.070 +0.944]

CWE: ~2 = 1200.0000 ¢, ~63/40 = 790.1542 ¢

error map: ⟨0.000 -0.413 -6.005 -1.447]

Optimal ET sequence: 3, 38, 41, 161c

Badness (Sintel): 2.71

Trismegistus

Main article: Mabilic and trismegistus

See also: No-threes subgroup temperaments § Mabilic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 -5 1 5], ⟨0 15 3 -5]]

mapping generators: ~2, ~168/125

Optimal tunings:

WE: ~2 = 1201.0799 ¢, ~168/125 = 527.1841 ¢

error map: ⟨+1.080 +0.408 -3.681 +0.653]

CWE: ~2 = 1200.0000 ¢, ~168/125 = 526.7349 ¢

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

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

Badness (Sintel): 2.49

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

WE: ~2 = 1200.8404 ¢, ~15/11 = 527.0289 ¢
CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6826 ¢

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

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273, 625/616

Mapping: [⟨1 -5 1 5 -4 -2], ⟨0 15 3 -5 17 13]]

Optimal tunings:

WE: ~2 = 1200.4759 ¢, ~15/11 = 526.8502 ¢
CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6548 ¢

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

Badness (Sintel): 1.37

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 -5 1 5 -4 -2 6], ⟨0 15 3 -5 17 13 -4]]

Optimal tunings:

WE: ~2 = 1200.5832 ¢, ~15/11 = 526.8804 ¢
CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6368 ¢

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

Badness (Sintel): 1.26

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 -15 -1 -3], ⟨0 20 4 7]]

mapping generators: ~2, ~25/14

Optimal tunings:

WE: ~2 = 1201.2708 ¢, ~25/14 = 996.0669 ¢

error map: ⟨+1.271 +0.322 -3.317 -0.170]

CWE: ~2 = 1200.0000 ¢, ~25/14 = 995.0515 ¢

error map: ⟨0.000 -0.926 -6.108 -3.466]

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

Badness (Sintel): 3.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 625/616

Mapping: [⟨1 -15 -1 -3 -4], ⟨0 20 4 7 9]]

Optimal tunings:

WE: ~2 = 1200.6716 ¢, ~25/14 = 995.6009 ¢
CWE: ~2 = 1200.0000 ¢, ~25/14 = 995.0633 ¢

Optimal ET sequence: 6, 35, 41

Badness (Sintel): 2.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/242, 625/616

Mapping: [⟨1 -15 -1 -3 -22], ⟨0 20 4 7 9 31]]

Optimal tunings:

WE: ~2 = 1200.6276 ¢, ~25/14 = 995.4920 ¢
CWE: ~2 = 1200.0000 ¢, ~25/14 = 994.9901 ¢

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

Badness (Sintel): 1.82

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:

WE: ~2 = 1201.3334 ¢, ~48/35 = 556.6311 ¢

error map: ⟨+1.333 +0.489 -3.158 -0.835]

CWE: ~2 = 1200.0000 ¢, ~48/35 = 556.0504 ¢

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

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

Badness (Sintel): 4.92

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:

WE: ~2 = 1200.4252 ¢, ~11/8 = 556.2831 ¢
CWE: ~2 = 1200.0000 ¢, ~11/8 = 556.0951 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 3.36

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:

WE: ~2 = 1199.8239 ¢, ~11/8 = 556.0389 ¢
CWE: ~2 = 1200.0000 ¢, ~11/8 = 556.1171 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 2.80

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:

WE: ~8/7 = 240.3877 ¢, ~5/4 = 380.4267 ¢ (~256/245 = 100.3488 ¢)

error map: ⟨+1.939 +0.178 -2.010 -3.398]

CWE: ~8/7 = 240.0000 ¢, ~5/4 = 379.9965 ¢ (~256/245 = 100.0035 ¢)

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

Optimal ET sequence: 25, 35, 60

Badness (Sintel): 7.27

@@ Line 5: / Line 5: @@
 | ja =
 }}
+{{Technical data page}}
 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 [[Rothenberg propriety|proper]] or nearly-proper [[mos scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.
@@ Line 21: / Line 22: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.499
+* [[WE]]: ~2 = 1201.2449{{c}}, ~5/4 = 380.4527{{c}}
-: [[error map]]: {{val| 0.000 +0.542 -5.814 }}
+: [[error map]]: {{val| +1.245 +0.309 -3.371 }}
-* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.058
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
-: error map: {{val| 0.000 -1.663 -6.255 }}
+: error map: {{val| 0.000 -0.858 -6.094 }}
 [[Minimax tuning]]:
 * [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 [[Tuning ranges]]:
@@ Line 38: / Line 39: @@
 {{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}
-[[Badness]] (Smith): 0.039163
+[[Badness]] (Sintel): 0.919
 === Overview to extensions ===
@@ Line 56: / Line 57: @@
 /224 is the [[marvel family|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the [[sensamagic family|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, {{multival| 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.
+By adding [[100/99]] and [[105/104]] to the list of commas, magic can be extended to the 11-limit and 13-limit. 11-limit magic 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. 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. For the 13-limit, 41edo makes for a recommendable tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 65: / Line 66: @@
 : mapping generators: ~2, ~5/4
-{{Multival|legend=1| 5 1 12 -10 5 25 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.651
+* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
-: [[error map]]: {{val| 0.000 +1.301 -5.662 -1.011 }}
+: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 }}
-* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.352
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
-: error map: {{val| 0.000 -0.195 -5.962 -4.602 }}
+: error map: {{val| 0.000 +0.333 -5.856 -3.335 }}
 [[Minimax tuning]]:
 * 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 [[Tuning ranges]]:
@@ Line 86: / Line 85: @@
 {{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}
-[[Badness]] (Smith): 0.018918
+[[Badness]] (Sintel): 0.479
 === 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
@@ Line 96: / Line 93: @@
 Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
-Wedgie: {{Multival| 5 1 12 -8 -10 5 -30 25 -22 -64 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.720
+* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.696
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}
 Minimax tuning:
 * 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
-: eigenmonzo (unchanged-interval) basis: 2.11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Tuning ranges:
@@ Line 113: / Line 108: @@
 {{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}
-Badness (Smith): 0.020352
+Badness (Sintel): 0.673
 ==== 13-limit ====
@@ Line 125: / Line 120: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.435
+* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.427
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}
 Tuning ranges:
@@ Line 134: / Line 129: @@
 {{Optimal ET sequence|legend=0| 19, 22f, 41 }}
-Badness (Smith): 0.021509
+Badness (Sintel): 0.889
 ===== Magical =====
@@ Line 144: / Line 139: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.510
+* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.604
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}
 {{Optimal ET sequence|legend=0| 19, 22f, 41 }}
-Badness (Smith): 0.020633
+Badness (Sintel): 1.05
-====== 19-limit ======
+====== Magicus ======
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 159: / Line 154: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.451
+* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}
 {{Optimal ET sequence|legend=0| 19, 41 }}
-Badness (Smith): 0.020881
+Badness (Sintel): 1.27
 ====== Magica ======
@@ Line 173: / Line 169: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.573
+* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}
 {{Optimal ET sequence|legend=0| 22fh, 41 }}
-Badness (Smith): 0.019945
+Badness (Sintel): 1.21
 ===== Magia =====
@@ Line 187: / Line 184: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.271
+* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}
 {{Optimal ET sequence|legend=0| 19g, 41, 60 }}
-Badness (Smith): 0.026232
+Badness (Sintel): 1.34
 ====== 19-limit ======
@@ Line 201: / Line 199: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.360
+* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}
 {{Optimal ET sequence|legend=0| 19gh, 41 }}
-Badness (Smith): 0.023709
+Badness (Sintel): 1.44
 ===== Evening =====
-Evening is a remarkable subgroup temperament of 19 &amp; 41 with prime harmonics of 29 and 31.
+Evening is a remarkable subgroup temperament of {{nowrap| 19 & 22f }} with prime harmonics of 29 and 31.
 Subgroup: 2.3.5.7.11.13.29.31
@@ Line 214: / Line 213: @@
 Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
-Sval mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
+Subgroup-val mapping: {{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
+* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.416
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}
 {{Optimal ET sequence|legend=0| 19, 22f, 41 }}
+Badness (Sintel): 0.807
 ==== Sorcery ====
@@ Line 230: / Line 231: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.674
+* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.477
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
 {{Optimal ET sequence|legend=0| 19, 22, 41f }}
-Badness (Smith): 0.025829
+Badness (Sintel): 1.07
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 52/51, 65/64, 78/77, 91/90, 100/99
-Mapping: {{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|legend=0| 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: {{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|legend=0| 19, 22 }}
-Badness (Smith): 0.023232
 ==== Necromancy ====
@@ Line 275: / Line 246: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.788
+* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.787
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}
 {{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
-Badness (Smith): 0.025275
+Badness (Sintel): 1.04
 ===== 17-limit =====
@@ Line 290: / Line 261: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.837
+* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.827
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}
 {{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}
-Badness (Smith): 0.022032
+Badness (Sintel): 1.12
-===== 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: {{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|legend=0| 19f, 22, 41 }}
-Badness (Smith): 0.021101
 ==== Soothsaying ====
@@ Line 320: / Line 276: @@
 Optimal tunings:
-* CTE: ~55/39 = 600.000, ~5/4 = 380.539
+* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
-* POTE: ~55/39 = 600.000, ~5/4 = 380.508
+* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}
 {{Optimal ET sequence|legend=0| 22, 60, 82 }}
-Badness (Smith): 0.055443
+Badness (Sintel): 2.29
 ===== 17-limit =====
@@ Line 335: / Line 291: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~5/4 = 380.553
+* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
-* POTE: ~17/12 = 600.000, ~5/4 = 380.508
+* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}
 {{Optimal ET sequence|legend=0| 22, 60, 82 }}
-Badness (Smith): 0.035654
+Badness (Sintel): 1.82
 ===== 19-limit =====
@@ Line 350: / Line 306: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~5/4 = 380.470
+* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
-* CWE: ~17/12 = 600.000, ~5/4 = 380.470
+* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}
 {{Optimal ET sequence|legend=0| 22, 60, 82 }}
-Badness (Smith): 0.031291
+Badness (Sintel): 1.90
 === Telepathy ===
@@ Line 363: / Line 319: @@
 Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
-Wedgie: {{Multival| 5 1 12 14 -10 5 5 25 29 -2 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 381.231
+* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 381.019
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}
 {{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}
-Badness (Smith): 0.027109
+Badness (Sintel): 0.896
 ==== 13-limit ====
@@ Line 382: / Line 336: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 381.196
+* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.520
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}
 {{Optimal ET sequence|legend=0| 19e, 22, 41ef }}
-Badness (Smith): 0.025522
+Badness (Sintel): 1.05
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 55/54, 65/64, 85/84, 91/90, 99/98
-Mapping: {{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|legend=0| 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: {{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|legend=0| 19egh, 22, 41efghh }}
-Badness (Smith): 0.019004
 ==== Intuition ====
@@ Line 427: / Line 351: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.816
+* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.483
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}
 {{Optimal ET sequence|legend=0| 19e, 22f }}
-Badness (Smith): 0.026089
+Badness (Sintel): 1.08
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 55/54, 66/65, 85/84, 99/98, 105/104
-Mapping: {{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|legend=0| 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: {{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|legend=0| 19egh, 22fh }}
-Badness (Smith): 0.019518
 === Horcrux ===
@@ Line 472: / Line 366: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 379.796
+* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 379.642
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}
 {{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
-Badness (Smith): 0.039282
+Badness (Sintel): 1.30
 ==== 13-limit ====
@@ Line 487: / Line 381: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 379.885
+* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 379.791
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}
 {{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}
-Badness (Smith): 0.031938
+Badness (Sintel): 1.32
 ===== 17-limit =====
@@ Line 502: / Line 396: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.216
+* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
-* CWE: ~2 = 1200.000, ~5/4 = 380.148
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}
 {{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}
-Badness (Smith): 0.028074
+Badness (Sintel): 1.43
 ===== Horcruxic =====
@@ Line 517: / Line 411: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 379.524
+* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
-* CWE: ~2 = 1200.000, ~5/4 = 379.571
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}
 {{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}
-Badness (Smith): 0.029556
+Badness (Sintel): 1.51
 ==== Glamour ====
@@ Line 532: / Line 426: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 379.757
+* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 379.116
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}
 {{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
-Badness (Smith): 0.033317
+Badness (Sintel): 1.38
 === Witchcraft ===
@@ Line 547: / Line 441: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.416
+* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.232
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{c}}
 {{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}
-Badness (Smith): 0.030706
+Badness (Sintel): 1.02
 ==== 13-limit ====
@@ Line 562: / Line 456: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.352
+* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.189
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{c}}
 {{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}
-Badness (Smith): 0.023547
+Badness (Sintel): 0.973
 ==== 17-limit ====
@@ Line 577: / Line 471: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 380.281
+* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 380.114
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{c}}
 {{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}
-Badness (Smith): 0.020756
+Badness (Sintel): 1.06
-==== 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: {{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|legend=0| 19egh, 41, 60eh }}
-Badness (Smith): 0.018625
 === Divination ===
@@ Line 607: / Line 486: @@
 Optimal tunings:
-* CTE: ~99/70 = 600.000, ~5/4 = 380.732
+* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
-* POTE: ~99/70 = 600.000, ~5/4 = 380.223
+* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}
 {{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}
-Badness (Smith): 0.035864
+Badness (Sintel): 1.19
 ==== 13-limit ====
@@ Line 622: / Line 501: @@
 Optimal tunings:
-* CTE: ~99/70 = 600.000, ~5/4 = 380.417
+* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
-* POTE: ~99/70 = 600.000, ~5/4 = 379.920
+* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}
 {{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
-Badness (Smith): 0.034551
+Badness (Sintel): 1.43
 ==== 17-limit ====
@@ Line 637: / Line 516: @@
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~5/4 = 380.433
+* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
-* CWE: ~17/12 = 600.000, ~5/4 = 380.067
+* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{c}}
 {{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
-Badness (Smith): 0.023775
+Badness (Sintel): 1.21
 === Hocus ===
@@ Line 649: / Line 528: @@
 Comma list: 225/224, 243/242, 245/242
-Mapping: {{mapping| 1 5 3 11 12 | 0 -10 -2 -24 -25 }}
+Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~14/11 = 409.760
+* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
-* POTE: ~2 = 1200.000, ~14/11 = 409.910
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}
 {{Optimal ET sequence|legend=0| 38d, 41, 120cd }}
-Badness (Smith): 0.038519
+Badness (Sintel): 1.27
 ==== 13-limit ====
@@ Line 664: / Line 543: @@
 Comma list: 105/104, 196/195, 243/242, 245/242
-Mapping: {{mapping| 1 5 3 11 12 16 | 0 -10 -2 -24 -25 -36 }}
+Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~14/11 = 409.842
+* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
-* POTE: ~2 = 1200.000, ~14/11 = 410.004
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}
 {{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}
-Badness (Smith): 0.030280
+Badness (Sintel): 1.25
 ==== 17-limit ====
@@ Line 679: / Line 558: @@
 Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
-Mapping: {{mapping| 1 5 3 11 12 16 14 | 0 -10 -2 -24 -25 -36 -29 }}
+Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~14/11 = 409.896
+* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
-* CWE: ~2 = 1200.000, ~14/11 = 409.994
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}
 {{Optimal ET sequence|legend=0| 38df, 41, 79d }}
-Badness (Smith): 0.025491
+Badness (Sintel): 1.30
 ==== 19-limit ====
@@ Line 694: / Line 573: @@
 Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
-Mapping: {{mapping| 1 5 3 11 12 16 14 8 | 0 -10 -2 -24 -25 -36 -29 -11 }}
+Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~14/11 = 409.884
+* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
-* CWE: ~2 = 1200.000, ~14/11 = 410.012
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}
 {{Optimal ET sequence|legend=0| 38df, 41, 79dh }}
-Badness (Smith): 0.020277
+Badness (Sintel): 1.23
 == Muggles ==
@@ Line 714: / Line 593: @@
 {{Mapping|legend=1| 1 0 2 5 | 0 5 1 -7 }}
-{{Multival|legend=1| 5 1 -7 -10 -25 -19 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~5/4 = 378.744
+* [[WE]]: ~2 = 1203.9554{{c}}, ~5/4 = 379.7269{{c}}
-: [[error map]]: {{val| 0.000 -8.235 -7.570 -20.034 }}
+: [[error map]]: {{val| +3.955 -3.321 +1.324 -7.137 }}
-* [[POTE]]: ~2 = 1200.000, ~5/4 = 378.479
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 378.5328{{c}}
-: error map: {{val| 0.000 -9.558 -7.834 -18.181 }}
+: error map: {{val| 0.000 -9.291 -7.781 -18.555 }}
 [[Tuning ranges]]:
@@ Line 730: / Line 607: @@
 {{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}
-[[Badness]] (Smith): 0.056206
+[[Badness]] (Sintel): 1.42
 === 11-limit ===
@@ Line 740: / Line 617: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 378.228
+* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 377.724
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}
 Tuning ranges:
@@ Line 749: / Line 626: @@
 {{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}
-Badness (Smith): 0.048038
+Badness (Sintel): 1.59
 ==== 13-limit ====
@@ Line 759: / Line 636: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 378.176
+* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 377.653
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}
 {{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}
-Badness (Smith): 0.030386
+Badness (Sintel): 1.26
 === Muggloid ===
@@ Line 774: / Line 651: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 377.724
+* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 377.832
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}
 {{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}
-Badness (Smith): 0.046970
+Badness (Sintel): 1.55
 ==== 13-limit ====
@@ Line 789: / Line 666: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 377.652
+* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
-* POTE: ~2 = 1200.000, ~5/4 = 377.838
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}
 {{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
-Badness (Smith): 0.028732
+Badness (Sintel): 1.19
-== Darkstone ==
+== Brightstone ==
-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 [http://www.tonalsoft.com/enc/m/magic.aspx ''Encyclopedia of Microtonal Music Theory'', Tonalsoft], this temperament is given a name ''witch''.
+Brightstone tempers out 64/63 and may be described as {{nowrap| 22 & 25 }}. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 36/35, 1875/1792
+[[Comma list]]: 64/63, 3125/3024
-{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
+{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
-{{Multival|legend=1| 5 1 9 -10 0 18 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~5/4 = 377.385
+* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
-: [[error map]]: {{val| 0.000 -15.028 -8.928 +27.643 }}
+: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
-* [[CWE]]: ~2 = 1200.000, ~5/4 = 376.963
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
-: error map: {{val| 0.000 -18.198 -9.562 +21.937 }}
+: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}
-{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}
+{{Optimal ET sequence|legend=1| 3, 19d, 22 }}
-[[Badness]] (Smith): 0.084213
+[[Badness]] (Sintel): 2.23
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 36/35, 45/44, 363/343
+Comma list: 55/54, 64/63, 625/616
-Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
+Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 377.388
+* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
-* CWE: ~2 = 1200.000, ~5/4 = 376.973
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}
-{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}
+{{Optimal ET sequence|legend=0| 22, 69b }}
-Badness (Smith): 0.046775
+Badness (Sintel): 1.85
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 27/26, 36/35, 45/44, 363/343
+Comma list: 40/39, 55/54, 64/63, 625/616
-Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
+Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 376.914
+* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
-* CWE: ~2 = 1200.000, ~5/4 = 376.422
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}
-{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}
+{{Optimal ET sequence|legend=0| 22f, 47bff }}
+Badness (Sintel): 1.99
+== Darkstone ==
+Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.
-Badness (Smith): 0.038328
+This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].
-== Brightstone ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 64/63, 3125/3024
+[[Comma list]]: 36/35, 1875/1792
-{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
+{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
-{{Multival|legend=1| 5 1 -10 -10 -30 -26 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~5/4 = 381.955
+* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
-: [[error map]]: {{val| 0.000 +7.818 -4.359 +11.627 }}
+: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
-* [[CWE]]: ~2 = 1200.000, ~5/4 = 381.956
+* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 376.9630{{c}}
-: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}
+: error map: {{val| 0.000 -17.140 -9.351 +23.841 }}
-{{Optimal ET sequence|legend=1| 3, 19d, 22 }}
+{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}
-[[Badness]] (Smith): 0.088072
+[[Badness]] (Sintel): 2.13
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 64/63, 100/99, 605/588
+Comma list: 36/35, 45/44, 363/343
-Mapping: {{mapping| 1 0 2 6 6 | 0 5 1 -10 -8 }}
+Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 381.790
+* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
-* CWE: ~2 = 1200.000, ~5/4 = 381.712
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}
-{{Optimal ET sequence|legend=0| 3, 19d, 22 }}
+{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}
-Badness (Smith): 0.047379
+Badness (Sintel): 1.55
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 64/63, 65/63, 100/99, 169/165
+Comma list: 27/26, 36/35, 45/44, 363/343
-Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 -8 -1 }}
+Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~5/4 = 381.732
+* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
-* CWE: ~2 = 1200.000, ~5/4 = 381.736
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}
+{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}
+Badness (Sintel): 1.58
-{{Optimal ET sequence|legend=0| 3, 22 }}
+Scales: [[User:BudjarnLambeth/Volcanic glass]]
-Badness (Smith): 0.039703
+Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)
 == Hocum ==
@@ Line 901: / Line 782: @@
 [[Comma list]]: 3125/3072, 4000/3969
-{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}
+{{Mapping|legend=1| 1 -5 1 14 | 0 10 2 -17 }}
-: mapping generators: ~2, ~63/50
-{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
+: mapping generators: ~2, ~63/40
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~63/50 = 409.836
+* [[WE]]: ~2 = 1200.8375{{c}}, ~63/40 = 790.7032{{c}}
-: [[error map]]: {{val| 0.000 -0.316 -5.986 -1.612 }}
+: [[error map]]: {{val| +0.838 +0.890 -4.070 +0.944 }}
-* [[POTE]]: ~2 = 1200.000, ~63/50 = 410.108
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
-: error map: {{val| 0.000 -0.437 -6.010 -1.406 }}
+: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}
 {{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}
-[[Badness]] (Smith): 0.107115
+[[Badness]] (Sintel): 2.71
 == Trismegistus ==
+{{Main| Mabilic and trismegistus }}
 {{See also| No-threes subgroup temperaments #Mabilic }}
@@ Line 924: / Line 804: @@
 [[Comma list]]: 1029/1024, 3125/3072
-{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}
+{{Mapping|legend=1| 1 -5 1 5 | 0 15 3 -5 }}
-: mapping generators: ~2, ~147/100
-{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
+: mapping generators: ~2, ~168/125
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~147/100 = 673.187
+* [[WE]]: ~2 = 1201.0799{{c}}, ~168/125 = 527.1841{{c}}
-: [[error map]]: {{val| 0.000 +0.240 -5.875 -2.891 }}
+: [[error map]]: {{val| +1.080 +0.408 -3.681 +0.653 }}
-* [[POTE]]: ~2 = 1200.000, ~147/100 = 673.290
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~168/125 = 526.7349{{c}}
 : error map: {{val| 0.000 -0.932 -6.109 -2.500 }}
 {{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}
-[[Badness]] (Smith): 0.098334
+[[Badness]] (Sintel): 2.49
 === 11-limit ===
@@ Line 945: / Line 823: @@
 Comma list: 385/384, 441/440, 625/616
-Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}
+Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/15 = 673.241
+* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
-* POTE: ~2 = 1200.000, ~22/15 = 673.340
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}
 {{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
-Badness (Smith): 0.045623
+Badness (Sintel): 1.51
 === 13-limit ===
@@ Line 960: / Line 838: @@
 Comma list: 105/104, 144/143, 275/273, 625/616
-Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}
+Mapping: {{mapping| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/15 = 673.294
+* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
-* POTE: ~2 = 1200.000, ~22/15 = 673.359
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6548{{c}}
 {{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
-Badness (Smith): 0.033081
+Badness (Sintel): 1.37
+=== 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: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
+Optimal tunings:
+* WE: ~2 = 1200.5832{{c}}, ~15/11 = 526.8804{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6368{{c}}
+{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
+Badness (Sintel): 1.26
 == Quadrimage ==
@@ Line 975: / Line 868: @@
 [[Comma list]]: 2401/2400, 3125/3072
-{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}
+{{Mapping|legend=1| 1 -15 -1 -3 | 0 20 4 7 }}
-: mapping generators: ~2, ~28/25
+: mapping generators: ~2, ~25/14
-{{Multival|legend=1| 20 4 7 -40 -45 5 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~28/25 = 204.860
+* [[WE]]: ~2 = 1201.2708{{c}}, ~25/14 = 996.0669{{c}}
-: [[error map]]: {{val| 0.000 +0.853 -5.752 -2.843 }}
+: [[error map]]: {{val| +1.271 +0.322 -3.317 -0.170 }}
-* [[POTE]]: ~2 = 1200.000, ~28/25 = 204.987
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
-: error map: {{val| 0.000 -1.691 -6.261 -3.733 }}
+: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}
-{{Optimal ET sequence|legend=1| 6, 29b, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
+{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
-[[Badness]] (Smith): 0.127422
+[[Badness]] (Sintel): 3.22
 === 11-limit ===
@@ Line 996: / Line 887: @@
 Comma list: 245/242, 385/384, 625/616
-Mapping: {{mapping| 1 5 3 4 5 | 0 -20 -4 -7 -9 }}
+Mapping: {{mapping| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~28/25 = 204.881
+* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
-* POTE: ~2 = 1200.000, ~28/25 = 204.956
+* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}
 {{Optimal ET sequence|legend=0| 6, 35, 41 }}
-Badness (Smith): 0.061572
+Badness (Sintel): 2.04
 === 13-limit ===
@@ Line 1,011: / Line 902: @@
 Comma list: 105/104, 144/143, 245/242, 625/616
-Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}
+Mapping: {{mapping| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~28/25 = 204.956
+* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
-* POTE: ~2 = 1200.000, ~28/25 = 205.028
+* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}
 {{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}
-Badness (Smith): 0.044047
+Badness (Sintel): 1.82
 == Quinmage ==
@@ Line 1,029: / Line 920: @@
 : mapping generators: ~2, ~48/35
-{{Multival|legend=1| 25 5 19 -50 -40 30 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~48/35 = 556.123
+* [[WE]]: ~2 = 1201.3334{{c}}, ~48/35 = 556.6311{{c}}
-: [[error map]]: {{val| 0.000 +1.132 -5.696 -2.480 }}
+: error map: {{val| +1.333 +0.489 -3.158 -0.835 }}
-* [[CWE]]: ~2 = 1200.000, ~48/35 = 556.050
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 556.0504{{c}}
 : error map: {{val| 0.000 -0.695 -6.062 -3.868 }}
 {{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}
-[[Badness]] (Smith): 0.194548
+[[Badness]] (Sintel): 4.92
 === 11-limit ===
@@ Line 1,050: / Line 939: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/8 = 556.122
+* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
-* CWE: ~2 = 1200.000, ~11/8 = 556.095
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}
 {{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
-Badness (Smith): 0.101724
+Badness (Sintel): 3.36
 === 13-limit ===
@@ Line 1,065: / Line 954: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/8 = 556.106
+* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
-* CWE: ~2 = 1200.000, ~11/8 = 556.117
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}
 {{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
-Badness (Smith): 0.067742
+Badness (Sintel): 2.80
 == Warlock ==
@@ Line 1,082: / Line 971: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~8/7 = 240.000, ~5/4 = 380.499 (~256/245 = 99.501)
+* [[WE]]: ~8/7 = 240.3877{{c}}, ~5/4 = 380.4267{{c}} (~256/245 = 100.3488{{c}})
-: [[error map]]: {{val| 0.000 +0.542 -5.814 -8.826 }}
+: error map: {{val| +1.939 +0.178 -2.010 -3.398 }}
-* [[CWE]]: ~8/7 = 240.000, ~5/4 = 379.997 (~256/245 = 100.003)
+* [[CWE]]: ~8/7 = 240.0000{{c}}, ~5/4 = 379.9965{{c}} (~256/245 = 100.0035{{c}})
 : error map: {{val| 0.000 -1.972 -6.317 -8.826 }}
 {{Optimal ET sequence|legend=1| 25, 35, 60 }}
-[[Badness]] (Smith): 0.287190
+[[Badness]] (Sintel): 7.27
 [[Category:Temperament families]]

Magic family: Difference between revisions

Latest revision as of 01:34, 12 May 2026

Magic

Overview to extensions

Septimal magic

11-limit

13-limit

Magical

Magicus

Magica

Magia

19-limit

Evening

Sorcery

Necromancy

17-limit

Soothsaying

17-limit

19-limit

Telepathy

13-limit

Intuition

Horcrux

13-limit

17-limit

Horcruxic

Glamour

Witchcraft

13-limit

17-limit

Divination

13-limit

17-limit

Hocus

13-limit

17-limit

19-limit

Muggles

11-limit

13-limit

Muggloid

13-limit

Brightstone

11-limit

13-limit

Darkstone

11-limit

13-limit

Hocum

Trismegistus

11-limit

13-limit

2.3.5.7.11.13.19 subgroup

Quadrimage

11-limit

13-limit

Quinmage

11-limit

13-limit

Warlock

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