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{{Mapping|legend=1| 1 0 2 | 0 5 1 }}
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}


: mapping generators: ~2, ~5/4
: Mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.4994
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.4994
Line 38: Line 38:


=== Overview to extensions ===
=== Overview to extensions ===
Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|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.  
Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|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, and warlock. Discussed elsewhere are
Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
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{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}


: mapping generators: ~2, ~5/4
: Mapping generators: ~2, ~5/4


{{Multival|legend=1| 5 1 12 -10 5 25 }}
{{Multival|legend=1| 5 1 12 -10 5 25 }}
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== Darkstone ==
== 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 [http://www.tonalsoft.com/enc/m/magic.aspx ''Encyclopedia of Microtonal Music Theory'', Tonalsoft], this temperament is given a name ''witch''.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Brightstone ==
== Brightstone ==
{{see also| Archytas clan }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 381.955
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 381.955


{{Optimal ET sequence|legend=1| 19d, 22 }}
{{Optimal ET sequence|legend=1| 3, …, 19d, 22 }}


[[Badness]]: 0.088072
[[Badness]]: 0.088072
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Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.790
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.790


Optimal ET sequence: {{Optimal ET sequence| 19d, 22 }}
Optimal ET sequence: {{Optimal ET sequence| 3, …, 19d, 22 }}


Badness: 0.047379
Badness: 0.047379
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 65/63, 100/99, 169/165
Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 -8 -1 }}
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.732
Optimal ET sequence: {{Optimal ET sequence| 3, …, 19d, 22 }}
Badness: 0.039703


== Hocum ==
== Hocum ==
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{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}
{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}


: mapping generators: ~2, ~63/50
: Mapping generators: ~2, ~63/50


{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~63/50 = 410.108
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~63/50 = 409.836


{{Optimal ET sequence|legend=1| 38, 41, 161c, 202c, 243c, 284c }}
{{Optimal ET sequence|legend=1| 3, …, 35d, 38, 41, 202cc, 243cc, 284cc }}


[[Badness]]: 0.107115
[[Badness]]: 0.107115
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{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}
{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}


: mapping generators: ~2, ~147/100
: Mapping generators: ~2, ~147/100


{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/100 = 673.290
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~147/100 = 673.187


{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}
{{Optimal ET sequence|legend=1| 16, 25, 41 }}


[[Badness]]: 0.098334
[[Badness]]: 0.098334
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Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}
Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.340
Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.241


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139c, 180c }}
Optimal ET sequence: {{Optimal ET sequence| 16, 25e, 41, 221cc, 262ccde }}


Badness: 0.045623
Badness: 0.045623
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Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}
Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.359
Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.294


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139cf }}
Optimal ET sequence: {{Optimal ET sequence| 16, 25e, 41, 139cf, 180cf, 221ccf }}


Badness: 0.033081
Badness: 0.033081
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{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}
{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}


: mapping generators: ~2, ~28/25
: Mapping generators: ~2, ~28/25


{{Multival|legend=1| 20 4 7 -40 -45 5 }}
{{Multival|legend=1| 20 4 7 -40 -45 5 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 204.987
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 204.860


{{Optimal ET sequence|legend=1| 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
{{Optimal ET sequence|legend=1| 6, …, 29b, 35, 41 }}


[[Badness]]: 0.127422
[[Badness]]: 0.127422
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Mapping: {{mapping| 1 5 3 4 5 | 0 -20 -4 -7 -9 }}
Mapping: {{mapping| 1 5 3 4 5 | 0 -20 -4 -7 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 204.956
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.881


{{Optimal ET sequence|legend=1| 35, 41 }}
Optimal ET sequence: {{Optimal ET sequence| 6, …, 29b, 35, 41 }}


Badness: 0.061572
Badness: 0.061572
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Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}
Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 205.028
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.956


{{Optimal ET sequence|legend=1| 35f, 41, 117c }}
Optimal ET sequence: {{Optimal ET sequence| 35f, 41, 199ccdef, 240ccdef, 281ccdeff }}


Badness: 0.044047
Badness: 0.044047
== Quinmage ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3125/3072, 16875/16807
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}
{{Multival|legend=1| 25 5 19 -50 -40 30 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~48/35 = 556.123
{{Optimal ET sequence|legend=1| 13b, 28b, 41 }}
[[Badness]]: 0.194548
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 625/616, 2401/2376
Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}
Optimal tuning (CTE): ~2 = 1\1, ~11/8 = 556.122
Optimal ET sequence: {{Optimal ET sequence| 13b, 28b, 41 }}
Badness: 0.101724
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 364/363, 385/384, 625/616
Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
Optimal tuning (CTE): ~2 = 1\1, ~11/8 = 556.106
Optimal ET sequence: {{Optimal ET sequence| 13b, 28b, 41 }}
Badness: 0.067742


== Warlock ==
== Warlock ==
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{{Mapping|legend=1| 5 0 10 14 | 0 5 1 0 }}
{{Mapping|legend=1| 5 0 10 14 | 0 5 1 0 }}


: mapping generators: ~8/7, ~5/4
: Mapping generators: ~8/7, ~5/4


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~5/4 = 379.7131
[[Optimal tuning]] ([[CTE]]): ~8/7 = 1\5, ~5/4 = 380.499


{{Optimal ET sequence|legend=1| 25, 35, 60 }}
{{Optimal ET sequence|legend=1| 25, 35, 60 }}

Revision as of 02:27, 8 October 2023

The magic family of temperaments tempers out 3125/3072, the small diesis or magic comma. The septimal version of magic is optimal, for some searches, in the 9-odd-limit. It has slightly higher complexity than meantone and is also closer to just intonation. It is the simplest rank-2 temperament that tunes every 9-odd-limit interval better than is possible in 12edo. The most prominent deficiency is that it lacks proper or nearly-proper mos scales in the 5- to 10-note region. Properties may depend on tuning and extension.

Magic

Main article: Magic

The monzo of the magic comma is [-10 -1 5⟩, and flipping that yields ⟨⟨ 5 1 -10 ]] for the wedgie. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)5 = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the optimal patent val generator, also makes a lot of sense, and using 19edo or 22edo is always possible.

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

Mapping generators: ~2, ~5/4

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.4994

Minimax tuning:

eigenmonzo (unchanged-interval) 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)
  • 5-odd-limit diamond monotone and tradeoff: ~5/4 = [378.910, 386.314]

Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.

Optimal ET sequence: 3, 13b, 16, 19, 22, 41

Badness: 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

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, ⟨⟨ 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 ]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.6512

Minimax tuning:

eigenmonzo (unchanged-interval) 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)
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

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

Optimal ET sequence: 19, 22, 41, 104, 145c, 186c

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7200

Minimax tuning:

  • 11-odd-limit: ~5/4 = [1/3 1/9 0 0 -1/18⟩
Eigenmonzo (unchanged-interval) 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)
  • 11-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

Optimal ET sequence: 19, 22, 41, 104, 145c, 249cce

Badness: 0.020352

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.4354

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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

Optimal ET sequence: 19, 22f, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.5103

Optimal ET sequence: 19, 22f, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.4507

Optimal ET sequence: 19, 22f, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.5733

Optimal ET sequence: 19h, 22fh, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.2706

Optimal ET sequence: 19g, 41, 101cde, 142cdefg

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.3604

Optimal ET sequence: 19g, 41

Badness: 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 tuning (POTE): ~2 = 1\1, ~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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.6741

Optimal ET sequence: 19, 22, 41f

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7839

Optimal ET sequence: 19, 22, 41f

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.6248

Optimal ET sequence: 19, 22, 41f

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7876

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

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.8373

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

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7411

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

Badness: 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 tuning (CTE): ~55/39 = 1\2, ~5/4 = 380.5385

Optimal ET sequence: 22, 60, 82

Badness: 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 tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.5533

Optimal ET sequence: 22, 60, 82

Badness: 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 tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.4704

Optimal ET sequence: 22, 60, 82

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 381.2311

Optimal ET sequence: 19e, 22, 63e, 85ee

Badness: 0.027109

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.1957

Optimal ET sequence: 19e, 22, 63eff

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 381.2884

Optimal ET sequence: 19eg, 22, 63effg

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 381.4641

Optimal ET sequence: 19egh, 22

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.8157

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

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.9518

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

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.8658

Optimal ET sequence: 19egh, 22fh, 41eg, 63efg

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 379.7959

Optimal ET sequence: 3de, …, 16d, 19, 41ee, 60ee

Badness: 0.039282

Charisma

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 245/243

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 11 18]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.8847

Optimal ET sequence: 3def, …, 19, 41ee, 60ee

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.2159

Optimal ET sequence: 3defg, …, 19g, 41eeg

Badness: 0.028074

Charismic

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 tuning (CTE): ~2 = 1\1, ~5/4 = 379.5237

Optimal ET sequence: 3defg, …, 19

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 379.7574

Optimal ET sequence: 3de, …, 16d, 19, 41eef, 60eeff

Badness: 0.033317

Witchcraft

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 441/440

Mapping: [⟨1 0 2 -1 -7], ⟨0 5 1 12 33]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.4163

Optimal ET sequence: 19e, 22e, 41

Badness: 0.030706

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243, 275/273

Mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.3523

Optimal ET sequence: 19e, 41, 142cdf

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.2809

Optimal ET sequence: 19eg, 41, 101cd, 142cdfg

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.3389

Optimal ET sequence: 19egh, 41, 101cdhh, 142cdfghh

Badness: 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 tuning (CTE): ~99/70 = 1\2, ~5/4 = 380.7317

Optimal ET sequence: 22, 60e, 82e, 104e

Badness: 0.035864

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4], ⟨0 5 1 12 3 18]]

Optimal tuning (CTE): ~99/70 = 1\2, ~5/4 = 380.4171

Optimal ET sequence: 22f, 60e, 82ef

Badness: 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 tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.4329

Optimal ET sequence: 22f, 60e, 82ef

Badness: 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 tuning (CTE): ~2 = 1\1, ~14/11 = 409.7603

Optimal ET sequence: 38d, 41

Badness: 0.038519

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 3 11 12 16], ⟨0 -10 -2 -24 -25 -36]]

Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8421

Optimal ET sequence: 38df, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~14/11 = 409.8958

Optimal ET sequence: 38df, 41, 120cdg, 161cdg, 202ccddfgg

Badness: 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 tuning (CTE): ~2 = 1\1, ~14/11 = 409.8836

Optimal ET sequence: 38df, 41, 120cdgh, 161cdgh, 202ccddfgghh

Badness: 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]]

Wedgie: ⟨⟨ 5 1 -7 -10 -25 -19 ]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 378.744

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]
  • 7-odd-limit diamond monotone and tradeoff: ~5/4 = [375.882, 378.947]
  • 9-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947

Optimal ET sequence: 16, 19

Badness: 0.056206

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 126/125, 385/384

Mapping: [⟨1 0 2 5 0], ⟨0 5 1 -7 11]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 378.228

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
  • 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]
  • 11-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947

Optimal ET sequence: 16, 19, 54bd, 73bcd, 92bcdde

Badness: 0.048038

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 126/125

Mapping: [⟨1 0 2 5 0 4], ⟨0 5 1 -7 11 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.653

Optimal ET sequence: 16, 19, 54bdf, 73bcdff, 92bcddeff

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 377.724

Optimal ET sequence: 3, 10bd, 13bd, 16, 19e, 35ee

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 377.652

Optimal ET sequence: 3, 10bd, 13bd, 16, 19e, 35eef

Badness: 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]]

Wedgie: ⟨⟨ 5 1 9 -10 0 18 ]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.385

Optimal ET sequence: 3d, …, 13b, 16, 19d, 35d

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 377.388

Optimal ET sequence: 3de, …, 13be, 16, 19d, 35d

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 376.914

Optimal ET sequence: 3def, …, 16, 35d

Badness: 0.038328

Brightstone

See also: Archytas clan

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Wedgie: ⟨⟨ 5 1 -10 -10 -30 -26 ]]

Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.955

Optimal ET sequence: 3, …, 19d, 22

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 381.790

Optimal ET sequence: 3, …, 19d, 22

Badness: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 381.732

Optimal ET sequence: 3, …, 19d, 22

Badness: 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

Wedgie: ⟨⟨ 10 2 -17 -20 -55 -45 ]]

Optimal tuning (CTE): ~2 = 1\1, ~63/50 = 409.836

Optimal ET sequence: 3, …, 35d, 38, 41, 202cc, 243cc, 284cc

Badness: 0.107115

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

Wedgie: ⟨⟨ 15 3 -5 -30 -50 -20 ]]

Optimal tuning (CTE): ~2 = 1\1, ~147/100 = 673.187

Optimal ET sequence: 16, 25, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~22/15 = 673.241

Optimal ET sequence: 16, 25e, 41, 221cc, 262ccde

Badness: 0.045623

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 10 4 0 13 11], ⟨0 -15 -3 5 -17 -13]]

Optimal tuning (CTE): ~2 = 1\1, ~22/15 = 673.294

Optimal ET sequence: 16, 25e, 41, 139cf, 180cf, 221ccf

Badness: 0.033081

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

Wedgie: ⟨⟨ 20 4 7 -40 -45 5 ]]

Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.860

Optimal ET sequence: 6, …, 29b, 35, 41

Badness: 0.127422

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 3 4 5], ⟨0 -20 -4 -7 -9]]

Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.881

Optimal ET sequence: 6, …, 29b, 35, 41

Badness: 0.061572

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 3 4 5 9], ⟨0 -20 -4 -7 -9 -31]]

Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 204.956

Optimal ET sequence: 35f, 41, 199ccdef, 240ccdef, 281ccdeff

Badness: 0.044047

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

Wedgie: ⟨⟨ 25 5 19 -50 -40 30 ]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 556.123

Optimal ET sequence: 13b, 28b, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~11/8 = 556.122

Optimal ET sequence: 13b, 28b, 41

Badness: 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 tuning (CTE): ~2 = 1\1, ~11/8 = 556.106

Optimal ET sequence: 13b, 28b, 41

Badness: 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 tuning (CTE): ~8/7 = 1\5, ~5/4 = 380.499

Optimal ET sequence: 25, 35, 60

Badness: 0.287190

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