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


= Five limit magic =
== Five limit magic ==
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 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)<sup>5</sup> = 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|19EDO]] or [[22edo|22EDO]] is always possible.
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 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)<sup>5</sup> = 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|19EDO]] or [[22edo|22EDO]] is always possible.


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[[Badness]]: 0.039163
[[Badness]]: 0.039163


== Seven-limit extensions ==
=== Seven-limit 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 magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.
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 magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.


= Magic =
== Magic ==
{{main| Magic }}
{{main| Magic }}


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[[Badness]]: 0.018918
[[Badness]]: 0.018918


== 11-limit ==
=== 11-limit ===
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)


Line 102: Line 102:
Badness: 0.020352
Badness: 0.020352


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 120: Line 120:
Badness: 0.021509
Badness: 0.021509


=== Sorcery ===
==== Sorcery ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.025829
Badness: 0.025829


=== Necromancy ===
==== Necromancy ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.025275
Badness: 0.025275


=== Soothsaying ===
==== Soothsaying ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.055443
Badness: 0.055443


== Telepathy ==
=== Telepathy ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 182: Line 182:
Badness: 0.027109
Badness: 0.027109


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.025522
Badness: 0.025522


== Horcrux ==
=== Horcrux ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Badness: 0.039282
Badness: 0.039282


== Divination ==
=== Divination ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 221: Line 221:
Badness: 0.035864
Badness: 0.035864


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 234: Line 234:
Badness: 0.034551
Badness: 0.034551


== Witchcraft ==
=== Witchcraft ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 247: Line 247:
Badness: 0.030706
Badness: 0.030706


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.023547
Badness: 0.023547


== Hocus ==
=== Hocus ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Badness: 0.038519
Badness: 0.038519


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 286: Line 286:
Badness: 0.030280
Badness: 0.030280


= Muggles =
== Muggles ==
Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo|19EDO]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices.  
Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo|19EDO]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices.  


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[[Badness]]: 0.056206
[[Badness]]: 0.056206


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 316: Line 316:
Badness: 0.048038
Badness: 0.048038


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.030386
Badness: 0.030386


= Astrology =
== Astrology ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 344: Line 344:
[[Badness]]: 0.082673
[[Badness]]: 0.082673


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 357: Line 357:
Badness: 0.039151
Badness: 0.039151


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor]


=== Horoscope ===
==== Horoscope ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.035284
Badness: 0.035284


= Spell =
== Spell ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 401: Line 401:
[[Badness]]: 0.080958
[[Badness]]: 0.080958


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 414: Line 414:
Badness: 0.059791
Badness: 0.059791


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 427: Line 427:
Badness: 0.045591
Badness: 0.045591


=== Cantrip ===
==== Cantrip ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 440: Line 440:
Badness: 0.041603
Badness: 0.041603


= Hocum =
== Hocum ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 455: Line 455:
[[Badness]]: 0.107115
[[Badness]]: 0.107115


= Trismegistus =
== Trismegistus ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 470: Line 470:
[[Badness]]: 0.098334
[[Badness]]: 0.098334


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 483: Line 483:
Badness: 0.045623
Badness: 0.045623


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 496: Line 496:
Badness: 0.033081
Badness: 0.033081


= Quadrimage =
== Quadrimage ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


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[[Badness]]: 0.127422
[[Badness]]: 0.127422


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 524: Line 524:
Badness: 0.061572
Badness: 0.061572


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Revision as of 21:45, 1 June 2021

The magic family of temperaments tempers out 3125/3072, the small diesis or magic comma. A magic temperament is optimal, for some searches, in the 9-odd-limit. It has slightly higher complexity than meantone and is also closer to just intonation. It is the simplest rank-2 temperament that tunes every 9-odd-limit interval better than is possible in 12EDO. Properties may depend on tuning and extension.

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

Five limit magic

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

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

Mapping generators: ~2, ~5/4

POTE generator: ~5/4 = 380.058

Minimax tuning:

[[1 0 0⟩, [0 1 0⟩, [2 1/5 0⟩]
Eigenmonzos: 2, 3

Tuning ranges:

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

Template:Val list

Badness: 0.039163

Seven-limit extensions

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

Magic

Main article: Magic

Magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. 41EDO is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.

Magic, with its accurate fifths, works well with 9-odd-limit harmony. It is more accurate than meantone and simpler than garibaldi. It is a little tricky to work with because in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.

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

245/243, the sensamagic comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.

By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, ⟨⟨ 5 1 12 -8 … ]]. For this, 104EDO provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1], ⟨0 5 1 12]]

Mapping generators: ~2, ~5/4

Wedgie: ⟨⟨ 5 1 12 -10 5 25 ]]

POTE generator: ~5/4 = 380.352

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [2 1/5 0 0⟩, [-1 12/5 0 0⟩]
Eigenmonzos: 2, 3

Tuning ranges:

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

Template:Val list

Badness: 0.018918

11-limit

Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 100/99

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

POTE generator: ~5/4 = 380.696

Tuning ranges:

Vals: Template:Val list

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]]

POTE generator: ~5/4 = 380.427

Tuning ranges:

Vals: Template:Val list

Badness: 0.021509

Sorcery

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 100/99

Mapping: [⟨1 0 2 -1 6 4], ⟨0 5 1 12 -8 -1]]

POTE generator: ~5/4 = 380.477

Tuning ranges:

Vals: Template:Val list

Badness: 0.025829

Necromancy

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 275/273

Mapping: [⟨1 0 2 -1 6 11], ⟨0 5 1 12 -8 -23]]

POTE generator: ~5/4 = 380.787

Tuning ranges:

Vals: Template:Val list

Badness: 0.025275

Soothsaying

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 1352/1331

Mapping: [⟨2 0 4 -2 12 15], ⟨0 5 1 12 -8 -12]]

POTE generator: ~5/4 = 380.508

Vals: Template:Val list

Badness: 0.055443

Telepathy

Subgroup: 2.3.5.7.11

Comma list: 55/54, 99/98, 176/175

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

POTE generator: ~5/4 = 381.019

Vals: Template:Val list

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]]

POTE generator: ~5/4 = 380.520

Vals: Template:Val list

Badness: 0.025522

Horcrux

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~5/4 = 379.642

Vals: Template:Val list

Badness: 0.039282

Divination

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 245/243

Mapping: [⟨2 0 4 -2 5], ⟨0 5 1 12 3]]

POTE generator: ~5/4 = 380.233

Vals: Template:Val list

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]]

POTE generator: ~5/4 = 379.920

Vals: Template:Val list

Badness: 0.034551

Witchcraft

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~5/4 = 380.232

Vals: Template:Val list

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]]

POTE generator: ~5/4 = 380.189

Vals: Template:Val list

Badness: 0.023547

Hocus

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~14/11 = 409.910

Vals: Template:Val list

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]]

POTE generator: ~14/11 = 410.004

Vals: Template:Val list

Badness: 0.030280

Muggles

Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is 19EDO, in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices.

Subgroup: 2.3.5.7

Comma list: 126/125, 525/512

Mapping: [⟨1 0 2 5], ⟨0 5 1 -7]]

Wedgie: ⟨⟨ 5 1 -7 -10 -25 -19 ]]

POTE generator: ~5/4 = 378.479

Template:Val list

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]]

POTE generator: ~5/4 = 377.724

Vals: Template:Val list

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]]

POTE generator: ~5/4 = 377.724

Vals: Template:Val list

Badness: 0.030386

Astrology

Subgroup: 2.3.5.7

Comma list: 50/49, 3125/3072

Mapping: [⟨2 0 4 5], ⟨0 5 1 1]]

Wedgie: ⟨⟨ 10 2 2 -20 -25 -1 ]]

POTE generator: ~5/4 = 380.578

Template:Val list

Badness: 0.082673

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 176/175

Mapping: [⟨2 0 4 5 5], ⟨0 5 1 1 3]]

POTE generator: ~5/4 = 380.530

Vals: Template:Val list

Badness: 0.039151

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/64, 78/77, 121/120

Mapping: [⟨2 0 4 5 5 8], ⟨0 5 1 1 3 -1]]

POTE generator: ~5/4 = 379.787

Vals: Template:Val list

Badness: 0.034376

Music

Horoscope

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 66/65, 105/104, 121/120

Mapping: [⟨2 0 4 5 5 3], ⟨0 5 1 1 3 7]]

POTE generator: ~5/4 = 379.837

Vals: Template:Val list

Badness: 0.035284

Spell

Subgroup: 2.3.5.7

Comma list: 49/48, 3125/3072

Mapping: [⟨1 0 2 2], ⟨0 10 2 5]]

Wedgie: ⟨⟨ 10 2 5 -20 -20 6 ]]

POTE generator: ~28/25 = 189.927

Template:Val list

Badness: 0.080958

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 125/121

Mapping: [⟨1 0 2 2 3], ⟨0 10 2 5 3]]

POTE generator: ~11/10 = 190.285

Vals: Template:Val list

Badness: 0.059791

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: [⟨1 0 2 2 3 4], ⟨0 10 2 5 3 -2]]

POTE generator: ~11/10 = 189.928

Vals: Template:Val list

Badness: 0.045591

Cantrip

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: [⟨1 0 2 2 3 1], ⟨0 10 2 5 3 17]]

POTE generator: ~11/10 = 190.360

Vals: Template:Val list

Badness: 0.041603

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 5 3 -3], ⟨0 -10 -2 17]]

Wedgie: ⟨⟨ 10 2 -17 -20 -55 -45 ]]

POTE generator: ~63/50 = 400.108

Template:Val list

Badness: 0.107115

Trismegistus

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 10 4 0], ⟨0 -15 -3 5]]

Wedgie: ⟨⟨ 15 3 -5 -30 -50 -20 ]]

POTE generator: ~147/100 = 673.290

Template:Val list

Badness: 0.098334

11-limit

Subgroup: 2.3.5.7.11

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

POTE generator: ~22/15 = 673.340

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

Vals: Template:Val list

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]]

POTE generator: ~22/15 = 673.359

Vals: Template:Val list

Badness: 0.033081

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 5 3 4], ⟨0 -20 -4 -7]]

Wedgie: ⟨⟨ 20 4 7 -40 -45 5 ]]

POTE generator: ~28/25 = 204.987

Template:Val list

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]]

POTE generator: ~28/25 = 204.956

Vals: Template:Val list

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]]

POTE generator: ~28/25 = 205.028

Vals: Template:Val list

Badness: 0.044047

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