Magic family: Difference between revisions

Revision as of 17:08, 11 October 2021

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.

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. Weak extensions include astrology, spell, hocum, trismegistus, and quadrimage.

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

POTE generator: ~5/4 = 380.058

Minimax tuning:

5-odd-limit

[[1 0 0⟩, [0 1 0⟩, [2 1/5 0⟩]

Eigenmonzos (unchanged intervals): 2, 3

Tuning ranges:

Diamond monotone range: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
Diamond tradeoff range: ~5/4 = [378.910, 386.314]
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.

Template:Val list

Badness: 0.039163

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 MOSes 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 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:

7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [2 1/5 0 0⟩, [-1 12/5 0 0⟩]

Eigenmonzos (unchanged intervals): 2, 3

Tuning ranges:

Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
Diamond tradeoff range: ~5/4 = [378.910, 386.314]
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.

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

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

POTE generator: ~5/4 = 380.696

Tuning ranges:

Diamond monotone range: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
Diamond tradeoff range: ~5/4 = [378.910, 386.314]
Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]

@@ Line 5: / Line 5: @@
 | ja =
 }}
-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 family|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 [[12 EDO]]. Properties may depend on tuning and extension.
+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 [[Rothenberg propriety|proper]] or nearly-proper [[MOS scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.
-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.
+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. Weak extensions include astrology, spell, hocum, trismegistus, and quadrimage.
-== Five-limit magic ==
+== 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 [[19 EDO]] or [[22 EDO]] is always possible.
+{{Main| Magic }}
+The monzo of the magic comma 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]] or [[22edo]] is always possible.
 Subgroup: 2.3.5
@@ Line 38: / Line 40: @@
 [[Badness]]: 0.039163
-=== Extensions ===
+== Septimal magic ==
-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 ==
 {{main| Magic }}
-Magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41 EDO]] 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.
+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 MOSes 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 temperament|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.
+This temperament, with its accurate fifths, works well with [[9-odd-limit]] harmony. It is more accurate than [[meantone]] and simpler than [[Garibaldi temperament|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.
-/224 is the [[Marvel temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.
+/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.
-/243, the [[Sensamagic clan|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.
+/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, [[104 EDO]] 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]] 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.
 Subgroup: 2.3.5.7
@@ Line 313: / Line 312: @@
 == Muggles ==
-Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19 EDO]], 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, 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 & more melodically effective.
+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, 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

Magic family: Difference between revisions

Revision as of 17:08, 11 October 2021

Magic

Septimal magic

11-limit

13-limit

Sorcery

Necromancy

Soothsaying

Telepathy

13-limit

Horcrux

Charisma

Glamour

Witchcraft

13-limit

Divination

13-limit

Hocus

13-limit

Muggles

11-limit

13-limit

Astrology

11-limit

13-limit

Horoscope

Spell

11-limit

13-limit

Cantrip

Hocum

Trismegistus

11-limit

13-limit

Quadrimage

11-limit

13-limit

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