Magic family: Difference between revisions

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 [[12edo]]. Properties may depend on tuning and extension.

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)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.

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.

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.

{{Val list|legend=1| 19, 22, 41, 104, 145c }}

Badness: 0.0204

{{Val list|legend=1| 19, 22f, 41, 265cdef }}

Badness: 0.0215

{{Val list|legend=1| 19, 22, 41f }}

Badness: 0.0258

{{Val list|legend=1| 19f, 22, 41, 63, 104 }}

Badness: 0.0253

{{Val list|legend=1| 22, 60, 82 }}

Badness: 0.0554

{{Val list|legend=1| 19e, 22, 41e, 63e }}

Badness: 0.0271

{{Val list|legend=1| 19e, 22, 41ef }}

Badness: 0.0255

{{Val list|legend=1| 19, 41e, 60e }}

Badness: 0.0393

{{Val list|legend=1| 22, 38d, 60e, 142cde }}

Badness: 0.0359

{{Val list|legend=1| 22f, 60e }}

Badness: 0.0346

{{Val list|legend=1| 41, 60e, 101cd, 243cde }}

Badness: 0.0307

{{Val list|legend=1| 41, 60e, 101cd }}

Badness: 0.0235

{{Val list|legend=1| 38d, 41, 120cd, 161cd, 202cd }}

Badness: 0.0385

{{Val list|legend=1| 41, 79d, 120cd }}

Badness: 0.0303

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.  

[[Badness]]: 0.0562

{{Val list|legend=1| 16, 19, 35, 54bd }}

Badness: 0.0480

{{Val list|legend=1| 16, 19, 35f, 54bdf }}

Badness: 0.0309

[[Badness]]: 0.0827

{{Val list|legend=1| 6, 16, 22, 60de, 82de }}

Badness: 0.0392

{{Val list|legend=1| 6, 16, 22, 38f }}

Badness: 0.0344

{{Val list|legend=1| 16, 22f, 38 }}

Badness: 0.0353

{{Val list|legend=1| 6, 19, 82d }}

[[Badness]]: 0.0810

{{Val list|legend=1| 6, 19, 44de, 63de, 82de }}

Badness: 0.0598

{{Val list|legend=1| 6, 19, 82def }}

Badness: 0.0456

{{Val list|legend=1| 19, 44de, 63de, 82de }}

Badness: 0.0416

[[Badness]]: 0.1071

[[Badness]]: 0.0983

{{Val list|legend=1| 16, 25e, 41, 98c, 139c, 180c }}

Badness: 0.0456

{{Val list|legend=1| 16, 25e, 41, 98c, 139cf }}

Badness: 0.0331

[[Badness]]: 0.1274

{{Val list|legend=1| 6, 35, 41, 199cde, 240cde, 281cde }}

Badness: 0.0616

{{Val list|legend=1| 41, 117c, 158cd, 199cdef }}

Badness: 0.0440