Magic
Magic is a linear temperament in which the ~380 cent generator represents 5/4, and five of those make a 3/1. This implies that the magic comma 3125/3072 is tempered out, making it a member of the Magic family. This article also assumes the default mapping for the prime 7, which tempers out 225/224 and makes two generators equivalent to 14/9. 7/4 can be reached by 12 generators in this mapping. (There is an alternative mapping for 7 known as muggles, but there's basically no reason to use it unless you're using 19edo, in which case it's identical to magic anyway.)
EDOs that contain good magic scales include 19edo, 22edo, 41edo, 60edo and 104edo.
Magic has certain properties that commend it as a step up in complexity from traditional harmony:
- Every non-trivial 7-limit interval is better tuned than in 12edo.
- It is the simplest mapping with the above property.
- It is only slightly more complex than meantone (both work well with a 19 note gamut).
- 5-limit intervals are simpler than other 7-limit intervals.
It fails to be a panacea because:
- It has no proper MOS scales of between 3 and 16 notes.
- It is more complex than meantone
- The 3/2 approximation is 5 times as complex as the 5/4 approximation (the generator) so modulation by fifths is more constrained than you may be used to.
Because the generator is so close to 1\3 of an octave, and the interval left over (which represents both 128/125 and 25/24) is accordingly so small, all small magic MOSes consist of three large intervals alternating with three groups of this small interval. Specifically, there are the following MOSes, where s always represents the characteristic small interval of 128/125~25/24.
- 3L 4s: LsLsLss where L = 6/5
- 3L 7s: LssLssLsss where L = 7/6
- 3L 10s: LsssLsssLssss where L = 9/8
- 3L 13s: LssssLssssLsssss where L is a neutral second, which can be taken to represent 12/11 (in magic temperament) or 11/10 (in the related telepathy temperament). In 22edo they are identical.
Interval chain[edit]
| 0. | 380.352 | 760.704 | 1141.056 | 321.408 | 701.76 | 1082.112 | 262.464 | 642.816 | 1023.168 | 203.52 | 583.872 | 964.224 | 144.576 |
| 1/1 | 5/4 | 14/9 | 48/25~125/64 | 6/5 | 3/2 | 15/8 | 7/6 | (16/11) | 9/5 | 9/8 | 7/5 | 7/4 | (12/11) |
The generator chain val for 13-limit magic is <0 5 1 12 -8 18|, so that five generators give an approximate 3, twelve 14, minus eight 11/64, and eighteen 52.
Spectrum of Magic Tunings by Eigenmonzos[edit]
| Eigenmonzo | Major Third |
|---|---|
| 6/5 | 378.910 |
| 10/9 | 379.733 |
| 7/5 | 380.228 |
| 4/3 | 380.391 (5, 7 and 9 limit minimax) |
| 11/9 | 380.700 (11 limit minimax) |
| 8/7 | 380.735 |
| 12/11 | 380.818 |
| 14/11 | 380.875 |
| 7/6 | 380.982 |
| 11/8 | 381.085 |
| 11/10 | 381.666 |
| 9/7 | 382.458 |
| 5/4 | 386.314 |
Chords of magic[edit]
Magic Tetrachords[edit]
Music[edit]
The Magic of Belief Magic[19] in 41et tuning
Little Magical Object play Magic[19] in 41et tuning by Jake Freivald
Andrew Milne; magic with 379.8 cent generator
A brief demonstration of the near-Just musical temperament which flattens the pure major third of 5:4 by a few cents, such that 5 major thirds does not exceed 3:1 (a pure fifth + 1 octave), but meets it precisely. In a purely tuned system, the thirds would exceed 3:1 by what is known as the small diesis, (a ratio 3125/3072, about thirty cents). This temperament, then, brings (almost) pure thirds and pure fifths together. Cameron Bobro
The earliest implementation (by happy accident, it seems) of this temperament was, to my knowledge, by Paul von Janko over a century ago. More recently, an online tuning community has elaborated many precise variations, calling the temperament "magic".. This piece is a demonstration of the array of pitches created by using 22 generators (the slightly tempered 5:4) within the octave, an approach which creates a "moment of symmetry", with all pitches separated by the same two intervals. This has many curious repercussions, creating some musical possibilities and restricting others. Cameron Bobro
Golden Age disco involving magic comma pumps.
Extravagant Food a single magic comma pump in under 60 seconds in 60-equal.
Gene's Jitterbug 9-limit harmony, may not require magic.