Magic

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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, which may be better melodically for small mos scales due to the smaller generator making the small step a bit larger, but there is little reason to use it unless you are using 19edo, in which case it is 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:

  • It is the simplest mapping capable of tuning every 9-odd-limit interval better than in 12edo.
  • 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 is not a panacea because:

  • It has no proper mos scales with between 3 and 16 notes over a single period per octave.
  • It is more complex than meantone (higher complexity and badness).
  • 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 is accordingly so small, all small magic mos scales consist of three large intervals alternating with three groups of this small interval. Specifically, there are the following scales, where s always represents the characteristic small interval, which simultaneously represents 128/125, 36/35, 28/27, and 25/24.

  • 3L 4s: LsLsLss, where L represents 6/5
  • 3L 7s: LssLssLsss, where L represents 7/6
  • 3L 10s: LsssLsssLssss, where L represents 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.

For technical information, see Magic family #Magic. For a discussion on alternative 11- and 13-limit extensions, see Magic extensions.

Interval chain

In the following table, odd harmonics 1–13 and their inverses are in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 380.5 5/4
2 760.9 14/9
3 1141.4 27/14
4 321.8 6/5
5 702.3 3/2
6 1082.7 15/8, 28/15
7 263.2 7/6
8 643.7 (16/11)
9 1024.1 9/5
10 204.6 9/8
11 585.0 7/5
12 965.5 7/4
13 145.9 (12/11)

* In 7-limit CWE tuning

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.

Chords and harmony

The fundamental otonal consonance of magic, voiced in a roughly tertian manner, is 4:5:6:7:9, available in a 13-tone mos. To start with, consider the the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2). In terms of generator steps, they are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. Two approaches to functional harmony thus arise.

First, we can use the triads above as the basis of harmony, but swapping the roles of 3 and 5 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. This leads to an approach closely adherent to mos scales. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. The suspended chord of meantone is made of two generators stacked, and doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

Second, we can use the same triads as the basis of harmony, and keeping the role of the chain of fifths as the spine on which the functions are defined. This means dominant is still 3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. This is essentially working in JI, but using the commas tempered out in some way to lock into the identity of the temperament.

Scales

Mos scales
Transversal scales
Others

Tunings

Tuning spectrum

Edo
generator
Eigenmonzo
(unchanged-interval)
Generator (¢) Comments
21/13 369.747
13/7 378.617
5/3 378.910
6\19 378.947 Lower bound of 9- to 15-odd-limit diamond monotone
15/13 379.355
13/9 379.577
13/10 379.660
9/5 379.733
13/12 379.890
27/20 379.968 5-odd-limit least squares
19\60 380.000
13/8 380.029
15/14 380.093
32\101 380.198 101cde val
7/5 380.228
21/20 380.279
13/11 380.354 13- and 15-odd-limit minimax
[0 56 -31 46 -94 88 380.377 13-odd-limit least squares
[0 36 -23 32 380.384 9-odd-limit least squares
[0 58 -29 52 -108 100 380.389 15-odd-limit least squares
3/2 380.391 5-, 7- and 9-odd-limit minimax
13\41 380.488
[0 1 -7 15 380.506 7-odd-limit least squares
21/16 380.634
11/9 380.700 11-odd-limit minimax
[0 85 -14 52 -68 380.714 11-odd-limit least squares
7/4 380.735
33\104 380.769 104ff val
21/11 380.779
11/6 380.818
11/7 380.875
20\63 380.952 63f val
7/6 380.982
11/8 381.085
15/11 381.211
15/8 381.378
11/10 381.666
7\22 381.818 22f val, upper bound of 9- to 15-odd-limit diamond monotone
9/7 382.458
5/4 386.314

Music

Cameron Bobro
  • Magical DaydreamA 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.
  • Evening HorizonThe 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.
Graham Breed
Jake Freivald
  • Little Magical Object (2013) – play | SoundCloud – Magic[19] in 41edo tuning
Andrew Milne
Chris Vaisvil (site)
Xenllium

See also

External links