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 MOSes due to the smaller generator making the small step a bit larger, but there is little 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 9-odd-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 (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 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, 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.

Interval chain

Cents* 0 380.352 760.704 1141.056 321.408 701.760 1082.112 262.464 642.816 1023.168 203.520 583.872 964.224 144.576
Ratios 1/1 5/4 14/9 27/14 6/5 3/2 15/8 7/6 (16/11) 9/5 9/8 7/5 7/4 (12/11)

* in 7-limit POTE 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

List of chords

Main article: Chords of magic
See also: Magic Tetrachords

Functional harmony

In terms of generator steps, the just major triad (1-5/4-3/2) and just minor triad (1-6/5-3/2) 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 suggest using 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. We also have the augmented triad (1-5/4-14/9) as the magic analog of the suspended chord of meantone.

Second, we suggest using the same triads as the basis of harmony, and keeping the roles of 3 and 5 as in meantone. 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. Try starting unlimited and thinking directly about ratios and comma pumps. You might end up with a diatonic-like scale with comma-level inflections here and there, but it is also possible to slap whatever scale you like, using inflections to reach the ratios.

Scales

MOS scales
Transversal scales
Others

Tuning spectra

Magic

Gencom: [2 5/4; 100/99 105/104 144/143 196/195]

Gencom mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

ET
generator 		eigenmonzo
(unchanged-interval) 		generator
(¢) 		comments
14/13 378.617
6/5 378.910
6\19 378.947
15/13 379.355
18/13 379.577
13/10 379.660
10/9 379.733
13/12 379.890
27/20 379.968 5-odd-limit least squares
19\60 380.000
16/13 380.029
15/14 380.093
32\101 380.198
7/5 380.228
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
4/3 380.391 5, 7 and 9-odd-limit minimax
13\41 380.488
[0 1 -7 15⟩ 380.506 7-odd-limit least squares
11/9 380.700 11-odd-limit minimax
[0 85 -14 52 -68⟩ 380.714 11-odd-limit least squares
8/7 380.735
33\104 380.769
12/11 380.818
14/11 380.875
20\63 380.952
7/6 380.982
11/8 381.085
15/11 381.211
16/15 381.378
11/10 381.666
7\22 381.818
9/7 382.458
5/4 386.314

Necromancy

Gencom: [2 5/4; 100/99 225/224 245/243 275/273]

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

ET
generator 		eigenmonzo
(unchanged-interval) 		generator
(¢) 		comments
6/5 378.910
6\19 378.947
10/9 379.733
27/20 379.968 5-odd-limit least squares
19\60 380.000
15/14 380.093
32\101 380.198
7/5 380.228
[0 36 -23 32⟩ 380.384 9-odd-limit least squares
4/3 380.391 5, 7 and 9-odd-limit minimax
13\41 380.488
[0 1 -7 15⟩ 380.506 7-odd-limit least squares
11/9 380.700 11-odd-limit minimax
18/13 380.709 13 and 15-odd-limit minimax
[0 85 -14 52 -68⟩ 380.714 11-odd-limit least squares
13/11 380.719
8/7 380.735
13/12 380.765
33\104 380.769
[0 -179 -10 -87 53 158⟩ 380.785 13-odd-limit least squares
14/13 380.809
[0 -222 -53 -93 67 187⟩ 380.817 15-odd-limit least squares
12/11 380.818
16/13 380.847
14/11 380.875
20\63 380.952
15/13 380.957
7/6 380.982
13/10 381.074
11/8 381.085
15/11 381.211
16/15 381.378
11/10 381.666
7\22 381.818
9/7 382.458
5/4 386.314

Sorcery

Gencom: [2 5/4; 65/64 78/77 91/90 100/99]

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

ET
generator 		eigenmonzo
(unchanged-interval) 		generator
(¢) 		comments
16/13 359.472
13/10 372.893
13/12 376.905
15/13 378.249
18/13 378.489
6/5 378.910
6\19 378.947
14/13 379.100
10/9 379.733
27/20 379.968 5-odd-limit least squares
19\60 380.000
15/14 380.093
32\101 380.198
7/5 380.228
[0 36 -23 32⟩ 380.384 9-odd-limit least squares
4/3 380.391 5, 7 and 9-odd-limit minimax
[0 -113 12 -65 75 26⟩ 380.427 13-odd-limit least squares
[0 134 9 71 -89 -33⟩ 380.457 15-odd-limit least squares
13\41 380.488
[0 1 -7 15⟩ 380.506 7-odd-limit least squares
11/9 380.700 11, 13 and 15-odd-limit minimax
[0 85 -14 52 -68⟩ 380.714 11-odd-limit least squares
8/7 380.735
33\104 380.769
12/11 380.818
14/11 380.875
20\63 380.952
7/6 380.982
11/8 381.085
15/11 381.211
16/15 381.378
11/10 381.666
7\22 381.818
9/7 382.458
13/11 384.173
5/4 386.314

Telepathy

Gencom: [2 5/4; 55/54 65/64 91/90 99/98]

Gencom mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

ET
generator 		eigenmonzo
(unchanged-interval) 		generator
(¢) 		comments
16/13 359.472
13/10 372.893
13/12 376.905
15/13 378.249
18/13 378.489
6/5 378.910
6\19 378.947
14/13 379.100
10/9 379.733
27/20 379.968 5-odd-limit least squares
19\60 380.000
15/14 380.093
32\101 380.198
7/5 380.228
[0 36 -23 32⟩ 380.384 9-odd-limit least squares
4/3 380.391 5, 7 and 9-odd-limit minimax
13\41 380.488
[0 1 -7 15⟩ 380.506 7-odd-limit least squares
[0 47 -34 43 57 -48⟩ 380.676 13-odd-limit least squares
[0 46 -35 49 65 -55⟩ 380.691 15-odd-limit least squares
13/11 380.719 13 and 15-odd-limit minimax
8/7 380.735
33\104 380.769
20\63 380.952
7/6 380.982
16/15 381.378
[0 19 -36 30 42⟩ 381.380 11-odd-limit least squares
7\22 381.818
11/10 381.923 11-odd-limit minimax
11/8 382.237
9/7 382.458
15/11 382.881
12/11 383.263
5/4 386.314
11/9 386.852
14/11 391.246

Intuition

Gencom: [2 5/4; 55/54 66/65 99/98 105/104]

Gencom mapping: [⟨1 0 2 -1 -1 -2], ⟨0 5 1 12 14 18]]

ET
generator 		eigenmonzo
(unchanged-interval) 		generator
(¢) 		comments
13/11 372.302
14/13 378.617
6/5 378.910
6\19 378.947
15/13 379.355
18/13 379.577
13/10 379.660
10/9 379.733
13/12 379.890
27/20 379.968 5-odd-limit least squares
19\60 380.000
16/13 380.029 13 and 15-odd-limit minimax
15/14 380.093
32\101 380.198
7/5 380.228
[0 36 -23 32⟩ 380.384 9-odd-limit least squares
4/3 380.391 5, 7 and 9-odd-limit minimax
13\41 380.488
[0 1 -7 15⟩ 380.506 7-odd-limit least squares
[0 -30 -73 30 46 78⟩ 380.562 15-odd-limit least squares
[0 -10 -53 24 38 66⟩ 380.568 13-odd-limit least squares
8/7 380.735
33\104 380.769
20\63 380.952
7/6 380.982
16/15 381.378
[0 19 -36 30 42⟩ 381.380 11-odd-limit least squares
7\22 381.818
11/10 381.923 11-odd-limit minimax
11/8 382.237
9/7 382.458
15/11 382.881
12/11 383.263
5/4 386.314
11/9 386.852
14/11 391.246

Witchcraft

Gencom: [2 5/4; 105/104 196/195 245/243 275/273]

Gencom mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

ET
generator 		eigenmonzo
(unchanged-interval) 		generator
(¢) 		comments
14/13 378.617
6/5 378.910
6\19 378.947
15/13 379.355
18/13 379.577
13/10 379.660
10/9 379.733
27/20 379.968 5-odd-limit least squares
13/12 379.890
19\60 380.000
16/13 380.029
15/14 380.093
15/11 380.113 15-odd-limit minimax
14/11 380.119 13-odd-limit minimax
11/10 380.156
[0 -106 -111 11 179 59⟩ 380.193 15-odd-limit least squares
32\101 380.198
[0 -67 -72 5 152 47⟩ 380.218 13-odd-limit least squares
7/5 380.228
[0 -38 -55 11 137⟩ 380.278 11-odd-limit least squares
11/9 380.322
12/11 380.334
11/8 380.343 11-odd-limit minimax
[0 36 -23 32⟩ 380.384 9-odd-limit least squares
4/3 380.391 5, 7 and 9-odd-limit minimax
13\41 380.488
13/11 380.719
8/7 380.735
33\104 380.769
20\63 380.952
7/6 380.982
16/15 381.378
7\22 381.818
9/7 382.458
5/4 386.314

Music

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

See also

External links

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