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 is 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 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
• 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 = 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

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

Main article: Chords of magic

See also: Magic Tetrachords

Scales

MOS scales

• Magic7
• Magic10
• Magic13
• Magic16

Transversal scales

• Magic19trans37
• Magic19trans37ex
• Magic22trans37
• Magic22trans37ex

Others

• Paulsmagic

Tuning Spectra

Spectrum of Magic Tunings by Eigenmonzos

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

Gencom map: [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]

Eigenmonzo	Major Third
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 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 limit minimax)
|0 56 -31 46 -94 88>	380.377 (13 limit least squares)
|0 36 -23 32>	380.384 (9 limit least squares)
|0 58 -29 52 -108 100>	380.389 (15 limit least squares)
4/3	380.391 (5, 7 and 9 limit minimax)
13\41	380.488
|0 1 -7 15>	380.506 (7 limit least squares)
11/9	380.700 (11 limit minimax)
|0 85 -14 52 -68>	380.714 (11 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

Spectrum of Necromancy Tunings by Eigenmonzos

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

Gencom map: [<1 0 2 -1 6 11|, <0 5 1 12 -8 -23|]

Eigenmonzo	Major Third
6/5	378.910
6\19	378.947
10/9	379.733
27/20	379.968 (5 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 limit least squares)
4/3	380.391 (5, 7 and 9 limit minimax)
13\41	380.488
|0 1 -7 15>	380.506 (7 limit least squares)
11/9	380.700 (11 limit minimax)
18/13	380.709 (13 and 15 limit minimax)
|0 85 -14 52 -68>	380.714 (11 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 limit least squares)
14/13	380.809
|0 -222 -53 -93 67 187>	380.817 (15 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

Spectrum of Sorcery Tunings by Eigenmonzos

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

Gencom map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|]

Eigenmonzo	Major Third
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 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 limit least squares)
4/3	380.391 (5, 7 and 9 limit minimax)
|0 -113 12 -65 75 26>	380.427 (13 limit least squares)
|0 134 9 71 -89 -33>	380.457 (15 limit least squares)
13\41	380.488
|0 1 -7 15>	380.506 (7 limit least squares)
11/9	380.700 (11, 13 and 15 limit minimax)
|0 85 -14 52 -68>	380.714 (11 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

Spectrum of Telepathy Tunings by Eigenmonzos

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

Gencom map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]

Eigenmonzo	Major Third
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 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 limit least squares)
4/3	380.391 (5, 7 and 9 limit minimax)
13\41	380.488
|0 1 -7 15>	380.506 (7 limit least squares)
|0 47 -34 43 57 -48>	380.676 (13 limit least squares)
|0 46 -35 49 65 -55>	380.691 (15 limit least squares)
13/11	380.719 (13 and 15 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 limit least squares)
7\22	381.818
11/10	381.923 (11 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

Spectrum of Witchcraft Tunings by Eigenmonzos

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

Gencom map: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]

Eigenmonzo	Major Third
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 limit least squares)
13/12	379.890
19\60	380.000
16/13	380.029
15/14	380.093
15/11	380.113 (15 limit minimax)
14/11	380.119 (13 limit minimax)
11/10	380.156
|0 -106 -111 11 179 59>	380.193 (15 limit least squares)
32\101	380.198
|0 -67 -72 5 152 47>	380.218 (13 limit least squares)
7/5	380.228
|0 -38 -55 11 137>	380.278 (11 limit least squares)
11/9	380.322
12/11	380.334
11/8	380.343 (11 limit minimax)
|0 36 -23 32>	380.384 (9 limit least squares)
4/3	380.391 (5, 7 and 9 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

Spectrum of Muggles Tunings by Eigenmonzos

Gencom: [2 5/4; 45/44 65/64 78/77 126/125]

Gencom map: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|]

Eigenmonzo	Major Third
11/9	347.408
16/13	359.472
15/11	372.610
13/10	372.893
12/11	374.894
8/7	375.882
13/11	375.899
11/10	376.500
14/11	376.805
13/12	376.905
7/5	377.186
11/8	377.393 (11, 13 and 15 limit minimax)
|0 113 -12 -68 58 -26>	377.630 (13 limit least squares)
|0 -21 -5 27>	377.640 (7 limit least squares)
|0 134 9 -81 63 -33>	377.718 (15 limit least squares)
|0 85 -14 -62 46>	377.758 (11 limit least squares)
7/6	377.761 (7 limit minimax)
15/13	378.249
15/14	378.419
18/13	378.489
9/7	378.534 (9 limit minimax)
|0 93 -4 -44>	378.554 (9 limit least squares)
14/13	378.617
6/5	378.910
6\19	378.947
10/9	379.733
27/20	379.968 (5 limit least squares)
19\60	380.000
32\101	380.198
4/3	380.391 (5 limit minimax)
13\41	380.488
33\104	380.769
20\63	380.952
16/15	381.378
7\22	381.818
5/4	386.314

Music

• Chromatic piece in magic 16 in magic16
• A Piece in Paulsmagic in paulsmagic
• The Magic of Belief Magic[19] in 41edo tuning by Chris Vaisvil
• Little Magical Object play Magic[19] in 41edo tuning by Jake Freivald
• Magic Traveller magic with 379.8 cent generator by Andrew Milne
• 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. by Cameron Bobro
• 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. by Cameron Bobro
• Golden Age disco involving magic comma pumps.
• Extravagant Food a single magic comma pump in under 60 seconds in 60edo.
• Gene's Jitterbug 9-odd-limit harmony, may not require magic.