Magic extensions

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Magic has various competing extensions to the 11- and 13-limit. The following temperaments are discussed in this article:

  • Tridecimal magic (19 & 22f) – tempering out 100/99, 105/104, 144/143, and 196/195;
  • Necromancy (19f & 22) – tempering out 100/99, 225/224, 245/243, and 275/273;
  • Sorcery (19 & 22) – tempering out 65/64, 78/77, 91/90, and 100/99;
  • Witchcraft (19e & 22ef) – tempering out 105/104, 196/195, 245/243, and 275/273;
  • Telepathy (19e & 22) – tempering out 55/54, 65/64, 91/90, and 99/98.

Interval chain

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

# Cents* Approximate ratios
7-limit 13-limit extensions
Magic Necromancy Sorcery Witchcraft Telepathy
0 0.0 1/1
1 380.4 5/4 26/21 16/13 16/13
2 760.9 14/9 20/13 11/7, 20/13
3 1141.4 27/14 52/27, 64/33 64/33 64/33 52/27 88/45
4 321.8 6/5 40/33 40/33 40/33 11/9
5 702.3 3/2
6 1082.7 15/8, 28/15 13/7 24/13 13/7 24/13
7 263.2 7/6 13/11
8 643.7 35/24 13/9, 16/11 16/11 16/11 13/9 22/15
9 1024.1 9/5 20/11 20/11 20/11 11/6
10 204.6 9/8
11 585.0 7/5 18/13 18/13
12 965.5 7/4 26/15 26/15
13 146.0 35/32 12/11, 13/12 12/11 12/11, 14/13 13/12 11/10, 14/13
14 526.4 27/20 15/11 15/11 15/11 11/8
15 906.9 27/16, 42/25 22/13 22/13
16 87.3 21/20 27/26 22/21 27/26
17 467.8 21/16 13/10 13/10
18 848.2 49/30 13/8, 18/11 18/11 18/11, 21/13 13/8 21/13
19 28.7 49/48, 81/80, 126/125 45/44 45/44 45/44 33/32
20 409.2 63/50 14/11 14/11 14/11
21 789.6 63/40 11/7
22 1170.1 49/25, 63/32 39/20 39/20

* In 7-limit CWE tuning

Tuning spectra

Magic

Edo
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 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

Necromancy

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
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

Sorcery

Edo
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 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

Telepathy

Edo
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 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

Witchcraft

Edo
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 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

Muggles

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
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
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