Magic extensions: Difference between revisions

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VisualWikitext
-muggles (not in the scope of this article)
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| 5 limit least squares
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| 380.029
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| 13/11
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| 380.354
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| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 56 -31 46 -94 88}}
| {{monzo| 0 56 -31 46 -94 88 }}
| 380.377
| 380.377
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| {{monzo| 0 36 -23 32 }}
| 380.384
| 380.384
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 58 -29 52 -108 100}}
| {{monzo| 0 58 -29 52 -108 100 }}
| 380.389
| 380.389
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 380.391
| 380.391
| 5, 7 and 9 limit minimax
| 5-, 7- and 9-odd-limit minimax
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| 13\41
| 13\41
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|  
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| {{vector|0 1 -7 15}}
| {{monzo| 0 1 -7 15 }}
| 380.506
| 380.506
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| 11/9
| 11/9
| 380.700
| 380.700
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 85 -14 52 -68}}
| {{monzo| 0 85 -14 52 -68 }}
| 380.714
| 380.714
| 11 limit least squares
| 11-odd-limit least squares
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| 380.735
| 380.735
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| 12/11
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| 380.818
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| 14/11
| 11/7
| 380.875
| 380.875
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| 15/88
| 381.378
| 381.378
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| 378.910
| 378.910
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| 379.733
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| 379.968
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| 5 limit least squares
| 5-odd-limit least squares
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| 19\60
| 19\60
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|  
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| {{vector|0 36 -23 32}}
| {{monzo| 0 36 -23 32 }}
| 380.384
| 380.384
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 380.391
| 380.391
| 5, 7 and 9 limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|-
| 13\41
| 13\41
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|-
|  
|  
| {{vector|0 1 -7 15}}
| {{monzo| 0 1 -7 15 }}
| 380.506
| 380.506
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
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|  
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| 18/13
| 13/9
| 380.709
| 380.709
| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 85 -14 52 -68}}
| {{monzo| 0 85 -14 52 -68 }}
| 380.714
| 380.714
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
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| 8/7
| 7/4
| 380.735
| 380.735
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| {{vector|0 -179 -10 -87 53 158}}
| {{monzo| 0 -179 -10 -87 53 158 }}
| 380.785
| 380.785
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| 14/13
| 13/7
| 380.809
| 380.809
|  
|  
|-
|-
|  
|  
| {{vector|0 -222 -53 -93 67 187}}
| {{monzo| 0 -222 -53 -93 67 187 }}
| 380.817
| 380.817
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 12/11
| 11/6
| 380.818
| 380.818
|  
|  
|-
|-
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| 16/13
| 13/8
| 380.847
| 380.847
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| 14/11
| 11/7
| 380.875
| 380.875
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| 16/15
| 15/8
| 381.378
| 381.378
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| 16/13
| 13/8
| 359.472
| 359.472
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| 18/13
| 13/9
| 378.489
| 378.489
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| 6/5
| 5/3
| 378.910
| 378.910
|  
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| 14/13
| 13/7
| 379.100
| 379.100
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| 10/9
| 9/5
| 379.733
| 379.733
|  
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| 27/20
| 27/20
| 379.968
| 379.968
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 695: Line 695:
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| {{monzo| 0 36 -23 32 }}
| 380.384
| 380.384
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 380.391
| 380.391
| 5, 7 and 9 limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 -113 12 -65 75 26}}
| {{monzo| 0 -113 12 -65 75 26 }}
| 380.427
| 380.427
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 134 9 71 -89 -33}}
| {{monzo| 0 134 9 71 -89 -33 }}
| 380.457
| 380.457
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
| 13\41
| 13\41
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|-
|-
|  
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| {{vector|0 1 -7 15}}
| {{monzo| 0 1 -7 15 }}
| 380.506
| 380.506
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| 11/9
| 11/9
| 380.700
| 380.700
| 11, 13 and 15 limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 85 -14 52 -68}}
| {{monzo| 0 85 -14 52 -68 }}
| 380.714
| 380.714
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
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| 8/7
| 7/4
| 380.735
| 380.735
|  
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| 12/11
| 11/6
| 380.818
| 380.818
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| 14/11
| 11/7
| 380.875
| 380.875
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| 16/15
| 15/8
| 381.378
| 381.378
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| 16/13
| 13/8
| 359.472
| 359.472
|  
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| 18/13
| 13/9
| 378.489
| 378.489
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|-
|-
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| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
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| 14/13
| 13/7
| 379.100
| 379.100
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| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
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| 27/20
| 27/20
| 379.968
| 379.968
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
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|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| {{monzo| 0 36 -23 32 }}
| 380.384
| 380.384
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 380.391
| 380.391
| 5, 7 and 9 limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|-
| 13\41
| 13\41
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|-
|-
|  
|  
| {{vector|0 1 -7 15}}
| {{monzo| 0 1 -7 15 }}
| 380.506
| 380.506
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 47 -34 43 57 -48}}
| {{monzo| 0 47 -34 43 57 -48 }}
| 380.676
| 380.676
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 46 -35 49 65 -55}}
| {{monzo| 0 46 -35 49 65 -55 }}
| 380.691
| 380.691
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 13/11
| 13/11
| 380.719
| 380.719
| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 8/7
| 7/4
| 380.735
| 380.735
|  
|  
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| 16/15
| 15/8
| 381.378
| 381.378
|  
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|-
|-
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| {{vector|0 19 -36 30 42}}
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 381.380
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
| 7\22
| 7\22
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| 11/10
| 11/10
| 381.923
| 381.923
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
|  
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Line 974: Line 974:
|-
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| 12/11
| 11/6
| 383.263
| 383.263
|  
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|-
|-
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| 14/11
| 11/7
| 391.246
| 391.246
|  
|  
Line 1,003: Line 1,003:
|-
|-
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| 14/13
| 13/7
| 378.617
| 378.617
|  
|  
|-
|-
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|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
Line 1,023: Line 1,023:
|-
|-
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| 18/13
| 13/9
| 379.577
| 379.577
|  
|  
Line 1,033: Line 1,033:
|-
|-
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|  
| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
Line 1,040: Line 1,040:
| 27/20
| 27/20
| 379.968
| 379.968
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
Line 1,053: Line 1,053:
|-
|-
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|  
| 16/13
| 13/8
| 380.029
| 380.029
|  
|  
Line 1,065: Line 1,065:
| 15/11
| 15/11
| 380.113
| 380.113
| 15 limit minimax
| 15-odd-limit minimax
|-
|-
|  
|  
| 14/11
| 11/7
| 380.119
| 380.119
| 13 limit minimax
| 13-odd-limit minimax
|-
|-
|  
|  
Line 1,078: Line 1,078:
|-
|-
|  
|  
| {{vector|0 -106 -111 11 179 59}}
| {{monzo| 0 -106 -111 11 179 59 }}
| 380.193
| 380.193
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
| 32\101
| 32\101
Line 1,088: Line 1,088:
|-
|-
|  
|  
| {{vector|0 -67 -72 5 152 47}}
| {{monzo| 0 -67 -72 5 152 47 }}
| 380.218
| 380.218
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
Line 1,098: Line 1,098:
|-
|-
|  
|  
| {{vector|0 -38 -55 11 137}}
| {{monzo| 0 -38 -55 11 137 }}
| 380.278
| 380.278
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
Line 1,108: Line 1,108:
|-
|-
|  
|  
| 12/11
| 11/6
| 380.334
| 380.334
|  
|  
Line 1,115: Line 1,115:
| 11/8
| 11/8
| 380.343
| 380.343
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| {{monzo| 0 36 -23 32 }}
| 380.384
| 380.384
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 380.391
| 380.391
| 5, 7 and 9 limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|-
| 13\41
| 13\41
Line 1,138: Line 1,138:
|-
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|  
|  
| 8/7
| 7/4
| 380.735
| 380.735
|  
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Line 1,158: Line 1,158:
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| 16/15
| 15/8
| 381.378
| 381.378
|  
|  

Revision as of 13:18, 29 December 2024

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.

The most important of these is tridecimal magic, where primes 11 and 13 are mapped such that 10/9 and 11/10 are tempered together and that 14/13 and 16/15 are tempered together. Necromancy takes the same mapping of 11 as magic, but swaps in a more complex mapping for 13, enabling the possibility of a higher-accuracy harmony. Similarly, witchcraft takes the same mapping of 13 as magic, but swaps in a more complex mapping for 11.

Sorcery swaps in a less complex and less accurate mapping for prime 13, as it finds ~16/13 where magic will find ~26/21, which is of course conflated with ~5/4. Telepathy adopts less complex and less accurate mapping for both primes 11 and 13.

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
13/7 378.617
5/3 378.910
6\19 378.947
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
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
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
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
11/6 380.818
11/7 380.875
20\63 380.952
7/6 380.982
11/8 381.085
15/11 381.211
15/88 381.378
11/10 381.666
7\22 381.818
9/7 382.458
5/4 386.314

Necromancy

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
5/3 378.910
6\19 378.947
9/5 379.733
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
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
11/9 380.700 11 limit minimax
13/9 380.709 13- and 15-odd-limit minimax
[0 85 -14 52 -68⟩ 380.714 11-odd-limit least squares
13/11 380.719
7/4 380.735
13/12 380.765
33\104 380.769
[0 -179 -10 -87 53 158⟩ 380.785 13-odd-limit least squares
13/7 380.809
[0 -222 -53 -93 67 187⟩ 380.817 15-odd-limit least squares
11/6 380.818
13/8 380.847
11/7 380.875
20\63 380.952
15/13 380.957
7/6 380.982
13/10 381.074
11/8 381.085
15/11 381.211
15/8 381.378
11/10 381.666
7\22 381.818
9/7 382.458
5/4 386.314

Sorcery

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
13/8 359.472
13/10 372.893
13/12 376.905
15/13 378.249
13/9 378.489
5/3 378.910
6\19 378.947
13/7 379.100
9/5 379.733
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
3/2 380.391 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
7/4 380.735
33\104 380.769
11/6 380.818
11/7 380.875
20\63 380.952
7/6 380.982
11/8 381.085
15/11 381.211
15/8 381.378
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
13/8 359.472
13/10 372.893
13/12 376.905
15/13 378.249
13/9 378.489
5/3 378.910
6\19 378.947
13/7 379.100
9/5 379.733
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
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
[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
7/4 380.735
33\104 380.769
20\63 380.952
7/6 380.982
15/8 381.378
[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
11/6 383.263
5/4 386.314
11/9 386.852
11/7 391.246

Witchcraft

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
13/7 378.617
5/3 378.910
6\19 378.947
15/13 379.355
13/9 379.577
13/10 379.660
9/5 379.733
27/20 379.968 5-odd-limit least squares
13/12 379.890
19\60 380.000
13/8 380.029
15/14 380.093
15/11 380.113 15-odd-limit minimax
11/7 380.119 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
11/6 380.334
11/8 380.343 11-odd-limit minimax
[0 36 -23 32⟩ 380.384 9-odd-limit least squares
3/2 380.391 5-, 7- and 9-odd-limit minimax
13\41 380.488
13/11 380.719
7/4 380.735
33\104 380.769
20\63 380.952
7/6 380.982
15/8 381.378
7\22 381.818
9/7 382.458
5/4 386.314
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