Magic extensions

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;
Witchcraft (19e & 22ef) – tempering out 105/104, 196/195, 245/243, and 275/273;
Sorcery (19 & 22) – tempering out 65/64, 78/77, 91/90, and 100/99;
Intuition (19e & 22f) – tempering out 55/54, 66/65, 99/98, and 105/104;
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. Likewise, intuitive uses a less complex and less accurate mapping for prime 11. Finally, telepathy adopts the changes of both sorcery and intuition, resulting in a very low-accuracy temperament.

Interval chain

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

#	Cents*	Approximate ratios
		7-limit	13-limit extensions
		7-limit	Magic	Necromancy	Witchcraft	Sorcery	Intuition	Telepathy
0	0.0	1/1
1	380.4	5/4	26/21		26/21	16/13	26/21	16/13
2	760.9	14/9				20/13	11/7	11/7, 20/13
3	1141.4	27/14	52/27, 64/33	64/33	52/27	64/33	21/11, 52/27	21/11
4	321.8	6/5	40/33	40/33		40/33	11/9, 13/11	11/9
5	702.3	3/2
6	1082.7	15/8, 28/15	13/7		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	13/9	16/11	13/9, 22/15	22/15
9	1024.1	9/5	20/11	20/11		20/11	11/6	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		26/15
13	146.0	35/32	12/11, 13/12	12/11	13/12	12/11, 14/13	11/10, 13/12	11/10, 14/13
14	526.4	27/20	15/11	15/11		15/11	11/8	11/8
15	906.9	27/16, 42/25		22/13	22/13			22/13
16	87.3	21/20			22/21	27/26		27/26
17	467.8	21/16	13/10		13/10		13/10
18	848.2	49/30	13/8, 18/11	18/11	13/8	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	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		39/20

* In 7-limit CWE tuning

Tuning spectra

Magic

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

Necromancy

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	5/3	378.910
6\19		378.947	19f val, lower bound of 9- and 11-odd-limit diamond monotone
	9/5	379.733
	27/20	379.968	5-odd-limit least squares
19\60		380.000	60f val
	15/14	380.093
32\101		380.198	101cdeff val
	7/5	380.228
	21/20	380.279
	[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	Lower bound of 13- and 15-odd-limit diamond monotone
	[0 1 -7 15⟩	380.506	7-odd-limit least squares
	21/16	380.634
	11/9	380.700	11-odd-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
	21/13	380.756
	13/12	380.765
33\104		380.769
	21/11	380.779
	[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	Upper bound of 13- and 15-odd-limit diamond monotone
	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	Upper bound of 9- and 11-odd-limit diamond monotone
	9/7	382.458
	5/4	386.314

Witchcraft

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	21/13	369.747
	13/7	378.617
	5/3	378.910
6\19		378.947	19e val, lower bound of 9-odd-limit diamond monotone
	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	60e val, lower bound of 11- to 15-odd-limit diamond monotone
	13/8	380.029
	21/11	380.034
	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	101cd val
	[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
	21/20	380.279
	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	Upper bound of 11- to 15-odd-limit diamond monotone
	21/16	380.634
	13/11	380.719
	7/4	380.735
33\104		380.769	104eeff val
20\63		380.952	63eef val
	7/6	380.982
	15/8	381.378
7\22		381.818	22ef val, upper bound of 9-odd-limit diamond monotone
	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	Lower bound of 9- and 11-odd-limit diamond monotone 13- and 15-odd-limit diamond monotone (singleton)
	13/7	379.100
	21/13	379.459
	9/5	379.733
	27/20	379.968	5-odd-limit least squares
19\60		380.000	60ff val
	15/14	380.093
32\101		380.198	101cdefff val
	7/5	380.228
	21/20	380.279
	[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	41f val
	[0 1 -7 15⟩	380.506	7-odd-limit least squares
	21/16	380.634
	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	104fff val
	21/11	380.779
	11/6	380.818
	11/7	380.875
20\63		380.952	63ff 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	Upper bound of 9- and 11-odd-limit diamond monotone
	9/7	382.458
	13/11	384.173
	5/4	386.314

Intuition

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	21/13	369.747
	13/11	372.302
	21/11	373.154
	13/7	378.617
	5/3	378.910
6\19		378.947	19e val, lower bound of 9-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	60eee val
	13/8	380.029	13- and 15-odd-limit minimax
	15/14	380.093
32\101		380.198	101cdeeee val
	7/5	380.228
	21/20	380.279
	[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	41e val
	[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
	21/16	380.634
	7/4	380.735
33\104		380.769	104eeeff val
20\63		380.952	63ef val
	7/6	380.982
	15/8	381.378
	[0 19 -36 30 42⟩	381.380	11-odd-limit least squares
7\22		381.818	22f val, upper bound of 9-odd-limit diamond monotone 11- to 15-odd-limit diamond monotone (singleton)
	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

Telepathy

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	13/8	359.472
	13/10	372.893
	21/11	373.154
	13/12	376.905
	15/13	378.249
	13/9	378.489
	5/3	378.910
6\19		378.947	19e val, lower bound of 9-odd-limit diamond monotone 2.3.5.7.13-subgroup 13-odd-limit diamond monotone (singleton)
	13/7	379.100
	21/13	379.459
	9/5	379.733
	27/20	379.968	5-odd-limit least squares
19\60		380.000	60eeeff val
	15/14	380.093
32\101		380.198	101cdeeeefff val
	7/5	380.228
	21/20	380.279
	[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	41ef val
	[0 1 -7 15⟩	380.506	7-odd-limit least squares
	21/16	380.634
	[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	104eeefff val
20\63		380.952	63eff val
	7/6	380.982
	15/8	381.378
	[0 19 -36 30 42⟩	381.380	11-odd-limit least squares
7\22		381.818	Upper bound of 9-odd-limit diamond monotone 11-odd-limit diamond monotone (singleton)
	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