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. Necromancy is the 41 & 63 temperament supported by 104edo, whereas witchcraft is the 41 & 60e temperament supported by 101edo in the 101cd val.
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, intuition 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.
See Magic family #Magic, #Necromany, #Witchcraft, #Sorcery, #Intuition, and #Telepathy for technical data.
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 | 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 |