Magic extensions: Difference between revisions

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[[Magic]] has various competing [[extension]]s to the 11- and 13-limit. The following temperaments are discussed in this article:
{{Breadcrumb|Magic}}


* '''Tridecimal magic''' (19 & 22f) – tempering out 100/99, 105/104, 144/143, and 196/195;
[[Magic]] has various competing [[extension]]s to the [[11-limit|11-]] and [[13-limit]]. The following temperaments are discussed in this article:
* '''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.  
* '''Tridecimal magic''' ({{nowrap| 19 & 22f }}) – tempering out 100/99, 105/104, 144/143, and 196/195;
* '''Necromancy''' ({{nowrap| 19f & 22 }}) – tempering out 100/99, 225/224, 245/243, and 275/273;
* '''Witchcraft''' ({{nowrap| 19e & 22ef }}) – tempering out 105/104, 196/195, 245/243, and 275/273;
* '''Sorcery''' ({{nowrap| 19 & 22 }}) – tempering out 65/64, 78/77, 91/90, and 100/99;
* '''Intuition''' ({{nowrap| 19e & 22f }}) – tempering out 55/54, 66/65, 99/98, and 105/104;
* '''Telepathy''' ({{nowrap| 19e & 22 }}) – tempering out 55/54, 65/64, 91/90, and 99/98.  


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.  
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 {{nowrap| 41 & 63 }} temperament [[support]]ed by [[104edo]], whereas witchcraft is the {{nowrap| 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]], [[Magic family #Necromancy|#Necromany]], [[Magic family #Witchcraft|#Witchcraft]], [[Magic family #Sorcery|#Sorcery]], [[Magic family #Intuition|#Intuition]], and [[Magic family #Telepathy|#Telepathy]] for technical data.  


== Interval chain ==
== Interval chain ==
Line 15: Line 20:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
! colspan="6" | Approximate ratios
! colspan="7" | Approximate ratios
|-
|-
! rowspan="2" | 7-limit
! rowspan="2" | 7-limit
! colspan="5" | 13-limit extensions
! colspan="6" | 13-limit extensions
|-
|-
! Magic
! Magic
! Necromancy
! Necromancy
! Witchcraft
! Sorcery
! Sorcery
! Witchcraft
! Intuition
! Telepathy
! Telepathy
|-
|-
Line 31: Line 38:
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|  
|  
|  
|  
Line 42: Line 50:
| 26/21
| 26/21
|  
|  
| 26/21
| '''16/13'''
| '''16/13'''
|  
| 26/21
| '''16/13'''
| '''16/13'''
|-
|-
Line 49: Line 58:
| 760.9
| 760.9
| 14/9
| 14/9
|
|  
|  
|  
|  
| 20/13
| 20/13
|  
| 11/7
| 11/7, 20/13
| 11/7, 20/13
|-
|-
Line 60: Line 70:
| 52/27, 64/33
| 52/27, 64/33
| 64/33
| 64/33
| 52/27
| 64/33
| 64/33
| 52/27
| 21/11, 52/27
| 88/45
| 21/11
|-
|-
| 4
| 4
Line 69: Line 80:
| 40/33
| 40/33
| 40/33
| 40/33
|
| 40/33
| 40/33
|  
| 11/9, 13/11
| 11/9
| 11/9
|-
|-
Line 76: Line 88:
| 702.3
| 702.3
| '''3/2'''
| '''3/2'''
|
|  
|  
|  
|  
Line 87: Line 100:
| 13/7
| 13/7
|  
|  
| 13/7
| 24/13
| 24/13
| 13/7
| 13/7
Line 94: Line 108:
| 263.2
| 263.2
| 7/6
| 7/6
|
|  
|  
|  
|  
Line 105: Line 120:
| 13/9, '''16/11'''
| 13/9, '''16/11'''
| '''16/11'''
| '''16/11'''
| 13/9
| '''16/11'''
| '''16/11'''
| 13/9
| 13/9, 22/15
| 22/15
| 22/15
|-
|-
Line 114: Line 130:
| 20/11
| 20/11
| 20/11
| 20/11
|
| 20/11
| 20/11
|  
| 11/6
| 11/6
| 11/6
|-
|-
Line 121: Line 138:
| 204.6
| 204.6
| '''9/8'''
| '''9/8'''
|
|  
|  
|  
|  
Line 130: Line 148:
| 585.0
| 585.0
| 7/5
| 7/5
|
|  
|  
|  
|  
Line 141: Line 160:
| 26/15
| 26/15
|  
|  
| 26/15
|  
|  
| 26/15
| 26/15
Line 150: Line 170:
| 12/11, 13/12
| 12/11, 13/12
| 12/11
| 12/11
| 13/12
| 12/11, 14/13
| 12/11, 14/13
| 13/12
| 11/10, 13/12
| 11/10, 14/13
| 11/10, 14/13
|-
|-
Line 159: Line 180:
| 15/11
| 15/11
| 15/11
| 15/11
|
| 15/11
| 15/11
|  
| '''11/8'''
| '''11/8'''
| '''11/8'''
|-
|-
Line 167: Line 189:
| 27/16, 42/25
| 27/16, 42/25
|  
|  
| 22/13
| 22/13
| 22/13
|
|  
|  
| 22/13
| 22/13
Line 176: Line 200:
|  
|  
|  
|  
| 22/21
| 27/26
| 27/26
| 22/21
|  
| 27/26
| 27/26
|-
|-
Line 185: Line 210:
| 13/10
| 13/10
|  
|  
| 13/10
|  
|  
| 13/10
| 13/10
Line 194: Line 220:
| '''13/8''', 18/11
| '''13/8''', 18/11
| 18/11
| 18/11
| '''13/8'''
| 18/11, 21/13
| 18/11, 21/13
| '''13/8'''
| '''13/8'''
Line 203: Line 230:
| 45/44
| 45/44
| 45/44
| 45/44
|
| 45/44
| 45/44
|  
| 33/32
| 33/32
| 33/32
|-
|-
Line 212: Line 240:
| 14/11
| 14/11
| 14/11
| 14/11
|
| 14/11
| 14/11
|  
|  
Line 220: Line 249:
| 63/40
| 63/40
|  
|  
|
| 11/7
|  
|  
|  
|  
| 11/7
|  
|  
|-
|-
Line 230: Line 260:
| 39/20
| 39/20
|  
|  
| 39/20
|  
|  
| 39/20
| 39/20
Line 236: Line 267:
<nowiki/>* In 7-limit [[CWE tuning]]
<nowiki/>* In 7-limit [[CWE tuning]]


== Tuning spectra ==
== Tunings ==
=== Magic ===
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (magic)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~5/4 = 380.700{{c}}
| 11/9
| ~5/4 = 380.714{{c}}
| {{Monzo| 0 85 -14 52 -68 }}
|-
| 13-odd-limit
| ~5/4 = 380.354{{c}}
| 13/11
| ~5/4 = 380.377{{c}}
| {{Monzo| 0 56 -31 46 -94 88 }}
|-
| 15-odd-limit
| ~5/4 = 380.354{{c}}
| 13/11
| ~5/4 = 380.389{{c}}
| {{Monzo| 0 58 -29 52 -108 100 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (necromancy)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~5/4 = 380.709{{c}}
| 13/9
| ~5/4 = 380.705{{c}}
| {{Monzo| 0 -179 -10 -87 53 158 }}
|-
| 15-odd-limit
| ~5/4 = 380.709{{c}}
| 13/9
| ~5/4 = 380.817{{c}}
| {{Monzo| 0 -222 -53 -93 67 187 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (witchcraft)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~5/4 = 380.343{{c}}
| 11/8
| ~5/4 = 380.278{{c}}
| {{Monzo| 0 -38 -55 11 137 }}
|-
| 13-odd-limit
| ~5/4 = 380.119{{c}}
| 11/7
| ~5/4 = 380.218{{c}}
| {{Monzo| 0 -67 -72 5 152 47 }}
|-
| 15-odd-limit
| ~5/4 = 380.113{{c}}
| 15/11
| ~5/4 = 380.193{{c}}
| {{Monzo| 0 -106 -111 11 179 59 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (sorcery)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~5/4 = 380.700{{c}}
| 11/9
| ~5/4 = 380.427{{c}}
| {{Monzo| 0 -113 12 -65 75 26 }}
|-
| 15-odd-limit
| ~5/4 = 380.700{{c}}
| 11/9
| ~5/4 = 380.457{{c}}
| {{Monzo| 0 134 9 71 -89 -33 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (intuition)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~5/4 = 380.029{{c}}
| 13/8
| ~5/4 = 380.568{{c}}
| {{Monzo| 0 -10 -53 24 38 66 }}
|-
| 15-odd-limit
| ~5/4 = 380.029{{c}}
| 13/8
| ~5/4 = 380.562{{c}}
| {{Monzo| 0 -30 -73 30 46 78 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (telepathy)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~5/4 = 381.923{{c}}
| 11/10
| ~5/4 = 381.380{{c}}
| {{Monzo| 0 19 -36 30 42 }}
|-
| 13-odd-limit
| ~5/4 = 380.719{{c}}
| 13/11
| ~5/4 = 380.676{{c}}
| {{Monzo| 0 47 -34 43 57 -48 }}
|-
| 15-odd-limit
| ~5/4 = 380.719{{c}}
| 13/11
| ~5/4 = 380.691{{c}}
| {{Monzo| 0 46 -35 49 65 -55 }}
|}
 
=== Tuning spectra ===
==== Magic ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 14/13
| 21/13
| 369.747
|
|-
|
| 13/7
| 378.617
| 378.617
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
Line 258: Line 464:
|  
|  
| 378.947
| 378.947
|  
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 266: Line 472:
|-
|-
|  
|  
| 18/13
| 13/9
| 379.577
| 379.577
|  
|  
Line 276: Line 482:
|-
|-
|  
|  
| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
Line 284: Line 490:
| 379.890
| 379.890
|  
|  
|-
|
| 27/20
| 379.968
| 5 limit least squares
|-
|-
| 19\60
| 19\60
Line 296: Line 497:
|-
|-
|  
|  
| 16/13
| 13/8
| 380.029
| 380.029
|  
|  
Line 308: Line 509:
|  
|  
| 380.198
| 380.198
|  
| 101cde val
|-
|-
|  
|  
Line 316: Line 517:
|-
|-
|  
|  
| 13/11
| 21/20
| 380.354
| 380.279
| 13 and 15 limit minimax
|-
|  
|  
| {{vector|0 56 -31 46 -94 88}}
| 380.377
| 13 limit least squares
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| 13/11
| 380.384
| 380.354
| 9 limit least squares
| 13- and 15-odd-limit minimax
|-
|
| {{vector|0 58 -29 52 -108 100}}
| 380.389
| 15 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 346: Line 537:
|-
|-
|  
|  
| {{vector|0 1 -7 15}}
| 21/16
| 380.506
| 380.634
| 7 limit least squares
|  
|-
|-
|  
|  
| 11/9
| 11/9
| 380.700
| 380.700
| 11 limit minimax
| 11-odd-limit minimax
|-
|
| {{vector|0 85 -14 52 -68}}
| 380.714
| 11 limit least squares
|-
|-
|  
|  
| 8/7
| 7/4
| 380.735
| 380.735
|  
|  
Line 368: Line 554:
|  
|  
| 380.769
| 380.769
| 104ff val
|-
|
| 21/11
| 380.779
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 380.818
| 380.818
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 380.875
| 380.875
|  
|  
Line 383: Line 574:
|  
|  
| 380.952
| 380.952
|  
| 63f val
|-
|-
|  
|  
Line 401: Line 592:
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
Line 413: Line 604:
|  
|  
| 381.818
| 381.818
|  
| 22f val, upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 426: Line 617:
|}
|}


=== Necromancy ===
==== Necromancy ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
Line 442: Line 633:
|  
|  
| 378.947
| 378.947
|  
| 19f val, lower bound of 9- and 11-odd-limit diamond monotone
|-
|-
|  
|  
| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5 limit least squares
|-
|-
| 19\60
| 19\60
|  
|  
| 380.000
| 380.000
|  
| 60f val
|-
|-
|  
|  
Line 467: Line 653:
|  
|  
| 380.198
| 380.198
|  
| 101cdeff val
|-
|-
|  
|  
Line 475: Line 661:
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| 21/20
| 380.384
| 380.279
| 9 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
|  
|  
| 380.488
| 380.488
| Lower bound of 13- and 15-odd-limit diamond monotone
|-
|  
|  
|-
| 21/16
| 380.634
|  
|  
| {{vector|0 1 -7 15}}
| 380.506
| 7 limit least squares
|-
|-
|  
|  
| 11/9
| 11/9
| 380.700
| 380.700
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
| 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}}
| 380.714
| 11 limit least squares
|-
|-
|  
|  
Line 515: Line 696:
|-
|-
|  
|  
| 8/7
| 7/4
| 380.735
| 380.735
|
|-
|
| 21/13
| 380.756
|  
|  
|-
|-
Line 530: Line 716:
|-
|-
|  
|  
| {{vector|0 -179 -10 -87 53 158}}
| 21/11
| 380.785
| 380.779
| 13 limit least squares
|  
|-
|-
|  
|  
| 14/13
| 13/7
| 380.809
| 380.809
|  
|  
|-
|-
|  
|  
| {{vector|0 -222 -53 -93 67 187}}
| 11/6
| 380.817
| 15 limit least squares
|-
|
| 12/11
| 380.818
| 380.818
|  
|  
|-
|-
|  
|  
| 16/13
| 13/8
| 380.847
| 380.847
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 380.875
| 380.875
|  
|  
Line 562: Line 743:
|  
|  
| 380.952
| 380.952
|  
| Upper bound of 13- and 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 590: Line 771:
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
Line 602: Line 783:
|  
|  
| 381.818
| 381.818
| Upper bound of 9- and 11-odd-limit diamond monotone
|-
|
| 9/7
| 382.458
|
|-
|
| 5/4
| 386.314
|
|}
==== Witchcraft ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! 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
|
|-
|
| 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
|
|-
| 32\101
|
| 380.198
| 101cd val
|-
|
| 7/5
| 380.228
|
|-
|
| 21/20
| 380.279
|  
|  
|-
|
| 11/9
| 380.322
|
|-
|
| 11/6
| 380.334
|
|-
|
| 11/8
| 380.343
| 11-odd-limit minimax
|-
|
| 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
|-
|-
|  
|  
Line 615: Line 975:
|}
|}


=== Sorcery ===
==== Sorcery ====
 
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/13
| 13/8
| 359.472
| 359.472
|  
|  
Line 645: Line 1,004:
|-
|-
|  
|  
| 18/13
| 13/9
| 378.489
| 378.489
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
Line 657: Line 1,016:
|  
|  
| 378.947
| 378.947
|  
| Lower bound of 9- and 11-odd-limit diamond monotone<br>13- and 15-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
| 14/13
| 13/7
| 379.100
| 379.100
|  
|  
|-
|-
|  
|  
| 10/9
| 21/13
| 379.733
| 379.459
|  
|  
|-
|-
|  
|  
| 27/20
| 9/5
| 379.968
| 379.733
| 5 limit least squares
|  
|-
|-
| 19\60
| 19\60
|  
|  
| 380.000
| 380.000
|  
| 60ff val
|-
|-
|  
|  
Line 687: Line 1,046:
|  
|  
| 380.198
| 380.198
|  
| 101cdefff val
|-
|-
|  
|  
Line 695: Line 1,054:
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| 21/20
| 380.384
| 380.279
| 9 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}}
| 380.427
| 13 limit least squares
|-
|
| {{vector|0 134 9 71 -89 -33}}
| 380.457
| 15 limit least squares
|-
|-
| 13\41
| 13\41
|  
|  
| 380.488
| 380.488
| 41f val
|-
|  
|  
|-
| 21/16
| 380.634
|  
|  
| {{vector|0 1 -7 15}}
| 380.506
| 7 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}}
| 7/4
| 380.714
| 11 limit least squares
|-
|
| 8/7
| 380.735
| 380.735
|  
|  
Line 742: Line 1,086:
|  
|  
| 380.769
| 380.769
| 104fff val
|-
|
| 21/11
| 380.779
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 380.818
| 380.818
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 380.875
| 380.875
|  
|  
Line 757: Line 1,106:
|  
|  
| 380.952
| 380.952
|  
| 63ff val
|-
|-
|  
|  
Line 775: Line 1,124:
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
Line 787: Line 1,136:
|  
|  
| 381.818
| 381.818
|  
| Upper bound of 9- and 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 805: Line 1,154:
|}
|}


=== Telepathy ===
==== Intuition ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/13
| 21/13
| 359.472
| 369.747
|  
|  
|-
|-
|  
|  
| 13/10
| 13/11
| 372.893
| 372.302
|  
|  
|-
|-
|  
|  
| 13/12
| 21/11
| 376.905
| 373.154
|  
|  
|-
|-
|  
|  
| 15/13
| 13/7
| 378.249
| 378.617
|  
|  
|-
|-
|  
|  
| 18/13
| 5/3
| 378.489
| 378.910
|  
|  
|-
| 6\19
|
| 378.947
| 19e val, lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 6/5
| 15/13
| 378.910
| 379.355
|  
|  
|-
|-
| 6\19
|  
|  
| 378.947
| 13/9
| 379.577
|  
|  
|-
|-
|  
|  
| 14/13
| 13/10
| 379.100
| 379.660
|  
|  
|-
|-
|  
|  
| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
|-
|-
|  
|  
| 27/20
| 13/12
| 379.968
| 379.890
| 5 limit least squares
|  
|-
|-
| 19\60
| 19\60
|  
|
| 380.000
| 380.000
| 60eee val
|-
|  
|  
| 13/8
| 380.029
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 874: Line 1,233:
|-
|-
| 32\101
| 32\101
|  
|
| 380.198
| 380.198
|  
| 101cdeeee val
|-
|-
|  
|  
Line 884: Line 1,243:
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| 21/20
| 380.384
| 380.279
| 9 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
|  
|
| 380.488
| 380.488
|  
| 41e val
|-
|-
|  
|  
| {{vector|0 1 -7 15}}
| 21/16
| 380.506
| 380.634
| 7 limit least squares
|-
|  
|  
| {{vector|0 47 -34 43 57 -48}}
| 380.676
| 13 limit least squares
|-
|-
|  
|  
| {{vector|0 46 -35 49 65 -55}}
| 7/4
| 380.691
| 15 limit least squares
|-
|
| 13/11
| 380.719
| 13 and 15 limit minimax
|-
|
| 8/7
| 380.735
| 380.735
|  
|  
|-
|-
| 33\104
| 33\104
|  
|
| 380.769
| 380.769
|  
| 104eeeff val
|-
|-
| 20\63
| 20\63
|  
|
| 380.952
| 380.952
|  
| 63ef val
|-
|-
|  
|  
Line 939: Line 1,283:
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
|-
|
| {{vector|0 19 -36 30 42}}
| 381.380
| 11 limit least squares
|-
|-
| 7\22
| 7\22
|  
|
| 381.818
| 381.818
|  
| 22f val, upper bound of 9-odd-limit diamond monotone<br>11- to 15-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
| 11/10
| 11/10
| 381.923
| 381.923
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
Line 974: Line 1,313:
|-
|-
|  
|  
| 12/11
| 11/6
| 383.263
| 383.263
|  
|  
Line 989: Line 1,328:
|-
|-
|  
|  
| 14/11
| 11/7
| 391.246
| 391.246
|  
|  
|}
|}


=== Witchcraft ===
==== Telepathy ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 14/13
| 13/8
| 378.617
| 359.472
|  
|  
|-
|-
|  
|  
| 6/5
| 13/10
| 378.910
| 372.893
|  
|  
|-
|-
| 6\19
|  
|  
| 378.947
| 21/11
| 373.154
|  
|  
|-
|-
|  
|  
| 15/13
| 13/12
| 379.355
| 376.905
|  
|  
|-
|-
|  
|  
| 18/13
| 15/13
| 379.577
| 378.249
|  
|  
|-
|-
|  
|  
| 13/10
| 13/9
| 379.660
| 378.489
|  
|  
|-
|-
|  
|  
| 10/9
| 5/3
| 379.733
| 378.910
|  
|  
|-
|-
| 6\19
|  
|  
| 27/20
| 378.947
| 379.968
| 19e val, lower bound of 9-odd-limit diamond monotone<br>2.3.5.7.13-subgroup 13-odd-limit diamond monotone (singleton)
| 5 limit least squares
|-
|-
|  
|  
| 13/12
| 13/7
| 379.890
| 379.100
|  
|  
|-
|-
| 19\60
|  
|  
| 380.000
| 21/13
| 379.459
|  
|  
|-
|-
|  
|  
| 16/13
| 9/5
| 380.029
| 379.733
|
|-
| 19\60
|  
|  
| 380.000
| 60eeeff val
|-
|-
|  
|  
Line 1,061: Line 1,405:
| 380.093
| 380.093
|  
|  
|-
|
| 15/11
| 380.113
| 15 limit minimax
|-
|
| 14/11
| 380.119
| 13 limit minimax
|-
|
| 11/10
| 380.156
|
|-
|
| {{vector|0 -106 -111 11 179 59}}
| 380.193
| 15 limit least squares
|-
|-
| 32\101
| 32\101
|  
|  
| 380.198
| 380.198
|  
| 101cdeeeefff val
|-
|
| {{vector|0 -67 -72 5 152 47}}
| 380.218
| 13 limit least squares
|-
|-
|  
|  
Line 1,098: Line 1,417:
|-
|-
|  
|  
| {{vector|0 -38 -55 11 137}}
| 21/20
| 380.278
| 380.279
| 11 limit least squares
|-
|  
|  
| 11/9
| 380.322
|
|-
|
| 12/11
| 380.334
|
|-
|
| 11/8
| 380.343
| 11 limit minimax
|-
|
| {{vector|0 36 -23 32}}
| 380.384
| 9 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
|  
|  
| 380.488
| 380.488
| 41ef val
|-
|
| 21/16
| 380.634
|  
|  
|-
|-
Line 1,135: Line 1,439:
| 13/11
| 13/11
| 380.719
| 380.719
|  
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 8/7
| 7/4
| 380.735
| 380.735
|  
|  
Line 1,145: Line 1,449:
|  
|  
| 380.769
| 380.769
|  
| 104eeefff val
|-
|-
| 20\63
| 20\63
|  
|  
| 380.952
| 380.952
|  
| 63eff val
|-
|-
|  
|  
Line 1,158: Line 1,462:
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
Line 1,165: Line 1,469:
|  
|  
| 381.818
| 381.818
| Upper bound of 9-odd-limit diamond monotone<br>11-odd-limit diamond monotone (singleton)
|-
|
| 11/10
| 381.923
| 11-odd-limit minimax
|-
|
| 11/8
| 382.237
|  
|  
|-
|-
Line 1,170: Line 1,484:
| 9/7
| 9/7
| 382.458
| 382.458
|
|-
|
| 15/11
| 382.881
|
|-
|
| 11/6
| 383.263
|  
|  
|-
|-
Line 1,175: Line 1,499:
| 5/4
| 5/4
| 386.314
| 386.314
|
|-
|
| 11/9
| 386.852
|
|-
|
| 11/7
| 391.246
|  
|  
|}
|}
Line 1,180: Line 1,514:
[[Category:Magic]]
[[Category:Magic]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Latest revision as of 14:08, 26 September 2025

< Magic

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

Tunings

Target tunings

Target tunings (magic)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
11-odd-limit ~5/4 = 380.700 ¢ 11/9 ~5/4 = 380.714 ¢ [0 85 -14 52 -68⟩
13-odd-limit ~5/4 = 380.354 ¢ 13/11 ~5/4 = 380.377 ¢ [0 56 -31 46 -94 88⟩
15-odd-limit ~5/4 = 380.354 ¢ 13/11 ~5/4 = 380.389 ¢ [0 58 -29 52 -108 100⟩
Target tunings (necromancy)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~5/4 = 380.709 ¢ 13/9 ~5/4 = 380.705 ¢ [0 -179 -10 -87 53 158⟩
15-odd-limit ~5/4 = 380.709 ¢ 13/9 ~5/4 = 380.817 ¢ [0 -222 -53 -93 67 187⟩
Target tunings (witchcraft)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
11-odd-limit ~5/4 = 380.343 ¢ 11/8 ~5/4 = 380.278 ¢ [0 -38 -55 11 137⟩
13-odd-limit ~5/4 = 380.119 ¢ 11/7 ~5/4 = 380.218 ¢ [0 -67 -72 5 152 47⟩
15-odd-limit ~5/4 = 380.113 ¢ 15/11 ~5/4 = 380.193 ¢ [0 -106 -111 11 179 59⟩
Target tunings (sorcery)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~5/4 = 380.700 ¢ 11/9 ~5/4 = 380.427 ¢ [0 -113 12 -65 75 26⟩
15-odd-limit ~5/4 = 380.700 ¢ 11/9 ~5/4 = 380.457 ¢ [0 134 9 71 -89 -33⟩
Target tunings (intuition)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~5/4 = 380.029 ¢ 13/8 ~5/4 = 380.568 ¢ [0 -10 -53 24 38 66⟩
15-odd-limit ~5/4 = 380.029 ¢ 13/8 ~5/4 = 380.562 ¢ [0 -30 -73 30 46 78⟩
Target tunings (telepathy)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
11-odd-limit ~5/4 = 381.923 ¢ 11/10 ~5/4 = 381.380 ¢ [0 19 -36 30 42⟩
13-odd-limit ~5/4 = 380.719 ¢ 13/11 ~5/4 = 380.676 ¢ [0 47 -34 43 57 -48⟩
15-odd-limit ~5/4 = 380.719 ¢ 13/11 ~5/4 = 380.691 ¢ [0 46 -35 49 65 -55⟩

Tuning spectra

Magic

Edo
generator 		Unchanged interval
(eigenmonzo) 		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
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
3/2 380.391 5-, 7- and 9-odd-limit minimax
13\41 380.488
21/16 380.634
11/9 380.700 11-odd-limit minimax
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 		Unchanged interval
(eigenmonzo) 		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
19\60 380.000 60f val
15/14 380.093
32\101 380.198 101cdeff val
7/5 380.228
21/20 380.279
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
21/16 380.634
11/9 380.700 11-odd-limit minimax
13/9 380.709 13- and 15-odd-limit minimax
13/11 380.719
7/4 380.735
21/13 380.756
13/12 380.765
33\104 380.769
21/11 380.779
13/7 380.809
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 		Unchanged interval
(eigenmonzo) 		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
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
32\101 380.198 101cd val
7/5 380.228
21/20 380.279
11/9 380.322
11/6 380.334
11/8 380.343 11-odd-limit minimax
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 		Unchanged interval
(eigenmonzo) 		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
19\60 380.000 60ff val
15/14 380.093
32\101 380.198 101cdefff val
7/5 380.228
21/20 380.279
3/2 380.391 5-, 7- and 9-odd-limit minimax
13\41 380.488 41f val
21/16 380.634
11/9 380.700 11-, 13- and 15-odd-limit minimax
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 		Unchanged interval
(eigenmonzo) 		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
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
3/2 380.391 5-, 7- and 9-odd-limit minimax
13\41 380.488 41e val
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
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 		Unchanged interval
(eigenmonzo) 		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
19\60 380.000 60eeeff val
15/14 380.093
32\101 380.198 101cdeeeefff val
7/5 380.228
21/20 380.279
3/2 380.391 5-, 7- and 9-odd-limit minimax
13\41 380.488 41ef val
21/16 380.634
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
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
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