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


* '''Tridecimal magic''' ({{nowrap|19 & 22f}}) – tempering out 100/99, 105/104, 144/143, and 196/195;  
* '''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;  
* '''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;  
* '''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;  
* '''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;  
* '''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.  
* '''Telepathy''' ({{nowrap| 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 {{nowrap| 41 & 63 }} temperament [[support]]ed by [[104edo]], whereas witchcraft is the {{nowrap| 41 & 60e }} temperament supported by [[101edo]] in the 101cd [[val]].  
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.  
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.  
Line 267: 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
Line 320: Line 490:
| 379.890
| 379.890
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 360: Line 525:
| 380.354
| 380.354
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 56 -31 46 -94 88 }}
| 380.377
| 13-odd-limit least squares
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|
| {{monzo| 0 58 -29 52 -108 100 }}
| 380.389
| 15-odd-limit least squares
|-
|-
|  
|  
Line 385: Line 535:
| 380.488
| 380.488
|  
|  
|-
|
| {{monzo| 0 1 -7 15 }}
| 380.506
| 7-odd-limit least squares
|-
|-
|  
|  
Line 400: Line 545:
| 380.700
| 380.700
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|
| {{monzo| 0 85 -14 52 -68 }}
| 380.714
| 11-odd-limit least squares
|-
|-
|  
|  
Line 477: 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
Line 499: Line 639:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 529: Line 664:
| 380.279
| 380.279
|  
|  
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|-
|  
|  
Line 544: Line 674:
| 380.488
| 380.488
| Lower bound of 13- and 15-odd-limit diamond monotone
| Lower bound of 13- and 15-odd-limit diamond monotone
|-
|
| {{monzo| 0 1 -7 15 }}
| 380.506
| 7-odd-limit least squares
|-
|-
|  
|  
Line 564: Line 689:
| 380.709
| 380.709
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 85 -14 52 -68 }}
| 380.714
| 11-odd-limit least squares
|-
|-
|  
|  
Line 599: Line 719:
| 380.779
| 380.779
|  
|  
|-
|
| {{monzo| 0 -179 -10 -87 53 158 }}
| 380.785
| 13-odd-limit least squares
|-
|-
|  
|  
Line 609: Line 724:
| 380.809
| 380.809
|  
|  
|-
|
| {{monzo| 0 -222 -53 -93 67 187 }}
| 380.817
| 15-odd-limit least squares
|-
|-
|  
|  
Line 686: Line 796:
|}
|}


=== Witchcraft ===
==== Witchcraft ====
{| 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
Line 733: Line 843:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
|  
|  
Line 778: Line 883:
| 380.156
| 380.156
|  
|  
|-
|
| {{monzo| 0 -106 -111 11 179 59 }}
| 380.193
| 15-odd-limit least squares
|-
|-
| 32\101
| 32\101
Line 788: Line 888:
| 380.198
| 380.198
| 101cd val
| 101cd val
|-
|
| {{monzo| 0 -67 -72 5 152 47 }}
| 380.218
| 13-odd-limit least squares
|-
|-
|  
|  
Line 798: Line 893:
| 380.228
| 380.228
|  
|  
|-
|
| {{monzo| 0 -38 -55 11 137 }}
| 380.278
| 11-odd-limit least squares
|-
|-
|  
|  
Line 823: Line 913:
| 380.343
| 380.343
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|-
|  
|  
Line 890: 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
Line 947: Line 1,032:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 977: Line 1,057:
| 380.279
| 380.279
|  
|  
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|-
|  
|  
Line 987: Line 1,062:
| 380.391
| 380.391
| 5-, 7- and 9-odd-limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|
| {{monzo| 0 -113 12 -65 75 26 }}
| 380.427
| 13-odd-limit least squares
|-
|
| {{monzo| 0 134 9 71 -89 -33 }}
| 380.457
| 15-odd-limit least squares
|-
|-
| 13\41
| 13\41
Line 1,002: Line 1,067:
| 380.488
| 380.488
| 41f val
| 41f val
|-
|
| {{monzo| 0 1 -7 15 }}
| 380.506
| 7-odd-limit least squares
|-
|-
|  
|  
Line 1,017: Line 1,077:
| 380.700
| 380.700
| 11-, 13- and 15-odd-limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 85 -14 52 -68 }}
| 380.714
| 11-odd-limit least squares
|-
|-
|  
|  
Line 1,099: Line 1,154:
|}
|}


=== Intuition ===
==== Intuition ====
{| 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
Line 1,161: Line 1,216:
| 379.890
| 379.890
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 1,196: Line 1,246:
| 380.279
| 380.279
|  
|  
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|-
|  
|  
Line 1,211: Line 1,256:
| 380.488
| 380.488
| 41e val
| 41e val
|-
|
| {{monzo| 0 1 -7 15 }}
| 380.506
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -30 -73 30 46 78 }}
| 380.562
| 15-odd-limit least squares
|-
|
| {{monzo| 0 -10 -53 24 38 66 }}
| 380.568
| 13-odd-limit least squares
|-
|-
|  
|  
Line 1,256: Line 1,286:
| 381.378
| 381.378
|  
|  
|-
|
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 11-odd-limit least squares
|-
|-
| 7\22
| 7\22
Line 1,308: Line 1,333:
|}
|}


=== Telepathy ===
==== 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
Line 1,370: Line 1,395:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 1,400: Line 1,420:
| 380.279
| 380.279
|  
|  
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|-
|  
|  
Line 1,415: Line 1,430:
| 380.488
| 380.488
| 41ef val
| 41ef val
|-
|
| {{monzo| 0 1 -7 15 }}
| 380.506
| 7-odd-limit least squares
|-
|-
|  
|  
Line 1,425: Line 1,435:
| 380.634
| 380.634
|  
|  
|-
|
| {{monzo| 0 47 -34 43 57 -48 }}
| 380.676
| 13-odd-limit least squares
|-
|
| {{monzo| 0 46 -35 49 65 -55 }}
| 380.691
| 15-odd-limit least squares
|-
|-
|  
|  
Line 1,465: Line 1,465:
| 381.378
| 381.378
|  
|  
|-
|
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 11-odd-limit least squares
|-
|-
| 7\22
| 7\22
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