Magic extensions: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
→Tuning spectra: +tuning ranges; fix some warts
Target tunings
 
(9 intermediate revisions by 4 users not shown)
Line 1: Line 1:
[[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;
* '''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.  
* '''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. 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.  
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 16: Line 20:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
Line 262: 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
|-
|
| 21/13
| 369.747
|
|-
|-
|  
|  
Line 310: Line 490:
| 379.890
| 379.890
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 339: Line 514:
| 7/5
| 7/5
| 380.228
| 380.228
|
|-
|
| 21/20
| 380.279
|  
|  
|-
|-
Line 345: 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 372: Line 537:
|-
|-
|  
|  
| {{monzo| 0 1 -7 15 }}
| 21/16
| 380.506
| 380.634
| 7-odd-limit least squares
|  
|-
|-
|  
|  
Line 380: 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 395: Line 555:
| 380.769
| 380.769
| 104ff val
| 104ff val
|-
|
| 21/11
| 380.779
|
|-
|-
|  
|  
Line 427: Line 592:
|-
|-
|  
|  
| 15/88
| 15/8
| 381.378
| 381.378
|  
|  
Line 452: 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 474: Line 639:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 501: Line 661:
|-
|-
|  
|  
| {{monzo| 0 36 -23 32 }}
| 21/20
| 380.384
| 380.279
| 9-odd-limit least squares
|  
|-
|-
|  
|  
Line 516: Line 676:
|-
|-
|  
|  
| {{monzo| 0 1 -7 15 }}
| 21/16
| 380.506
| 380.634
| 7-odd-limit least squares
|  
|-
|-
|  
|  
Line 529: 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 543: Line 698:
| 7/4
| 7/4
| 380.735
| 380.735
|
|-
|
| 21/13
| 380.756
|  
|  
|-
|-
Line 556: Line 716:
|-
|-
|  
|  
| {{monzo| 0 -179 -10 -87 53 158 }}
| 21/11
| 380.785
| 380.779
| 13-odd-limit least squares
|  
|-
|-
|  
|  
Line 564: Line 724:
| 380.809
| 380.809
|  
|  
|-
|
| {{monzo| 0 -222 -53 -93 67 187 }}
| 380.817
| 15-odd-limit least squares
|-
|-
|  
|  
Line 641: 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
|-
|
| 21/13
| 369.747
|
|-
|-
|  
|  
Line 683: Line 843:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
|  
|  
Line 702: Line 857:
| 13/8
| 13/8
| 380.029
| 380.029
|
|-
|
| 21/11
| 380.034
|  
|  
|-
|-
Line 723: 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 733: 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 745: Line 895:
|-
|-
|  
|  
| {{monzo| 0 -38 -55 11 137 }}
| 21/20
| 380.278
| 380.279
| 11-odd-limit least squares
|  
|-
|-
|  
|  
Line 763: 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 778: Line 923:
| 380.488
| 380.488
| Upper bound of 11- to 15-odd-limit diamond monotone
| Upper bound of 11- to 15-odd-limit diamond monotone
|-
|
| 21/16
| 380.634
|
|-
|-
|  
|  
Line 825: 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 871: Line 1,021:
| 13/7
| 13/7
| 379.100
| 379.100
|
|-
|
| 21/13
| 379.459
|  
|  
|-
|-
Line 877: Line 1,032:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 904: Line 1,054:
|-
|-
|  
|  
| {{monzo| 0 36 -23 32 }}
| 21/20
| 380.384
| 380.279
| 9-odd-limit least squares
|  
|-
|-
|  
|  
Line 912: 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 929: Line 1,069:
|-
|-
|  
|  
| {{monzo| 0 1 -7 15 }}
| 21/16
| 380.506
| 380.634
| 7-odd-limit least squares
|  
|-
|-
|  
|  
Line 937: 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 952: Line 1,087:
| 380.769
| 380.769
| 104fff val
| 104fff val
|-
|
| 21/11
| 380.779
|
|-
|-
|  
|  
Line 1,014: 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
|-
|-
|
|
| 21/13
| 369.747
|
|-
|  
| 13/11
| 13/11
| 372.302
| 372.302
|
|
|-
|
| 21/11
| 373.154
|  
|-
|-
|
|  
| 13/7
| 13/7
| 378.617
| 378.617
|
|  
|-
|-
|
|  
| 5/3
| 5/3
| 378.910
| 378.910
Line 1,042: Line 1,192:
| 19e val, lower bound of 9-odd-limit diamond monotone
| 19e val, lower bound of 9-odd-limit diamond monotone
|-
|-
|
|  
| 15/13
| 15/13
| 379.355
| 379.355
|
|  
|-
|-
|
|  
| 13/9
| 13/9
| 379.577
| 379.577
|
|  
|-
|-
|
|  
| 13/10
| 13/10
| 379.660
| 379.660
|
|  
|-
|-
|
|  
| 9/5
| 9/5
| 379.733
| 379.733
|
|  
|-
|-
|
|  
| 13/12
| 13/12
| 379.890
| 379.890
|
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 1,077: Line 1,222:
| 60eee val
| 60eee val
|-
|-
|
|  
| 13/8
| 13/8
| 380.029
| 380.029
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|
|  
| 15/14
| 15/14
| 380.093
| 380.093
Line 1,092: Line 1,237:
| 101cdeeee val
| 101cdeeee val
|-
|-
|
|  
| 7/5
| 7/5
| 380.228
| 380.228
|  
|  
|-
|-
|
|  
| {{monzo| 0 36 -23 32 }}
| 21/20
| 380.384
| 380.279
| 9-odd-limit least squares
|  
|-
|-
|
|  
| 3/2
| 3/2
| 380.391
| 380.391
Line 1,112: Line 1,257:
| 41e val
| 41e val
|-
|-
|
|  
| {{monzo| 0 1 -7 15 }}
| 21/16
| 380.506
| 380.634
| 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
|-
|
| 7/4
| 7/4
| 380.735
| 380.735
Line 1,142: Line 1,277:
| 63ef val
| 63ef val
|-
|-
|
|  
| 7/6
| 7/6
| 380.982
| 380.982
|  
|  
|-
|-
|
|  
| 15/8
| 15/8
| 381.378
| 381.378
|  
|  
|-
|
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 11-odd-limit least squares
|-
|-
| 7\22
| 7\22
Line 1,162: Line 1,292:
| 22f val, upper bound of 9-odd-limit diamond monotone<br>11- to 15-odd-limit diamond monotone (singleton)
| 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-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|
|  
| 11/8
| 11/8
| 382.237
| 382.237
|  
|  
|-
|-
|
|  
| 9/7
| 9/7
| 382.458
| 382.458
|  
|  
|-
|-
|
|  
| 15/11
| 15/11
| 382.881
| 382.881
|  
|  
|-
|-
|
|  
| 11/6
| 11/6
| 383.263
| 383.263
|  
|  
|-
|-
|
|  
| 5/4
| 5/4
| 386.314
| 386.314
|  
|  
|-
|-
|
|  
| 11/9
| 11/9
| 386.852
| 386.852
|  
|  
|-
|-
|
|  
| 11/7
| 11/7
| 391.246
| 391.246
Line 1,203: 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 1,219: Line 1,349:
| 13/10
| 13/10
| 372.893
| 372.893
|
|-
|
| 21/11
| 373.154
|  
|  
|-
|-
Line 1,249: Line 1,384:
| 13/7
| 13/7
| 379.100
| 379.100
|
|-
|
| 21/13
| 379.459
|  
|  
|-
|-
Line 1,255: Line 1,395:
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|-
| 19\60
| 19\60
Line 1,282: Line 1,417:
|-
|-
|  
|  
| {{monzo| 0 36 -23 32 }}
| 21/20
| 380.384
| 380.279
| 9-odd-limit least squares
|  
|-
|-
|  
|  
Line 1,297: Line 1,432:
|-
|-
|  
|  
| {{monzo| 0 1 -7 15 }}
| 21/16
| 380.506
| 380.634
| 7-odd-limit least squares
|-
|  
|  
| {{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 1,340: Line 1,465:
| 381.378
| 381.378
|  
|  
|-
|
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 11-odd-limit least squares
|-
|-
| 7\22
| 7\22
Line 1,394: 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
Retrieved from "https://en.xen.wiki/index.php?title=Magic_extensions&oldid=210952"