Xenharmonic Wiki

Magic extensions: Difference between revisions

← Older edit
VisualWikitext
Expand with explanations
Target tunings
 
(15 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;
* '''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 168: Line 190:
|  
|  
| 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}}
| 7/4
| 380.714
| 11 limit least squares
|-
|
| 8/7
| 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
|-
|-
|  
|  
Line 615: Line 796:
|}
|}


=== Sorcery ===
==== 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
|-
|-
|  
|  
| 16/13
| 21/13
| 359.472
| 369.747
|  
|  
|-
|-
|  
|  
| 13/10
| 13/7
| 372.893
| 378.617
|  
|  
|-
|-
|  
|  
| 13/12
| 5/3
| 376.905
| 378.910
|
|-
| 6\19
|  
|  
| 378.947
| 19e val, lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 15/13
| 15/13
| 378.249
| 379.355
|  
|  
|-
|-
|  
|  
| 18/13
| 13/9
| 378.489
| 379.577
|  
|  
|-
|-
|  
|  
| 6/5
| 13/10
| 378.910
| 379.660
|  
|  
|-
|-
| 6\19
|  
|  
| 378.947
| 9/5
| 379.733
|  
|  
|-
|-
|  
|  
| 14/13
| 13/12
| 379.100
| 379.890
|  
|  
|-
|-
| 19\60
|  
|  
| 10/9
| 380.000
| 379.733
| 60e val, lower bound of 11- to 15-odd-limit diamond monotone
|-
|  
|  
|-
| 13/8
| 380.029
|  
|  
| 27/20
| 379.968
| 5 limit least squares
|-
|-
| 19\60
|  
|  
| 380.000
| 21/11
| 380.034
|  
|  
|-
|-
Line 682: Line 867:
| 15/14
| 15/14
| 380.093
| 380.093
|
|-
|
| 15/11
| 380.113
| 15-odd-limit minimax
|-
|
| 11/7
| 380.119
| 13-odd-limit minimax
|-
|
| 11/10
| 380.156
|  
|  
|-
|-
Line 687: Line 887:
|  
|  
| 380.198
| 380.198
|  
| 101cd val
|-
|-
|  
|  
Line 695: Line 895:
|-
|-
|  
|  
| {{vector|0 36 -23 32}}
| 21/20
| 380.384
| 380.279
| 9 limit least squares
|-
|  
|  
| 4/3
| 380.391
| 5, 7 and 9 limit minimax
|-
|-
|  
|  
| {{vector|0 -113 12 -65 75 26}}
| 11/9
| 380.427
| 380.322
| 13 limit least squares
|-
|  
|  
| {{vector|0 134 9 71 -89 -33}}
| 380.457
| 15 limit least squares
|-
|-
| 13\41
|  
|  
| 380.488
| 11/6
| 380.334
|  
|  
|-
|-
|  
|  
| {{vector|0 1 -7 15}}
| 11/8
| 380.506
| 380.343
| 7 limit least squares
| 11-odd-limit minimax
|-
|-
|  
|  
| 11/9
| 3/2
| 380.700
| 380.391
| 11, 13 and 15 limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|-
| 13\41
|  
|  
| {{vector|0 85 -14 52 -68}}
| 380.488
| 380.714
| Upper bound of 11- to 15-odd-limit diamond monotone
| 11 limit least squares
|-
|-
|  
|  
| 8/7
| 21/16
| 380.735
| 380.634
|  
|  
|-
|-
| 33\104
|  
|  
| 380.769
| 13/11
| 380.719
|  
|  
|-
|-
|  
|  
| 12/11
| 7/4
| 380.818
| 380.735
|  
|  
|-
|-
| 33\104
|  
|  
| 14/11
| 380.769
| 380.875
| 104eeff val
|  
|-
|-
| 20\63
| 20\63
|  
|  
| 380.952
| 380.952
|  
| 63eef val
|-
|-
|  
|  
Line 765: Line 955:
|-
|-
|  
|  
| 11/8
| 15/8
| 381.085
|
|-
|
| 15/11
| 381.211
|
|-
|
| 16/15
| 381.378
| 381.378
|
|-
|
| 11/10
| 381.666
|  
|  
|-
|-
Line 787: Line 962:
|  
|  
| 381.818
| 381.818
|  
| 22ef val, upper bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 9/7
| 9/7
| 382.458
| 382.458
|
|-
|
| 13/11
| 384.173
|  
|  
|-
|-
Line 805: Line 975:
|}
|}


=== Telepathy ===
==== 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 834: Line 1,004:
|-
|-
|  
|  
| 18/13
| 13/9
| 378.489
| 378.489
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
Line 846: 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 876: Line 1,046:
|  
|  
| 380.198
| 380.198
|  
| 101cdefff val
|-
|-
|  
|  
Line 884: 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
|-
|-
| 13\41
| 13\41
|  
|  
| 380.488
| 380.488
| 41f val
|-
|
| 21/16
| 380.634
|  
|  
|-
|-
|  
|  
| {{vector|0 1 -7 15}}
| 11/9
| 380.506
| 380.700
| 7 limit least squares
| 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 47 -34 43 57 -48}}
| 7/4
| 380.676
| 380.735
| 13 limit least squares
|  
|-
|-
| 33\104
|  
|  
| {{vector|0 46 -35 49 65 -55}}
| 380.769
| 380.691
| 104fff val
| 15 limit least squares
|-
|-
|  
|  
| 13/11
| 21/11
| 380.719
| 380.779
| 13 and 15 limit minimax
|  
|-
|-
|  
|  
| 8/7
| 11/6
| 380.735
| 380.818
|  
|  
|-
|-
| 33\104
|  
|  
| 380.769
| 11/7
| 380.875
|  
|  
|-
|-
Line 931: Line 1,106:
|  
|  
| 380.952
| 380.952
|  
| 63ff val
|-
|-
|  
|  
Line 939: Line 1,114:
|-
|-
|  
|  
| 16/15
| 11/8
| 381.378
| 381.085
|  
|  
|-
|-
|  
|  
| {{vector|0 19 -36 30 42}}
| 15/11
| 381.380
| 381.211
| 11 limit least squares
|  
|-
|-
| 7\22
|  
|  
| 381.818
| 15/8
| 381.378
|  
|  
|-
|-
|  
|  
| 11/10
| 11/10
| 381.923
| 381.666
| 11 limit minimax
|  
|-
|-
| 7\22
|  
|  
| 11/8
| 381.818
| 382.237
| Upper bound of 9- and 11-odd-limit diamond monotone
|  
|-
|-
|  
|  
Line 969: Line 1,144:
|-
|-
|  
|  
| 15/11
| 13/11
| 382.881
| 384.173
|  
|  
|-
|-
|  
|  
| 12/11
| 5/4
| 383.263
| 386.314
|  
|  
|}
==== Intuition ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|-
|  
|  
| 5/4
| 21/13
| 386.314
| 369.747
|  
|  
|-
|-
|  
|  
| 11/9
| 13/11
| 386.852
| 372.302
|  
|  
|-
|-
|  
|  
| 14/11
| 21/11
| 391.246
| 373.154
|  
|  
|}
=== Witchcraft ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
|-
|  
|  
| 14/13
| 13/7
| 378.617
| 378.617
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 378.910
| 378.910
|  
|  
|-
|-
| 6\19
| 6\19
|  
|
| 378.947
| 378.947
|  
| 19e val, lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
Line 1,023: Line 1,198:
|-
|-
|  
|  
| 18/13
| 13/9
| 379.577
| 379.577
|  
|  
Line 1,033: Line 1,208:
|-
|-
|  
|  
| 10/9
| 9/5
| 379.733
| 379.733
|  
|  
|-
|
| 27/20
| 379.968
| 5 limit least squares
|-
|-
|  
|  
Line 1,048: Line 1,218:
|-
|-
| 19\60
| 19\60
|  
|
| 380.000
| 380.000
|  
| 60eee val
|-
|-
|  
|  
| 16/13
| 13/8
| 380.029
| 380.029
|  
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 1,061: Line 1,231:
| 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
|  
| 101cdeeee val
|-
|
| {{vector|0 -67 -72 5 152 47}}
| 380.218
| 13 limit least squares
|-
|-
|  
|  
Line 1,098: Line 1,243:
|-
|-
|  
|  
| {{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}}
| 3/2
| 380.384
| 9 limit least squares
|-
|
| 4/3
| 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
|-
|-
|  
|  
| 13/11
| 21/16
| 380.719
| 380.634
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 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 1,158: Line 1,283:
|-
|-
|  
|  
| 16/15
| 15/8
| 381.378
| 381.378
|  
|  
|-
|-
| 7\22
| 7\22
|
| 381.818
| 22f val, upper bound of 9-odd-limit diamond monotone<br>11- to 15-odd-limit diamond monotone (singleton)
|-
|  
|  
| 381.818
| 11/10
| 381.923
| 11-odd-limit minimax
|-
|
| 11/8
| 382.237
|  
|  
|-
|-
Line 1,170: Line 1,305:
| 9/7
| 9/7
| 382.458
| 382.458
|
|-
|
| 15/11
| 382.881
|
|-
|
| 11/6
| 383.263
|  
|  
|-
|-
Line 1,175: Line 1,320:
| 5/4
| 5/4
| 386.314
| 386.314
|
|-
|
| 11/9
| 386.852
|
|-
|
| 11/7
| 391.246
|  
|  
|}
|}


=== Muggles ===
==== 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
|-
|-
|  
|  
| 11/9
| 13/8
| 347.408
| 359.472
|  
|  
|-
|-
|  
|  
| 16/13
| 13/10
| 359.472
| 372.893
|  
|  
|-
|-
|  
|  
| 15/11
| 21/11
| 372.610
| 373.154
|  
|  
|-
|-
|  
|  
| 13/10
| 13/12
| 372.893
| 376.905
|  
|  
|-
|-
|  
|  
| 12/11
| 15/13
| 374.894
| 378.249
|  
|  
|-
|-
|  
|  
| 8/7
| 13/9
| 375.882
| 378.489
|  
|  
|-
|-
|  
|  
| 13/11
| 5/3
| 375.899
| 378.910
|  
|  
|-
|-
| 6\19
|  
|  
| 11/10
| 378.947
| 376.500
| 19e val, lower bound of 9-odd-limit diamond monotone<br>2.3.5.7.13-subgroup 13-odd-limit diamond monotone (singleton)
|
|-
|-
|  
|  
| 14/11
| 13/7
| 376.805
| 379.100
|  
|  
|-
|-
|  
|  
| 13/12
| 21/13
| 376.905
| 379.459
|  
|  
|-
|-
|  
|  
| 7/5
| 9/5
| 377.186
| 379.733
|  
|  
|-
|-
| 19\60
|  
|  
| 11/8
| 380.000
| 377.393
| 60eeeff val
| 11, 13 and 15 limit minimax
|-
|-
|  
|  
| {{vector|0 113 -12 -68 58 -26}}
| 15/14
| 377.630
| 380.093
| 13 limit least squares
|-
|  
|  
| {{vector|0 -21 -5 27}}
| 377.640
| 7 limit least squares
|-
|-
| 32\101
|  
|  
| {{vector|0 134 9 -81 63 -33}}
| 380.198
| 377.718
| 101cdeeeefff val
| 15 limit least squares
|-
|-
|  
|  
| {{vector|0 85 -14 -62 46}}
| 7/5
| 377.758
| 380.228
| 11 limit least squares
|-
|  
|  
| 7/6
| 377.761
| 7 limit minimax
|-
|-
|  
|  
| 15/13
| 21/20
| 378.249
| 380.279
|  
|  
|-
|-
|  
|  
| 15/14
| 3/2
| 378.419
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
| 13\41
|  
|  
| 380.488
| 41ef val
|-
|-
|  
|  
| 18/13
| 21/16
| 378.489
| 380.634
|  
|  
|-
|-
|  
|  
| 9/7
| 13/11
| 378.534
| 380.719
| 9 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 93 -4 -44}}
| 7/4
| 378.554
| 380.735
| 9 limit least squares
|  
|-
|-
| 33\104
|  
|  
| 14/13
| 380.769
| 378.617
| 104eeefff val
|  
|-
|-
| 20\63
|  
|  
| 6/5
| 380.952
| 378.910
| 63eff val
|  
|-
|-
| 6\19
|  
|  
| 378.947
| 7/6
| 380.982
|  
|  
|-
|-
|  
|  
| 10/9
| 15/8
| 379.733
| 381.378
|  
|  
|-
|-
| 7\22
|  
|  
| 27/20
| 381.818
| 379.968
| Upper bound of 9-odd-limit diamond monotone<br>11-odd-limit diamond monotone (singleton)
| 5 limit least squares
|-
|-
| 19\60
|
| 380.000
|  
|  
| 11/10
| 381.923
| 11-odd-limit minimax
|-
|-
| 32\101
|  
|  
| 380.198
| 11/8
| 382.237
|  
|  
|-
|-
|  
|  
| 4/3
| 9/7
| 380.391
| 382.458
| 5 limit minimax
|-
| 13\41
|
| 380.488
|  
|  
|-
|-
| 33\104
|  
|  
| 380.769
| 15/11
| 382.881
|  
|  
|-
|-
| 20\63
|  
|  
| 380.952
| 11/6
| 383.263
|  
|  
|-
|-
|  
|  
| 16/15
| 5/4
| 381.378
| 386.314
|  
|  
|-
|-
| 7\22
|  
|  
| 381.818
| 11/9
| 386.852
|  
|  
|-
|-
|  
|  
| 5/4
| 11/7
| 386.314
| 391.246
|  
|  
|}
|}
Line 1,369: Line 1,514:
[[Category:Magic]]
[[Category:Magic]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Magic_extensions"