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=Tuning Spectra=
{{Breadcrumb|Magic}}


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


Gencom: [2 5/4; 100/99 105/104 144/143 196/195]
* '''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;
* '''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.


Gencom map: [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]
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]].


{| class="wikitable"
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 ==
In the following table, odd harmonics and subharmonics 1–13 are in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
|-
! | Eigenmonzo
! rowspan="3" | #
! | Major Third
! rowspan="3" | Cents*
! colspan="7" | Approximate ratios
|-
|-
| | 14/13
! rowspan="2" | 7-limit
| | 378.617
! colspan="6" | 13-limit extensions
|-
|-
| | 6/5
! Magic
| | 378.910
! Necromancy
! Witchcraft
! Sorcery
! Intuition
! Telepathy
|-
|-
| | 6\19
| 0
| | 378.947
| 0.0
| '''1/1'''
|
|
|  
|  
|
|  
|-
|-
| | 15/13
| 1
| | 379.355
| 380.4
| '''5/4'''
| 26/21
|
| 26/21
| '''16/13'''
| 26/21
| '''16/13'''
|-
|-
| | 18/13
| 2
| | 379.577
| 760.9
| 14/9
|
|
|
| 20/13
| 11/7
| 11/7, 20/13
|-
|-
| | 13/10
| 3
| | 379.660
| 1141.4
| 27/14
| 52/27, 64/33
| 64/33
| 52/27
| 64/33
| 21/11, 52/27
| 21/11
|-
|-
| | 10/9
| 4
| | 379.733
| 321.8
| 6/5
| 40/33
| 40/33
|
| 40/33
| 11/9, 13/11
| 11/9
|-
|-
| | 13/12
| 5
| | 379.890
| 702.3
| '''3/2'''
|
|
|
|  
|
|  
|-
|-
| | 27/20
| 6
| | 379.968 (5 limit least squares)
| 1082.7
| 15/8, 28/15
| 13/7
|
| 13/7
| 24/13
| 13/7
| 24/13
|-
|-
| | 19\60
| 7
| | 380.000
| 263.2
| 7/6
|
|
|  
| 13/11
|
|  
|-
|-
| | 16/13
| 8
| | 380.029
| 643.7
| 35/24
| 13/9, '''16/11'''
| '''16/11'''
| 13/9
| '''16/11'''
| 13/9, 22/15
| 22/15
|-
|-
| | 15/14
| 9
| | 380.093
| 1024.1
| 9/5
| 20/11
| 20/11
|  
| 20/11
| 11/6
| 11/6
|-
|-
| | 32\101
| 10
| | 380.198
| 204.6
| '''9/8'''
|
|
|  
|  
|
|  
|-
|-
| | 7/5
| 11
| | 380.228
| 585.0
| 7/5
|  
|
|
| 18/13
|
| 18/13
|-
|-
| | 13/11
| 12
| | 380.354 (13 and 15 limit minimax)
| 965.5
| '''7/4'''
| 26/15
|  
| 26/15
|
| 26/15
|
|-
|-
| | |0 56 -31 46 -94 88>
| 13
| | 380.377 (13 limit least squares)
| 146.0
| 35/32
| 12/11, 13/12
| 12/11
| 13/12
| 12/11, 14/13
| 11/10, 13/12
| 11/10, 14/13
|-
|-
| | |0 36 -23 32>
| 14
| | 380.384 (9 limit least squares)
| 526.4
| 27/20
| 15/11
| 15/11
|
| 15/11
| '''11/8'''
| '''11/8'''
|-
|-
| | |0 58 -29 52 -108 100>
| 15
| | 380.389 (15 limit least squares)
| 906.9
| 27/16, 42/25
|  
| 22/13
| 22/13
|
|
| 22/13
|-
|-
| | 4/3
| 16
| | 380.391 (5, 7 and 9 limit minimax)
| 87.3
| 21/20
|
|
| 22/21
| 27/26
|  
| 27/26
|-
|-
| | 13\41
| 17
| | 380.488
| 467.8
| 21/16
| 13/10
|
| 13/10
|
| 13/10
|  
|-
|-
| | |0 1 -7 15>
| 18
| | 380.506 (7 limit least squares)
| 848.2
| 49/30
| '''13/8''', 18/11
| 18/11
| '''13/8'''
| 18/11, 21/13
| '''13/8'''
| 21/13
|-
|-
| | 11/9
| 19
| | 380.700 (11 limit minimax)
| 28.7
| 49/48, 81/80, 126/125
| 45/44
| 45/44
|
| 45/44
| 33/32
| 33/32
|-
|-
| | |0 85 -14 52 -68>
| 20
| | 380.714 (11 limit least squares)
| 409.2
| 63/50
| 14/11
| 14/11
|  
| 14/11
|
|
|-
|-
| | 8/7
| 21
| | 380.735
| 789.6
| 63/40
|
|
| 11/7
|  
|
|  
|-
|-
| | 33\104
| 22
| | 380.769
| 1170.1
| 49/25, 63/32
| 39/20
|
| 39/20
|
| 39/20
|
|}
<nowiki/>* In 7-limit [[CWE tuning]]
 
== Tunings ==
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (magic)
|-
|-
| | 12/11
! rowspan="2" | Target
| | 380.818
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
| | 14/11
! Generator
| | 380.875
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
|-
| | 20\63
| 11-odd-limit
| | 380.952
| ~5/4 = 380.700{{c}}
| 11/9
| ~5/4 = 380.714{{c}}
| {{Monzo| 0 85 -14 52 -68 }}
|-
|-
| | 7/6
| 13-odd-limit
| | 380.982
| ~5/4 = 380.354{{c}}
| 13/11
| ~5/4 = 380.377{{c}}
| {{Monzo| 0 56 -31 46 -94 88 }}
|-
|-
| | 11/8
| 15-odd-limit
| | 381.085
| ~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)
|-
|-
| | 15/11
! rowspan="2" | Target
| | 381.211
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
| | 16/15
! Generator
| | 381.378
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
|-
| | 11/10
| 13-odd-limit
| | 381.666
| ~5/4 = 380.709{{c}}
| 13/9
| ~5/4 = 380.705{{c}}
| {{Monzo| 0 -179 -10 -87 53 158 }}
|-
|-
| | 7\22
| 15-odd-limit
| | 381.818
| ~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)
|-
|-
| | 9/7
! rowspan="2" | Target
| | 382.458
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
| | 5/4
! Generator
| | 386.314
! 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 }}
|}
|}


==Spectrum of Necromancy Tunings by Eigenmonzos==
{| 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 }}
|}


Gencom: [2 5/4; 100/99 225/224 245/243 275/273]
{| 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 }}
|}


Gencom map: [&lt;1 0 2 -1 6 11|, &lt;0 5 1 12 -8 -23|]
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
 
|+ style="white-space: nowrap;" | Target tunings (telepathy)
{| class="wikitable"
|-
! 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 }}
|-
|-
! | Eigenmonzo
| 13-odd-limit
! | Major Third
| ~5/4 = 380.719{{c}}
| 13/11
| ~5/4 = 380.676{{c}}
| {{Monzo| 0 47 -34 43 57 -48 }}
|-
|-
| | 6/5
| 15-odd-limit
| | 378.910
| ~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"
|-
|-
| | 6\19
! Edo<br>generator
| | 378.947
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|-
| | 10/9
|  
| | 379.733
| 21/13
| 369.747
|  
|-
|-
| | 27/20
|  
| | 379.968 (5 limit least squares)
| 13/7
| 378.617
|  
|-
|-
| | 19\60
|  
| | 380.000
| 5/3
| 378.910
|  
|-
|-
| | 15/14
| 6\19
| | 380.093
|  
| 378.947
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
| | 32\101
|  
| | 380.198
| 15/13
| 379.355
|  
|-
|-
| | 7/5
|  
| | 380.228
| 13/9
| 379.577
|  
|-
|-
| | |0 36 -23 32&gt;
|  
| | 380.384 (9 limit least squares)
| 13/10
| 379.660
|  
|-
|-
| | 4/3
|  
| | 380.391 (5, 7 and 9 limit minimax)
| 9/5
| 379.733
|  
|-
|-
| | 13\41
|  
| | 380.488
| 13/12
| 379.890
|  
|-
|-
| | |0 1 -7 15&gt;
| 19\60
| | 380.506 (7 limit least squares)
|  
| 380.000
|
|-
|-
| | 11/9
|  
| | 380.700 (11 limit minimax)
| 13/8
| 380.029
|
|-
|-
| | 18/13
|  
| | 380.709 (13 and 15 limit minimax)
| 15/14
| 380.093
|
|-
|-
| | |0 85 -14 52 -68&gt;
| 32\101
| | 380.714 (11 limit least squares)
|  
| 380.198
| 101cde val
|-
|-
| | 13/11
|  
| | 380.719
| 7/5
| 380.228
|
|-
|-
| | 8/7
|  
| | 380.735
| 21/20
| 380.279
|
|-
|-
| | 13/12
|  
| | 380.765
| 13/11
| 380.354
| 13- and 15-odd-limit minimax
|-
|-
| | 33\104
|  
| | 380.769
| 3/2
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
|-
| | |0 -179 -10 -87 53 158&gt;
| 13\41
| | 380.785 (13 limit least squares)
|  
| 380.488
|
|-
|-
| | 14/13
|  
| | 380.809
| 21/16
| 380.634
|
|-
|-
| | |0 -222 -53 -93 67 187&gt;
|  
| | 380.817 (15 limit least squares)
| 11/9
| 380.700
| 11-odd-limit minimax
|-
|-
| | 12/11
|  
| | 380.818
| 7/4
| 380.735
|
|-
|-
| | 16/13
| 33\104
| | 380.847
|  
| 380.769
| 104ff val
|-
|-
| | 14/11
|  
| | 380.875
| 21/11
| 380.779
|
|-
|-
| | 20\63
|  
| | 380.952
| 11/6
| 380.818
|
|-
|-
| | 15/13
|  
| | 380.957
| 11/7
| 380.875
|
|-
|-
| | 7/6
| 20\63
| | 380.982
|  
| 380.952
| 63f val
|-
|-
| | 13/10
|  
| | 381.074
| 7/6
| 380.982
|  
|-
|-
| | 11/8
|  
| | 381.085
| 11/8
| 381.085
|
|-
|-
| | 15/11
|  
| | 381.211
| 15/11
| 381.211
|
|-
|-
| | 16/15
|  
| | 381.378
| 15/8
| 381.378
|
|-
|-
| | 11/10
|  
| | 381.666
| 11/10
| 381.666
|
|-
|-
| | 7\22
| 7\22
| | 381.818
|  
| 381.818
| 22f val, upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
| | 9/7
|  
| | 382.458
| 9/7
| 382.458
|
|-
|-
| | 5/4
|  
| | 386.314
| 5/4
| 386.314
|
|}
|}


==Spectrum of Sorcery Tunings by Eigenmonzos==
==== Necromancy ====
 
{| class="wikitable center-all left-4"
Gencom: [2 5/4; 65/64 78/77 91/90 100/99]
|-
 
! Edo<br>generator
Gencom map: [&lt;1 0 2 -1 6 4|, &lt;0 5 1 12 -8 -1|]
! Unchanged interval<br>(eigenmonzo)
 
! Generator (¢)
{| class="wikitable"
! 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
|-
|-
! | Eigenmonzo
|  
! | Major Third
| 9/7
| 382.458
|
|-
|-
| | 16/13
|  
| | 359.472
| 5/4
| 386.314
|
|}
 
==== Witchcraft ====
{| class="wikitable center-all left-4"
|-
|-
| | 13/10
! Edo<br>generator
| | 372.893
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|-
| | 13/12
|  
| | 376.905
| 21/13
| 369.747
|  
|-
|-
| | 15/13
|  
| | 378.249
| 13/7
| 378.617
|
|-
|-
| | 18/13
|  
| | 378.489
| 5/3
| 378.910
|
|-
|-
| | 6/5
| 6\19
| | 378.910
|  
| 378.947
| 19e val, lower bound of 9-odd-limit diamond monotone
|-
|-
| | 6\19
|  
| | 378.947
| 15/13
| 379.355
|  
|-
|-
| | 14/13
|  
| | 379.100
| 13/9
| 379.577
|
|-
|-
| | 10/9
|  
| | 379.733
| 13/10
| 379.660
|
|-
|-
| | 27/20
|  
| | 379.968 (5 limit least squares)
| 9/5
| 379.733
|
|-
|-
| | 19\60
|  
| | 380.000
| 13/12
| 379.890
|  
|-
|-
| | 15/14
| 19\60
| | 380.093
|  
| 380.000
| 60e val, lower bound of 11- to 15-odd-limit diamond monotone
|-
|-
| | 32\101
|  
| | 380.198
| 13/8
| 380.029
|
|-
|-
| | 7/5
|  
| | 380.228
| 21/11
| 380.034
|
|-
|-
| | |0 36 -23 32&gt;
|  
| | 380.384 (9 limit least squares)
| 15/14
| 380.093
|
|-
|-
| | 4/3
|  
| | 380.391 (5, 7 and 9 limit minimax)
| 15/11
| 380.113
| 15-odd-limit minimax
|-
|-
| | |0 -113 12 -65 75 26&gt;
|  
| | 380.427 (13 limit least squares)
| 11/7
| 380.119
| 13-odd-limit minimax
|-
|-
| | |0 134 9 71 -89 -33&gt;
|  
| | 380.457 (15 limit least squares)
| 11/10
| 380.156
|
|-
|-
| | 13\41
| 32\101
| | 380.488
|  
| 380.198
| 101cd val
|-
|-
| | |0 1 -7 15&gt;
|  
| | 380.506 (7 limit least squares)
| 7/5
| 380.228
|
|-
|-
| | 11/9
|  
| | 380.700 (11, 13 and 15 limit minimax)
| 21/20
| 380.279
|
|-
|-
| | |0 85 -14 52 -68&gt;
|  
| | 380.714 (11 limit least squares)
| 11/9
| 380.322
|
|-
|-
| | 8/7
|  
| | 380.735
| 11/6
| 380.334
|
|-
|-
| | 33\104
|  
| | 380.769
| 11/8
| 380.343
| 11-odd-limit minimax
|-
|-
| | 12/11
|  
| | 380.818
| 3/2
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
|-
| | 14/11
| 13\41
| | 380.875
|  
| 380.488
| Upper bound of 11- to 15-odd-limit diamond monotone
|-
|-
| | 20\63
|  
| | 380.952
| 21/16
| 380.634
|
|-
|-
| | 7/6
|  
| | 380.982
| 13/11
| 380.719
|
|-
|-
| | 11/8
|  
| | 381.085
| 7/4
| 380.735
|  
|-
|-
| | 15/11
| 33\104
| | 381.211
|  
| 380.769
| 104eeff val
|-
|-
| | 16/15
| 20\63
| | 381.378
|  
| 380.952
| 63eef val
|-
|-
| | 11/10
|  
| | 381.666
| 7/6
| 380.982
|  
|-
|-
| | 7\22
|  
| | 381.818
| 15/8
| 381.378
|
|-
|-
| | 9/7
| 7\22
| | 382.458
|  
| 381.818
| 22ef val, upper bound of 9-odd-limit diamond monotone
|-
|-
| | 13/11
|  
| | 384.173
| 9/7
| 382.458
|  
|-
|-
| | 5/4
|  
| | 386.314
| 5/4
| 386.314
|
|}
|}


==Spectrum of Telepathy Tunings by Eigenmonzos==
==== Sorcery ====
 
{| class="wikitable center-all left-4"
Gencom: [2 5/4; 55/54 65/64 91/90 99/98]
 
Gencom map: [&lt;1 0 2 -1 -1 4|, &lt;0 5 1 12 14 -1|]
 
{| class="wikitable"
|-
! | Eigenmonzo
! | Major Third
|-
| | 16/13
| | 359.472
|-
|-
| | 13/10
! Edo<br>generator
| | 372.893
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|-
| | 13/12
|  
| | 376.905
| 13/8
| 359.472
|  
|-
|-
| | 15/13
|  
| | 378.249
| 13/10
| 372.893
|  
|-
|-
| | 18/13
|  
| | 378.489
| 13/12
| 376.905
|  
|-
|-
| | 6/5
|  
| | 378.910
| 15/13
| 378.249
|
|-
|-
| | 6\19
|  
| | 378.947
| 13/9
| 378.489
|
|-
|-
| | 14/13
|  
| | 379.100
| 5/3
| 378.910
|  
|-
|-
| | 10/9
| 6\19
| | 379.733
|  
| 378.947
| Lower bound of 9- and 11-odd-limit diamond monotone<br>13- and 15-odd-limit diamond monotone (singleton)
|-
|-
| | 27/20
|  
| | 379.968 (5 limit least squares)
| 13/7
| 379.100
|
|-
|-
| | 19\60
|  
| | 380.000
| 21/13
| 379.459
|  
|-
|-
| | 15/14
|  
| | 380.093
| 9/5
| 379.733
|  
|-
|-
| | 32\101
| 19\60
| | 380.198
|  
| 380.000
| 60ff val
|-
|-
| | 7/5
|  
| | 380.228
| 15/14
| 380.093
|
|-
|-
| | |0 36 -23 32&gt;
| 32\101
| | 380.384 (9 limit least squares)
|  
| 380.198
| 101cdefff val
|-
|-
| | 4/3
|  
| | 380.391 (5, 7 and 9 limit minimax)
| 7/5
| 380.228
|
|-
|-
| | 13\41
|  
| | 380.488
| 21/20
| 380.279
|
|-
|-
| | |0 1 -7 15&gt;
|  
| | 380.506 (7 limit least squares)
| 3/2
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
|-
| | |0 47 -34 43 57 -48&gt;
| 13\41
| | 380.676 (13 limit least squares)
|  
| 380.488
| 41f val
|-
|-
| | |0 46 -35 49 65 -55&gt;
|  
| | 380.691 (15 limit least squares)
| 21/16
| 380.634
|
|-
|-
| | 13/11
|  
| | 380.719 (13 and 15 limit minimax)
| 11/9
| 380.700
| 11-, 13- and 15-odd-limit minimax
|-
|-
| | 8/7
|  
| | 380.735
| 7/4
| 380.735
|
|-
|-
| | 33\104
| 33\104
| | 380.769
|  
| 380.769
| 104fff val
|-
|-
| | 20\63
|  
| | 380.952
| 21/11
| 380.779
|
|-
|-
| | 7/6
|  
| | 380.982
| 11/6
| 380.818
|
|-
|-
| | 16/15
|  
| | 381.378
| 11/7
| 380.875
|  
|-
|-
| | |0 19 -36 30 42&gt;
| 20\63
| | 381.380 (11 limit least squares)
|  
| 380.952
| 63ff val
|-
|-
| | 7\22
|  
| | 381.818
| 7/6
| 380.982
|  
|-
|-
| | 11/10
|  
| | 381.923 (11 limit minimax)
| 11/8
| 381.085
|
|-
|-
| | 11/8
|  
| | 382.237
| 15/11
| 381.211
|  
|-
|-
| | 9/7
|  
| | 382.458
| 15/8
| 381.378
|  
|-
|-
| | 15/11
|  
| | 382.881
| 11/10
| 381.666
|  
|-
|-
| | 12/11
| 7\22
| | 383.263
|  
| 381.818
| Upper bound of 9- and 11-odd-limit diamond monotone
|-
|-
| | 5/4
|  
| | 386.314
| 9/7
| 382.458
|  
|-
|-
| | 11/9
|  
| | 386.852
| 13/11
| 384.173
|  
|-
|-
| | 14/11
|  
| | 391.246
| 5/4
| 386.314
|  
|}
|}


==Spectrum of Witchcraft Tunings by Eigenmonzos==
==== Intuition ====
 
{| class="wikitable center-all left-4"
Gencom: [2 5/4; 105/104 196/195 245/243 275/273]
 
Gencom map: [&lt;1 0 2 -1 -7 -2|, &lt;0 5 1 12 33 18|]
 
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Edo<br>generator
! | Major Third
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|-
| | 14/13
|  
| | 378.617
| 21/13
| 369.747
|  
|-
|-
| | 6/5
|  
| | 378.910
| 13/11
| 372.302
|  
|-
|-
| | 6\19
|  
| | 378.947
| 21/11
| 373.154
|  
|-
|-
| | 15/13
|  
| | 379.355
| 13/7
| 378.617
|  
|-
|-
| | 18/13
|  
| | 379.577
| 5/3
| 378.910
|  
|-
|-
| | 13/10
| 6\19
| | 379.660
|
| 378.947
| 19e val, lower bound of 9-odd-limit diamond monotone
|-
|-
| | 10/9
|  
| | 379.733
| 15/13
| 379.355
|
|-
|-
| | 27/20
|  
| | 379.968 (5 limit least squares)
| 13/9
| 379.577
|
|-
|-
| | 13/12
|  
| | 379.890
| 13/10
| 379.660
|
|-
|-
| | 19\60
|  
| | 380.000
| 9/5
| 379.733
|  
|-
|-
| | 16/13
|  
| | 380.029
| 13/12
| 379.890
|  
|-
|-
| | 15/14
| 19\60
| | 380.093
|
| 380.000
| 60eee val
|-
|-
| | 15/11
|  
| | 380.113 (15 limit minimax)
| 13/8
| 380.029
| 13- and 15-odd-limit minimax
|-
|-
| | 14/11
|  
| | 380.119 (13 limit minimax)
| 15/14
| 380.093
|
|-
|-
| | 11/10
| 32\101
| | 380.156
|
| 380.198
| 101cdeeee val
|-
|-
| | |0 -106 -111 11 179 59&gt;
|  
| | 380.193 (15 limit least squares)
| 7/5
| 380.228
|
|-
|-
| | 32\101
|  
| | 380.198
| 21/20
| 380.279
|
|-
|-
| | |0 -67 -72 5 152 47&gt;
|  
| | 380.218 (13 limit least squares)
| 3/2
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
|-
| | 7/5
| 13\41
| | 380.228
|
| 380.488
| 41e val
|-
|-
| | |0 -38 -55 11 137&gt;
|  
| | 380.278 (11 limit least squares)
| 21/16
| 380.634
|
|-
|-
| | 11/9
|  
| | 380.322
| 7/4
| 380.735
|
|-
|-
| | 12/11
| 33\104
| | 380.334
|
| 380.769
| 104eeeff val
|-
|-
| | 11/8
| 20\63
| | 380.343 (11 limit minimax)
|
| 380.952
| 63ef val
|-
|-
| | |0 36 -23 32&gt;
|  
| | 380.384 (9 limit least squares)
| 7/6
| 380.982
|
|-
|-
| | 4/3
|  
| | 380.391 (5, 7 and 9 limit minimax)
| 15/8
| 381.378
|  
|-
|-
| | 13\41
| 7\22
| | 380.488
|
| 381.818
| 22f val, upper bound of 9-odd-limit diamond monotone<br>11- to 15-odd-limit diamond monotone (singleton)
|-
|-
| | 13/11
|  
| | 380.719
| 11/10
| 381.923
| 11-odd-limit minimax
|-
|-
| | 8/7
|  
| | 380.735
| 11/8
| 382.237
|  
|-
|-
| | 33\104
|  
| | 380.769
| 9/7
| 382.458
|  
|-
|-
| | 20\63
|  
| | 380.952
| 15/11
| 382.881
|  
|-
|-
| | 7/6
|  
| | 380.982
| 11/6
| 383.263
|  
|-
|-
| | 16/15
|  
| | 381.378
| 5/4
| 386.314
|  
|-
|-
| | 7\22
|  
| | 381.818
| 11/9
| 386.852
|  
|-
|-
| | 9/7
|  
| | 382.458
| 11/7
|-
| 391.246
| | 5/4
|  
| | 386.314
|}
|}


==Spectrum of Muggles Tunings by Eigenmonzos==
==== Telepathy ====
 
{| class="wikitable center-all left-4"
Gencom: [2 5/4; 45/44 65/64 78/77 126/125]
 
Gencom map: [&lt;1 0 2 5 0 4|, &lt;0 5 1 -7 11 -1|]
 
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Edo<br>generator
! | Major Third
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|-
| | 11/9
|  
| | 347.408
| 13/8
| 359.472
|  
|-
|-
| | 16/13
|  
| | 359.472
| 13/10
| 372.893
|  
|-
|-
| | 15/11
|  
| | 372.610
| 21/11
| 373.154
|  
|-
|-
| | 13/10
|  
| | 372.893
| 13/12
| 376.905
|  
|-
|-
| | 12/11
|  
| | 374.894
| 15/13
| 378.249
|  
|-
|-
| | 8/7
|  
| | 375.882
| 13/9
| 378.489
|  
|-
|-
| | 13/11
|  
| | 375.899
| 5/3
| 378.910
|  
|-
|-
| | 11/10
| 6\19
| | 376.500
|  
| 378.947
| 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
|  
| | 376.805
| 13/7
| 379.100
|  
|-
|-
| | 13/12
|  
| | 376.905
| 21/13
| 379.459
|  
|-
|-
| | 7/5
|  
| | 377.186
| 9/5
| 379.733
|  
|-
|-
| | 11/8
| 19\60
| | 377.393 (11, 13 and 15 limit minimax)
|  
| 380.000
| 60eeeff val
|-
|-
| | |0 113 -12 -68 58 -26&gt;
|  
| | 377.630 (13 limit least squares)
| 15/14
| 380.093
|  
|-
|-
| | |0 -21 -5 27&gt;
| 32\101
| | 377.640 (7 limit least squares)
|  
| 380.198
| 101cdeeeefff val
|-
|-
| | |0 134 9 -81 63 -33&gt;
|  
| | 377.718 (15 limit least squares)
| 7/5
| 380.228
|  
|-
|-
| | |0 85 -14 -62 46&gt;
|  
| | 377.758 (11 limit least squares)
| 21/20
| 380.279
|  
|-
|-
| | 7/6
|  
| | 377.761 (7 limit minimax)
| 3/2
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
|-
| | 15/13
| 13\41
| | 378.249
|  
| 380.488
| 41ef val
|-
|-
| | 15/14
|  
| | 378.419
| 21/16
| 380.634
|  
|-
|-
| | 18/13
|  
| | 378.489
| 13/11
| 380.719
| 13- and 15-odd-limit minimax
|-
|-
| | 9/7
|  
| | 378.534 (9 limit minimax)
| 7/4
| 380.735
|  
|-
|-
| | |0 93 -4 -44&gt;
| 33\104
| | 378.554 (9 limit least squares)
|  
| 380.769
| 104eeefff val
|-
|-
| | 14/13
| 20\63
| | 378.617
|  
| 380.952
| 63eff val
|-
|-
| | 6/5
|  
| | 378.910
| 7/6
| 380.982
|  
|-
|-
| | 6\19
|  
| | 378.947
| 15/8
| 381.378
|  
|-
|-
| | 10/9
| 7\22
| | 379.733
|  
| 381.818
| Upper bound of 9-odd-limit diamond monotone<br>11-odd-limit diamond monotone (singleton)
|-
|-
| | 27/20
|  
| | 379.968 (5 limit least squares)
| 11/10
| 381.923
| 11-odd-limit minimax
|-
|-
| | 19\60
|  
| | 380.000
| 11/8
| 382.237
|  
|-
|-
| | 32\101
|  
| | 380.198
| 9/7
| 382.458
|  
|-
|-
| | 4/3
|  
| | 380.391 (5 limit minimax)
| 15/11
| 382.881
|  
|-
|-
| | 13\41
|  
| | 380.488
| 11/6
| 383.263
|  
|-
|-
| | 33\104
|  
| | 380.769
| 5/4
| 386.314
|  
|-
|-
| | 20\63
|  
| | 380.952
| 11/9
| 386.852
|  
|-
|-
| | 16/15
|  
| | 381.378
| 11/7
|-
| 391.246
| | 7\22
|  
| | 381.818
|-
| | 5/4
| | 386.314
|}
|}
[[Category:Magic]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Magic_extensions"