Xenharmonic Wiki

Magic extensions: Difference between revisions

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+intuition (also an important extension)
Line 3: Line 3:
* '''Tridecimal magic''' (19 & 22f) – tempering out 100/99, 105/104, 144/143, and 196/195;  
* '''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;  
* '''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;  
* '''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;  
* '''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.  
* '''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.  
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.  


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.  
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.  


== Interval chain ==
== Interval chain ==
Line 17: Line 18:
! 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 33:
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|  
|  
|  
|  
Line 41: Line 44:
| '''5/4'''
| '''5/4'''
| 26/21
| 26/21
|
|  
|  
| '''16/13'''
| '''16/13'''
Line 49: Line 53:
| 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 65:
| 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 75:
| 40/33
| 40/33
| 40/33
| 40/33
|
| 40/33
| 40/33
|  
| 11/9, 13/11
| 11/9
| 11/9
|-
|-
Line 76: Line 83:
| 702.3
| 702.3
| '''3/2'''
| '''3/2'''
|
|  
|  
|  
|  
Line 87: Line 95:
| 13/7
| 13/7
|  
|  
| 13/7
| 24/13
| 24/13
| 13/7
| 13/7
Line 94: Line 103:
| 263.2
| 263.2
| 7/6
| 7/6
|
|  
|  
|  
|  
Line 105: Line 115:
| 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 125:
| 20/11
| 20/11
| 20/11
| 20/11
|
| 20/11
| 20/11
|  
| 11/6
| 11/6
| 11/6
|-
|-
Line 121: Line 133:
| 204.6
| 204.6
| '''9/8'''
| '''9/8'''
|
|  
|  
|  
|  
Line 130: Line 143:
| 585.0
| 585.0
| 7/5
| 7/5
|
|  
|  
|  
|  
Line 141: Line 155:
| 26/15
| 26/15
|  
|  
| 26/15
|  
|  
| 26/15
| 26/15
Line 150: Line 165:
| 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 175:
| 15/11
| 15/11
| 15/11
| 15/11
|
| 15/11
| 15/11
|  
| '''11/8'''
| '''11/8'''
| '''11/8'''
|-
|-
Line 167: Line 184:
| 27/16, 42/25
| 27/16, 42/25
|  
|  
| 22/13
| 22/13
| 22/13
|
|  
|  
| 22/13
| 22/13
Line 176: Line 195:
|  
|  
|  
|  
| 22/21
| 27/26
| 27/26
| 22/21
|  
| 27/26
| 27/26
|-
|-
Line 186: Line 206:
|  
|  
|  
|  
| 13/10
| 13/10
| 13/10
|  
|  
Line 194: Line 215:
| '''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 225:
| 45/44
| 45/44
| 45/44
| 45/44
|
| 45/44
| 45/44
|  
| 33/32
| 33/32
| 33/32
|-
|-
Line 212: Line 235:
| 14/11
| 14/11
| 14/11
| 14/11
|
| 14/11
| 14/11
|  
|  
Line 220: Line 244:
| 63/40
| 63/40
|  
|  
|
| 11/7
|  
|  
|  
|  
| 11/7
|  
|  
|-
|-
Line 228: Line 253:
| 1170.1
| 1170.1
| 49/25, 63/32
| 49/25, 63/32
| 39/20
|
| 39/20
| 39/20
|  
|  
|  
|  
| 39/20
|  
|  
|}
|}
Line 603: Line 629:
| 381.818
| 381.818
|  
|  
|-
|
| 9/7
| 382.458
|
|-
|
| 5/4
| 386.314
|
|}
=== Witchcraft ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
|
| 13/7
| 378.617
|
|-
|
| 5/3
| 378.910
|
|-
| 6\19
|
| 378.947
| 19e val
|-
|
| 15/13
| 379.355
|
|-
|
| 13/9
| 379.577
|
|-
|
| 13/10
| 379.660
|
|-
|
| 9/5
| 379.733
|
|-
|
| 27/20
| 379.968
| 5-odd-limit least squares
|-
|
| 13/12
| 379.890
|
|-
| 19\60
|
| 380.000
| 60e val
|-
|
| 13/8
| 380.029
|
|-
|
| 15/14
| 380.093
|
|-
|
| 15/11
| 380.113
| 15-odd-limit minimax
|-
|
| 11/7
| 380.119
| 13-odd-limit minimax
|-
|
| 11/10
| 380.156
|
|-
|
| {{monzo| 0 -106 -111 11 179 59 }}
| 380.193
| 15-odd-limit least squares
|-
| 32\101
|
| 380.198
| 101cdf val
|-
|
| {{monzo| 0 -67 -72 5 152 47 }}
| 380.218
| 13-odd-limit least squares
|-
|
| 7/5
| 380.228
|
|-
|
| {{monzo| 0 -38 -55 11 137 }}
| 380.278
| 11-odd-limit least squares
|-
|
| 11/9
| 380.322
|
|-
|
| 11/6
| 380.334
|
|-
|
| 11/8
| 380.343
| 11-odd-limit minimax
|-
|
| {{monzo| 0 36 -23 32 }}
| 380.384
| 9-odd-limit least squares
|-
|
| 3/2
| 380.391
| 5-, 7- and 9-odd-limit minimax
|-
| 13\41
|
| 380.488
|
|-
|
| 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
|-
|-
|  
|  
Line 804: Line 1,014:
|}
|}


=== 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|Eigenmonzo<br>(unchanged-interval)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|
| 13/8
| 13/11
| 359.472
| 372.302
|  
|
|-
|
| 13/7
| 378.617
|
|-
|-
|  
|
| 13/10
| 5/3
| 372.893
| 378.910
|  
|  
|-
|-
|  
| 6\19
| 13/12
|
| 376.905
| 378.947
|  
|  
|-
|-
|  
|
| 15/13
| 15/13
| 378.249
| 379.355
|  
|
|-
|-
|  
|
| 13/9
| 13/9
| 378.489
| 379.577
|  
|
|-
|-
|
|
| 5/3
| 13/10
| 378.910
| 379.660
|
|
|-
| 6\19
|
| 378.947
| 19e val
|-
|  
| 13/7
| 379.100
|  
|-
|-
|  
|
| 9/5
| 9/5
| 379.733
| 379.733
|  
|
|-
|
| 13/12
| 379.890
|
|-
|-
|  
|
| 27/20
| 27/20
| 379.968
| 379.968
Line 863: Line 1,073:
|-
|-
| 19\60
| 19\60
|
| 380.000
|  
|  
| 380.000
| 60eeeff val
|-
|-
|  
|
| 13/8
| 380.029
| 13 and 15-odd-limit minimax
|-
|
| 15/14
| 15/14
| 380.093
| 380.093
Line 873: Line 1,088:
|-
|-
| 32\101
| 32\101
|
| 380.198
|  
|  
| 380.198
| 101cdeeeefff val
|-
|-
|  
|
| 7/5
| 7/5
| 380.228
| 380.228
|  
|  
|-
|-
|  
|
| {{monzo| 0 36 -23 32 }}
| {{monzo| 0 36 -23 32 }}
| 380.384
| 380.384
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|  
|
| 3/2
| 3/2
| 380.391
| 380.391
Line 893: Line 1,108:
|-
|-
| 13\41
| 13\41
|
| 380.488
|  
|  
| 380.488
| 41ef val
|-
|-
|  
|
| {{monzo| 0 1 -7 15 }}
| {{monzo| 0 1 -7 15 }}
| 380.506
| 380.506
| 7-odd-limit least squares
| 7-odd-limit least squares
|-
|-
|  
|
| {{monzo| 0 47 -34 43 57 -48 }}
| {{monzo| 0 -30 -73 30 46 78 }}
| 380.676
| 380.562
| 13-odd-limit least squares
|-
|
| {{monzo| 0 46 -35 49 65 -55 }}
| 380.691
| 15-odd-limit least squares
| 15-odd-limit least squares
|-
|-
|  
|
| 13/11
| {{monzo| 0 -10 -53 24 38 66 }}
| 380.719
| 380.568
| 13- and 15-odd-limit minimax
| 13-odd-limit least squares
|-
|-
|  
|
| 7/4
| 7/4
| 380.735
| 380.735
Line 923: Line 1,133:
|-
|-
| 33\104
| 33\104
|
| 380.769
|  
|  
| 380.769
| 104eeefff val
|-
|-
| 20\63
| 20\63
|
| 380.952
|  
|  
| 380.952
| 63eff val
|-
|-
|  
|
| 7/6
| 7/6
| 380.982
| 380.982
|  
|  
|-
|-
|  
|
| 15/8
| 15/8
| 381.378
| 381.378
|  
|  
|-
|-
|  
|
| {{monzo| 0 19 -36 30 42 }}
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 381.380
Line 948: Line 1,158:
|-
|-
| 7\22
| 7\22
|  
|
| 381.818
| 381.818
|  
|  
|-
|-
|  
|
| 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 993: Line 1,203:
|}
|}


=== Witchcraft ===
=== Telepathy ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
Line 1,002: Line 1,212:
|-
|-
|  
|  
| 13/7
| 13/8
| 378.617
| 359.472
|  
|  
|-
|-
|  
|  
| 5/3
| 13/10
| 378.910
| 372.893
|  
|  
|-
|-
| 6\19
|  
|  
| 378.947
| 13/12
| 19e val
| 376.905
|  
|-
|-
|  
|  
| 15/13
| 15/13
| 379.355
| 378.249
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 379.577
| 378.489
|
|-
|
| 5/3
| 378.910
|  
|  
|-
|-
| 6\19
|  
|  
| 13/10
| 378.947
| 379.660
| 19e val
|-
|
| 13/7
| 379.100
|  
|  
|-
|-
Line 1,040: Line 1,260:
| 379.968
| 379.968
| 5-odd-limit least squares
| 5-odd-limit least squares
|-
|
| 13/12
| 379.890
|
|-
|-
| 19\60
| 19\60
|  
|  
| 380.000
| 380.000
| 60e val
| 60eeeff val
|-
|
| 13/8
| 380.029
|
|-
|-
|  
|  
Line 1,060: Line 1,270:
| 380.093
| 380.093
|  
|  
|-
|
| 15/11
| 380.113
| 15-odd-limit minimax
|-
|
| 11/7
| 380.119
| 13-odd-limit minimax
|-
|
| 11/10
| 380.156
|
|-
|
| {{monzo| 0 -106 -111 11 179 59 }}
| 380.193
| 15-odd-limit least squares
|-
|-
| 32\101
| 32\101
|  
|  
| 380.198
| 380.198
| 101cdf val
| 101cdeeeefff val
|-
|
| {{monzo| 0 -67 -72 5 152 47 }}
| 380.218
| 13-odd-limit least squares
|-
|-
|  
|  
Line 1,095: Line 1,280:
| 380.228
| 380.228
|  
|  
|-
|
| {{monzo| 0 -38 -55 11 137 }}
| 380.278
| 11-odd-limit least squares
|-
|
| 11/9
| 380.322
|
|-
|
| 11/6
| 380.334
|
|-
|
| 11/8
| 380.343
| 11-odd-limit minimax
|-
|-
|  
|  
Line 1,129: Line 1,294:
|  
|  
| 380.488
| 380.488
| 41ef val
|-
|  
|  
| {{monzo| 0 1 -7 15 }}
| 380.506
| 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
|-
|-
|  
|  
| 13/11
| 13/11
| 380.719
| 380.719
|  
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 1,144: Line 1,324:
|  
|  
| 380.769
| 380.769
| 104eeff val
| 104eeefff val
|-
|-
| 20\63
| 20\63
|  
|  
| 380.952
| 380.952
| 63eef val
| 63eff val
|-
|-
|  
|  
Line 1,160: Line 1,340:
| 381.378
| 381.378
|  
|  
|-
|
| {{monzo| 0 19 -36 30 42 }}
| 381.380
| 11-odd-limit least squares
|-
|-
| 7\22
| 7\22
|  
|  
| 381.818
| 381.818
| 22ef val
|  
|-
|
| 11/10
| 381.923
| 11-odd-limit minimax
|-
|
| 11/8
| 382.237
|
|-
|-
|  
|  
| 9/7
| 9/7
| 382.458
| 382.458
|
|-
|
| 15/11
| 382.881
|
|-
|
| 11/6
| 383.263
|  
|  
|-
|-
Line 1,174: Line 1,379:
| 5/4
| 5/4
| 386.314
| 386.314
|
|-
|
| 11/9
| 386.852
|
|-
|
| 11/7
| 391.246
|  
|  
|}
|}
Retrieved from "https://en.xen.wiki/w/Magic_extensions"