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Magic family: Difference between revisions

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Mapping generators: 2, 5/4
Mapping generators: 2, 5/4


{{EDOs|legend=1| 6, 16, 19, 22, 41, 60, 221c, 281c }}
{{EDOs|legend=1| 16, 19, 41, 60, 221c, 281c }}


== Seven limit children ==
== Seven limit children ==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.


=Magic=
= Magic =
(See also [[Magic|Magic]].)
{{main| Magic }}


Magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. [[41edo|41edo]] is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.
Magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41edo]] is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.


Magic, with its accurate fifths, works well with 9-limit harmony. It's more accurate than [[Meantone_family|meantone]] and simpler than [[Garibaldi_temperament|garibaldi]]. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is <<5 1 12 -10 5 25||.
Magic, with its accurate fifths, works well with [[9-odd-limit]] harmony. It's more accurate than [[Meantone family|meantone]] and simpler than [[Garibaldi temperament|garibaldi]]. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is <<5 1 12 -10 5 25||.


225/224 is the [[Marvel_temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.
225/224 is the [[Marvel temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.


245/243, the sensamagic comma, leads to another essentially tempered 9-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9. (This is shared with [[Sensi|sensi]], [[Semaphore_and_Godzilla|godzilla]], [[Superpyth|superpyth]], [[Tetracot_family|octacot]], [[Gamelismic_clan|rodan]], [[Shrutar|shrutar]], [[Porcupine_family|hedgehog]], [[Clyde_node|clyde]], and [[Sensamagic_clan|bohpier]]. See [http://x31eq.com/cgi-bin/uv.cgi?uvs=245:243 temperament finder].)
245/243, the [[Sensamagic clan|sensamagic]] comma, leads to another essentially tempered 9-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.  


By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, <<5 1 12 -8 ... ||. For this, [[104edo|104edo]] provides an excellent tuning, as it does also for the rank three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.
By adding [[100/99]] to the list of commas, magic can be extended to an 11-limit version, <<5 1 12 -8 ... ||. For this, [[104edo]] provides an excellent tuning, as it does also for the rank three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.


Commas: 225/224, 245/243
Commas: 225/224, 245/243
Line 55: Line 55:
[|1 0 0 0>, |0 1 0 0>, |2 1/5 0 0>, |-1 12/5 0 0>]
[|1 0 0 0>, |0 1 0 0>, |2 1/5 0 0>, |-1 12/5 0 0>]


[[Eigenmonzo|Eigenmonzos]]: 2, 3
[[Eigenmonzo]]s: 2, 3


valid range: [378.947, 381.818] (19 to 22)
valid range: [378.947, 381.818] (19 to 22)
Line 63: Line 63:
strict range: [378.947, 381.818]
strict range: [378.947, 381.818]


[[POTE_tuning|POTE generator]]: 380.352
[[POTE generator]]: 380.352


Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.
Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.
Line 69: Line 69:
Map: [<1 0 2 -1|, <0 5 1 12|]
Map: [<1 0 2 -1|, <0 5 1 12|]


[[generator|Generators]]: 2, 5/4
Mapping generators: 2, 5/4


EDOs: 41, 142cd, 183cd, 224cd
{{EDOs|legend=1| 19, 41, 142cd, 183cd, 224cd }}


==11-limit==
== 11-limit ==
{{see also| Chords of magic }}


Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4)=36/25. (16/11)/(26/25)=100/99.)
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
 
See also [[Chords_of_magic|Chords of magic]]


Commas: 225/224, 245/243, 100/99
Commas: 225/224, 245/243, 100/99
Line 87: Line 86:
strict range: [378.947, 381.818]
strict range: [378.947, 381.818]


[[POTE_tuning|POTE generator]]: 380.696
[[POTE generator]]: 380.696


Map: [<1 0 2 -1 6|, <0 5 1 12 -8|]
Map: [<1 0 2 -1 6|, <0 5 1 12 -8|]


EDOs: 19, 22, 41, 104, 145c
{{EDOs|legend=1| 19, 22, 41, 104, 145c }}


Badness: 0.0204
Badness: 0.0204


==13-limit==
=== 13-limit ===
Commas: 100/99, 105/104, 144/143, 196/195
Commas: 100/99, 105/104, 144/143, 196/195


Line 108: Line 107:
Map: [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]
Map: [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]


EDOS: 19, 41, 265cdef
{{EDOs|legend=1| 19, 22f, 41, 265cdef }}


Badness: 0.0215
Badness: 0.0215


===Sorcery===
=== Sorcery ===
Commas: 65/64, 78/77, 91/90, 100/99
Commas: 65/64, 78/77, 91/90, 100/99


Line 125: Line 124:
Map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|]
Map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|]


EDOs: 19, 22, 31f, 41f
{{EDOs|legend=1| 19, 22, 41f }}


Badness: 0.0258
Badness: 0.0258


===Necromancy===
=== Necromancy ===
Commas: 100/99, 225/224, 245/243, 275/273
Commas: 100/99, 225/224, 245/243, 275/273


Line 142: Line 141:
Map: [<1 0 2 -1 6 11|, <0 5 1 12 -8 -23|]
Map: [<1 0 2 -1 6 11|, <0 5 1 12 -8 -23|]


EDOs: 19, 22, 41, 63, 104
{{EDOs| 19f, 22, 41, 63, 104 }}


Badness: 0.0253
Badness: 0.0253


==Telepathy==
== Telepathy ==
Commas: 55/54, 99/98, 176/175
Commas: 55/54, 99/98, 176/175


Line 153: Line 152:
Map: [<1 0 2 -1 -1|, <0 5 1 12 14|]
Map: [<1 0 2 -1 -1|, <0 5 1 12 14|]


EDOs: 19e, 22, 41e, 63e
{{EDOs|legend=1| 19e, 22, 41e, 63e }}


Badness: 0.0271
Badness: 0.0271


===13-limit telepathy===
=== 13-limit ===
Commas: 55/54, 65/64, 91/90, 99/98
Commas: 55/54, 65/64, 91/90, 99/98


Line 164: Line 163:
Map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]
Map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]


EDOs: 19e, 22, 41ef
{{EDOs|legend=1| 19e, 22, 41ef }}


Badness: 0.0255
Badness: 0.0255


==Horcrux==
== Horcrux ==
Commas: 45/44, 56/55, 245/243
Commas: 45/44, 56/55, 245/243


Line 175: Line 174:
Map: [<1 0 2 -1 0|, <0 5 1 12 11|]
Map: [<1 0 2 -1 0|, <0 5 1 12 11|]


EDOs: 19, 60e
{{EDOs|legend=1| 19, 41e, 60e }}


Badness: 0.0393
Badness: 0.0393


=Divination=
== Divination ==
Commas: 121/120, 225/224, 245/243
Commas: 121/120, 225/224, 245/243


Line 186: Line 185:
Map: [<2 0 4 -2 5|, <0 5 1 12 3|]
Map: [<2 0 4 -2 5|, <0 5 1 12 3|]


EDOs: 22, 38d, 60e, 142cde
{{EDOs|legend=1| 22, 38d, 60e, 142cde }}


Badness: 0.0359
Badness: 0.0359


==13-limit==
=== 13-limit ===
Commas: 105/104, 121/120, 196/195, 245/243
Commas: 105/104, 121/120, 196/195, 245/243


Line 197: Line 196:
Map: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|]
Map: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|]


EDOs: 22f, 60e
{{EDOs|legend=1| 22f, 60e }}


Badness: 0.0346
Badness: 0.0346


=Soothsaying=
== Soothsaying ==
Commas: 100/99, 225/224, 245/243, 1352/1331
Commas: 100/99, 225/224, 245/243, 1352/1331


Line 208: Line 207:
Map: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|]
Map: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|]


EDOs: 22, 60, 82
{{EDOs|legend=1| 22, 60, 82 }}


Badness: 0.0554
Badness: 0.0554


=Witchcraft=
== Witchcraft ==
Commas: 225/224, 245/243, 441/440
Commas: 225/224, 245/243, 441/440


Line 219: Line 218:
Map: [<1 0 2 -1 -7|, <0 5 1 12 33|]
Map: [<1 0 2 -1 -7|, <0 5 1 12 33|]


EDOs: 41, 60e, 101cd, 243cde
{{EDOs|legend=1| 41, 60e, 101cd, 243cde }}


Badness: 0.0307
Badness: 0.0307


==13-limit==
=== 13-limit ===
Commas: 105/104, 196/195, 245/243, 275/273
Commas: 105/104, 196/195, 245/243, 275/273


Line 230: Line 229:
Map: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]
Map: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]


EDOs: 41, 60e, 101cd
{{EDOs|legend=1| 41, 60e, 101cd }}


Badness: 0.0235
Badness: 0.0235


=Muggles=
= Muggles =
Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is [[19edo|19edo]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. The muggles wedgie is <<5 1 -7 -10 -25 -19||.
Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. The muggles wedgie is <<5 1 -7 -10 -25 -19||.


Commas: 126/125, 525/512
Commas: 126/125, 525/512


[[POTE_tuning|POTE generator]]: ~5/4 = 378.479
[[POTE generator]]: ~5/4 = 378.479


Map: [<1 0 2 5|, <0 5 1 -7|]
Map: [<1 0 2 5|, <0 5 1 -7|]


EDOs: 19, 73bcd, 92bcd
{{EDOs|legend=1| 16, 19, 73bcd, 92bcd }}


Badness: 0.0562
Badness: 0.0562


==11-limit==
== 11-limit ==
Commas: 45/44, 126/125, 385/384
Commas: 45/44, 126/125, 385/384


[[POTE_tuning|POTE generator]]: ~5/4 = 377.724
[[POTE generator]]: ~5/4 = 377.724


Map: [<1 0 2 5 0|, <0 5 1 -7 11|]
Map: [<1 0 2 5 0|, <0 5 1 -7 11|]


EDOs: 16, 19, 35, 54bd
{{EDOs|legend=1| 16, 19, 35, 54bd }}


Badness: 0.0480
Badness: 0.0480


==13-limit==
== 13-limit ==
Commas: 45/44, 65/64, 78/77, 126/125
Commas: 45/44, 65/64, 78/77, 126/125


Line 265: Line 264:
Map: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|]
Map: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|]


EDOs: 16, 19, 35f, 54bdf
{{EDOs|legend=1| 16, 19, 35f, 54bdf }}


Badness: 0.0309
Badness: 0.0309


=Astrology=
= Astrology =
Commas: 50/49, 3125/3072
Commas: 50/49, 3125/3072


Line 278: Line 277:
Wedgie: <<10 2 2 -20 -25 -1||
Wedgie: <<10 2 2 -20 -25 -1||


EDOs: 6, 16, 22, 60d, 82d
{{EDOs|legend=1| 6, 16, 22, 60d, 82d }}


Badness: 0.0827
Badness: 0.0827


==11-limit==
== 11-limit ==
Commas: 50/49, 121/120, 176/175
Commas: 50/49, 121/120, 176/175


Line 289: Line 288:
Map: [<2 0 4 5 5|, <0 5 1 1 3|]
Map: [<2 0 4 5 5|, <0 5 1 1 3|]


EDOs: 6, 16, 22, 60de, 82de
{{EDOs|legend=1| 6, 16, 22, 60de, 82de }}


Badness: 0.0392
Badness: 0.0392


==13-limit==
=== 13-limit ===
Commas: 50/49, 65/64, 78/77, 121/120
Commas: 50/49, 65/64, 78/77, 121/120


Line 300: Line 299:
Map: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|]
Map: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|]


EDOs: 6, 16, 22, 38f
{{EDOs|legend=1| 6, 16, 22, 38f }}


Badness: 0.0344
Badness: 0.0344
Line 306: Line 305:
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor]
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor]


==Horoscope==
=== Horoscope ===
Commas: 50/49, 66/65, 105/104, 121/120
Commas: 50/49, 66/65, 105/104, 121/120


Line 313: Line 312:
Map: [<2 0 4 5 5 3|, <0 5 1 1 3 7|]
Map: [<2 0 4 5 5 3|, <0 5 1 1 3 7|]


EDOs: 16, 22f, 38
{{EDOs|legend=1| 16, 22f, 38 }}


Badness: 0.0353
Badness: 0.0353


=Spell=
= Spell =
Commas: 49/48, 3125/3072
Commas: 49/48, 3125/3072


Line 326: Line 325:
Wedgie: <<10 2 5 -20 -20 6||
Wedgie: <<10 2 5 -20 -20 6||


EDOs: 6, 19, 82d
{{EDOs|legend=1| 6, 19, 82d }}


Badness: 0.0810
Badness: 0.0810


==11-limit==
== 11-limit ==
Commas: 49/48, 56/55, 125/121
Commas: 49/48, 56/55, 125/121


Line 337: Line 336:
Map: [<1 0 2 2 3|, <0 10 2 5 3|]
Map: [<1 0 2 2 3|, <0 10 2 5 3|]


EDOs: 6, 19, 44de, 63de, 82de
{{EDOs|legend=1| 6, 19, 44de, 63de, 82de }}


Badness: 0.0598
Badness: 0.0598


==13-limit==
=== 13-limit ===
Commas: 49/48, 56/55, 78/77, 125/121
Commas: 49/48, 56/55, 78/77, 125/121


Line 348: Line 347:
Map: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|]
Map: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|]


EDOs: 6, 19, 82def
{{EDOs|legend=1| 6, 19, 82def }}


Badness: 0.0456
Badness: 0.0456


==Cantrip==
=== Cantrip ===
Commas: 49/48, 56/55, 91/90, 125/121
Commas: 49/48, 56/55, 91/90, 125/121


Line 359: Line 358:
Map: [<1 0 2 2 3 1|, <0 10 2 5 3 17|]
Map: [<1 0 2 2 3 1|, <0 10 2 5 3 17|]


EDOs: 19, 44de, 63de, 82de
{{EDOs|legend=1| 19, 44de, 63de, 82de }}


Badness: 0.0416
Badness: 0.0416


=Hocum=
= Hocum =
Commas: 3125/3072, 4000/3969
Commas: 3125/3072, 4000/3969


Line 372: Line 371:
Wedgie: <<10 2 -17 -20 -55 -45||
Wedgie: <<10 2 -17 -20 -55 -45||


EDOs: 38, 41, 161c, 202c, 243c, 284c
{{EDOs|legend=1| 38, 41, 161c, 202c, 243c, 284c }}


Badness: 0.1071
Badness: 0.1071


=Hocus=
== Hocus ==
Commas: 225/224, 243/242, 245/242
Commas: 225/224, 243/242, 245/242


Line 383: Line 382:
Map: [<1 5 3 11 12|, <0 -10 -2 -24 -25|]
Map: [<1 5 3 11 12|, <0 -10 -2 -24 -25|]


EDOs: 38d, 41, 120cd, 161cd, 202cd
{{EDOs|legend=1| 38d, 41, 120cd, 161cd, 202cd }}


Badness: 0.0385
Badness: 0.0385


==13-limit==
== 13-limit ==
Commas: 105/104, 196/195, 243/242, 245/242
Commas: 105/104, 196/195, 243/242, 245/242


Line 394: Line 393:
Map: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|]
Map: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|]


EDOs: 41, 79d, 120cd
{{EDOs|legend=1| 41, 79d, 120cd }}


Badness: 0.0303
Badness: 0.0303


=Trismegistus=
= Trismegistus =
Commas: 3125/3072, 1029/1024
Commas: 3125/3072, 1029/1024


Line 407: Line 406:
Wedgie: <<15 3 -5 -30 -50 -20||
Wedgie: <<15 3 -5 -30 -50 -20||


EDOs: 16, 25, 41, 139c, 180c, 221c, 262c
{{EDOs|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}


Badness: 0.0983
Badness: 0.0983


==11-limit==
== 11-limit ==
Commas: 385/384, 441/440, 625/616
Commas: 385/384, 441/440, 625/616


Line 418: Line 417:
Map: [<1 10 4 0 13|, <0 -15 -3 5 -17|]
Map: [<1 10 4 0 13|, <0 -15 -3 5 -17|]


EDOs: 16, 25e, 41, 98c, 139c, 180c
{{EDOs|legend=1| 16, 25e, 41, 98c, 139c, 180c }}


Badness: 0.0456
Badness: 0.0456
Line 429: Line 428:
Map: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|]
Map: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|]


EDOs: 16, 25e, 41, 98c, 139cf
{{EDOs|legend=1| 16, 25e, 41, 98c, 139cf }}


Badness: 0.0331
Badness: 0.0331


=Quadrimage=
= Quadrimage =
Commas: 2401/2400, 3125/3072
Commas: 2401/2400, 3125/3072


Line 442: Line 441:
Wedgie: <<20 4 7 -40 -45 5||
Wedgie: <<20 4 7 -40 -45 5||


EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd
{{EDOs|legend=1| 6, 35, 41, 158cd, 199cd, 240cd, 281cd }}


Badness: 0.1274
Badness: 0.1274


==11-limit==
== 11-limit ==
Commas: 245/242, 385/384, 625/616
Commas: 245/242, 385/384, 625/616


Line 453: Line 452:
Map: [<1 5 3 4 5|, <0 -20 -4 -7 -9|]
Map: [<1 5 3 4 5|, <0 -20 -4 -7 -9|]


EDOs: 6, 35, 41, 199cde, 240cde, 281cde
{{EDOs|legend=1| 6, 35, 41, 199cde, 240cde, 281cde }}


Badness: 0.0616
Badness: 0.0616


==13-limit==
== 13-limit ==
Commas: 105/104, 144/143, 245/242, 625/616
Commas: 105/104, 144/143, 245/242, 625/616


Line 464: Line 463:
Map: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|]
Map: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|]


EDOs: 41, 117c, 158cd, 199cdef
{{EDOs|legend=1| 41, 117c, 158cd, 199cdef }}


Badness: 0.0440
Badness: 0.0440
Line 471: Line 470:
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Magic]]
[[Category:Magic]]
 
[[Category:Rank 2]]
{{todo|cleanup|improve layout|review}}
Retrieved from "https://en.xen.wiki/w/Magic_family"