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}}

Revision as of 07:37, 16 December 2020

A magic temperament is optimal, for some searches, in the 9-limit. It has slightly higher complexity than meantone and is also closer to just intonation. It is the simplest rank 2 temperament that tunes every 9-limit interval better than is possible in 12edo. Properties may depend on tuning and extension.

The most prominent deficiency of magic temperaments is that they lack proper or nearly-proper MOS scales in the 5 to 10 note "diatonic" region.

Five limit magic

The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is [-10 -1 5⟩, and flipping that yields <<5 1 -10|| for the wedgie. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)5 = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the optimal patent val generator, also makes a lot of sense and using 19edo or 22edo is always possible.

Comma: 3125/3072

5-limit minimax

[<1 0 0|, <0 1 0|, <2 1/5 0|]

Eigenmonzos: 2, 3

valid range: [360.000, 400.000] (10 to 3)

nice range: [378.910, 386.314]

strict range: [378.910, 386.314]

POTE generator: ~5/4 = 380.058

Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.

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

Mapping generators: 2, 5/4

EDOs: 16, 19, 41, 60, 221c, 281c

Seven limit children

The second comma of the 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

Main article: Magic

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-odd-limit harmony. It's more accurate than meantone and simpler than 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 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.

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

7 and 9 limit minimax

[|1 0 0 0>, |0 1 0 0>, |2 1/5 0 0>, |-1 12/5 0 0>]

Eigenmonzos: 2, 3

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

nice range: [378.910, 386.314]

strict range: [378.947, 381.818]

POTE generator: 380.352

Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.

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

Mapping generators: 2, 5/4

EDOs: 19, 41, 142cd, 183cd, 224cd

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)/(36/25) = 100/99.)

Commas: 225/224, 245/243, 100/99

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

nice range: [378.910, 386.314]

strict range: [378.947, 381.818]

POTE generator: 380.696

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

EDOs: 19, 22, 41, 104, 145c

Badness: 0.0204

13-limit

Commas: 100/99, 105/104, 144/143, 196/195

valid range: [378.947, 381.818] (19 to 22f)

nice range: [378.617, 386.314]

strict range: [378.947, 381.818]

POTE generator: ~5/4 = 380.427

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

EDOs: 19, 22f, 41, 265cdef

Badness: 0.0215

Sorcery

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

valid range: 378.947 (19)

nice range: [359.472, 386.314]

strict range: 378.947

POTE generator: ~5/4 = 380.477

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

EDOs: 19, 22, 41f

Badness: 0.0258

Necromancy

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

valid range: [380.488, 380.952] (41 to 63)

nice range: [378.910, 386.314]

strict range: [380.488, 380.952]

POTE generator: ~5/4 = 380.787

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

19f, 22, 41, 63, 104

Badness: 0.0253

Telepathy

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

POTE generator: ~5/4 = 381.019

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

EDOs: 19e, 22, 41e, 63e

Badness: 0.0271

13-limit

Commas: 55/54, 65/64, 91/90, 99/98

POTE generator: ~5/4 = 380.520

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

EDOs: 19e, 22, 41ef

Badness: 0.0255

Horcrux

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

POTE generator: ~5/4 = 379.642

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

EDOs: 19, 41e, 60e

Badness: 0.0393

Divination

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

POTE generator: ~5/4 = 380.233

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

EDOs: 22, 38d, 60e, 142cde

Badness: 0.0359

13-limit

Commas: 105/104, 121/120, 196/195, 245/243

POTE generator: ~5/4 = 379.920

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

EDOs: 22f, 60e

Badness: 0.0346

Soothsaying

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

POTE generator: ~5/4 = 380.508

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

EDOs: 22, 60, 82

Badness: 0.0554

Witchcraft

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

POTE generator: ~5/4 = 380.232

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

EDOs: 41, 60e, 101cd, 243cde

Badness: 0.0307

13-limit

Commas: 105/104, 196/195, 245/243, 275/273

POTE generator: ~5/4 = 380.189

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

EDOs: 41, 60e, 101cd

Badness: 0.0235

Muggles

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

POTE generator: ~5/4 = 378.479

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

EDOs: 16, 19, 73bcd, 92bcd

Badness: 0.0562

11-limit

Commas: 45/44, 126/125, 385/384

POTE generator: ~5/4 = 377.724

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

EDOs: 16, 19, 35, 54bd

Badness: 0.0480

13-limit

Commas: 45/44, 65/64, 78/77, 126/125

POTE generator: ~5/4 = 377.724

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

EDOs: 16, 19, 35f, 54bdf

Badness: 0.0309

Astrology

Commas: 50/49, 3125/3072

POTE generator: ~5/4 = 380.578

Map: [<2 0 4 5|, <0 5 1 1|]

Wedgie: <<10 2 2 -20 -25 -1||

EDOs: 6, 16, 22, 60d, 82d

Badness: 0.0827

11-limit

Commas: 50/49, 121/120, 176/175

POTE generator: ~5/4 = 380.530

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

EDOs: 6, 16, 22, 60de, 82de

Badness: 0.0392

13-limit

Commas: 50/49, 65/64, 78/77, 121/120

POTE generator: ~5/4 = 379.787

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

EDOs: 6, 16, 22, 38f

Badness: 0.0344

Astrology Percussion Quintet No 1 by Joel Taylor

Horoscope

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

POTE generator: ~5/4 = 379.837

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

EDOs: 16, 22f, 38

Badness: 0.0353

Spell

Commas: 49/48, 3125/3072

POTE generator: ~28/25 = 189.927

Map: [<1 0 2 2|, <0 10 2 5|]

Wedgie: <<10 2 5 -20 -20 6||

EDOs: 6, 19, 82d

Badness: 0.0810

11-limit

Commas: 49/48, 56/55, 125/121

POTE generator: ~11/10 = 190.285

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

EDOs: 6, 19, 44de, 63de, 82de

Badness: 0.0598

13-limit

Commas: 49/48, 56/55, 78/77, 125/121

POTE generator: ~11/10 = 189.928

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

EDOs: 6, 19, 82def

Badness: 0.0456

Cantrip

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

POTE generator: ~11/10 = 190.360

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

EDOs: 19, 44de, 63de, 82de

Badness: 0.0416

Hocum

Commas: 3125/3072, 4000/3969

POTE generator: ~63/50 = 400.108

Map: [<1 5 3 -3|, <0 -10 -2 17|]

Wedgie: <<10 2 -17 -20 -55 -45||

EDOs: 38, 41, 161c, 202c, 243c, 284c

Badness: 0.1071

Hocus

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

POTE generator: ~14/11 = 409.910

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

EDOs: 38d, 41, 120cd, 161cd, 202cd

Badness: 0.0385

13-limit

Commas: 105/104, 196/195, 243/242, 245/242

POTE generator: ~14/11 = 410.004

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

EDOs: 41, 79d, 120cd

Badness: 0.0303

Trismegistus

Commas: 3125/3072, 1029/1024

POTE generator: ~147/100 = 673.290

Map: [<1 10 4 0|, <0 -15 -3 5|]

Wedgie: <<15 3 -5 -30 -50 -20||

EDOs: 16, 25, 41, 139c, 180c, 221c, 262c

Badness: 0.0983

11-limit

Commas: 385/384, 441/440, 625/616

POTE generator: ~22/15 = 673.340

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

EDOs: 16, 25e, 41, 98c, 139c, 180c

Badness: 0.0456

13-limit

Commas: 105/104, 144/143, 275/273, 625/616

POTE generator: ~22/15 = 673.359

Map: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|]

EDOs: 16, 25e, 41, 98c, 139cf

Badness: 0.0331

Quadrimage

Commas: 2401/2400, 3125/3072

POTE generator: ~28/25 = 204.987

Map: [<1 5 3 4|, <0 -20 -4 -7|]

Wedgie: <<20 4 7 -40 -45 5||

EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd

Badness: 0.1274

11-limit

Commas: 245/242, 385/384, 625/616

POTE generator: ~28/25 = 204.956

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

EDOs: 6, 35, 41, 199cde, 240cde, 281cde

Badness: 0.0616

13-limit

Commas: 105/104, 144/143, 245/242, 625/616

POTE generator: ~28/25 = 205.028

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

EDOs: 41, 117c, 158cd, 199cdef

Badness: 0.0440

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