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The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. A magic temperament is optimal, for some searches, in the [[9-odd-limit]]. It has slightly higher complexity than [[Meantone family|meantone]] and is also closer to just intonation. It is the simplest rank-2 temperament that tunes every 9-odd-limit interval better than is possible in [[12edo]]. Properties may depend on tuning and extension.
The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. A magic temperament is optimal, for some searches, in the [[9-odd-limit]]. It has slightly higher complexity than [[Meantone family|meantone]] and is also closer to just intonation. It is the simplest rank-2 temperament that tunes every 9-odd-limit interval better than is possible in [[12edo|12EDO]]. Properties may depend on tuning and extension.


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


= Five limit magic =
= Five limit magic =
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 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)<sup>5</sup> = 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.
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 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)<sup>5</sup> = 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|19EDO]] or [[22edo|22EDO]] is always possible.


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 35: Line 35:


{{Val list|legend=1| 19, 41, 60, 221c, 281c }}
{{Val list|legend=1| 19, 41, 60, 221c, 281c }}
[[Badness]]: 0.039163


== Seven-limit extensions ==
== Seven-limit extensions ==
Line 42: Line 44:
{{main| Magic }}
{{main| 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 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, with its accurate fifths, works well with [[9-odd-limit]] harmony. It is more accurate than [[meantone]] and simpler than [[Garibaldi temperament|garibaldi]]. It is a little tricky to work with because in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.  
Magic, with its accurate fifths, works well with [[9-odd-limit]] harmony. It is more accurate than [[meantone]] and simpler than [[Garibaldi temperament|garibaldi]]. It is a little tricky to work with because in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.  
Line 50: Line 52:
245/243, the [[Sensamagic clan|sensamagic]] comma, leads to another essentially tempered 9-odd-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.  
245/243, the [[Sensamagic clan|sensamagic]] comma, leads to another essentially tempered 9-odd-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, {{multival| 5 1 12 -8 … }}. For this, [[104edo]] provides an excellent tuning, as it does also for the rank-3 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, {{multival| 5 1 12 -8 … }}. For this, [[104edo|104EDO]] provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 77: Line 79:


{{Val list|legend=1| 19, 41, 142cd, 183cd, 224cd }}
{{Val list|legend=1| 19, 41, 142cd, 183cd, 224cd }}
[[Badness]]: 0.018918


== 11-limit ==
== 11-limit ==
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.)
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.)


Line 95: Line 98:
* strict range: [378.947, 381.818]
* strict range: [378.947, 381.818]


{{Val list|legend=1| 19, 22, 41, 104, 145c }}
Vals: {{Val list| 19, 22, 41, 104, 145c }}


Badness: 0.0204
Badness: 0.020352


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 114: Line 116:
* strict range: [378.947, 381.818]
* strict range: [378.947, 381.818]


{{Val list|legend=1| 19, 22f, 41, 265cdef }}
Vals: {{Val list| 19, 22f, 41, 265cdef }}


Badness: 0.0215
Badness: 0.021509


=== Sorcery ===
=== Sorcery ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 133: Line 134:
* strict range: 378.947
* strict range: 378.947


{{Val list|legend=1| 19, 22, 41f }}
Vals: {{Val list| 19, 22, 41f }}


Badness: 0.0258
Badness: 0.025829


=== Necromancy ===
=== Necromancy ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 152: Line 152:
* strict range: [380.488, 380.952]
* strict range: [380.488, 380.952]


{{Val list|legend=1| 19f, 22, 41, 63, 104 }}
Vals: {{Val list| 19f, 22, 41, 63, 104 }}


Badness: 0.0253
Badness: 0.025275


=== Soothsaying ===
=== Soothsaying ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 166: Line 165:
POTE generator: ~5/4 = 380.508
POTE generator: ~5/4 = 380.508


{{Val list|legend=1| 22, 60, 82 }}
Vals: {{Val list| 22, 60, 82 }}


Badness: 0.0554
Badness: 0.055443


== Telepathy ==
== Telepathy ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 180: Line 178:
POTE generator: ~5/4 = 381.019
POTE generator: ~5/4 = 381.019


{{Val list|legend=1| 19e, 22, 41e, 63e }}
Vals: {{Val list| 19e, 22, 41e, 63e }}


Badness: 0.0271
Badness: 0.027109


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 194: Line 191:
POTE generator: ~5/4 = 380.520
POTE generator: ~5/4 = 380.520


{{Val list|legend=1| 19e, 22, 41ef }}
Vals: {{Val list| 19e, 22, 41ef }}


Badness: 0.0255
Badness: 0.025522


== Horcrux ==
== Horcrux ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 208: Line 204:
POTE generator: ~5/4 = 379.642
POTE generator: ~5/4 = 379.642


{{Val list|legend=1| 19, 41e, 60e }}
Vals: {{Val list| 19, 41ee, 60ee }}


Badness: 0.0393
Badness: 0.039282


== Divination ==
== Divination ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 222: Line 217:
POTE generator: ~5/4 = 380.233
POTE generator: ~5/4 = 380.233


{{Val list|legend=1| 22, 38d, 60e, 142cde }}
Vals: {{Val list| 22, 38d, 60e, 142cde }}


Badness: 0.0359
Badness: 0.035864


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 236: Line 230:
POTE generator: ~5/4 = 379.920
POTE generator: ~5/4 = 379.920


{{Val list|legend=1| 22f, 60e }}
Vals: {{Val list| 22f, 60e }}


Badness: 0.0346
Badness: 0.034551


== Witchcraft ==
== Witchcraft ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 250: Line 243:
POTE generator: ~5/4 = 380.232
POTE generator: ~5/4 = 380.232


{{Val list|legend=1| 41, 60e, 101cd, 243cde }}
Vals: {{Val list| 41, 60e, 101cd, 243cde }}


Badness: 0.0307
Badness: 0.030706


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 264: Line 256:
POTE generator: ~5/4 = 380.189
POTE generator: ~5/4 = 380.189


{{Val list|legend=1| 41, 60e, 101cd }}
Vals: {{Val list| 41, 60e, 101cd }}


Badness: 0.0235
Badness: 0.023547


== Hocus ==
== Hocus ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 278: Line 269:
POTE generator: ~14/11 = 409.910
POTE generator: ~14/11 = 409.910


{{Val list|legend=1| 38d, 41, 120cd, 161cd, 202cd }}
Vals: {{Val list| 38d, 41, 120cd, 161cd, 202cd }}


Badness: 0.0385
Badness: 0.038519


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 292: Line 282:
POTE generator: ~14/11 = 410.004
POTE generator: ~14/11 = 410.004


{{Val list|legend=1| 41, 79d, 120cd }}
Vals: {{Val list| 41, 79d, 120cd }}


Badness: 0.0303
Badness: 0.030280


= Muggles =
= 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.  
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.  


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 311: Line 301:
{{Val list|legend=1| 16, 19, 73bcd, 92bcd }}
{{Val list|legend=1| 16, 19, 73bcd, 92bcd }}


[[Badness]]: 0.0562
[[Badness]]: 0.056206


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 323: Line 312:
POTE generator: ~5/4 = 377.724
POTE generator: ~5/4 = 377.724


{{Val list|legend=1| 16, 19, 35, 54bd }}
Vals: {{Val list| 16, 19, 35, 54bd }}


Badness: 0.0480
Badness: 0.048038


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 337: Line 325:
POTE generator: ~5/4 = 377.724
POTE generator: ~5/4 = 377.724


{{Val list|legend=1| 16, 19, 35f, 54bdf }}
Vals: {{Val list| 16, 19, 35f, 54bdf }}


Badness: 0.0309
Badness: 0.030386


= Astrology =
= Astrology =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 355: Line 342:
{{Val list|legend=1| 6, 16, 22, 60d, 82d }}
{{Val list|legend=1| 6, 16, 22, 60d, 82d }}


[[Badness]]: 0.0827
[[Badness]]: 0.082673


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 367: Line 353:
POTE generator: ~5/4 = 380.530
POTE generator: ~5/4 = 380.530


{{Val list|legend=1| 6, 16, 22, 60de, 82de }}
Vals: {{Val list| 6, 16, 22, 60de, 82de }}


Badness: 0.0392
Badness: 0.039151


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 381: Line 366:
POTE generator: ~5/4 = 379.787
POTE generator: ~5/4 = 379.787


{{Val list|legend=1| 6, 16, 22, 38f }}
Vals: {{Val list| 6, 16, 22, 38f }}


Badness: 0.0344
Badness: 0.034376


; Music
; Music
Line 389: Line 374:


=== Horoscope ===
=== Horoscope ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 398: Line 382:
POTE generator: ~5/4 = 379.837
POTE generator: ~5/4 = 379.837


{{Val list|legend=1| 16, 22f, 38 }}
Vals: {{Val list| 16, 22f, 38 }}


Badness: 0.0353
Badness: 0.035284


= Spell =
= Spell =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 414: Line 397:
[[POTE generator]]: ~28/25 = 189.927
[[POTE generator]]: ~28/25 = 189.927


{{Val list|legend=1| 6, 19, 82d }}
{{Val list|legend=1| 6, 19, 82dd }}


[[Badness]]: 0.0810
[[Badness]]: 0.080958


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 428: Line 410:
POTE generator: ~11/10 = 190.285
POTE generator: ~11/10 = 190.285


{{Val list|legend=1| 6, 19, 44de, 63de, 82de }}
Vals: {{Val list| 6, 19, 44de, 63dee, 82ddee }}


Badness: 0.0598
Badness: 0.059791


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 442: Line 423:
POTE generator: ~11/10 = 189.928
POTE generator: ~11/10 = 189.928


{{Val list|legend=1| 6, 19, 82def }}
Vals: {{Val list| 6, 19, 82ddeeff }}


Badness: 0.0456
Badness: 0.045591


=== Cantrip ===
=== Cantrip ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 456: Line 436:
POTE generator: ~11/10 = 190.360
POTE generator: ~11/10 = 190.360


{{Val list|legend=1| 19, 44de, 63de, 82de }}
Vals: {{Val list| 19, 44de, 63dee, 82ddee }}


Badness: 0.0416
Badness: 0.041603


= Hocum =
= Hocum =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 474: Line 453:
{{Val list|legend=1| 38, 41, 161c, 202c, 243c, 284c }}
{{Val list|legend=1| 38, 41, 161c, 202c, 243c, 284c }}


[[Badness]]: 0.1071
[[Badness]]: 0.107115


= Trismegistus =
= Trismegistus =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 490: Line 468:
{{Val list|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}
{{Val list|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}


[[Badness]]: 0.0983
[[Badness]]: 0.098334


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 502: Line 479:
Mapping: [{{val| 1 10 4 0 13 }}, {{val| 0 -15 -3 5 -17 }}]
Mapping: [{{val| 1 10 4 0 13 }}, {{val| 0 -15 -3 5 -17 }}]


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


Badness: 0.0456
Badness: 0.045623


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 516: Line 492:
POTE generator: ~22/15 = 673.359
POTE generator: ~22/15 = 673.359


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


Badness: 0.0331
Badness: 0.033081


= Quadrimage =
= Quadrimage =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 534: Line 509:
{{Val list|legend=1| 6, 35, 41, 158cd, 199cd, 240cd, 281cd }}
{{Val list|legend=1| 6, 35, 41, 158cd, 199cd, 240cd, 281cd }}


[[Badness]]: 0.1274
[[Badness]]: 0.127422


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 546: Line 520:
POTE generator: ~28/25 = 204.956
POTE generator: ~28/25 = 204.956


{{Val list|legend=1| 6, 35, 41, 199cde, 240cde, 281cde }}
Vals: {{Val list| 6, 35, 41, 199cde, 240cde, 281cde }}


Badness: 0.0616
Badness: 0.061572


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 560: Line 533:
POTE generator: ~28/25 = 205.028
POTE generator: ~28/25 = 205.028


{{Val list|legend=1| 41, 117c, 158cd, 199cdef }}
Vals: {{Val list| 41, 117c, 158cd, 199cdef }}


Badness: 0.0440
Badness: 0.044047


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 11:06, 16 May 2021

The magic family of temperaments tempers out 3125/3072, the small diesis or magic comma. A magic temperament is optimal, for some searches, in the 9-odd-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-odd-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.

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

Mapping generators: ~2, ~5/4

POTE generator: ~5/4 = 380.058

Minimax tuning:

[[1 0 0⟩, [0 1 0⟩, [2 1/5 0⟩]
Eigenmonzos: 2, 3

Tuning ranges:

  • valid range: [360.000, 400.000] (3\10 to 1\3)
  • nice range: [378.910, 386.314]
  • strict range: [378.910, 386.314]

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

Template:Val list

Badness: 0.039163

Seven-limit extensions

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 is more accurate than meantone and simpler than garibaldi. It is a little tricky to work with because in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.

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-odd-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-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1], ⟨0 5 1 12]]

Mapping generators: ~2, ~5/4

Wedgie: ⟨⟨ 5 1 12 -10 5 25 ]]

POTE generator: ~5/4 = 380.352

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [2 1/5 0 0⟩, [-1 12/5 0 0⟩]
Eigenmonzos: 2, 3

Tuning ranges:

  • valid range: [378.947, 381.818] (6\19 to 7\22)
  • nice range: [378.910, 386.314]
  • strict range: [378.947, 381.818]

Algebraic generator: Tirzbirat or Septimage, the real root of 5x5 + 4x - 20, 380.7604 cents.

Template:Val list

Badness: 0.018918

11-limit

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.)

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 100/99

Mapping: [⟨1 0 2 -1 6], ⟨0 5 1 12 -8]]

POTE generator: ~5/4 = 380.696

Tuning ranges:

  • valid range: [378.947, 381.818] (6\19 to 7\22)
  • nice range: [378.910, 386.314]
  • strict range: [378.947, 381.818]

Vals: Template:Val list

Badness: 0.020352

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

POTE generator: ~5/4 = 380.427

Tuning ranges:

  • valid range: [378.947, 381.818] (6\19 to 7\22)
  • nice range: [378.617, 386.314]
  • strict range: [378.947, 381.818]

Vals: Template:Val list

Badness: 0.021509

Sorcery

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 100/99

Mapping: [⟨1 0 2 -1 6 4], ⟨0 5 1 12 -8 -1]]

POTE generator: ~5/4 = 380.477

Tuning ranges:

  • valid range: 378.947 (6\19)
  • nice range: [359.472, 386.314]
  • strict range: 378.947

Vals: Template:Val list

Badness: 0.025829

Necromancy

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 275/273

Mapping: [⟨1 0 2 -1 6 11], ⟨0 5 1 12 -8 -23]]

POTE generator: ~5/4 = 380.787

Tuning ranges:

  • valid range: [380.488, 380.952] (13\41 to 20\63)
  • nice range: [378.910, 386.314]
  • strict range: [380.488, 380.952]

Vals: Template:Val list

Badness: 0.025275

Soothsaying

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 1352/1331

Mapping: [⟨2 0 4 -2 12 15], ⟨0 5 1 12 -8 -12]]

POTE generator: ~5/4 = 380.508

Vals: Template:Val list

Badness: 0.055443

Telepathy

Subgroup: 2.3.5.7.11

Comma list: 55/54, 99/98, 176/175

Mapping: [⟨1 0 2 -1 -1], ⟨0 5 1 12 14]]

POTE generator: ~5/4 = 381.019

Vals: Template:Val list

Badness: 0.027109

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

POTE generator: ~5/4 = 380.520

Vals: Template:Val list

Badness: 0.025522

Horcrux

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 11]]

POTE generator: ~5/4 = 379.642

Vals: Template:Val list

Badness: 0.039282

Divination

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 245/243

Mapping: [⟨2 0 4 -2 5], ⟨0 5 1 12 3]]

POTE generator: ~5/4 = 380.233

Vals: Template:Val list

Badness: 0.035864

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4], ⟨0 5 1 12 3 18]]

POTE generator: ~5/4 = 379.920

Vals: Template:Val list

Badness: 0.034551

Witchcraft

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 441/440

Mapping: [⟨1 0 2 -1 -7], ⟨0 5 1 12 33]]

POTE generator: ~5/4 = 380.232

Vals: Template:Val list

Badness: 0.030706

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243, 275/273

Mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

POTE generator: ~5/4 = 380.189

Vals: Template:Val list

Badness: 0.023547

Hocus

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 245/242

Mapping: [⟨1 5 3 11 12], ⟨0 -10 -2 -24 -25]]

POTE generator: ~14/11 = 409.910

Vals: Template:Val list

Badness: 0.038519

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 243/242, 245/242

Mapping: [⟨1 5 3 11 12 16], ⟨0 -10 -2 -24 -25 -36]]

POTE generator: ~14/11 = 410.004

Vals: Template:Val list

Badness: 0.030280

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.

Subgroup: 2.3.5.7

Comma list: 126/125, 525/512

Mapping: [⟨1 0 2 5], ⟨0 5 1 -7]]

Wedgie: ⟨⟨ 5 1 -7 -10 -25 -19 ]]

POTE generator: ~5/4 = 378.479

Template:Val list

Badness: 0.056206

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 126/125, 385/384

Mapping: [⟨1 0 2 5 0], ⟨0 5 1 -7 11]]

POTE generator: ~5/4 = 377.724

Vals: Template:Val list

Badness: 0.048038

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 126/125

Mapping: [⟨1 0 2 5 0 4], ⟨0 5 1 -7 11 -1]]

POTE generator: ~5/4 = 377.724

Vals: Template:Val list

Badness: 0.030386

Astrology

Subgroup: 2.3.5.7

Comma list: 50/49, 3125/3072

Mapping: [⟨2 0 4 5], ⟨0 5 1 1]]

Wedgie: ⟨⟨ 10 2 2 -20 -25 -1 ]]

POTE generator: ~5/4 = 380.578

Template:Val list

Badness: 0.082673

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 176/175

Mapping: [⟨2 0 4 5 5], ⟨0 5 1 1 3]]

POTE generator: ~5/4 = 380.530

Vals: Template:Val list

Badness: 0.039151

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/64, 78/77, 121/120

Mapping: [⟨2 0 4 5 5 8], ⟨0 5 1 1 3 -1]]

POTE generator: ~5/4 = 379.787

Vals: Template:Val list

Badness: 0.034376

Music

Horoscope

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 66/65, 105/104, 121/120

Mapping: [⟨2 0 4 5 5 3], ⟨0 5 1 1 3 7]]

POTE generator: ~5/4 = 379.837

Vals: Template:Val list

Badness: 0.035284

Spell

Subgroup: 2.3.5.7

Comma list: 49/48, 3125/3072

Mapping: [⟨1 0 2 2], ⟨0 10 2 5]]

Wedgie: ⟨⟨ 10 2 5 -20 -20 6 ]]

POTE generator: ~28/25 = 189.927

Template:Val list

Badness: 0.080958

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 125/121

Mapping: [⟨1 0 2 2 3], ⟨0 10 2 5 3]]

POTE generator: ~11/10 = 190.285

Vals: Template:Val list

Badness: 0.059791

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: [⟨1 0 2 2 3 4], ⟨0 10 2 5 3 -2]]

POTE generator: ~11/10 = 189.928

Vals: Template:Val list

Badness: 0.045591

Cantrip

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: [⟨1 0 2 2 3 1], ⟨0 10 2 5 3 17]]

POTE generator: ~11/10 = 190.360

Vals: Template:Val list

Badness: 0.041603

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 5 3 -3], ⟨0 -10 -2 17]]

Wedgie: ⟨⟨ 10 2 -17 -20 -55 -45 ]]

POTE generator: ~63/50 = 400.108

Template:Val list

Badness: 0.107115

Trismegistus

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 10 4 0], ⟨0 -15 -3 5]]

Wedgie: ⟨⟨ 15 3 -5 -30 -50 -20 ]]

POTE generator: ~147/100 = 673.290

Template:Val list

Badness: 0.098334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 625/616

POTE generator: ~22/15 = 673.340

Mapping: [⟨1 10 4 0 13], ⟨0 -15 -3 5 -17]]

Vals: Template:Val list

Badness: 0.045623

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273, 625/616

Mapping: [⟨1 10 4 0 13 11], ⟨0 -15 -3 5 -17 -13]]

POTE generator: ~22/15 = 673.359

Vals: Template:Val list

Badness: 0.033081

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 5 3 4], ⟨0 -20 -4 -7]]

Wedgie: ⟨⟨ 20 4 7 -40 -45 5 ]]

POTE generator: ~28/25 = 204.987

Template:Val list

Badness: 0.127422

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 625/616

Mapping: [⟨1 5 3 4 5], ⟨0 -20 -4 -7 -9]]

POTE generator: ~28/25 = 204.956

Vals: Template:Val list

Badness: 0.061572

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/242, 625/616

Mapping: [⟨1 5 3 4 5 9], ⟨0 -20 -4 -7 -9 -31]]

POTE generator: ~28/25 = 205.028

Vals: Template:Val list

Badness: 0.044047

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