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Marvel

The head of the marvel family is marvel, which tempers out 225/224, the septimal kleisma or marvel comma. Marvel has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by 197edo.

Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to catakleismic temperament. Another temperament which does little damage to tuning accuracy is compton temperament, for which 240edo may be used.

See also Marvel temperaments.

Vital statistics

Comma c = 225/224
Related linear temperament: catakleismic temperament

7-limit minimax: 3 and 5 1/4c flat, 7 just
[|1 0 0 0>, |5/4 1/2 -1/2 1/4>, |5/4 -1/2 1/2 1/4>, |0 0 0 1>]
Eigenmonzo subgroup: 2.5/3.7

9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just
[|1 0 0 0>, |5/6 2/3 -1/3 1/6>, |5/3 -2/3 1/3 1/3>, |0 0 0 1>]
Eigenmonzo subgroup: 2.9/5.7

Lattice basis: secor length 1.256, 3/2 length 1.369
Angle(secor, 3/2) = 106.958 degrees
Map to lattice: [<0 0 -1 -2|, <0 1 -1 0|]

Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]
Generators: 2, 3, 5
EDOs: 10, 12, 19, 22, 31, 41,72, 197, 269c
Badness: 0.0000365

Projection pairs: 7 225/32
Spectrum: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9

Scales: marvel9, marvel10, marvel11, marvel12, marvel19, marvel22, pump12_1, pump12_2, pump13, pump14, pump15, pump16, pump17, pump18

Minkowski blocks

{2, 3, 5} subgroup

8: 16/15, 250/243
9: 135/128, 128/125
10: 25/24, 2048/2025
11: 135/128, 2048/1875
12: 2048/2025, 128/125
15: 128/125, 32768/30375
17: 25/24, 2278125/2097152
19: 16875/16384, 81/80
21: 128/125, 273375/262144
22: 2048/2025, 3125/3072
29: 16875/16384, 32805/32768
31: 81/80, 34171875/33554432
41: 34171875/33554432, 3125/3072

Music

Semimarvelous Blue Drawf by Chris Vaisvil
notJoelTaylorsTreeSpirit-Marvel10-2 by Billy Stiltner

11-limit (Unimarv)

Commas: 225/224, 385/384
Related linear temperament: catakleismic temperament

Minimax tuning:
[|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>,
|3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]
Eigenmonzo subgroup: 2.9/5.11/9

Lattice basis: secor length 1.0364 5/4 length 1.0759
Angle(secor, 5/4) = 104.028 degrees
Map to lattice: [<0 -1 0 -2 1|, <0 -1 1 0 -2|]

Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]
Generators: 2, 3, 5
Edos: 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 269ce, 341ce
Badness: 0.000255

Projection pairs: 7 225/32 11 4096/375
Spectrum: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11
Scales: marvel22_11, unimarv19, unimarv22

Hobbit bases

{2, 3, 5} subgroup

12: 128/125, 2048/2025
15: 128/125, 32768/30375
19: 16875/16384, 81/80
22: 2048/2025, 2109375/2097152
31: 2109375/2097152, 81/80
41: 3125/3072, 34171875/33554432

13-limit

Commas: 225/224, 385/384, 351/350

13-limit eigenmonzo subgroup: 2.11/9.13/9
15-limit eigenmonzo subgroup: 2.15/11.15/13

Map: [<1 0 0 -5 12 -4|, <0 1 0 2 -1 -1|, <0 0 1 2 -3 4|]
EDOs: 19, 22, 31, 50, 53, 72, 103, 175f, 300cef, 403bcef
Badness: 0.000690

Spectrum: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11

Hecate

Commas: 225/224, 385/384, 325/324

13-limit eigenmonzo subgroup: 2.7.13/5
15-limit eigenmonzo subgroup: 2.7.15/13

Map: [<1 0 0 -5 12 2|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -2|]
EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef
Badness: 0.000721

Projection pairs: 7 225/32 11 4096/375 13 324/25
Spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11

17-limit

Commas: 225/224, 385/384, 325/324, 595/594

Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 -6|]
EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg
Badness: 0.000869

Enodia

Commas: 225/224, 385/384, 325/324, 375/374

Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 6|]
EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg
Badness: .000917

Marvelcat

Commas: 169/168, 225/224, 385/384

Map: [<1 0 0 -5 12 -|, <0 2 0 4 -2 3|, <0 0 1 2 -3 1|]
EDOs: 9, 10, 19, 44, 53, 72, 125f, 197ef, 269cef
Badness: 0.0009997

Marvell

Commas: 225/224, 385/384, 1573/1568

13-limit eigenmonzo subgroup: 2.9/5.11/9
15-limit eigenmonzo subgroup: 2.7.15/13

Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]
EDOs: 9, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcef, 610bcef
Badness: 0.000862

Isis

Commas: 225/224, 385/384, 275/273

Map: [<1 0 0 -5 12 17|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -3|]
EDOs: 10, 22, 31, 41, 53, 94
Badness: 0.000866

Projection pairs: 7 225/32 11 4096/375 13 131072/10125

Deecee

Commas: 225/224, 385/384, 364/363

13-limit eigenmonzo subgroup: 2.9/5.13/9
15-limit eigenmonzo subgroup: 2.3.13/5

Map: [<1 0 0 -5 12 27|, <0 1 0 2 -1 -3|, <0 0 1 2 -3 -8|]
EDOs: 9, 22, 41, 63, 72, 185cf, 257cf
Badness: 0.000920

Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875

Mirage

Commas: 225/224, 243/242, 385/384

Map: [<1 1 3 3 2 0|, <0 6 -7 -2 15 0|, <0 0 0 0 0 1|]
EDOs: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef
Badness: 0.000738

Minerva

Commas: 99/98, 176/175
Related linear temperament: orwell

Eigenmonzo subgroup: 2.7/5.11/9

Lattice basis: 16/15 length 0.8997 5/4 length 1.0457
Angle(16/15, 5/4) = 98.6044 degrees
Map to lattice: [<0 -1 0 -2 -2|, <0 -1 1 0 2|]

Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]
Generators: 2, 3, 5
EDOs: 9, 12, 21, 22, 31, 43, 53, 74, 75, 96, 127
Badness: 0.000381

Projection pairs: 7 225/32 11 5625/512

Scales: minerva12, minerva22x

Athene

Commas: 99/98, 176/175, 275/273

13-limit eigenmonzo subgroup: 2.11/9.13/7
15-limit eigenmonzo subgroup: 2.11/9.13/7

Map: [<1 0 0 -5 -9 -4|, <0 1 0 2 2 -1|, <0 0 1 2 4 4|]
EDOs: 22, 31, 53, 84e, 118d, 149df, 171de, 202def
Badness: 0.000818

Projection pairs: 7 225/32 11 5625/512 13 625/48

Other eleven limit marvel children

The second comma of the normal comma list defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.

Spectacle

Commas: 225/224, 243/242
Related linear temperament: marvo

Minimax tuning:
[|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>]
Eigenmonzo subgroup: 2.9/5.11

Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|]
Generators: 2, 11/9, 5
EDOs: 10, 31, 41, 72, 240, 259b, 269ce, 310c, 331bc, 353c, 497bc, 569bc
Badness: 0.000499

Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9

Scales: spectacle31

13-limit

Commas: 225/224, 243/242, 351/350

Map: [<1 1 0 -3 2 -5|, <0 2 0 4 5 -2|, <0 0 1 2 0 4|]
EDOs: 31, 72, 103, 175f, 209, 240
Badness: 0.001009

Apollo

Commas: 100/99, 225/224
Related linear temperament: Magic

Eigenmonzo subgroup: 2.7/5.11/9

Map: [<1 0 0 -5 2|, <0 1 0 2 -2|, <0 0 1 2 2|]
EDOs: 12, 19, 22, 41, 104edo, 157ce, 198ce, 220ce, 261ce

Projection pairs: 7 225/32 11 100/9

13-limit

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

Eigenmonzo subgroup: 2.11/9.13/9

Map: [<1 0 0 -5 2 7|, <0 1 0 2 -2 -5|, <0 0 1 2 2 2|]
EDOs: 22, 29, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef

Projection pairs: 7 225/32 11 100/9 13 3200/243

Potassium

Commas: 45/44, 56/55

Eigenmonzo subgroup: 2.9/7.11

Map: [<1 0 0 -5 -2|, <0 1 0 2 2|, <0 0 1 2 1|]
EDOs: 9, 10, 12, 19, 31e, 50e
Badness: 0.000464

Projection pairs: 7 225/32 11 45/4

13-limit

Commas: 45/44, 56/55, 78/77

13-limit eigenmonzo subgroup: 2.9/7.13/9
15-limit eigenmonzo subgroup: 2.9/7.13/9

Map: [<1 0 0 -5 -2 -8|, <0 1 0 2 2 3|, <0 0 1 2 1 3|]
EDOs: 9, 10, 19, 31e, 50e
Badness: 0.000733

Projection pairs: 7 225/32 11 45/4 13 3375/256

Fantastic

Commas: 225/224, 4375/4356

Map: [<2 0 0 -10 -7|, <0 1 0 2 0|, <0 0 1 2 3|]
EDOs: 12, 22, 50, 72, 166, 238c, 310c
Badness: 0.000743

Catakleismoid

Commas: 225/224, 4375/4374

Map: [<1 0 1 -3 0|, <0 6 5 22 0|, <0 0 0 0 1|]
EDOs: 19, 53, 72, 197e, 269ce
Badness: 0.001275

13-limit

Commas: 169/168, 225/224, 325/324

Map: [<1 0 1 -3 0 0|, <0 6 5 22 0 14|, <0 0 0 0 1 0|]
EDOs: 19, 53, 72, 125f, 197ef, 269cef
Badness: 0.000916

Hestia

Commas: 225/224, 125000/124509

Map: [<1 0 0 -5 9|, <0 2 0 4 -7|, <0 0 1 2 0|]
EDOs: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce
Badness: 0.00154

13-limit

Commas: 169/168, 225/224, 1001/1000

Map: [<1 0 0 -5 9 -1|, <0 2 0 4 -7 3|, <0 0 1 2 0 1|]
EDOs: 19, 29, 43, 53, 72, 125f, 197ef, 269cef
Badness: 0.001062

Malcolm

Commas: 225/224, 2200/2187

Map: [<1 0 0 -5 -3|, <0 1 0 2 7|, <0 0 1 2 -2|]
EDOs: 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce
Badness: 0.001250

13-limit

commas: 225/224, 275/273, 325/324

Map: [<1 0 0 -5 -3 2|, <0 1 0 2 7 4|, <0 0 1 2 -2 -2|]
EDOs: 41, 53, 94, 429cdef, 523cdef
Badness: 0.001075

Tripod

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

13-limit eigenmonzo subgroup: 2.9/7.13/11
15-limit eigenmonzo subgroup: 2.5/3.13/11

Map: [<1 0 0 -5 12 -8|, <0 1 0 2 -1 3|, <0 0 1 2 -3 3|]
EDOs: 9, 10, 19, 31, 41, 72f, 81, 91, 122f, 163df
Badness: 0.000745

Projection pairs: 7 225/32 11 4096/375 13 3375/256

Prodigy

Prodigy shrinks 1024/1029, 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to miracle temperament. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.

Commas: 225/224, 441/440
Related linear temperament: miracle

Minimax tuning:
[|1 0 0 0 0>, |13/12 1/2 -1/4 0 1/12>,
|13/6 -1 1/2 0 1/6>,
|3/2 -1 1/2 0 1/2>, |0 0 0 0 1>]
Eigenmonzo subgroup: 2.9/5.11

Lattice basis: secor length 0.9111, 3/2 length 0.9477
Angle(secor, 3/2) = 65.933
Map to lattice: [<0 0 -1 -2 -3|, <0 1 -1 0 3|]

Map: [<1 0 0 -5 -13|, <0 1 0 2 6|, <0 0 1 2 3|]
Generators: 2, 3, 5
EDOs: 10, 12, 29, 31, 41, 72, 247c, 319bcde, 391bcde, 463bcde
Badness: 0.000344

Projection pairs: 7 225/32 11 91125/8192

Scales: prodigy11, prodigy12, prodigy29

Hobbit bases

{2, 3, 5} subgroup

31: 81/80, 34171875/33554432
41: 34171875/33554432, 32805/32768

13-limit

Commas: 105/104, 196/195, 352/351

Map: [<1 0 0 -5 -13 -8|, <0 1 0 2 6 3|, <0 0 1 2 3 3|]
EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef
Badness: 0.000736

Prodigious

Commas: 225/224, 441/440, 364/363

Map: [<1 0 0 -5 -13 -23|, <0 1 0 2 6 11|, <0 0 1 2 3 4|]
EDOs: 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf
Badness: 0.000900

Prodigal

Commas: 225/224, 441/440, 351/350

Map: [<1 0 0 -5 -13 -4|, <0 1 0 2 6 -1|, <0 0 1 2 3 4|]
EDOs: 31, 72, 103, 175f
Badness: 0.000889