Marvel family

The marvel family is the set of temperaments that temper out the 7-limit marvel comma (225/224 = [-5 2 2 -1⟩) which is also named septimal kleisma. These temperaments hence equate 16/15 and 15/14, or equivalently they equate two 5/4's and one 14/9. The marvel comma is noteworthy in that it is tempered out by many common edos and rank-2 temperaments.

The marvel comma can also be viewed as a comma of the 2.9.25.7 subgroup. Hence it is tempered out by any subset edos of marvel-supporting edos that have this subgroup, such as 11edo and 17edo which are subsets of 22edo and 34edo which temper out the marvel comma.

Marvel

Main article: Marvel

The head of the marvel family is marvel, which tempers out 225/224. 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 marvel temperaments for some other rank-2 temperaments.

Period: 1\1

Optimal (POTE) generators:

EDO generators:

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

Associated temperament: catakleismic

Technical data

Comma list: c = 225/224

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

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 0 -1 -2], ⟨0 1 -1 0]]

Lattice basis:

secor length = 1.256, 3/2 length = 1.369

Angle (secor, 3/2) = 106.958 degrees

Minimax tuning:

• 7-odd-limit: 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-odd-limit: 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

Badness: 0.0000365

Projection pairs: 7 225/32

Complexity spectrum: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9

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

Eleven-limit extensions

The second comma of the normal comma list defines which 11-limit family member we are looking at.

• 4125/4096 gives unidecimal marvel,
• 91125/90112 gives prodigy,
• 5632/5625 gives minerva, and
• 243/242 gives spectacle.

11-limit (unimarv)

Period: 1\1

Optimal (POTE) generators:

EDO generators:

Scales (Scala files): marvel22_11, unimarv19, unimarv22

Associated temperament: catakleismic

Technical data

Comma list: 225/224, 385/384

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

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 -1 0 -2 1], ⟨0 -1 1 0 -2]]

Lattice basis:

secor length = 1.0364, 5/4 length = 1.0759

Angle (secor, 5/4) = 104.028 degrees

Minimax tuning:

• 11-odd-limit

[[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

Badness: 0.000255

Projection pairs: 7 225/32 11 4096/375

Complexity 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

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

Comma list: 225/224, 385/384, 351/350

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

Minimax tuning:

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

Badness: 0.000690

Complexity 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

Comma list: 225/224, 385/384, 325/324

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

Minimax tuning:

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

Badness: 0.000721

Projection pairs: 7 225/32 11 4096/375 13 324/25

Complexity 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

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

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

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, 104, 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

Artemis

Commas: 121/120, 225/224

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

EDOs: 9, 20, 22, 31, 53

13-limit

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

Map: [<1 0 1 -3 2 -5|, <0 1 1 4 1 6|, <0 0 2 4 1 6|]

EDOs: 9, 20, 29, 31

Diana

Commas: 121/120, 225/224, 275/273

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

EDOs: 22, 29, 31, 53

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

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