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: ~3/2 = 700.4075, ~5/4 = 383.6376

EDO generators: (11, 6)\19, (18, 10)\31, (24, 13)\41

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

Associated temperament: catakleismic

Technical data

Subgroup: 2.3.5.7

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: 0.0365 × 10^-3

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

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: ~3/2 = 700.3887, ~5/4 = 383.5403

EDO generators: (13, 7)\22, (18, 10)\31, (24, 13)\41

Scales (Scala files): marvel22_11, unimarv19, unimarv22

Associated temperament: catakleismic

Technical data

Subgroup: 2.3.5.7.11

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: 0.255 × 10^-3

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

Period: 1\1

Optimal (POTE) generators: ~3/2 = 699.7367, ~5/4 = 384.0613

EDO generators: (18, 10)\31, (31, 17)\53, (42, 23)\72

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

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: 0.690 × 10^-3

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

Period: 1\1

Optimal (POTE) generators: ~3/2 = 700.9779, ~5/4 = 383.1622

EDO generators: (24, 13)\41, (31, 17)\53, (42, 23)\72

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

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: 0.721 × 10^-3

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

Period: 1\1

Optimal (POTE) generators: ~3/2 = 700.9619, ~5/4 = 383.0310

EDO generators: (13, 7)\22, (24, 13)\41, (42, 23)\72

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 385/384, 325/324, 595/594

Mapping: [⟨1 0 0 -5 12 2 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 -6]]

Badness: 0.869 × 10^-3

Enodia

Period: 1\1

Optimal (POTE) generators: ~3/2 = 700.9658, ~5/4 = 383.3063

EDO generators: (13, 7)\22, (31, 17)\53, (42, 23)\72

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 385/384, 325/324, 375/374

Mapping: [⟨1 0 0 -5 12 2 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 6]]

Badness: 0.917 × 10^-3

Marvelcat

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 385/384

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

Badness: 0.9997 × 10^-3

Marvell

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 385/384, 1573/1568

Mapping: [⟨1 0 0 -5 12 -29], ⟨0 1 0 2 -1 6], ⟨0 0 1 2 -3 10]]

Minimax tuning:

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

Badness: 0.862 × 10^-3

Isis

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 385/384, 275/273

Mapping: [⟨1 0 0 -5 12 17], ⟨0 1 0 2 -1 4], ⟨0 0 1 2 -3 -3]]

Badness: 0.866 × 10^-3

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

Deecee

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 385/384, 364/363

Mapping: [⟨1 0 0 -5 12 27], ⟨0 1 0 2 -1 -3], ⟨0 0 1 2 -3 -8]]

Minimax tuning:

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

Badness: 0.920 × 10^-3

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

Tripod

Subgroup: 2.3.5.7.11.13

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

Minimax tuning:

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

Mapping: [⟨1 0 0 -5 12 -8], ⟨0 1 0 2 -1 3], ⟨0 0 1 2 -3 3]]

Badness: 0.745 × 10^-3

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

Mirage

Comma list: 225/224, 243/242, 385/384

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

Badness: 0.738 × 10^-3

Minerva

Comma list: 99/98, 176/175

Subgroup: 2.3.5.7.11

Associated temperament: orwell

Mapping: [⟨1 0 0 -5 -9], ⟨0 1 0 2 2], ⟨0 0 1 2 4]]

Mapping generators: ~2, ~3, ~5

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

Lattice basis:

16/15 length = 0.8997, 5/4 length = 1.0457

Angle (16/15, 5/4) = 98.6044 degrees

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7/5.11/9

Badness: 0.381 × 10^-3

Projection pairs: 7 225/32 11 5625/512

Scales: minerva12, minerva22x

Athene

Comma list: 99/98, 176/175, 275/273

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 0 -5 -9 -4], ⟨0 1 0 2 2 -1], ⟨0 0 1 2 4 4]]

Minimax tuning:

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

Badness: 0.818 × 10^-3

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

Spectacle

Comma list: 225/224, 243/242

Subgroup: 2.3.5.7.11

Associated temperament: marvo

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

Mapping generators: ~2, ~11/9, ~5

Minimax tuning:

• 11-odd-limit

[[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

Badness: 0.499 × 10^-3

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

Scales: spectacle31

13-limit

Comma list: 225/224, 243/242, 351/350

Subgroup: 2.3.5.7.11.13

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

Badness: 1.009 × 10^-3

Apollo

Comma list: 100/99, 225/224

Subgroup: 2.3.5.7.11

Associated temperament: magic

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

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7/5.11/9

Projection pairs: 7 225/32 11 100/9

13-limit

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

Subgroup: 2.3.5.7.11.13

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

Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9

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

Artemis

Comma list: 121/120, 225/224

Subgroup: 2.3.5.7.11

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

13-limit

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

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 1 -3 2 -5], ⟨0 1 1 4 1 6], ⟨0 0 2 4 1 6]]

Diana

Comma list: 121/120, 225/224, 275/273

Subgroup: 2.3.5.7.11.13

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

Potassium

Comma list: 45/44, 56/55

Subgroup: 2.3.5.7.11

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

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.9/7.11

Badness: 0.464 × 10^-3

Projection pairs: 7 225/32 11 45/4

13-limit

Comma list: 45/44, 56/55, 78/77

Subgroup: 2.3.5.7.11.13

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

Minimax tuning:

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

Badness: 0.733 × 10^-3

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

Fantastic

Comma list: 225/224, 4375/4356

Subgroup: 2.3.5.7.11

Mapping: [⟨2 0 0 -10 -7], ⟨0 1 0 2 0], ⟨0 0 1 2 3]]

Badness: 0.743 × 10^-3

Catakleismoid

Comma list: 225/224, 4375/4374

Subgroup: 2.3.5.7.11

Mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 0], ⟨0 0 0 0 1]]

Badness: 1.275 × 10^-3

13-limit

Comma list: 169/168, 225/224, 325/324

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 1 -3 0 0], ⟨0 6 5 22 0 14], ⟨0 0 0 0 1 0]]

Badness: 0.916 × 10^-3

Hestia

Comma list: 225/224, 125000/124509

Subgroup: 2.3.5.7.11

Mapping: [⟨1 0 0 -5 9], ⟨0 2 0 4 -7], ⟨0 0 1 2 0]]

Badness: 1.54 × 10^-3

13-limit

Comma list: 169/168, 225/224, 1001/1000

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 0 -5 9 -1], ⟨0 2 0 4 -7 3], ⟨0 0 1 2 0 1]]

Badness: 1.062 × 10^-3

Malcolm

Comma list: 225/224, 2200/2187

Subgroup: 2.3.5.7.11

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

Badness: 1.250 × 10^-3

13-limit

Comma list: 225/224, 275/273, 325/324

Subgroup: 2.3.5.7.11.13

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

Badness: 1.075 × 10^-3

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.

Comma list: 225/224, 441/440

Subgroup: 2.3.5.7.11

Associated temperament: miracle

Mapping: [⟨1 0 0 -5 -13], ⟨0 1 0 2 6], ⟨0 0 1 2 3]]

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 0 -1 -2 -3], ⟨0 1 -1 0 3]]

Lattice basis:

secor length = 0.9111, 3/2 length = 0.9477

Angle (secor, 3/2) = 65.933

Minimax tuning:

• 11-odd-limit

[[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

Badness: 0.344 × 10^-3

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

Comma list: 105/104, 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 0 -5 -13 -8], ⟨0 1 0 2 6 3], ⟨0 0 1 2 3 3]]

Badness: 0.736 × 10^-3

Prodigious

Comma list: 225/224, 441/440, 364/363

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 0 -5 -13 -23], ⟨0 1 0 2 6 11], ⟨0 0 1 2 3 4]]

Badness: 0.900 × 10^-3

Prodigal

Comma list: 225/224, 441/440, 351/350

Subgroup: 2.3.5.7.11.13

Mapping: [⟨1 0 0 -5 -13 -4], ⟨0 1 0 2 6 -1], ⟨0 0 1 2 3 4]]

Badness: 0.889 × 10^-3