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 known as 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 (when using the 34b val) 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.

Subgroup: 2.3.5.7

Comma list: 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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 700.974, ~5/4 = 384.208
• CWE: ~2 = 1\1, ~3/2 = 700.622, ~5/4 = 383.854

Minimax tuning:

• 7-odd-limit: 3 and 5 1/4-comma 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 (unchanged-interval) basis: 2.5/3.7

• 9-odd-limit: 3 1/6-comma flat, 5 1/3-comma 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 (unchanged-interval) basis: 2.9/5.7

Optimal ET sequence: 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c

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

Associated temperament: catakleismic

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

Overview to 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. These and many others considered below use the same generators as marvel.

Temperaments discussed elsewhere include

• Supernatural (+245/243) → Keemic family
• Artemis (+121/120) → Biyatismic clan
• Spectacle (+243/242) → Rastmic rank-3 clan
• Mirage (+243/242, +385/384) → Rastmic rank-3 clan

Undecimal marvel (unimarv)

Main article: Marvel

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 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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.373, ~5/4 = 383.146
• CWE: ~2 = 1\1, ~3/2 = 700.605, ~5/4 = 383.454

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 (unchanged-interval) basis: 2.9/5.11/9

Optimal ET sequence: 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce

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

Associated temperament: catakleismic

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

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 700.182, ~5/4 = 384.400
• CWE: ~2 = 1\1, ~3/2 = 699.811, ~5/4 = 384.118

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/9
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.15/11.15/13

Optimal ET sequence: 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff

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

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.563, ~5/4 = 383.015
• CWE: ~2 = 1\1, ~3/2 = 701.100, ~5/4 = 383.132

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.13/5
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.15/13

Optimal ET sequence: 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf

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

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 225/224, 325/324, 385/384, 400/399

Sval mapping: [⟨1 0 0 -5 12 2 9], ⟨0 1 0 2 -1 4 -3], ⟨0 0 1 2 -3 -2 0]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.424, ~5/4 = 383.029
• CWE: ~2 = 1\1, ~3/2 = 701.200, ~5/4 = 383.114

Optimal ET sequence: 41, 53, 72, 94, 113, 166

Badness: 0.773 × 10^-3

Apotropaia

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.586, ~5/4 = 382.733
• CWE: ~2 = 1\1, ~3/2 = 701.088, ~5/4 = 382.971

Optimal ET sequence: 41, 53g, 72, 166g, 238cfg

Badness: 0.869 × 10^-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 385/384, 400/399, 595/594

Mapping: [⟨1 0 0 -5 12 2 18 9], ⟨0 1 0 2 -1 4 0 -3], ⟨0 0 1 2 -3 -2 -6 0]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.443, ~5/4 = 382.739
• CWE: ~2 = 1\1, ~3/2 = 701.201, ~5/4 = 382.940

Optimal ET sequence: 41, 53g, 72, 94, 113, 166g

Badness: 0.978 × 10^-3

Enodia

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 0 -5 12 2 -13], ⟨0 1 0 2 -1 4 2], ⟨0 0 1 2 -3 -2 6]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.627, ~5/4 = 383.346
• CWE: ~2 = 1\1, ~3/2 = 701.095, ~5/4 = 383.314

Optimal ET sequence: 41g, 53, 72, 166g, 238cfg

Badness: 0.917 × 10^-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399

Mapping: [⟨1 0 0 -5 12 2 -13 9], ⟨0 1 0 2 -1 4 2 -3], ⟨0 0 1 2 -3 -2 6 0]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.479, ~5/4 = 383.379
• CWE: ~2 = 1\1, ~3/2 = 701.211, ~5/4 = 383.305

Optimal ET sequence: 41g, 53, 72, 94, 125f, 166g

Badness: 1.03 × 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]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.385, ~5/4 = 383.208
• CWE: ~2 = 1\1, ~3/2 = 700.611, ~5/4 = 383.493

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.11/9
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7.15/13

Optimal ET sequence: 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff

Badness: 0.862 × 10^-3

Isis

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.997, ~5/4 = 383.134
• CWE: ~2 = 1\1, ~3/2 = 701.928, ~5/4 = 383.228

Optimal ET sequence: 10, 19f, 22, 31, 41, 53, 94

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, 364/363, 385/384

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.702, ~5/4 = 382.074
• CWE: ~2 = 1\1, ~3/2 = 700.718, ~5/4 = 382.661

Minimax tuning:

• 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5.13/9
• 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5

Optimal ET sequence: 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 700.028, ~5/4 = 382.694
• CWE: ~2 = 1\1, ~3/2 = 699.404, ~5/4 = 382.917

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13/11
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/11

Optimal ET sequence: 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df

Badness: 0.745 × 10^-3

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

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]]

mapping generators: ~2, ~26/15, ~5

Optimal tunings:

• CTE: ~2 = 1\1, ~26/15 = 950.961, ~5/4 = 383.088
• CWE: ~2 = 1\1, ~26/15 = 950.426, ~5/4 = 383.479

Optimal ET sequence: 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff

Badness: 0.9997 × 10^-3

Minerva

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 700.379, ~5/4 = 386.617
• CWE: ~2 = 1\1, ~3/2 = 700.301, ~5/4 = 386.579

Minimax tuning: 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/5.11/9

Optimal ET sequence: 9, 12, 19e, 22, 31, 53, 84e, 96, 127

Badness: 0.381 × 10^-3

Projection pairs: 7 225/32 11 5625/512

Scales: minerva12, minerva22x

Associated temperament: würschmidt

Athene

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.179, ~5/4 = 385.888
• CWE: ~2 = 1\1, ~3/2 = 701.214, ~5/4 = 385.934

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/7
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/7

Optimal ET sequence: 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def

Badness: 0.818 × 10^-3

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

Apollo

See also: Ptolemismic clan § Apollo

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 703.418, ~5/4 = 381.331
• CWE: ~2 = 1\1, ~3/2 = 703.461, ~5/4 = 381.507

Minimax tuning: 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/5.11/9

Optimal ET sequence: 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce

Projection pairs: 7 225/32 11 100/9

Associated temperament: magic

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 703.959, ~5/4 = 381.001
• CWE: ~2 = 1\1, ~3/2 = 703.985, ~5/4 = 381.358

Minimax tuning: 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/9

Optimal ET sequence: 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef

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

Potassium

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 695.937, ~5/4 = 384.836
• CWE: ~2 = 1\1, ~3/2 = 696.059, ~5/4 = 384.947

Minimax tuning: 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.11

Optimal ET sequence: 7d, 9, 10, 12, 19, 31e, 50e

Badness: 0.464 × 10^-3

Projection pairs: 7 225/32 11 45/4

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 695.894, ~5/4 = 384.602
• CWE: ~2 = 1\1, ~3/2 = 695.948, ~5/4 = 384.637

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13/9
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7.13/9

Optimal ET sequence: 9, 10, 12f, 19, 31e, 50e

Badness: 0.733 × 10^-3

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

Malcolm

"Malcolm" redirects here. For Alexander Malcolm's JI scale, see Malcolm (scale).

Subgroup: 2.3.5.7.11

Comma list: 225/224, 2200/2187

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 702.322, ~5/4 = 382.976
• CWE: ~2 = 1\1, ~3/2 = 702.054, ~5/4 = 382.622

Optimal ET sequence: 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce

Badness: 1.250 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 702.235, ~5/4 = 383.205
• CWE: ~2 = 1\1, ~3/2 = 702.088, ~5/4 = 383.063

Optimal ET sequence: 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef

Badness: 1.075 × 10^-3

Scales: malco

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. 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.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 699.995, ~5/4 = 384.330
• CWE: ~2 = 1\1, ~3/2 = 699.861, ~5/4 = 383.772

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 (unchanged-interval) basis: 2.9/5.11

Optimal ET sequence: 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde

Badness: 0.344 × 10^-3

Projection pairs: 7 225/32 11 91125/8192

Scales: prodigy11, prodigy12, prodigy29

Associated temperament: miracle

Hobbit bases

2.3.5 subgroup

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

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 700.617, ~5/4 = 382.244
• CWE: ~2 = 1\1, ~3/2 = 700.469, ~5/4 = 381.669

Optimal ET sequence: 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff

Badness: 0.736 × 10^-3

Prodigious

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 700.305, ~5/4 = 384.075
• CWE: ~2 = 1\1, ~3/2 = 700.330, ~5/4 = 383.503

Optimal ET sequence: 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff

Badness: 0.900 × 10^-3

Prodigal

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 699.698, ~5/4 = 384.884
• CWE: ~2 = 1\1, ~3/2 = 699.554, ~5/4 = 384.350

Optimal ET sequence: 12f, 19e, 31, 53e, 60eff, 72, 103, 175f

Badness: 0.889 × 10^-3

Protannic

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000

Mapping: [⟨1 0 0 -5 -13 21], ⟨0 1 0 2 6 -8], ⟨0 0 1 2 3 -2]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 698.947, ~5/4 = 385.480
• CWE: ~2 = 1\1, ~3/2 = 699.422, ~5/4 = 383.983

Optimal ET sequence: 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff

Badness: 0.953 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440

Mapping: [⟨1 0 0 -5 -13 21 12], ⟨0 1 0 2 6 -8 -5], ⟨0 0 1 2 3 -2 0]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 698.969, ~5/4 = 385.440
• CWE: ~2 = 1\1, ~3/2 = 699.485, ~5/4 = 383.874

Optimal ET sequence: 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg

Badness: 0.772 × 10^-3

Fantastic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4356

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

mapping generators: ~99/70, ~3, ~5

Optimal tunings:

• CTE: ~99/70 = 1\2, ~3/2 = 701.132, ~5/4 = 383.926
• CWE: ~99/70 = 1\2, ~3/2 = 700.816, ~5/4 = 383.535

Optimal ET sequence: 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c

Badness: 0.743 × 10^-3

Hestia

Named by Graham Breed in 2011, hestia was found to be locally efficient in the higher limits among all rank-3 extensions of marvel, although it is a weak extension.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 125000/124509

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

mapping generators: ~2, ~400/231, ~5

Optimal tunings:

• CTE: ~2 = 1\1, ~400/231 = 950.043, ~5/4 = 384.858
• CWE: ~2 = 1\1, ~400/231 = 950.121, ~5/4 = 383.953

Optimal ET sequence: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce

Badness: 1.54 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~26/15 = 950.131, ~5/4 = 385.245
• CWE: ~2 = 1\1, ~26/15 = 950.211, ~5/4 = 383.949

Optimal ET sequence: 19, 29, 43, 53, 72, 125f, 197ef, 269cef

Badness: 1.062 × 10^-3

Morfil

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323

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

mapping generators: ~2, ~3, ~84/55

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.459, ~84/55 = 739.747
• CWE: ~2 = 1\1, ~3/2 = 701.111, ~84/55 = 739.451

Optimal ET sequence: 29, 31, 60e, 91e, 94, 125

Badness: 1.152 × 10^-3

Catakleismoid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4374

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

mapping generators: ~2, ~6/5, ~11

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 316.834, ~11/8 = 551.318
• CWE: ~2 = 1\1, ~6/5 = 316.771, ~11/8 = 550.034

Optimal ET sequence: 19, 34d, 53, 72, 197e, 269ce

Badness: 1.275 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 316.886, ~11/8 = 551.318
• CTE: ~2 = 1\1, ~6/5 = 316.794, ~11/8 = 549.590

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197ef, 269cef

Badness: 0.916 × 10^-3