Marvel family
The marvel family of rank-3 temperaments tempers 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 equal 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 34d val) which temper out the marvel comma.
Marvel
The head of the marvel family is marvel, which tempers out 225/224. Marvel has a normal generator list 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, with which it shares the optimal patent val. Another temperament which does little damage to tuning accuracy is compton, 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
- 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
Scales: marvel9, marvel10, marvel11, marvel12, marvel19, marvel22, pump12_1, pump12_2, pump13, pump14, pump15, pump16, pump17, pump18, marvel wholetone
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)
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
- [[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
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
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
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]]
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
Scales: apollo wholetone, indigo17
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]]
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 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
- [[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
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
Besides 4375/4356, fantastic also tempers out 9801/9800 and splits the octave in two.
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 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[1], 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 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 ET sequence: 29, 31, 60e, 91e, 94, 125
Badness: 1.152 × 10-3
Catakleismoid
Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.
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 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
- CWE: ~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