Marvel family

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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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.4075, ~5/4 = 383.6376

Minimax tuning:

[[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 sequence9, 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

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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.3887, ~5/4 = 383.5403

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

Optimal ET sequence9, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 699.7367, ~5/4 = 384.0613

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 sequence19, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.9779, ~5/4 = 383.1622

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 sequence19, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.4245, ~5/4 = 383.0293

Optimal ET sequence41, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.5860, ~5/4 = 382.7331

Optimal ET sequence41, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.4435, ~5/4 = 382.7395

Optimal ET sequence41, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.6270, ~5/4 = 383.3456

Optimal ET sequence41g, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.4791, ~5/4 = 383.3795

Optimal ET sequence41g, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.3937, ~5/4 = 383.5725

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 sequence9, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 701.9156, ~5/4 = 383.2445

Optimal ET sequence10, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.4560, ~5/4 = 382.8177

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 sequence9, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 699.2335, ~5/4 = 382.9775

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 sequence9, 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 tuning (POTE): ~2 = 1\1, ~15/13 = 249.7138, ~5/4 = 383.5816

Optimal ET sequence9, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.2593, ~5/4 = 386.5581

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

Optimal ET sequence9, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 701.2342, ~5/4 = 385.9594

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 sequence12f, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.4846, ~5/4 = 381.6033

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

Optimal ET sequence12, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.9984, ~5/4 = 381.5352

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

Optimal ET sequence12f, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 696.1714, ~5/4 = 385.0500

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

Optimal ET sequence7d, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 696.0103, ~5/4 = 384.6785

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 sequence9, 10, 12f, 19, 31e, 50e

Badness: 0.733 × 10-3

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

Malcolm

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 tuning (POTE): ~2 = 1\1, ~3/2 = 701.8913, ~5/4 = 382.4083

Optimal ET sequence12e, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 701.8913, ~5/4 = 382.4083

Optimal ET sequence12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef

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. 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 tuning (POTE): ~2 = 1\1, ~3/2 = 699.7981, ~5/4 = 383.5114

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 (unchanged-interval) basis: 2.9/5.11

Optimal ET sequence10, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.4006, ~5/4 = 381.4025

Optimal ET sequence10, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.3407, ~5/4 = 383.2592

Optimal ET sequence12f, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 699.4864, ~5/4 = 384.0998

Optimal ET sequence12f, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 699.5536, ~5/4 = 383.5696

Optimal ET sequence29, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 699.6262, ~5/4 = 383.4458

Optimal ET sequence29g, 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 tuning (POTE): ~99/70 = 1\2, ~3/2 = 700.6242, ~5/4 = 383.2978

Optimal ET sequence12, 22, 34d, 50, 60e, 72, 166, 238c, 310c

Badness: 0.743 × 10-3

Hestia

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 tuning (POTE): ~2 = 1\1, ~400/231 = 950.1474, ~5/4 = 383.6467

Optimal ET sequence19, 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 tuning (POTE): ~2 = 1\1, ~26/15 = 950.2349, ~5/4 = 383.5558

Optimal ET sequence19, 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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.8983, ~84/55 = 739.3812

Optimal ET sequence29, 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 tuning (POTE): ~2 = 1\1, ~6/5 = 316.7318, ~11/8 = 549.2528

Optimal ET sequence19, 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 tuning (POTE): ~2 = 1\1, ~6/5 = 316.7410, ~11/8 = 548.6028

Optimal ET sequence19, 34d, 53, 72, 125f, 197ef, 269cef

Badness: 0.916 × 10-3