Marvel family: Difference between revisions

Revision as of 14:20, 12 September 2022

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

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

Template:Val list

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,
243/242 gives spectacle,

as well as others considered below. Temperaments discussed elsewhere include supernatural (→ Keemic family).

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]]

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

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

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

Template:Val list

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): ~3/2 = 699.7367, ~5/4 = 384.0613

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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): ~3/2 = 700.9779, ~5/4 = 383.1622

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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

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

Optimal GPV sequence: Template:Val list

Badness: 0.869 × 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 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 6]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.917 × 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): ~3/2 = 700.3937, ~5/4 = 383.5725

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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): ~3/2 = 701.9156, ~5/4 = 383.2445

Optimal GPV sequence: Template:Val list

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): ~3/2 = 700.4560, ~5/4 = 382.8177

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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): ~3/2 = 699.2335, ~5/4 = 382.9775

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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): ~15/13 = 249.7138, ~5/4 = 383.5816

Optimal GPV sequence: Template:Val list

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): ~3/2 = 700.2593, ~5/4 = 386.5581

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

Template:Val list

Badness: 0.381 × 10^-3

Projection pairs: 7 225/32 11 5625/512

Scales (Scala files): 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): ~3/2 = 701.2342, ~5/4 = 385.9594

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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]]

Optimal tuning (POTE): ~3/2 = 703.4846, ~5/4 = 381.6033

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

Template:Val list

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): ~3/2 = 703.9984, ~5/4 = 381.5352

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

Optimal GPV sequence: Template:Val list

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): ~3/2 = 696.1714, ~5/4 = 385.0500

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

Template:Val list

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): ~3/2 = 696.0103, ~5/4 = 384.6785

Minimax tuning:

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

Optimal GPV sequence: Template:Val list

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): ~3/2 = 701.8913, ~5/4 = 382.4083

Template:Val list

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): ~3/2 = 701.8913, ~5/4 = 382.4083

Optimal GPV sequence: Template:Val list

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.

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): ~3/2 = 699.7981, ~5/4 = 383.5114

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

Template:Val list

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): ~3/2 = 700.4006, ~5/4 = 381.4025

Optimal GPV sequence: Template:Val list

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): ~3/2 = 700.3407, ~5/4 = 383.2592

Optimal GPV sequence: Template:Val list

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): ~3/2 = 699.4864, ~5/4 = 384.0998

Optimal GPV sequence: Template:Val list

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): ~3/2 = 699.5536, ~5/4 = 383.5696

Optimal GPV sequence: Template:Val list

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): ~3/2 = 699.6262, ~5/4 = 383.4458

Optimal GPV sequence: Template:Val list

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): ~3/2 = 700.6242, ~5/4 = 383.2978

Template:Val list

Badness: 0.743 × 10^-3

Spectacle

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

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

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

Optimal tuning (POTE): ~11/9 = 350.0570, ~5/4 = 383.9323

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

Template:Val list

Badness: 0.499 × 10^-3

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

Scales (Scala files): spectacle31

Associated temperament: marvo

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tuning (POTE): ~11/9 = 349.9247, ~5/4 = 384.3505

Optimal GPV sequence: Template:Val list *

* optimal patent val: 240

Badness: 1.009 × 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): ~231/200 = 249.8526, ~5/4 = 383.6467

Template:Val list

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]]

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

Optimal tuning (POTE): ~15/13 = 249.7651, ~5/4 = 383.5558

Optimal GPV sequence: Template:Val list

Badness: 1.062 × 10^-3

Artemis

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224

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

Mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~3/2 = 699.8719, ~11/10 = 158.3232

Template:Val list

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~3/2 = 698.7090, ~11/10 = 158.7117

Optimal GPV sequence: Template:Val list

Diana

Subgroup: 2.3.5.7.11.13

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

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

Mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~3/2 = 700.9789, ~11/10 = 159.0048

Optimal GPV sequence: Template:Val list

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): ~3/2 = 700.8983, ~84/55 = 739.3812

Template:Val list

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): ~6/5 = 316.7318, ~11/8 = 549.2528

Template:Val list

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]]

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

Optimal tuning (POTE): ~6/5 = 316.7410, ~11/8 = 548.6028

Optimal GPV sequence: Template:Val list

Badness: 0.916 × 10^-3

Mirage

Subgroup: 2.3.5.7.11.13

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

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

Mapping generators: ~2, ~15/14, ~13

Optimal tuning (POTE): ~15/14 = 116.6327, ~13/8 = 837.7040

Optimal GPV sequence: Template:Val list

Badness: 0.738 × 10^-3

@@ Line 16: / Line 16: @@
 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224
@@ Line 30: / Line 30: @@
 : Angle (secor, 3/2) = 106.958 degrees
-[[POTE generator]]s: ~3/2 = 700.4075, ~5/4 = 383.6376
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.4075, ~5/4 = 383.6376
 [[Minimax tuning]]:
@@ Line 69: / Line 69: @@
 }}
-=== Eleven-limit extensions  ===
+=== Overview to extensions ===
 The second comma of the [[Normal lists #Normal interval list|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
+* 5632/5625 gives minerva,
-* 243/242 gives spectacle.
+* 243/242 gives spectacle,
+as well as others considered below. Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]).
 == Undecimal marvel (unimarv) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 385/384
@@ Line 89: / Line 91: @@
 : Angle (secor, 5/4) = 104.028 degrees
-[[POTE generator]]s: ~3/2 = 700.3887, ~5/4 = 383.5403
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.3887, ~5/4 = 383.5403
 [[Minimax tuning]]:
@@ Line 125: / Line 127: @@
 Mapping: [{{val| 1 0 0 -5 12 -4 }}, {{val| 0 1 0 2 -1 -1 }}, {{val| 0 0 1 2 -3 4 }}]
-POTE generators: ~3/2 = 699.7367, ~5/4 = 384.0613
+Optimal tuning (POTE): ~3/2 = 699.7367, ~5/4 = 384.0613
 Minimax tuning:
@@ Line 131: / Line 133: @@
 * 15-odd-limit eigenmonzo subgroup: 2.15/11.15/13
-Vals: {{Val list| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
+Optimal GPV sequence: {{Val list| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
 Badness: 0.690 × 10<sup>-3</sup>
@@ Line 144: / Line 146: @@
 Mapping: [{{val| 1 0 0 -5 12 2 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -2 }}]
-POTE generators: ~3/2 = 700.9779, ~5/4 = 383.1622
+Optimal tuning (POTE): ~3/2 = 700.9779, ~5/4 = 383.1622
 Minimax tuning:
@@ Line 150: / Line 152: @@
 * 15-odd-limit eigenmonzo subgroup: 2.7.15/13
-Vals: {{Val list| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
+Optimal GPV sequence: {{Val list| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
 Badness: 0.721 × 10<sup>-3</sup>
@@ Line 165: / Line 167: @@
 Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 -6 }}]
-POTE generators: ~3/2 = 700.9619, ~5/4 = 383.0310
+Optimal tuning (POTE): ~3/2 = 700.9619, ~5/4 = 383.0310
-Vals: {{Val list| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
+Optimal GPV sequence: {{Val list| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
 Badness: 0.869 × 10<sup>-3</sup>
@@ Line 178: / Line 180: @@
 Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 6 }}]
-POTE generators: ~3/2 = 700.9658, ~5/4 = 383.3063
+Optimal tuning (POTE): ~3/2 = 700.9658, ~5/4 = 383.3063
-Vals: {{Val list| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}
+Optimal GPV sequence: {{Val list| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}
 Badness: 0.917 × 10<sup>-3</sup>
@@ Line 191: / Line 193: @@
 Mapping: [{{val| 1 0 0 -5 12 -29 }}, {{val| 0 1 0 2 -1 6 }}, {{val| 0 0 1 2 -3 10 }}]
-POTE generators: ~3/2 = 700.3937, ~5/4 = 383.5725
+Optimal tuning (POTE): ~3/2 = 700.3937, ~5/4 = 383.5725
 Minimax tuning:
@@ Line 197: / Line 199: @@
 * 15-odd-limit eigenmonzo subgroup: 2.7.15/13
-Vals: {{Val list| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
+Optimal GPV sequence: {{Val list| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
 Badness: 0.862 × 10<sup>-3</sup>
@@ Line 208: / Line 210: @@
 Mapping: [{{val| 1 0 0 -5 12 17 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -3 }}]
-POTE generators: ~3/2 = 701.9156, ~5/4 = 383.2445
+Optimal tuning (POTE): ~3/2 = 701.9156, ~5/4 = 383.2445
-Vals: {{Val list| 10, 19f, 22, 31, 41, 53, 94 }}
+Optimal GPV sequence: {{Val list| 10, 19f, 22, 31, 41, 53, 94 }}
 Badness: 0.866 × 10<sup>-3</sup>
@@ Line 223: / Line 225: @@
 Mapping: [{{val| 1 0 0 -5 12 27 }}, {{val| 0 1 0 2 -1 -3 }}, {{val| 0 0 1 2 -3 -8 }}]
-POTE generators: ~3/2 = 700.4560, ~5/4 = 382.8177
+Optimal tuning (POTE): ~3/2 = 700.4560, ~5/4 = 382.8177
 Minimax tuning:
@@ Line 229: / Line 231: @@
 * 15-odd-limit eigenmonzo subgroup: 2.3.13/5
-Vals: {{Val list| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
+Optimal GPV sequence: {{Val list| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
 Badness: 0.920 × 10<sup>-3</sup>
@@ Line 242: / Line 244: @@
 Mapping: [{{val| 1 0 0 -5 12 -8 }}, {{val| 0 1 0 2 -1 3 }}, {{val| 0 0 1 2 -3 3 }}]
-POTE generators: ~3/2 = 699.2335, ~5/4 = 382.9775
+Optimal tuning (POTE): ~3/2 = 699.2335, ~5/4 = 382.9775
 Minimax tuning:
@@ Line 248: / Line 250: @@
 * 15-odd-limit eigenmonzo subgroup: 2.5/3.13/11
-Vals: {{Val list| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
+Optimal GPV sequence: {{Val list| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
 Badness: 0.745 × 10<sup>-3</sup>
@@ Line 263: / Line 265: @@
 Mapping generators: ~2, ~26/15, ~5
-POTE generators: ~15/13 = 249.7138, ~5/4 = 383.5816
+Optimal tuning (POTE): ~15/13 = 249.7138, ~5/4 = 383.5816
-Vals: {{Val list| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
+Optimal GPV sequence: {{Val list| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
 Badness: 0.9997 × 10<sup>-3</sup>
 == Minerva ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 99/98, 176/175
@@ Line 282: / Line 284: @@
 : Angle (16/15, 5/4) = 98.6044 degrees
-[[POTE generator]]s: ~3/2 = 700.2593, ~5/4 = 386.5581
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.2593, ~5/4 = 386.5581
 [[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.7/5.11/9
@@ Line 303: / Line 305: @@
 Mapping: [{{val| 1 0 0 -5 -9 -4 }}, {{val| 0 1 0 2 2 -1 }}, {{val| 0 0 1 2 4 4 }}]
-POTE generators: ~3/2 = 701.2342, ~5/4 = 385.9594
+Optimal tuning (POTE): ~3/2 = 701.2342, ~5/4 = 385.9594
 Minimax tuning:
@@ Line 309: / Line 311: @@
 * 15-odd-limit eigenmonzo subgroup: 2.11/9.13/7
-Vals: {{Val list| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
+Optimal GPV sequence: {{Val list| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
 Badness: 0.818 × 10<sup>-3</sup>
@@ Line 316: / Line 318: @@
 == Apollo ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 100/99, 225/224
@@ Line 322: / Line 324: @@
 [[Mapping]]: [{{val| 1 0 0 -5 2 }}, {{val| 0 1 0 2 -2 }}, {{val| 0 0 1 2 2 }}]
-[[POTE generator]]s: ~3/2 = 703.4846, ~5/4 = 381.6033
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.4846, ~5/4 = 381.6033
 [[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.7/5.11/9
@@ Line 339: / Line 341: @@
 Mapping: [{{val| 1 0 0 -5 2 7 }}, {{val| 0 1 0 2 -2 -5 }}, {{val| 0 0 1 2 2 2 }}]
-POTE generators: ~3/2 = 703.9984, ~5/4 = 381.5352
+Optimal tuning (POTE): ~3/2 = 703.9984, ~5/4 = 381.5352
 Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9
-Vals: {{Val list| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
+Optimal GPV sequence: {{Val list| 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
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 45/44, 56/55
@@ Line 354: / Line 356: @@
 [[Mapping]]: [{{val| 1 0 0 -5 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 1 }}]
-[[POTE generator]]s: ~3/2 = 696.1714, ~5/4 = 385.0500
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 696.1714, ~5/4 = 385.0500
 [[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.9/7.11
@@ Line 371: / Line 373: @@
 Mapping: [{{val| 1 0 0 -5 -2 -8 }}, {{val| 0 1 0 2 2 3 }}, {{val| 0 0 1 2 1 3 }}]
-POTE generators: ~3/2 = 696.0103, ~5/4 = 384.6785
+Optimal tuning (POTE): ~3/2 = 696.0103, ~5/4 = 384.6785
 Minimax tuning:
@@ Line 377: / Line 379: @@
 * 15-odd-limit eigenmonzo subgroup: 2.9/7.13/9
-Vals: {{Val list| 9, 10, 12f, 19, 31e, 50e }}
+Optimal GPV sequence: {{Val list| 9, 10, 12f, 19, 31e, 50e }}
 Badness: 0.733 × 10<sup>-3</sup>
@@ Line 384: / Line 386: @@
 == Malcolm ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 2200/2187
@@ Line 390: / Line 392: @@
 [[Mapping]]: [{{val| 1 0 0 -5 -3 }}, {{val| 0 1 0 2 7 }}, {{val| 0 0 1 2 -2 }}]
-[[POTE generator]]s: ~3/2 = 701.8913, ~5/4 = 382.4083
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.8913, ~5/4 = 382.4083
 {{Val list|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce }}
@@ Line 403: / Line 405: @@
 Mapping: [{{val| 1 0 0 -5 -3 2 }}, {{val| 0 1 0 2 7 4 }}, {{val| 0 0 1 2 -2 -2 }}]
-POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083
+Optimal tuning (POTE): ~3/2 = 701.8913, ~5/4 = 382.4083
-Vals: {{Val list| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
+Optimal GPV sequence: {{Val list| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
 Badness: 1.075 × 10<sup>-3</sup>
@@ Line 412: / Line 414: @@
 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.
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 441/440
@@ Line 424: / Line 426: @@
 : Angle (secor, 3/2) = 65.933
-[[POTE generator]]s: ~3/2 = 699.7981, ~5/4 = 383.5114
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 699.7981, ~5/4 = 383.5114
 [[Minimax tuning]]:
@@ Line 454: / Line 456: @@
 Mapping: [{{val| 1 0 0 -5 -13 -8 }}, {{val| 0 1 0 2 6 3 }}, {{val| 0 0 1 2 3 3 }}]
-POTE generators: ~3/2 = 700.4006, ~5/4 = 381.4025
+Optimal tuning (POTE): ~3/2 = 700.4006, ~5/4 = 381.4025
-Vals: {{Val list| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
+Optimal GPV sequence: {{Val list| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
 Badness: 0.736 × 10<sup>-3</sup>
@@ Line 467: / Line 469: @@
 Mapping: [{{val| 1 0 0 -5 -13 -23 }}, {{val| 0 1 0 2 6 11 }}, {{val| 0 0 1 2 3 4 }}]
-POTE generators: ~3/2 = 700.3407, ~5/4 = 383.2592
+Optimal tuning (POTE): ~3/2 = 700.3407, ~5/4 = 383.2592
-Vals: {{Val list| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
+Optimal GPV sequence: {{Val list| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
 Badness: 0.900 × 10<sup>-3</sup>
@@ Line 480: / Line 482: @@
 Mapping: [{{val| 1 0 0 -5 -13 -4 }}, {{val| 0 1 0 2 6 -1 }}, {{val| 0 0 1 2 3 4 }}]
-POTE generators: ~3/2 = 699.4864, ~5/4 = 384.0998
+Optimal tuning (POTE): ~3/2 = 699.4864, ~5/4 = 384.0998
-Vals: {{Val list| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
+Optimal GPV sequence: {{Val list| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
 Badness: 0.889 × 10<sup>-3</sup>
@@ Line 493: / Line 495: @@
 Mapping: [{{val| 1 0 0 -5 -13 21 }}, {{val| 0 1 0 2 6 -8 }}, {{val| 0 0 1 2 3 -2 }}]
-POTE generators: ~3/2 = 699.5536, ~5/4 = 383.5696
+Optimal tuning (POTE): ~3/2 = 699.5536, ~5/4 = 383.5696
-Vals: {{val list| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
+Optimal GPV sequence: {{val list| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
 Badness: 0.953 × 10<sup>-3</sup>
@@ Line 506: / Line 508: @@
 Mapping: [{{val| 1 0 0 -5 -13 21 12 }}, {{val| 0 1 0 2 6 -8 -5 }}, {{val| 0 0 1 2 3 -2 0 }}]
-POTE generators: ~3/2 = 699.6262, ~5/4 = 383.4458
+Optimal tuning (POTE): ~3/2 = 699.6262, ~5/4 = 383.4458
-Vals: {{val list| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
+Optimal GPV sequence: {{val list| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
 Badness: 0.772 × 10<sup>-3</sup>
 == Fantastic ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 4375/4356
@@ Line 521: / Line 523: @@
 Mapping generators: ~99/70, ~3, ~5
-[[POTE generator]]s: ~3/2 = 700.6242, ~5/4 = 383.2978
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.6242, ~5/4 = 383.2978
 {{Val list|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
@@ Line 528: / Line 530: @@
 == Spectacle ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 243/242
@@ Line 536: / Line 538: @@
 Mapping generators: ~2, ~11/9, ~5
-[[POTE generator]]s: ~11/9 = 350.0570, ~5/4 = 383.9323
+[[Optimal tuning]] ([[POTE]]): ~11/9 = 350.0570, ~5/4 = 383.9323
 [[Minimax tuning]]:
@@ Line 562: / Line 564: @@
 Mapping generators: ~2, ~11/9, ~5
-POTE generators: ~11/9 = 349.9247, ~5/4 = 384.3505
+Optimal tuning (POTE): ~11/9 = 349.9247, ~5/4 = 384.3505
-Vals: {{Val list| 31, 72, 103, 175f }} <nowiki>*</nowiki>
+Optimal GPV sequence: {{Val list| 31, 72, 103, 175f }} <nowiki>*</nowiki>
 <nowiki>*</nowiki> optimal patent val: [[240edo|240]]
@@ Line 571: / Line 573: @@
 == Hestia ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 125000/124509
@@ Line 579: / Line 581: @@
 Mapping generators: ~2, ~400/231, ~5
-[[POTE generator]]s: ~231/200 = 249.8526, ~5/4 = 383.6467
+[[Optimal tuning]] ([[POTE]]): ~231/200 = 249.8526, ~5/4 = 383.6467
 {{Val list|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}
@@ Line 594: / Line 596: @@
 Mapping generators: ~2, ~26/15, ~5
-POTE generators: ~15/13 = 249.7651, ~5/4 = 383.5558
+Optimal tuning (POTE): ~15/13 = 249.7651, ~5/4 = 383.5558
-Vals: {{Val list| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
+Optimal GPV sequence: {{Val list| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
 Badness: 1.062 × 10<sup>-3</sup>
 == Artemis ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 121/120, 225/224
@@ Line 609: / Line 611: @@
 Mapping generators: ~2, ~3, ~11/10
-[[POTE generator]]s: ~3/2 = 699.8719, ~11/10 = 158.3232
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 699.8719, ~11/10 = 158.3232
 {{Val list|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
@@ Line 622: / Line 624: @@
 Mapping generators: ~2, ~3, ~11/10
-POTE generators: ~3/2 = 698.7090, ~11/10 = 158.7117
+Optimal tuning (POTE): ~3/2 = 698.7090, ~11/10 = 158.7117
-Vals: {{Val list| 9, 20, 22f, 29, 31 }}
+Optimal GPV sequence: {{Val list| 9, 20, 22f, 29, 31 }}
 === Diana ===
@@ Line 635: / Line 637: @@
 Mapping generators: ~2, ~3, ~11/10
-POTE generators: ~3/2 = 700.9789, ~11/10 = 159.0048
+Optimal tuning (POTE): ~3/2 = 700.9789, ~11/10 = 159.0048
-Vals: {{Val list| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
+Optimal GPV sequence: {{Val list| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
 == Morfil ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 1331/1323
@@ Line 648: / Line 650: @@
 Mapping generators: ~2, ~3, ~84/55
-[[POTE generator]]s: ~3/2 = 700.8983, ~84/55 = 739.3812
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 700.8983, ~84/55 = 739.3812
 {{Val list|legend=1| 29, 31, 60e, 91e, 94, 125 }}
@@ Line 655: / Line 657: @@
 == Catakleismoid ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 225/224, 4375/4374
@@ Line 663: / Line 665: @@
 Mapping generators: ~2, ~6/5, ~11
-[[POTE generator]]s: ~6/5 = 316.7318, ~11/8 = 549.2528
+[[Optimal tuning]] ([[POTE]]): ~6/5 = 316.7318, ~11/8 = 549.2528
 {{Val list|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
@@ Line 678: / Line 680: @@
 Mapping generators: ~2, ~6/5, ~11
-POTE generators: ~6/5 = 316.7410, ~11/8 = 548.6028
+Optimal tuning (POTE): ~6/5 = 316.7410, ~11/8 = 548.6028
-Vals: {{Val list| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
+Optimal GPV sequence: {{Val list| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
 Badness: 0.916 × 10<sup>-3</sup>
@@ Line 693: / Line 695: @@
 Mapping generators: ~2, ~15/14, ~13
-POTE generators: ~15/14 = 116.6327, ~13/8 = 837.7040
+Optimal tuning (POTE): ~15/14 = 116.6327, ~13/8 = 837.7040
-Vals: {{Val list| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
+Optimal GPV sequence: {{Val list| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
 Badness: 0.738 × 10<sup>-3</sup>

Marvel family: Difference between revisions

Revision as of 14:20, 12 September 2022

Marvel

Overview to extensions

Undecimal marvel (unimarv)

13-limit

Hecate

17-limit

Enodia

Marvell

Isis

Deecee

Tripod

Marvelcat

Minerva

Athene

Apollo

13-limit

Potassium

13-limit

Malcolm

13-limit

Prodigy

13-limit

Prodigious

Prodigal

Protannic

17-limit

Fantastic

Spectacle

13-limit

Hestia

13-limit

Artemis

13-limit

Diana

Morfil

Catakleismoid

13-limit

Mirage

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