Marvel family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The marvel family of rank-3 temperaments tempers out the 7-limit marvel comma (ratio: 225/224, monzo: [-5 2 2 -1⟩), 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

Main article: 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:

~15/14 length = 1.256, ~3/2 length = 1.369

angle (~15/14, ~3/2) = 106.958°

Optimal tunings:

• WE: ~2 = 1200.5971 ¢, ~3/2 = 700.7560 ¢, ~5/4 = 383.8285 ¢

error map: ⟨+0.597 -0.602 -1.291 +0.940]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6222 ¢, ~5/4 = 383.8540 ¢

error map: ⟨0.000 -1.333 -2.460 +0.127]

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⟩]

unchanged-interval (eigenmonzo) 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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.7

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

Badness (Sintel): 0.161

Projection pairs: 7 225/32

Scales: marvel9, marvel10, marvel11, marvel12, marvel19, marvel22, pump12_1, pump12_2, pump13, pump14, pump15, pump16, pump17, pump18, marvel wholetone

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
• Marvelpine (+4000/3993) → Wizardharry clan
• Mirage (+243/242, +385/384) → Rastmic rank-3 clan
• Catakleismoid (+4375/4374) → Kleismic rank-3 family

Undecimal marvel

Main article: Marvel

Undecimal marvel tempers out 385/384 as well as 540/539, and is loosely associated with wizard. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). 197edo remains useful as a tuning, with the 197e val, but 166edo, which among other things has a virtually pure 7, works as well.

In the 13-limit, 225/224 factors as (351/350)⋅(625/624) or (325/324)⋅(729/728). Tempering out 351/350 and 625/624 leads to helios, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of catakleismic.

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:

~15/14 length = 1.0364, ~5/4 length = 1.0759

angle (~15/14, ~5/4) = 104.028°

Optimal tunings:

• WE: ~2 = 1200.6395 ¢, ~3/2 = 700.7619 ¢, ~5/4 = 383.7447 ¢

error map: ⟨+0.639 -0.554 -1.290 +0.827 -0.117]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6048 ¢, ~5/4 = 383.4538 ¢

error map: ⟨0.000 -1.350 -2.860 -0.709 -2.284]

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⟩]

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

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

Badness (Sintel): 0.306

Projection pairs: 7 225/32 11 4096/375

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

Helios

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:

• WE: ~2 = 1200.8043 ¢, ~3/2 = 700.2057 ¢, ~5/4 = 384.3188 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.8109 ¢, ~5/4 = 384.1177 ¢

Minimax tuning:

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

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

Badness (Sintel): 0.645

Hecate

Hecate tempers out 325/324, the marveltwin comma, such that 16/13 is found by a stack of two ~10/9's, similar to how 8/7 is found by a stack of two 15/14~16/15's. Hecate has a natural extension to include prime 19, where it further tempers out 400/399 and 513/512, taking advantage of the factorization 225/224 = (400/399)⋅(513/512). For both of these cases 166edo remains an excellent tuning.

13-limit

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:

• WE: ~2 = 1200.5788 ¢, ~3/2 = 701.3161 ¢, ~5/4 = 383.3471 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1003 ¢, ~5/4 = 383.1315 ¢

Minimax tuning:

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

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

Badness (Sintel): 0.674

Projection pairs: 7 225/32 11 4096/375 13 324/25

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

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

Optimal tunings:

• WE: ~2 = 1200.4716 ¢, ~3/2 = 701.4395 ¢, ~5/4 = 383.2779 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2002 ¢, ~5/4 = 383.1136 ¢

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

Badness (Sintel): 0.756

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:

• WE: ~2 = 1200.6057 ¢, ~3/2 = 701.3157 ¢, ~5/4 = 383.2243 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0877 ¢, ~5/4 = 382.9709 ¢

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

Badness (Sintel): 0.827

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:

• WE: ~2 = 1200.4927 ¢, ~3/2 = 701.4506 ¢, ~5/4 = 383.1291 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2007 ¢, ~5/4 = 382.9404 ¢

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

Badness (Sintel): 1.01

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:

• WE: ~2 = 1200.6165 ¢, ~3/2 = 701.3259 ¢, ~5/4 = 383.5032 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0949 ¢, ~5/4 = 383.3140 ¢

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

Badness (Sintel): 0.872

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:

• WE: ~2 = 1200.5038 ¢, ~3/2 = 701.4654 ¢, ~5/4 = 383.4549 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2113 ¢, ~5/4 = 383.3052 ¢

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

Badness (Sintel): 1.05

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:

• WE: ~2 = 1200.6404 ¢, ~3/2 = 700.7675 ¢, ~5/4 = 383.7772 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6113 ¢, ~5/4 = 383.4925 ¢

Minimax tuning:

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

Optimal ET sequence: 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 341cef

Badness (Sintel): 0.806

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:

• WE: ~2 = 1200.1824 ¢, ~3/2 = 702.0223 ¢, ~5/4 = 383.3028 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9276 ¢, ~5/4 = 383.2283 ¢

Optimal ET sequence: 10, 19f, 22, 31, 41, 53, 84e, 94

Badness (Sintel): 0.810

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:

• WE: ~2 = 1200.7533 ¢, ~3/2 = 700.8957 ¢, ~5/4 = 383.0580 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.7184 ¢, ~5/4 = 382.6611 ¢

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 (Sintel): 0.861

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:

• WE: ~2 = 1200.4789 ¢, ~3/2 = 699.5125 ¢, ~5/4 = 383.1304 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.4041 ¢, ~5/4 = 382.9166 ¢

Minimax tuning:

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

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

Badness (Sintel): 0.697

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:

• WE: ~2 = 1200.7720 ¢, ~3/2 = 950.8976 ¢, ~5/4 = 383.8283 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.4265 ¢, ~5/4 = 383.4790 ¢

Optimal ET sequence: 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef

Badness (Sintel): 0.935

Prodigy

Prodigy tempers out 441/440 and shrinks 243/242, 384/385, 1024/1029 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:

~15/14 length = 0.9111, ~3/2 length = 0.9477

angle (~15/14, ~3/2) = 65.933°

Optimal tunings:

• WE: ~2 = 1200.7854 ¢, ~3/2 = 700.2562 ¢, ~5/4 = 383.7624 ¢

error map: ⟨+0.785 -0.913 -0.980 -0.003 +0.721]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.8610 ¢, ~5/4 = 383.7724 ¢

error map: ⟨0.000 -2.094 -2.541 -1.559 -0.835]

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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.11

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

Badness (Sintel): 0.402

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

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:

• WE: ~2 = 1200.8252 ¢, ~3/2 = 700.8823 ¢, ~5/4 = 381.6647 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4689 ¢, ~5/4 = 381.6687 ¢

Optimal ET sequence: 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd

Badness (Sintel): 0.689

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:

• WE: ~2 = 1200.6284 ¢, ~3/2 = 700.7075 ¢, ~5/4 = 383.4599 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3302 ¢, ~5/4 = 383.5030 ¢

Optimal ET sequence: 12f, 29, 31f, 41, 72, 185cf, 257cff

Badness (Sintel): 0.841

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:

• WE: ~2 = 1200.7798 ¢, ~3/2 = 699.9410 ¢, ~5/4 = 384.3494 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.5538 ¢, ~5/4 = 384.3496 ¢

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

Badness (Sintel): 0.831

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:

• WE: ~2 = 1200.9450 ¢, ~3/2 = 700.1045 ¢, ~5/4 = 383.8716 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.4224 ¢, ~5/4 = 383.9828 ¢

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

Badness (Sintel): 0.891

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:

• WE: ~2 = 1200.9342 ¢, ~3/2 = 700.1708 ¢, ~5/4 = 383.7444 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.4849 ¢, ~5/4 = 383.8744 ¢

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

Badness (Sintel): 0.734

Minerva

Minerva tempers out 99/98 as well as 176/175. It may be described as 12 & 22 & 31, and is loosely associated with würschmidt.

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°

Optimal tunings:

• WE: ~2 = 1200.1086 ¢, ~3/2 = 700.3226 ¢, ~5/4 = 386.5931 ¢

error map: ⟨+0.109 -1.524 +0.497 +5.114 -4.192]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3006 ¢, ~5/4 = 386.5785 ¢

error map: ⟨0.000 -1.654 +0.265 +4.932 -4.403]

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

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

Badness (Sintel): 0.458

Projection pairs: 7 225/32 11 5625/512

Scales: minerva12, minerva22x

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:

• WE: ~2 = 1199.9127 ¢, ~3/2 = 701.1832 ¢, ~5/4 = 385.9313 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2143 ¢, ~5/4 = 385.9336 ¢

Minimax tuning:

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

Optimal ET sequence: 12f, 19e, 22, 31, 53, 84e, 118d

Badness (Sintel): 0.765

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

Apollo

See also: Ptolemismic clan § Apollo

Lattice for apollo.

Apollo tempers out not only 100/99 but 896/891. Note that marvel tempers together 25/24 and 28/27, and apollo further equates it with 33/32 via the vanishing of 100/99. This makes it a weak extension of parapyth, and associates it with magic. The lattice structure is very compact, comparable to that of ares, from which apollo only differs in the mapping of prime 7.

The canonical 13-limit extension is implied by parapyth, tempering out 352/351 and 364/363, but there are a number of other extenions to consider, these being called phoebus and musagetes, after epithets of Apollo. Phoebus tempers out 105/104 and finds ~16/13 as a stack of three secors. Musagetes tempers out 144/143 and conflates 16/13 and 11/9, which in this case is simply a stack of two ~10/9's. These extensions are supported by 13-limit magic, unlike the canonical one.

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:

• WE: ~2 = 1199.8250 ¢, ~3/2 = 703.3820 ¢, ~5/4 = 381.5476 ¢

error map: ⟨-0.175 +1.252 -5.116 +0.858 +4.313]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4612 ¢, ~5/4 = 381.5071 ¢

error map: ⟨0.000 +1.506 -4.807 +1.111 +4.774]

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

Optimal ET sequence: 12, 19, 22, 34d, 41, 104

Badness (Sintel): 0.637

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:

• WE: ~2 = 1199.6919 ¢, ~3/2 = 703.8176 ¢, ~5/4 = 381.4372 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9853 ¢, ~5/4 = 381.3579 ¢

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

Optimal ET sequence: 12f, 19f, 22, 29, 34d, 41, 63, 104

Badness (Sintel): 0.964

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

Phoebus

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195

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

Optimal tunings:

• WE: ~2 = 1200.3724 ¢, ~3/2 = 702.5274 ¢, ~5/4 = 379.6345 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3357 ¢, ~5/4 = 379.6830 ¢

Optimal ET sequence: 12f, 19, 22f, 29, 41

Badness (Sintel): 0.886

Musagetes

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 225/224

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

Optimal tunings:

• WE: ~2 = 1199.2695 ¢, ~3/2 = 702.5589 ¢, ~5/4 = 382.7715 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7998 ¢, ~5/4 = 382.7740 ¢

Optimal ET sequence: 19, 22f, 34d, 41, 75e, 94e, 116ef

Badness (Sintel): 1.14

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:

• WE: ~2 = 1199.7106 ¢, ~3/2 = 696.0036 ¢, ~5/4 = 384.9572 ¢

error map: ⟨-0.289 -6.241 -1.935 -7.194 +25.068]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.0586 ¢, ~5/4 = 384.9472 ¢

error map: ⟨0.000 -5.896 -1.367 -6.814 +25.746]

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

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

Badness (Sintel): 0.557

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:

• WE: ~2 = 1199.8192 ¢, ~3/2 = 695.9054 ¢, ~5/4 = 384.6205 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 695.9480 ¢, ~5/4 = 384.6372 ¢

Minimax tuning:

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

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

Badness (Sintel): 0.686

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:

• WE: ~2 = 1199.5261 ¢, ~3/2 = 702.1990 ¢, ~5/4 = 382.5759 ¢

error map: ⟨+0.526 +0.770 -2.686 +1.250 -1.077]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0535 ¢, ~5/4 = 382.6222 ¢

error map: ⟨0.000 +0.098 -3.692 +0.525 -2.188]

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

Badness (Sintel): 1.50

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:

• WE: ~2 = 1200.2427 ¢, ~3/2 = 702.1575 ¢, ~5/4 = 383.0704 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0882 ¢, ~5/4 = 383.0630 ¢

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

Badness (Sintel): 1.00

Scales: malco

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 tunings:

• WE: ~99/70 = 600.2896 ¢, ~3/2 = 700.9624 ¢, ~5/4 = 383.4829 ¢

error map: ⟨+0.579 -0.413 -1.672 +0.644 +0.579]

• CWE: ~99/70 = 600.0000 ¢, ~3/2 = 700.8160 ¢, ~5/4 = 383.5350 ¢

error map: ⟨0.000 -1.139 -2.779 -0.124 -0.713]

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

Badness (Sintel): 0.893

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:

• WE: ~2 = 1200.6417 ¢, ~400/231 = 950.6555 ¢, ~5/4 = 383.8518 ¢

error map: ⟨+0.642 -0.644 -1.179 +0.858 -0.131]

• CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.1211 ¢, ~5/4 = 383.9530 ¢

error map: ⟨0.000 -1.712 -2.361 -0.436 -2.165]

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

Badness (Sintel): 1.845

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:

• WE: ~2 = 1200.8238 ¢, ~26/15 = 950.8873 ¢, ~5/4 = 383.8191 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.2108 ¢, ~5/4 = 383.9486 ¢

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

Badness (Sintel): 0.993

Morfil

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323

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

mapping generators: ~2, ~3, ~55/42

Optimal tunings:

• WE: ~2 = 1200.5931 ¢, ~3/2 = 701.2447 ¢, ~55/42 = 460.8465 ¢

error map: ⟨+0.593 -0.117 -1.285 +1.942 -2.302]

• CWE: ~2 = 1200.000 ¢, ~3/2 = 701.1108 ¢, ~55/42 = 460.5488 ¢

error map: ⟨0.000 -0.844 -2.446 +1.131 -3.569]

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

Badness (Sintel): 1.38

Subgroup extensions

Char

Subgroup: 2.3.5.7.17

Comma list: 120/119, 225/224

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

Optimal tunings:

• WE: ~2 = 1199.3681 ¢, ~3/2 = 701.3669 ¢, ~5/4 = 384.9641 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6286 ¢, ~5/4 = 385.0841 ¢

Optimal ET sequence: 9, 10, 12, 19, 22, 31, 41, 53

Badness (Sintel): 0.381

Devimarvel

Subgroup: 2.3.5.7.19

Comma list: 225/224, 400/399

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

Optimal tunings:

• WE: ~2 = 1200.3454 ¢, ~3/2 = 701.1728 ¢, ~5/4 = 383.7499 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9546 ¢, ~5/4 = 383.7912 ¢

Optimal ET sequence: 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c

Badness (Sintel): 0.310