# Marvel family

(Redirected from Apollo)

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

[[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
[[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

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

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

• 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

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

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

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

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: 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg

#### 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: 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg

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

### 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: 10, 19f, 22, 31, 41, 53, 94

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

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

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

## 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

Projection pairs: 7 225/32 11 5625/512

Scales (Scala files): 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 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: 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def

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

Projection pairs: 7 225/32 11 100/9

### 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: 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 tuning (POTE): ~3/2 = 696.1714, ~5/4 = 385.0500

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: 9, 10, 12f, 19, 31e, 50e

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

### 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: 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef

## 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

[[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

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

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

### 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: 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff

### 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: 12f, 19e, 31, 53e, 60eff, 72, 103, 175f

### 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: 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff

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

## 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

## 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

[[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

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

Scales (Scala files): spectacle31

### 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: 31, 72, 103, 175f *

* optimal patent val: 240

## 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): ~400/231 = 950.1474, ~5/4 = 383.6467

### 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): ~26/15 = 950.2349, ~5/4 = 383.5558

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

## 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

### 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: 9, 20, 22f, 29, 31

### 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: 22, 29, 31, 53, 82e, 84e, 113e, 166ee

## 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

## 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

### 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: 19, 34d, 53, 72, 125f, 197ef, 269cef

## 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: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef