Marvel temperaments

This page discusses some of the temperaments tempering out [-5 2 2 -1⟩ = 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila.

Since (5/4)² = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.

The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.

Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

Negri

Main article: Negri

Negri tempers out the negri comma in the 5-limit, 49/48 and 225/224 in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 to the comma list; this will be discussed below under the title of negra.

5-limit

Period: 1\1

Optimal (POTE) generator: ~16/15 = 125.7549

EDO generators: 1\9, 1\10, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

Wedgie: ⟨⟨4 -3 -14]]

7-limit

Period: 1\1

Optimal (POTE) generator: ~15/14 = 125.608

EDO generators: 1\9, 1\10, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]

Wedgie: ⟨⟨4 -3 2 -14 -8 13]]

Negra

This is the 2.3.5.7.13 extension of negri.

Period: 1\1

Optimal (POTE) generator: ~14/13 = 125.567

EDO generators: 1\9, 1\10, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.13

Comma list: 49/48, 65/64, 91/90

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

Gencom map: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3]]

Gencom: [2 14/13; 49/48 65/64 91/90]

11-limit

Period: 1\1

Optimal (POTE) generator: ~15/14 = 126.474

EDO generators: 1\9, 1\10, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 56/55

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

Badness: 0.0262

13-limit

Period: 1\1

Optimal (POTE) generator: ~14/13 = 126.431

EDO generators: 1\9, 1\10, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 56/55, 78/77

POTE generator: ~14/13 = 126.431

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

Negril

Period: 1\1

Optimal (POTE) generator: ~15/14 = 124.767

EDO generators: 2\19, 3\29

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 225/224

Mapping: [⟨1 2 2 3 2], ⟨0 -4 3 -2 14]]

Badness: 0.0387

13-limit

Period: 1\1

Optimal (POTE) generator: ~14/13 = 124.716

EDO generators: 2\19, 3\29

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90, 875/858

Mapping: [⟨1 2 2 3 2 4], ⟨0 -4 3 -2 14 -3]]

Badness: 0.0244

Negric

Period: 1\1

Optimal (POTE) generator: ~15/14 = 127.039

EDO generators: 1\9, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 33/32, 49/48, 77/75

Mapping: [⟨1 2 2 3 3], ⟨0 -4 3 -2 4]]

Badness: 0.0306

13-limit

Period: 1\1

Optimal (POTE) generator: ~14/13 = 127.039

EDO generators: 1\9, 2\19

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 49/48, 65/64, 91/90

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

Badness: 0.0202

Negroni

Period: 1\1

Optimal (POTE) generator: ~15/14 = 124.539

EDO generators: 1\10, 2\19, 3\29

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 225/224

Mapping: [⟨1 2 2 3 5], ⟨0 -4 3 -2 -15]]

Badness: 0.0353

13-limit

Period: 1\1

Optimal (POTE) generator: ~14/13 = 124.545

EDO generators: 1\10, 2\19, 3\29

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 65/64, 91/90

Mapping: [⟨1 2 2 3 5 4], ⟨0 -4 3 -2 -15 -3]]

Badness: 0.0216

Wilsec

Period: 1\1

Optimal (POTE) generator: ~11/8 = 537.186

EDO generators: 13\29, 17\38

Scales (Scala files):

Technical data

Comma list: 49/48, 121/120, 225/224

Mapping: [⟨1 6 -1 5 4], ⟨0 -8 6 -4 -1]]

Badness: 0.0419

13-limit

Period: 1\1

Optimal (POTE) generator: ~11/8 = 537.208

EDO generators: 13\29, 17\38

Scales (Scala files):

Technical data

Comma list: 49/48, 65/64, 91/90, 121/120

Mapping: [⟨1 6 -1 5 4 7], ⟨0 -8 6 -4 -1 -6]]

Badness: 0.0252

17-limit

Period: 1\1

Optimal (POTE) generator: ~11/8 = 537.230

EDO generators: 13\29, 17\38

Scales (Scala files):

Technical data

Comma list: 49/48, 65/64, 91/90, 121/120, 154/153

Mapping: [⟨1 6 -1 5 4 7 -2], ⟨0 -8 6 -4 -1 -6 11]]

Badness: 0.0218

19-limit

Period: 1\1

Optimal (POTE) generator: ~11/8 = 537.214

EDO generators: 13\29, 17\38

Scales (Scala files):

Technical data

Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153

Mapping: [⟨1 6 -1 5 4 7 -2 7], ⟨0 -8 6 -4 -1 -6 11 -5]]

Badness: 0.0168

Passive

See also: Archytas clan § Passion

Comma list: 225/224, 256/245

POTE generator: ~16/15 = 98.809

Mapping: [⟨1 2 2 3], ⟨0 -5 4 -2]]

Badness: 0.0751

Wizard

5-limit

Comma list: 2197265625/2147483648

POTE generator: ~5/4 = 383.212

Mapping: [⟨2 1 5], ⟨0 6 -1]]

Badness: 0.3864

7-limit

Comma list: 225/224, 118098/117649

POTE generator: ~5/4 = 383.256

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

Wedgie: ⟨⟨12 -2 20 -31 -2 52]]

Badness: 0.0408

11-limit

Comma list: 225/224, 385/384, 4000/3993

POTE generator: ~5/4 = 383.232

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Badness: 0.0185

Lizard

Comma list: 225/224, 351/350, 364/363, 385/384

POTE generator: ~5/4 = 383.389

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Badness: 0.0218

17-limit

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

POTE generator: ~5/4 = 383.381

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Badness: 0.0145

19-limit

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

POTE generator: ~5/4 = 383.477

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Badness: 0.0157

Gizzard

Comma list: 225/224, 385/384, 325/324, 1573/1568

POTE generator: ~5/4 = 383.170

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Badness: 0.0203

17-limit

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

POTE generator: ~5/4 = 383.175

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Badness: 0.0136

19-limit

Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399

POTE generator: ~5/4 = 383.138

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Badness: 0.0148

Mage

Comma list: 99/98, 176/175, 1331/1296

POTE generator: ~55/48 = 216.876

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Badness: 0.0578

Triton

Comma list: 225/224, 1029/1000

POTE generator: ~7/5 = 568.865

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

Wedgie: ⟨⟨3 -7 -8 -18 -21 1]]

Badness: 0.0592

11-limit

Comma list: 45/44, 56/55, 1029/1000

POTE generator: ~7/5 = 569.144

Mapping: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]

Badness: 0.0457

Tritonic

5-limit

Comma list: 553584375/536870912

POTE generator: ~45/32 = 580.219

Mapping: [⟨1 4 -3], ⟨0 -5 11]]

Badness: 0.3785

7-limit

Comma list: 225/224, 50421/50000

POTE generator: ~7/5 = 580.286

Mapping: [⟨1 4 -3 -3], ⟨0 -5 11 12]]

Wedgie: ⟨⟨5 -11 -12 -29 -33 3]]

11-limit

Comma list: 121/120, 225/224, 441/440

POTE generator: ~7/5 = 580.267

Mapping: [⟨1 4 -3 -3 2], ⟨0 -5 11 12 3]]

Badness: 0.0237

13-limit

Comma list: 105/104, 121/120, 196/195, 275/273

POTE generator: ~7/5 = 580.108

Mapping: [⟨1 4 -3 -3 2 -5], ⟨0 -5 11 12 3 18]]

Badness: 0.0230

Tritoni

Comma list: 225/224, 385/384, 27783/27500

POTE generator: ~7/5 = 580.389

Mapping: [⟨1 4 -3 -3 17], ⟨0 -5 11 12 -28]]

Badness: 0.0455

Merman

Comma list: 1121008359375/1099511627776

POTE generator: ~45/32 = 585.634

Mapping: [⟨1 5 -5], ⟨0 -7 15]]

Badness: 0.4538

7-limit

Comma list: 225/224, 2500000/2470629

POTE generator: ~7/5 = 585.585

Mapping: [⟨1 5 -5 -5], ⟨0 -7 15 16]]

Wedgie: ⟨⟨7 -15 -16 -40 -45 5]]

Badness: 0.0551

11-limit

Comma list: 225/224, 441/440, 1344/1331

POTE generator: ~7/5 = 585.606

Mapping: [⟨1 5 -5 -5 2], ⟨0 -7 15 16 3]]

Badness: 0.0364

13-limit

Comma list: 144/143, 225/224, 364/363, 441/440

POTE generator: ~7/5 = 585.657

Mapping: [⟨1 5 -5 -5 2 12], ⟨0 -7 15 16 3 -17]]

Badness: 0.0275

Septimin

Comma list: 35595703125/34359738368

POTE generator: ~75/64 = 263.658

Mapping: [⟨1 4 1], ⟨0 -11 6]]

Badness: 0.6709

7-limit

Comma list: 225/224, 84035/82944

POTE generator: ~7/6 = 263.632

Mapping: [⟨1 4 1 5], ⟨0 -11 6 -10]]

Wedgie: ⟨⟨11 -6 10 -35 -15 40]]

11-limit

Comma list: 225/224, 385/384, 2401/2376

POTE generator: ~7/6 = 263.634

Mapping: [⟨1 4 1 5 5], ⟨0 -11 6 -10 -7]]

13-limit

Comma list: 105/104, 144/143, 196/195, 245/242

POTE generator: ~7/6 = 263.700

Mapping: [⟨1 4 1 5 5 7], ⟨0 -11 6 -10 -7 -15]]

Slender

Comma list: 225/224, 589824/588245

POTE generator: ~49/48 = 38.413

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

Wedgie: ⟨⟨13 -10 6 -46 -27 42]]

Badness: 0.0569

11-limit

Comma list: 225/224, 385/384, 1331/1323

POTE generator: ~49/48 = 38.387

Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]

Badness: 25.342

13-limit

Comma list: 225/224, 275/273, 385/384, 1331/1323

POTE generator: ~49/48 = 38.314

Mapping: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]

Badness: 25.913

Marvo

See also: Gravity family

Comma list: 225/224, 78125000/78121827

POTE generator: ~27/20 = 516.694

Mapping: [⟨1 5 12 29], ⟨0 -6 -17 -46]]

Wedgie: ⟨⟨6 17 46 13 56 59]]

Badness: 0.0976

11-limit

Comma list: 225/224, 243/242, 4000/3993

POTE generator: ~27/20 = 516.699

Mapping: [⟨1 5 12 29 12], ⟨0 -6 -17 -46 -15]]

Badness: 0.0317

13-limit

Comma list: 225/224, 243/242, 351/350, 1625/1617

POTE generator: ~27/20 = 516.730

Mapping: [⟨1 5 12 29 12 39], ⟨0 -6 -17 -46 -15 -62]]

Badness: 0.0269

Marvolo

Comma list: 225/224, 156250000/155649627

POTE generator: ~21/20 = 83.348

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

Wedgie: ⟨⟨6 -19 -26 -44 -58 -7]]

Badness: 0.0833

11-limit

Comma list: 225/224, 441/440, 4000/3993

POTE generator: ~21/20 = 83.340

Mapping: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]

Badness: 0.0290

13-limit

Comma list: 169/168, 225/224, 364/363, 441/440

POTE generator: ~21/20 = 83.330

Mapping: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]

Badness: 0.0215

Amavil

5-limit (mabila)

Comma list: 268435456/263671875

POTE generator: ~512/375 = 529.6849

Mapping: [⟨1 6 1], ⟨0 -10 3]]

Badness: 0.2325

7-limit

Comma list: 225/224, 17496/16807

POTE generator: ~48/35 = 529.979

Mapping: [⟨1 6 1 9], ⟨0 -10 3 -14]]

Wedgie: ⟨⟨10 -3 14 -28 -6 41]]

Badness: 0.1096

11-limit

Comma list: 99/98, 176/175, 864/847

POTE generator: ~15/11 = 529.974

Mapping: [⟨1 6 1 9 7], ⟨0 -10 3 -14 -8]]

Badness: 0.0426

13-limit

Comma list: 78/77, 99/98, 144/143, 176/175

POTE generator: ~15/11 = 529.951

Mapping: [⟨1 6 1 9 7 9], ⟨0 -10 3 -14 -8 -12]]

Badness: 0.0258

Enneaportent

Comma list: 225/224, 40353607/40310784

POTE generator: ~5/4 = 383.165

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

Wedgie: ⟨⟨18 -9 18 -56 -22 67]]

Badness: 0.0937

11-limit

Comma list: 225/224, 385/384, 12005/11979

POTE generator: ~5/4 = 383.146

Mapping: [⟨9 0 28 11 24], ⟨0 2 -1 2 1]]

Badness: 0.0304

13-limit

Comma list: 169/168, 225/224, 364/363, 1716/1715

POTE generator: ~5/4 = 383.047

Mapping: [⟨9 0 28 11 24 19], ⟨0 2 -1 2 1 2]]

Badness: 0.0223

Submajor

5-limit

Comma list: 69198046875/68719476736

POTE generator: ~10125/8192 = 362.321

Mapping: [⟨1 4 -1], ⟨0 -8 11]]

Badness: 0.1302

7-limit

Comma list: 225/224, 51200/50421

POTE generator: ~49/40 = 362.255

Mapping: [⟨1 4 -1 1], ⟨0 -8 11 6]]

Wedgie: ⟨⟨8 -11 -6 -36 -32 17]]

Badness: 0.0605

11-limit

Comma list: 225/224, 385/384, 6655/6561

POTE generator: ~27/22 = 362.101

Mapping: [⟨1 4 -1 1 11], ⟨0 -8 11 6 -25]]

Badness: 0.0506

13-limit

Comma list: 169/168, 225/224, 275/273, 385/384

POTE generator: ~16/13 = 362.105

Mapping: [⟨1 4 -1 1 11 4], ⟨0 -8 11 6 -25 -1]]

Badness: 0.0277

Interpental

Comma list: 99/98, 176/175, 51200/50421

POTE generator: ~49/40 = 362.418

Mapping: [⟨1 4 -1 1 -5], ⟨0 -8 11 6 28]]

Badness: 0.0518

13-limit

Comma list: 99/98, 169/168, 176/175, 640/637

POTE generator: ~16/13 = 362.402

Mapping: [⟨1 4 -1 1 -5 4], ⟨0 -8 11 6 28 -1]]

Badness: 0.0297

Alphorn

Comma list: 225/224, 5764801/5668704

POTE generator: ~48/35 = 556.221

Mapping: [⟨1 9 0 13], ⟨0 -16 5 -22]]

Wedgie: ⟨⟨16 -5 22 -45 -10 65]]

Badness: 0.1293

11-limit

Comma list: 225/224, 385/384, 12250/11979

POTE generator: ~11/8 = 556.144

Mapping: [⟨1 9 0 13 3], ⟨0 -16 5 -22 1]]

Badness: 0.0735