Marvel temperaments

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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

Commas: 49/48, 225/224

POTE temperament: ~15/14 = 125.608

Map: [<1 2 2 3|, <0 -4 3 -2|]

Wedgie: <<4 -3 2 -14 -8 13||

EDOs: 9, 10, 19, 86

11-limit

Commas: 45/44, 49/48, 56/55

POTE generator: ~15/14 = 126.474

Map: [<1 2 2 3 4|, <0 -4 3 -2 -5|]

EDOs: 9, 10, 19

Badness: 0.0262

13-limit

Commas: 45/44, 49/48, 56/55, 78/77

POTE generator: ~14/13 = 126.431

Map: [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]

EDOs: 9, 10, 19

Negril

Commas: 49/48, 100/99, 225/224

POTE generator: ~15/14 = 124.767

Map: [<1 2 2 3 2|, <0 -4 3 -2 14|]

EDOs: 19, 29, 48df, 77cdf

Badness: 0.0387

13-limit

Commas: 49/48, 65/64, 91/90, 875/858

POTE generator: ~15/14 = 124.716

Map: [<1 2 2 3 2 4|, <0 -4 3 -2 14 -3|]

EDOs: 9, 10, 19, 29, 48, 77

Badness: 0.0244

Negric

Commas: 33/32, 49/48, 77/75

POTE generator: ~15/14 = 127.039

Map: [<1 2 2 3 3|, <0 -4 3 -2 4|]

EDOs: 9, 19, 85

Badness: 0.0306

13-limit

Commas: 33/32, 49/48, 65/64, 91/90

POTE generator: ~15/14 = 127.039

Map: [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|]

EDOs: 9, 19, 85

Badness: 0.0202

Negroni

Commas: 49/48, 55/54, 225/224

POTE generator: ~15/14 = 124.539

Map: [<1 2 2 3 5|, <0 -4 3 -2 -15|]

EDOs: 10, 19e, 29, 77cde

Badness: 0.0353

13-limit

Commas: 49/48, 55/54, 65/64, 91/90

POTE generator: ~15/14 = 124.545

Map: [<1 2 2 3 5 4|, <0 -4 3 -2 -15 -3|]

EDOs: 10, 19e, 29, 77cdef

Badness: 0.0216

Wizard

Comma: 2197265625/2147483648

POTE generator: ~5/4 = 383.212

Map: [<2 1 5|, <0 6 -1|]

EDOs: 22, 50, 72, 166, 238c, 310c, 548bc

Badness: 0.3864

7-limit

Commas: 225/224, 118098/117649

POTE temperament: 383.256

Map: [<2 1 5 2|, <0 6 -1 10|]

Wedgie: <<12 -2 20 -31 -2 52||

EDOs: 22, 50, 72, 94, 166, 238c, 310c, 382c

Badness: 0.0408

11-limit

Commas: 225/224, 385/384, 4000/3993

POTE temperament: 383.232

Map: [<2 1 5 2 8|, <0 6 -1 10 -3|]

EDOs: 22, 50, 72, 94, 166, 238c, 310c

Badness: 0.0185

Lizard

Commas: 225/224, 351/350, 364/363, 385/384

POTE temperament: 383.389

Map: [<2 1 5 2 8 11|, <0 6 -1 10 -3 -10|]

EDOs: 22, 50, 72, 122, 194df

Badness: 0.0218

17-limit

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

POTE temperament: 383.381

Map: [<2 1 5 2 8 11 6|, <0 6 -1 10 -3 -10 6|]

EDOs: 22, 50, 72, 122g, 194dfg

Badness: 0.0145

19-limit

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

POTE generator: ~5/4 = 383.477

Map: [<2 1 5 2 8 11 6 2|, <0 6 -1 10 -3 -10 6 18|]

EDOs: 22h, 50, 72, 122g, 194dfg

Badness: 0.0157

Gizzard

Commas: 225/224, 385/384, 325/324, 1573/1568

POTE generator: ~5/4 = 383.170

Map: [<2 1 5 2 8 -2|, <0 6 -1 10 -3 26|]

EDOs: 22, 72, 94, 166, 238cf

Badness: 0.0203

17-limit

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

POTE generator: ~5/4 = 383.175

Map: [<2 1 5 2 8 -2 6|, <0 6 -1 10 -3 26 6|]

EDOs: 22f, 72, 94, 166g, 238cfg

Badness: 0.0136

19-limit

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

POTE generator: ~5/4 = 383.138

Map: [<2 1 5 2 8 -2 6 15|, <0 6 -1 10 -3 26 6 -18|]

EDOs: 22f, 72, 94, 166g

Badness: 0.0148

Mage

Commas: 99/98, 176/175, 1331/1296

POTE generator: ~55/48 = 216.876

Map: [<2 1 5 2 4|, <0 6 -1 10 8|]

EDOs: 22, 72e, 94e

Badness: 0.0578

Triton

Commas: 225/224, 1029/1000

POTE temperament: ~7/5 = 568.865

Map: [<1 0 6 7|, <0 3 -7 -8|]

Wedgie: <<3 -7 -8 -18 -21 1||

EDOs: 17d, 19, 78bd, 97bd

Badness: 0.0592

11-limit

Commas: 45/44, 56/55, 1029/1000

POTE generator: ~7/5 = 569.144

Map: [<1 0 6 7 4|, <0 3 -7 -8 -1|]

EDOs: 19, 59bde, 78bde, 97bde

Badness: 0.0457

Tritonic

Comnma: 553584375/536870912

POTE generator: ~45/32 = 580.219

Map: [<1 4 -3|, <0 -5 11|]

EDOs: 29, 31, 60, 91

Badness: 0.3785

7-limit

Commas: 225/224, 50421/50000

POTE temperament: 580.286

Map: [<1 4 -3 -3|, <0 -5 11 12|]

Wedgie: <<5 -11 -12 -29 -33 3||

EDOs: 29, 31, 60, 91, 122, 213bcd

11-limit

Commas: 121/120, 225/224, 441/440

POTE generator: ~7/5 = 580.267

Map: [<1 4 -3 -3 2|, <0 -5 11 12 3|]

EDOs: 29, 31, 60e

Badness: 0.0237

13-limit

Commas: 105/104, 121/120, 196/195, 275/273

POTE generator: ~7/5 = 580.108

Map: [<1 4 -3 -3 2 -5|, <0 -5 11 12 3 18|]

EDOs: 29, 31, 60e, 151cde

Badness: 0.0230

Tritoni

Commas: 225/224, 385/384, 27783/27500

POTE generator: ~7/5 = 580.389

Map: [<1 4 -3 -3 17|, <0 -5 11 12 -28|]

EDOs: 31, 91, 122

Badness: 0.0455

Merman

Comma: 1121008359375/1099511627776

POTE generator: ~45/32 = 585.634

Map: [<1 5 -5|, <0 -7 15|]

EDOs: 41, 84, 125, 209, 334c, 543bc

Badness: 0.4538

7-limit

Commas: 225/224, 2500000/2470629

POTE generator: ~7/5 = 585.585

Map: [<1 5 -5 -5|, <0 -7 15 16|]

Wedgie: <<7 -15 -16 -40 -45 5||

EDOs: 41, 84, 125

Badness: 0.0551

11-limit

Commas: 225/224, 441/440, 1344/1331

POTE generator: ~7/5 = 585.606

Map: [<1 5 -5 -5 2|, <0 -7 15 16 3]]

EDOs: 41, 84, 125e

Badness: 0.0364

13-limit

Commas: 144/143, 225/224, 364/363, 441/440

POTE generator: ~7/5 = 585.657

Map: [<1 5 -5 -5 2 12|, <0 -7 15 16 3 -17|]

EDOs: 41, 84, 125e, 209ef, 293ef

Badness: 0.0275

Septimin

Comma: 35595703125/34359738368

POTE generator: ~75/64 = 263.658

Map: [<1 4 1|, <0 -11 6|]

EDOs: 9, 32, 41, 91, 132, 223c, 355bc

Badness: 0.6709

7-limit

Commas: 225/224, 84035/82944

POTE temperament: 263.632

Map: [<1 4 1 5|, <0 -11 6 -10|]

Wedgie: <<11 -6 10 -35 -15 40||

EDOs: 9, 32, 41, 91, 132

11-limit

Commas: 225/224, 385/384, 2401/2376

POTE temperament: 263.634

Map: [<1 4 1 5 5|, <0 -11 6 -10 -7|]

EDOs: 9, 32, 41, 91, 223cdef

13-limit

Commas: 105/104, 144/143, 196/195, 245/242

POTE temperament: 263.700

Map: [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|]

EDOs: 9, 41, 91

Slender

Commas: 225/224, 589824/588245

POTE temperament: ~49/48 = 38.413

Map: [<1 2 2 3|, <0 -13 10 -6|]

Wedgie: <<13 -10 6 -46 -27 42||

EDOs: 31, 94, 125

Badness: 0.0569

11-limit

Commas: 225/224, 385/384, 1331/1323

POTE temperament: ~49/48 = 38.387

Map: [<1 2 2 3 4|, <0 -13 10 -6 -17|]

EDOs: 31, 63, 94, 125

Badness: 25.342

13-limit

Commas: 225/224, 275/273, 385/384, 1331/1323

POTE generator: ~49/48 = 38.314

Map: [<1 2 2 3 4 3|, <0 -13 10 -6 -17 22|]

EDOs: 31, 63, 94

Badness: 25.913

Marvo

Commas: 225/224, 78125000/78121827

POTE temperament: ~27/20 = 516.694

Map: [<1 5 12 29|, <0 -6 -17 -46|]

Wedgie: <<6, 17, 46, 13, 56 59||

EDOs: 65, 72

11-limit

Commas: 225/224, 243/242, 4000/3993

POTE temperament: ~27/20 = 516.699

Map: [<1 5 12 29 12|, <0 -6 -17 -46 -15|]

EDOs: 65, 72, 281, 497

13-limit

Commas: 243/242, 225/224, 351/350, 1625/1617

POTE generator: ~27/20 = 516.730

Map: [<1 5 12 29 12 39|, <0 -6 -17 -46 -15 -62|]

EDOs: 72, 137, 209, 281f, 490bcf

Badness: 0.0269

Marvolo

Commas: 225/224, 156250000/155649627

POTE generator: ~21/20 = 83.348

Map: [<1 2 1 1|, <0 -6 19 26|]

Wedgie: <<6 -19 -26 -44 -58 -7||

EDOs: 29, 43, 72, 619bcd, 691bcd

Badness: 0.0833

11-limit

Commas: 225/224, 441/440, 4000/3993

POTE generator: ~21/20 = 83.340

Map: [<1 2 1 1 2|, <0 -6 19 26 21|]

EDOs: 29, 43, 72

Badness: 0.0290

13-limit

Commas: 169/168, 225/224, 364/363, 441/440

POTE generator: ~21/20 = 83.330

Map: [<1 2 1 1 2 3|, <0 -6 19 26 21 10|]

EDOs: 29, 43, 72, 115f

Badness: 0.0215

Amavil

Commas: 225/224, 17496/16807

POTE generator: ~48/35 = 529.979

Map: [<1 6 1 9|, <0 -10 3 -14|]

Wedgie: <<10 -3 14 -28 -6 41||

EDOs: 9, 34d, 43, 77d

Badness: 0.1096

11-limit

Commas: 99/98, 176/175, 864/847

POTE generator: ~15/11 = 529.974

Map: [<1 6 1 9 7|, <0 -10 3 -14 -8|]

EDOs: 9, 34d, 43, 77de

Badness: 0.0426

13-limit

Commas: 78/77, 99/98, 144/143, 176/175

POTE generator: ~15/11 = 529.951

Map: [<1 6 1 9 7 9|, <0 -10 3 -14 -8 -12|]

EDOs: 9, 34d, 43, 77de

Badness: 0.0258

Enneaportent

Commas: 225/224, 40353607/40310784

POTE generator: ~5/4 = 383.165

Map: [<9 0 28 11|, <0 2 -1 2|]

Wedgie: <<18 -9 18 -56 -22 67||

EDOs: 9, 54, 63, 72, 495bcd

Badness: 0.0937

11-limit

Commas: 225/224, 385/384, 12005/11979

POTE generator: ~5/4 = 383.146

Map: [<9 0 28 11 24|, <0 2 -1 2 1|]

EDOs: 9, 54, 63, 72, 423cd, 495bcd

Badness: 0.0304

13-limit

Commas: 169/168, 225/224, 364/363, 1716/1715

POTE generator: ~5/4 = 383.047

Map: [<9 0 28 11 24 19|, <0 2 -1 2 1 2|]

EDOs: 9, 54, 63, 72, 279cf

Badness: 0.0223

Submajor

Comma: 69198046875/68719476736

POTE generator: ~10125/8192 = 362.321

Map: [<1 4 -1|, <0 -8 11|]

EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc

Badness: 0.1302

7-limit

Commas: 225/224, 51200/50421

POTE generator: ~49/40 = 362.255

Map: [<1 4 -1 1|, <0 -8 11 6|]

Wedgie: <<8 -11 -6 -36 -32 17||

EDOs: 10, 33, 43, 53

Badness: 0.0605

11-limit

Commas: 225/224, 385/384, 6655/6561

POTE generator: ~27/22 = 362.101

Map: [<1 4 -1 1 11|, <0 -8 11 6 -25|]

EDOs: 10, 53, 116, 169de, 285cde

Badness: 0.0506

13-limit

Commas: 169/168, 225/224, 275/273, 385/384

POTE generator: ~16/13 = 362.105

Map: [<1 4 -1 1 11 4|, <0 -8 11 6 -25 -1|]

EDOs: 10, 53, 116, 169de, 285cdef

Badness: 0.0277

Interpental

Commas: 99/98, 176/175, 51200/50421

POTE generator: ~49/40 = 362.418

Map: [<1 4 -1 1 -5|, <0 -8 11 6 28|]

EDOs: 43, 53, 96, 149d

Badness: 0.0518

13-limit

Commas: 99/98, 169/168, 176/175, 640/637

POTE generator: ~16/13 = 362.402

Map: [<1 4 -1 1 -5 4|, <0 -8 11 6 28 -1|]

EDOs: 43, 53, 96, 149d

Badness: 0.0297

Alphorn

Commas: 225/224, 5764801/5668704

POTE generator: ~48/35 = 556.221

Map: [<1 9 0 13|, <0 -16 5 -22|]

Wedgie: <<16 -5 22 -45 -10 65||

EDOs: 28d, 41, 151cd, 192cd, 233cd

Badness: 0.1293

11-limit

Commas: 225/224, 385/384, 12250/11979

POTE generator: ~11/8 = 556.144

Map: [<1 9 0 13 3|, <0 -16 5 -22 1|]

EDOs: 41, 315cde

Badness: 0.0735