Marvel temperaments

This page discusses miscellaneous rank-2 temperaments tempering out [-5 2 2 -1⟩ = 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

• sharp, {25/24, 28/27} → Dicot family #Sharp
• pelogic, {21/20, 135/128} → Pelogic family #Pelogic
• august, {36/35, 128/125} → Augmented family #August
• pajara, {50/49, 64/63}, also in: diaschismic family / jubilismic clan / archytas clan
• septimal meantone, {81/80, 126/125} → Meantone family #Septimal meantone
• magic, {225/224, 245/243}, also in: magic family
• miracle, {225/224, 1029/1024}, also in: gamelismic clan
• orwell, {225/224, 1728/1715}, also in: semicomma family
• garibaldi, {225/224, 3125/3087}, also in: schismatic family
• catakleismic, {225/224, 4375/4374}, also in: kleismic family
• decic, {225/224, 16807/16384} → Cloudy clan #Decic
• escapade, {225/224, 65625/65536} → Escapade family #Escapade
• fog, {225/224, 156250/151263} → Misty family #Fog
• compton, {225/224, 250047/250000} → Compton family #Compton
• houborizic {225/224, 1250000/1240029} → Amity family #Houborizic
• qintosec, {225/224, 2560000/2470629} → Qintosec family #Qintosec
• maquila, {225/224, 30233088/28824005} → Maquila family #Maquila

Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvo, marvolo, amavil, enneaportent, submajor, and alphorn.

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.

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

Wedgie: ⟨⟨ 4 -3 -14 ]]

POTE generator: ~16/15 = 125.7549

Badness: 0.086856

7-limit

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 ]]

POTE generator: ~15/14 = 125.608

Badness: 0.026483

Negra

This is the 2.3.5.7.13 extension of negri.

Subgroup: 2.3.5.7.13

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

Sval mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -3]]

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

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

POTE generator: ~14/13 = 125.567

11-limit

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]]

POTE generator: ~15/14 = 126.474

Vals: Template:Val list

Badness: 0.026190

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~14/13 = 126.431

Vals: Template:Val list

Badness: 0.017833

Negril

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]]

POTE generator: ~15/14 = 124.767

Vals: Template:Val list

Badness: 0.038679

13-limit

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]]

POTE generator: ~14/13 = 124.716

Vals: Template:Val list

Badness: 0.024383

Negric

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]]

POTE generator: ~15/14 = 127.039

Vals: Template:Val list

Badness: 0.030617

13-limit

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]]

POTE generator: ~14/13 = 127.039

Vals: Template:Val list

Badness: 0.020205

Negroni

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]]

POTE generator: ~15/14 = 124.539

Vals: Template:Val list

Badness: 0.035296

13-limit

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]]

POTE generator: ~14/13 = 124.545

Vals: Template:Val list

Badness: 0.021559

Wilsec

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~11/8 = 537.186

Vals: Template:Val list

Badness: 0.041886

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~11/8 = 537.208

Vals: Template:Val list

Badness: 0.025192

17-limit

Subgroup: 2.3.5.7.11.13.17

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]]

POTE generator: ~11/8 = 537.230

Vals: Template:Val list

Badness: 0.021778

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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]]

POTE generator: ~11/8 = 537.214

Vals: Template:Val list

Badness: 0.016828

Passive

Passive (1&11) tempers out the same 5-limit comma as the passion temperament (12&49), but with the marvel (126/125) rather than 64/63 tempered out.

Subgroup: 2.3.5.7

Comma list: 225/224, 256/245

Mapping: [⟨1 2 2 3], ⟨0 -5 4 -2]]

POTE generator: ~16/15 = 98.809

Badness: 0.075055

Wizard

For the 5-limit version of this temperament, see High badness temperaments #Wizard.

See also: Wizard

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

Mapping generators: ~1225/864, ~245/216

POTE generator: ~5/4 = 383.256

Wedgie: ⟨⟨ 12 -2 20 -31 -2 52 ]]

Badness: 0.040846

Scales: wizard22

11-limit

Subgroup: 2.3.5.7.11

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

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

Mapping generators: ~99/70, ~25/22

POTE generator: ~5/4 = 383.232

Vals: Template:Val list

Badness: 0.018539

Scales: wizard22

Lizard

Subgroup: 2.3.5.7.11.13

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

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

Mapping generators: ~99/70, ~25/22

POTE generator: ~5/4 = 383.389

Vals: Template:Val list

Badness: 0.021781

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Mapping generators: ~17/12, ~17/15

POTE generator: ~5/4 = 383.381

Vals: Template:Val list

Badness: 0.014536

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Mapping generators: ~17/12, ~17/15

POTE generator: ~5/4 = 383.477

Vals: Template:Val list

Badness: 0.015702

Gizzard

Subgroup: 2.3.5.7.11.13

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

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

Mapping generators: ~99/70, ~25/22

POTE generator: ~5/4 = 383.170

Vals: Template:Val list

Badness: 0.020252

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Mapping generators: ~17/12, ~17/15

POTE generator: ~5/4 = 383.175

Vals: Template:Val list

Badness: 0.013624

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Mapping generators: ~17/12, ~17/15

POTE generator: ~5/4 = 383.138

Vals: Template:Val list

Badness: 0.014810

Mage

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Mapping generators: ~77/54, ~55/48

POTE generator: ~5/4 = 383.124

Vals: Template:Val list

Badness: 0.057799

Triton

For the 5-limit version of this temperament, see High badness temperaments #Stump.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

Wedgie: ⟨⟨ 3 -7 -8 -18 -21 1 ]]

POTE generator: ~7/5 = 568.865

Badness: 0.059245

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~7/5 = 569.144

Vals: Template:Val list

Badness: 0.045675

Tritonic

For the 5-limit version of this temperament, see High badness temperaments #Tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping: [⟨1 4 -3 -3], ⟨0 -5 11 12]]

Wedgie: ⟨⟨ 5 -11 -12 -29 -33 3 ]]

POTE generator: ~7/5 = 580.286

Badness: 0.047578

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~7/5 = 580.267

Vals: Template:Val list

Badness: 0.023659

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~7/5 = 580.108

Vals: Template:Val list

Badness: 0.022993

Tritoni

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~7/5 = 580.389

Vals: Template:Val list

Badness: 0.045456

Merman

For the 5-limit version of this temperament, see High badness temperaments #Merman.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 5 -5 -5], ⟨0 -7 15 16]]

Wedgie: ⟨⟨ 7 -15 -16 -40 -45 5 ]]

POTE generator: ~7/5 = 585.585

Badness: 0.055078

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~7/5 = 585.606

Vals: Template:Val list

Badness: 0.036383

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~7/5 = 585.657

Vals: Template:Val list

Badness: 0.027544

Septimin

For the 5-limit version of this temperament, see High badness temperaments #Septimin.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 4 1 5], ⟨0 -11 6 -10]]

Wedgie: ⟨⟨ 11 -6 10 -35 -15 40 ]]

POTE generator: ~7/6 = 263.632

Badness: 0.054502

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~7/6 = 263.634

Vals: Template:Val list

Badness: 0.031309

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~7/6 = 263.700

Vals: Template:Val list

Badness: 0.023117

Slender

Subgroup: 2.3.5.7

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

Wedgie: ⟨⟨ 13 -10 6 -46 -27 42 ]]

POTE generator: ~49/48 = 38.413

Badness: 0.056934

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~49/48 = 38.387

Vals: Template:Val list

Badness: 0.025342

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~49/48 = 38.314

Vals: Template:Val list

Badness: 0.025913

Marvo

See also: Gravity family

Subgroup: 2.3.5.7

Comma list: 225/224, 78125000/78121827

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

Wedgie: ⟨⟨ 6 17 46 13 56 59 ]]

POTE generator: ~27/20 = 516.694

Badness: 0.097627

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~27/20 = 516.699

Vals: Template:Val list

Badness: 0.031685

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~27/20 = 516.730

Vals: Template:Val list

Badness: 0.026882

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

Wedgie: ⟨⟨ 6 -19 -26 -44 -58 -7 ]]

POTE generator: ~21/20 = 83.348

Badness: 0.083338

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~21/20 = 83.340

Vals: Template:Val list

Badness: 0.028965

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~21/20 = 83.330

Vals: Template:Val list

Badness: 0.021470

Amavil

5-limit (Mabila)

Subgroup: 2.3.5

Comma list: 268435456/263671875

Mapping: [⟨1 6 1], ⟨0 -10 3]]

POTE generator: ~512/375 = 529.6849

Badness: 0.232481

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 17496/16807

Mapping: [⟨1 6 1 9], ⟨0 -10 3 -14]]

Wedgie: ⟨⟨ 10 -3 14 -28 -6 41 ]]

POTE generator: ~48/35 = 529.979

Badness: 0.109625

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~15/11 = 529.974

Vals: Template:Val list

Badness: 0.042649

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~15/11 = 529.951

Vals: Template:Val list

Badness: 0.025791

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

Wedgie: ⟨⟨ 18 -9 18 -56 -22 67 ]]

POTE generator: ~5/4 = 383.165

Badness: 0.093679

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~5/4 = 383.146

Vals: Template:Val list

Badness: 0.030426

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~5/4 = 383.047

Vals: Template:Val list

Badness: 0.022322

Submajor

Subgroup: 2.3.5

Comma list: 69198046875/68719476736

Mapping: [⟨1 4 -1], ⟨0 -8 11]]

POTE generator: ~10125/8192 = 362.321

Badness: 0.130236

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 51200/50421

Mapping: [⟨1 4 -1 1], ⟨0 -8 11 6]]

Wedgie: ⟨⟨ 8 -11 -6 -36 -32 17 ]]

POTE generator: ~49/40 = 362.255

Badness: 0.060533

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~27/22 = 362.101

Vals: Template:Val list

Badness: 0.050582

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~16/13 = 362.105

Vals: Template:Val list

Badness: 0.027689

Interpental

Subgroup: 2.3.5.7.11

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]]

POTE generator: ~49/40 = 362.418

Vals: Template:Val list

Badness: 0.051806

13-limit

Subgroup: 2.3.5.7.11.13

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]]

POTE generator: ~16/13 = 362.402

Vals: Template:Val list

Badness: 0.029680

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 9 0 13], ⟨0 -16 5 -22]]

Wedgie: ⟨⟨ 16 -5 22 -45 -10 65 ]]

POTE generator: ~48/35 = 556.221

Badness: 0.129258

11-limit

Subgroup: 2.3.5.7.11

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

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

POTE generator: ~11/8 = 556.144

Vals: Template:Val list

Badness: 0.073459

Tertiosec

Subgroup: 2.3.5

Comma list: [-89 21 24⟩

Mapping: [⟨3 7 5], ⟨0 -8 7]]

POTE generator: ~16/15 = 112.281

Badness: 0.851003

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 7 5 9], ⟨0 -8 7 -2]]

Wedgie: ⟨⟨ 24 -21 6 -89 -58 73 ]]

POTE generator: ~15/14 = 112.283

Badness: 0.431636

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 3840/3773, 12005/11979

Mapping: [⟨3 7 5 9 9], ⟨0 -8 7 -2 5]]

POTE generator: ~15/14 = 112.171

Vals: Template:Val list

Badness: 0.173485