Marvel temperaments: Difference between revisions

Revision as of 04:49, 8 January 2022

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
Pelogic, {21/20, 135/128} → Pelogic family
August, {36/35, 128/125} → Augmented family
Pajara, {50/49, 64/63} → Diaschismic family
Meantone, {81/80, 126/125} → Meantone family
Magic, {225/224, 245/243} → Magic family
Passive, {225/224, 256/245} → Passion family
Miracle, {225/224, 1029/1024} → Gamelismic clan
Orwell, {225/224, 1728/1715} → Semicomma family
Garibaldi, {225/224, 3125/3087} → Schismatic family
Catakleismic, {225/224, 4375/4374} → Kleismic family
Decic, {225/224, 16807/16384} → Cloudy clan
Escapade, {225/224, 65625/65536} → Escapade family
Fog, {225/224, 156250/151263} → Misty family
Compton, {225/224, 250047/250000} → Compton family
Betic, {225/224, 1071875/1062882} → Sycamore family
Houborizic {225/224, 1250000/1240029} → Amity family
Qintosec, {225/224, 2560000/2470629} → Qintosec family
Quintapole, {225/224, 7812500/7411887} → Quintaleap family
Maquila, {225/224, 30233088/28824005} → Maquila family
Marvo, {225/224, 78125000/78121827} → Gravity family

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

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: Template:Val list

Badness: 0.016828

Wizard

Main article: Wizard

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Template:Val list

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

Optimal GPV sequence: 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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: Template:Val list

Badness: 0.027544

Untriton

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

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 6 -7 -7], ⟨0 -9 19 20]]

Wedgie: ⟨⟨ 9 -19 -20 -51 -57 7 ]]

POTE generator: ~7/5 = 588.641

Template:Val list

Badness: 0.143976

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [⟨1 6 -7 -7 1], ⟨0 -9 19 20 5]]

POTE generator: ~7/5 = 588.626

Optimal GPV sequence: Template:Val list

Badness: 0.074295

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [⟨1 6 -7 -7 1 -12], ⟨0 -9 19 20 5 32]]

POTE generator: ~7/5 = 588.654

Optimal GPV sequence: Template:Val list

Badness: 0.047441

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: Template:Val list

Badness: 0.023117

Misneb

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

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 3 1 3], ⟨0 -15 14 -2]]

Wedgie: ⟨⟨ 15 -14 2 -57 -39 44 ]]

POTE generator: ~16/15 = 113.235

Template:Val list

Badness: 0.140970

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [⟨1 3 1 3 1], ⟨0 -15 14 -2 26]]

POTE generator: ~16/15 = 113.323

Optimal GPV sequence: Template:Val list

Badness: 0.085390

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [⟨1 3 1 3 1 2], ⟨0 -15 14 -2 26 18]]

POTE generator: ~16/15 = 113.323

Optimal GPV sequence: Template:Val list

Badness: 0.045569

Musneb

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 66550/64827

Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]

POTE generator: ~16/15 = 113.142

Optimal GPV sequence: Template:Val list

Badness: 0.087333

Slender

Slender (31&32) tempers out the hewuermera comma in addition to the marvel comma, this has a generator 49/48, 3 of which equal marvel's 16/15 - 15/14, and 10 generators is 5/4.

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: Template:Val list

Badness: 0.025913

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: Template:Val list

Badness: 0.021470

Amavil

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

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Template:Val list

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

Template:Val list

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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Optimal GPV sequence: 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

Template:Val list

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

Optimal GPV sequence: 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

Template:Val list

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

Template:Val list

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

Optimal GPV sequence: Template:Val list

Badness: 0.173485

Gwazy

See also: Very high accuracy temperaments § Kwazy

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

Wedgie: ⟨⟨ 16 -10 12 -53 -26 56 ]]

POTE generator: ~35/32 = 162.658

Template:Val list

Badness: 0.178826

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 65536/65219

Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]

POTE generator: ~11/10 = 162.592

Optimal GPV sequence: Template:Val list

Badness: 0.068410

Gracecordial

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

Wedgie: ⟨⟨ 1 -20 -38 -34 -63 -32 ]]

POTE generator: ~3/2 = 700.824

Template:Val list

Badness: 0.096279

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 236328125/234365481

Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]

POTE generator: ~3/2 = 700.834

Optimal GPV sequence: Template:Val list

Badness: 0.089588

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]

POTE generator: ~3/2 = 700.841

Optimal GPV sequence: Template:Val list

Badness: 0.052235

@@ Line 36: / Line 36: @@
 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
@@ Line 93: / Line 92: @@
 POTE generator: ~15/14 = 126.474
-Vals: {{Val list| 9, 10, 19 }}
+Optimal GPV sequence: {{Val list| 9, 10, 19 }}
 Badness: 0.026190
@@ Line 106: / Line 105: @@
 POTE generator: ~14/13 = 126.431
-Vals: {{Val list| 9, 10, 19 }}
+Optimal GPV sequence: {{Val list| 9, 10, 19 }}
 Badness: 0.017833
@@ Line 119: / Line 118: @@
 POTE generator: ~15/14 = 124.767
-Vals: {{Val list| 19, 29, 48d, 77cdd }}
+Optimal GPV sequence: {{Val list| 19, 29, 48d, 77cdd }}
 Badness: 0.038679
@@ Line 132: / Line 131: @@
 POTE generator: ~14/13 = 124.716
-Vals: {{Val list| 19, 29, 48df, 77cddf }}
+Optimal GPV sequence: {{Val list| 19, 29, 48df, 77cddf }}
 Badness: 0.024383
@@ Line 145: / Line 144: @@
 POTE generator: ~15/14 = 127.039
-Vals: {{Val list| 9, 19e }}
+Optimal GPV sequence: {{Val list| 9, 19e }}
 Badness: 0.030617
@@ Line 158: / Line 157: @@
 POTE generator: ~14/13 = 127.039
-Vals: {{Val list| 9, 19e }}
+Optimal GPV sequence: {{Val list| 9, 19e }}
 Badness: 0.020205
@@ Line 171: / Line 170: @@
 POTE generator: ~15/14 = 124.539
-Vals: {{Val list| 10, 19e, 29, 77cddee }}
+Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddee }}
 Badness: 0.035296
@@ Line 184: / Line 183: @@
 POTE generator: ~14/13 = 124.545
-Vals: {{Val list| 10, 19e, 29, 77cddeef }}
+Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddeef }}
 Badness: 0.021559
@@ Line 197: / Line 196: @@
 POTE generator: ~11/8 = 537.186
-Vals: {{Val list| 9, 20, 29, 38d, 67cdde }}
+Optimal GPV sequence: {{Val list| 9, 20, 29, 38d, 67cdde }}
 Badness: 0.041886
@@ Line 210: / Line 209: @@
 POTE generator: ~11/8 = 537.208
-Vals: {{Val list| 9, 20, 29, 38df, 67cddef }}
+Optimal GPV sequence: {{Val list| 9, 20, 29, 38df, 67cddef }}
 Badness: 0.025192
@@ Line 223: / Line 222: @@
 POTE generator: ~11/8 = 537.230
-Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
+Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
 Badness: 0.021778
@@ Line 236: / Line 235: @@
 POTE generator: ~11/8 = 537.214
-Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
+Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
 Badness: 0.016828
@@ Line 273: / Line 272: @@
 POTE generator: ~5/4 = 383.232
-Vals: {{Val list| 22, 50, 72, 166, 238c, 310c }}
+Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
 Badness: 0.018539
@@ Line 290: / Line 289: @@
 POTE generator: ~5/4 = 383.389
-Vals: {{Val list| 22, 50, 72, 122, 194df }}
+Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
 Badness: 0.021781
@@ Line 305: / Line 304: @@
 POTE generator: ~5/4 = 383.381
-Vals: {{Val list| 22, 50, 72, 122g, 194dfg }}
+Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
 Badness: 0.014536
@@ Line 320: / Line 319: @@
 POTE generator: ~5/4 = 383.477
-Vals: {{Val list| 22h, 50, 72, 122g, 194dfg }}
+Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
 Badness: 0.015702
@@ Line 335: / Line 334: @@
 POTE generator: ~5/4 = 383.170
-Vals: {{Val list| 72, 166, 238cf }}
+Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
 Badness: 0.020252
@@ Line 350: / Line 349: @@
 POTE generator: ~5/4 = 383.175
-Vals: {{Val list| 72, 166g, 238cfg }}
+Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
 Badness: 0.013624
@@ Line 365: / Line 364: @@
 POTE generator: ~5/4 = 383.138
-Vals: {{Val list| 72, 94, 166g }}
+Optimal GPV sequence: {{Val list| 72, 94, 166g }}
 Badness: 0.014810
@@ Line 380: / Line 379: @@
 POTE generator: ~5/4 = 383.124
-Vals: {{Val list| 22, 50e, 72ee, 94ee }}
+Optimal GPV sequence: {{Val list| 22, 50e, 72ee, 94ee }}
 Badness: 0.057799
@@ Line 410: / Line 409: @@
 POTE generator: ~7/5 = 569.144
-Vals: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
+Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
 Badness: 0.045675
@@ Line 440: / Line 439: @@
 POTE generator: ~7/5 = 580.267
-Vals: {{Val list| 29, 31, 60e }}
+Optimal GPV sequence: {{Val list| 29, 31, 60e }}
 Badness: 0.023659
@@ Line 453: / Line 452: @@
 POTE generator: ~7/5 = 580.108
-Vals: {{Val list| 29, 31, 60e, 151cde }}
+Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
 Badness: 0.022993
@@ Line 466: / Line 465: @@
 POTE generator: ~7/5 = 580.389
-Vals: {{Val list| 31, 91, 122, 153d }}
+Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
 Badness: 0.045456
@@ Line 496: / Line 495: @@
 POTE generator: ~7/5 = 585.606
-Vals: {{Val list| 41, 84, 125e }}
+Optimal GPV sequence: {{Val list| 41, 84, 125e }}
 Badness: 0.036383
@@ Line 509: / Line 508: @@
 POTE generator: ~7/5 = 585.657
-Vals: {{Val list| 41, 84, 125e, 209ef, 293ef }}
+Optimal GPV sequence: {{Val list| 41, 84, 125e, 209ef, 293ef }}
 Badness: 0.027544
@@ Line 539: / Line 538: @@
 POTE generator: ~7/5 = 588.626
-Vals: {{Val list| 51, 53 }}
+Optimal GPV sequence: {{Val list| 51, 53 }}
 Badness: 0.074295
@@ Line 552: / Line 551: @@
 POTE generator: ~7/5 = 588.654
-Vals: {{Val list| 51f, 53 }}
+Optimal GPV sequence: {{Val list| 51f, 53 }}
 Badness: 0.047441
@@ Line 582: / Line 581: @@
 POTE generator: ~7/6 = 263.634
-Vals: {{Val list| 41, 91, 223cdef }}
+Optimal GPV sequence: {{Val list| 41, 91, 223cdef }}
 Badness: 0.031309
@@ Line 595: / Line 594: @@
 POTE generator: ~7/6 = 263.700
-Vals: {{Val list| 41, 91 }}
+Optimal GPV sequence: {{Val list| 41, 91 }}
 Badness: 0.023117
@@ Line 625: / Line 624: @@
 POTE generator: ~16/15 = 113.323
-Vals: {{Val list| 21, 32e, 53, 127, 180de }}
+Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
 Badness: 0.085390
@@ Line 638: / Line 637: @@
 POTE generator: ~16/15 = 113.323
-Vals: {{Val list| 21, 32e, 53, 127, 180de }}
+Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
 Badness: 0.045569
@@ Line 651: / Line 650: @@
 POTE generator: ~16/15 = 113.142
-Vals: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
+Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
 Badness: 0.087333
@@ Line 657: / Line 656: @@
 == Slender ==
 Slender (31&amp;32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma, this has a generator [[49/48]], 3 of which equal marvel's 16/15 - 15/14, and 10 generators is 5/4.
 Subgroup: 2.3.5.7
@@ Line 682: / Line 680: @@
 POTE generator: ~49/48 = 38.387
-Vals: {{Val list| 31, 63, 94, 125 }}
+Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
 Badness: 0.025342
@@ Line 695: / Line 693: @@
 POTE generator: ~49/48 = 38.314
-Vals: {{Val list| 31, 63, 94 }}
+Optimal GPV sequence: {{Val list| 31, 63, 94 }}
 Badness: 0.025913
@@ Line 723: / Line 721: @@
 POTE generator: ~21/20 = 83.340
-Vals: {{Val list| 29, 43, 72 }}
+Optimal GPV sequence: {{Val list| 29, 43, 72 }}
 Badness: 0.028965
@@ Line 736: / Line 734: @@
 POTE generator: ~21/20 = 83.330
-Vals: {{Val list| 29, 43, 72, 115f }}
+Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
 Badness: 0.021470
@@ Line 766: / Line 764: @@
 POTE generator: ~15/11 = 529.974
-Vals: {{Val list| 9, 34d, 43, 77de }}
+Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
 Badness: 0.042649
@@ Line 779: / Line 777: @@
 POTE generator: ~15/11 = 529.951
-Vals: {{Val list| 9, 34d, 43, 77de }}
+Optimal GPV sequence: {{Val list| 9, 34d, 43, 77de }}
 Badness: 0.025791
@@ Line 807: / Line 805: @@
 POTE generator: ~5/4 = 383.146
-Vals: {{Val list| 9, 63, 72, 423cd, 495bcd }}
+Optimal GPV sequence: {{Val list| 9, 63, 72, 423cd, 495bcd }}
 Badness: 0.030426
@@ Line 820: / Line 818: @@
 POTE generator: ~5/4 = 383.047
-Vals: {{Val list| 9, 63, 72, 279cf }}
+Optimal GPV sequence: {{Val list| 9, 63, 72, 279cf }}
 Badness: 0.022322
@@ Line 861: / Line 859: @@
 POTE generator: ~27/22 = 362.101
-Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
+Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
 Badness: 0.050582
@@ Line 874: / Line 872: @@
 POTE generator: ~16/13 = 362.105
-Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
+Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
 Badness: 0.027689
@@ Line 889: / Line 887: @@
 POTE generator: ~49/40 = 362.418
-Vals: {{Val list| 43, 53, 96, 149d }}
+Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
 Badness: 0.051806
@@ Line 904: / Line 902: @@
 POTE generator: ~16/13 = 362.402
-Vals: {{Val list| 43, 53, 96, 149d }}
+Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
 Badness: 0.029680
@@ Line 932: / Line 930: @@
 POTE generator: ~11/8 = 556.144
-Vals: {{Val list| 28d, 41, 315cde }}
+Optimal GPV sequence: {{Val list| 28d, 41, 315cde }}
 Badness: 0.073459
@@ Line 973: / Line 971: @@
 POTE generator: ~15/14 = 112.171
-Vals: {{Val list| 21, 54, 75e }}
+Optimal GPV sequence: {{Val list| 21, 54, 75e }}
 Badness: 0.173485
@@ Line 1,003: / Line 1,001: @@
 POTE generator: ~11/10 = 162.592
-Vals: {{Val list| 22, 74, 96, 118d }}
+Optimal GPV sequence: {{Val list| 22, 74, 96, 118d }}
 Badness: 0.068410
@@ Line 1,031: / Line 1,029: @@
 POTE generator: ~3/2 = 700.834
-Vals: {{Val list| 12e, 113, 125, 238c }}
+Optimal GPV sequence: {{Val list| 12e, 113, 125, 238c }}
 Badness: 0.089588
@@ Line 1,044: / Line 1,042: @@
 POTE generator: ~3/2 = 700.841
-Vals: {{Val list| 12e, 113, 125f, 238cf }}
+Optimal GPV sequence: {{Val list| 12e, 113, 125f, 238cf }}
 Badness: 0.052235

Marvel temperaments: Difference between revisions

Revision as of 04:49, 8 January 2022

Negri

7-limit

Negra

11-limit

13-limit

Negril

13-limit

Negric

13-limit

Negroni

13-limit

Wilsec

13-limit

17-limit

19-limit

Wizard

11-limit

Lizard

17-limit

19-limit

Gizzard

17-limit

19-limit

Mage

Triton

11-limit

Tritonic

11-limit

13-limit

Tritoni

Merman

11-limit

13-limit

Untriton

11-limit

13-limit

Septimin

11-limit

13-limit

Misneb

11-limit

13-limit

Musneb

Slender

11-limit

13-limit

Marvolo

11-limit

13-limit

Amavil

11-limit

13-limit

Enneaportent

11-limit

13-limit

Submajor

7-limit

11-limit

13-limit

Interpental

13-limit

Alphorn

11-limit

Tertiosec

7-limit

11-limit

Gwazy

11-limit

Gracecordial

11-limit

13-limit

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