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
- 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
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
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.
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
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