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
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):
7-limit
Period: 1\1
Optimal (POTE) generator: ~15/14 = 125.608
EDO generators: 1\9, 1\10, 2\19
Scales (Scala files):
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):
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):
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):
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
Scales (Scala files):
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
Scales (Scala files):
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
Scales (Scala files):
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
Scales (Scala files):
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):
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):
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
Scales (Scala files):
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
Scales (Scala files):
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
Scales (Scala files):
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
Scales (Scala files):
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
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
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