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
5-limit
Note that this is technically not a marvel temperament until the 7-limit, but also placed here for reference.
Comma list: 16875/16384
POTE generator: ~15/14 = 125.7549
Mapping: [⟨1 2 2], ⟨0 -4 3]]
Wedgie: ⟨⟨4 -3 -14]]
7-limit
Comma list: 49/48, 225/224
POTE generator: ~15/14 = 125.608
Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]
Wedgie: ⟨⟨4 -3 2 -14 -8 13]]
11-limit
Comma list: 45/44, 49/48, 56/55
POTE generator: ~15/14 = 126.474
Mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -5]]
Badness: 0.0262
13-limit
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
Comma list: 49/48, 100/99, 225/224
POTE generator: ~15/14 = 124.767
Mapping: [⟨1 2 2 3 2], ⟨0 -4 3 -2 14]]
Badness: 0.0387
13-limit
Comma list: 49/48, 65/64, 91/90, 875/858
POTE generator: ~15/14 = 124.716
Mapping: [⟨1 2 2 3 2 4], ⟨0 -4 3 -2 14 -3]]
Badness: 0.0244
Negric
Comma list: 33/32, 49/48, 77/75
POTE generator: ~15/14 = 127.039
Mapping: [⟨1 2 2 3 3], ⟨0 -4 3 -2 4]]
Badness: 0.0306
13-limit
Comma list: 33/32, 49/48, 65/64, 91/90
POTE generator: ~15/14 = 127.039
Mapping: [⟨1 2 2 3 3 4], ⟨0 -4 3 -2 4 -3]]
Badness: 0.0202
Negroni
Comma list: 49/48, 55/54, 225/224
POTE generator: ~15/14 = 124.539
Mapping: [⟨1 2 2 3 5], ⟨0 -4 3 -2 -15]]
Badness: 0.0353
13-limit
Comma list: 49/48, 55/54, 65/64, 91/90
POTE generator: ~15/14 = 124.545
Mapping: [⟨1 2 2 3 5 4], ⟨0 -4 3 -2 -15 -3]]
Badness: 0.0216
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
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||
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|]
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|]
Badness: 25.913
Marvo
Commas: 225/224, 78125000/78121827
POTE generator: ~27/20 = 516.694
Map: [<1 5 12 29|, <0 -6 -17 -46|]
Wedgie: <<6 17 46 13 56 59||
EDOs: 7d, 65d, 72, 137, 209, 281, 569bcc
Badness: 0.0976
11-limit
Commas: 225/224, 243/242, 4000/3993
POTE generator: ~27/20 = 516.699
Map: [<1 5 12 29 12|, <0 -6 -17 -46 -15|]
EDOs: 7d, 65d, 72, 281, 353c, 425bc, 497bc
Badness: 0.0317
13-limit
Commas: 225/224, 243/242, 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|]
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|]
Badness: 0.0215
Amavil
5-limit (mabila)
Comma: 268435456/263671875
POTE generator: ~512/375 = 529.6849
Map: [<1 6 1|, <0 -10 3|]
EDOs: 9, 25, 34, 77, 111, 145, 256c
Badness: 0.2325
7-limit
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||
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|]
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|]
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||
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|]
Badness: 0.0223
Submajor
5-limit
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||
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|]
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|]
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