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[edit]
Commas: 49/48, 225/224
POTE temperament: ~15/14 = 125.608
Map: [<1 2 2 3|, <0 -4 3 -2|]
Wedgie: <<4 -3 2 -14 -8 13||
11-limit[edit]
Commas: 45/44, 49/48, 56/55
POTE generator: ~15/14 = 126.474
Map: [<1 2 2 3 4|, <0 -4 3 -2 -5|]
EDOs: 9, 10, 19
Badness: 0.0262
13-limit[edit]
Commas: 45/44, 49/48, 56/55, 78/77
POTE generator: ~14/13 = 126.431
Map: [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]
EDOs: 9, 10, 19
Negril[edit]
Commas: 49/48, 100/99, 225/224
POTE generator: ~15/14 = 124.767
Map: [<1 2 2 3 2|, <0 -4 3 -2 14|]
Badness: 0.0387
13-limit[edit]
Commas: 49/48, 65/64, 91/90, 875/858
POTE generator: ~15/14 = 124.716
Map: [<1 2 2 3 2 4|, <0 -4 3 -2 14 -3|]
EDOs: 9, 10, 19, 29, 48, 77
Badness: 0.0244
Negric[edit]
Commas: 33/32, 49/48, 77/75
POTE generator: ~15/14 = 127.039
Map: [<1 2 2 3 3|, <0 -4 3 -2 4|]
EDOs: 9, 19, 85
Badness: 0.0306
13-limit[edit]
Commas: 33/32, 49/48, 65/64, 91/90
POTE generator: ~15/14 = 127.039
Map: [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|]
EDOs: 9, 19, 85
Badness: 0.0202
Negroni[edit]
Commas: 49/48, 55/54, 225/224
POTE generator: ~15/14 = 124.539
Map: [<1 2 2 3 5|, <0 -4 3 -2 -15|]
EDOs: 10, 19e, 29, 77cde
Badness: 0.0353
13-limit[edit]
Commas: 49/48, 55/54, 65/64, 91/90
POTE generator: ~15/14 = 124.545
Map: [<1 2 2 3 5 4|, <0 -4 3 -2 -15 -3|]
EDOs: 10, 19e, 29, 77cdef
Badness: 0.0216
Wizard[edit]
Comma: 2197265625/2147483648
POTE generator: ~5/4 = 383.212
Map: [<2 1 5|, <0 6 -1|]
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Badness: 0.3864
7-limit[edit]
Commas: 225/224, 118098/117649
POTE temperament: 383.256
Map: [<2 1 5 2|, <0 6 -1 10|]
Wedgie: <<12 -2 20 -31 -2 52||
EDOs: 22, 50, 72, 94, 166, 238c, 310c, 382c
Badness: 0.0408
11-limit[edit]
Commas: 225/224, 385/384, 4000/3993
POTE temperament: 383.232
Map: [<2 1 5 2 8|, <0 6 -1 10 -3|]
EDOs: 22, 50, 72, 94, 166, 238c, 310c
Badness: 0.0185
Lizard[edit]
Commas: 225/224, 351/350, 364/363, 385/384
POTE temperament: 383.389
Map: [<2 1 5 2 8 11|, <0 6 -1 10 -3 -10|]
EDOs: 22, 50, 72, 122, 194df
Badness: 0.0218
17-limit[edit]
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
POTE temperament: 383.381
Map: [<2 1 5 2 8 11 6|, <0 6 -1 10 -3 -10 6|]
EDOs: 22, 50, 72, 122g, 194dfg
Badness: 0.0145
19-limit[edit]
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
POTE generator: ~5/4 = 383.477
Map: [<2 1 5 2 8 11 6 2|, <0 6 -1 10 -3 -10 6 18|]
EDOs: 22h, 50, 72, 122g, 194dfg
Badness: 0.0157
Gizzard[edit]
Commas: 225/224, 385/384, 325/324, 1573/1568
POTE generator: ~5/4 = 383.170
Map: [<2 1 5 2 8 -2|, <0 6 -1 10 -3 26|]
Badness: 0.0203
17-limit[edit]
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
POTE generator: ~5/4 = 383.175
Map: [<2 1 5 2 8 -2 6|, <0 6 -1 10 -3 26 6|]
EDOs: 22f, 72, 94, 166g, 238cfg
Badness: 0.0136
19-limit[edit]
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
POTE generator: ~5/4 = 383.138
Map: [<2 1 5 2 8 -2 6 15|, <0 6 -1 10 -3 26 6 -18|]
EDOs: 22f, 72, 94, 166g
Badness: 0.0148
Mage[edit]
Commas: 99/98, 176/175, 1331/1296
POTE generator: ~55/48 = 216.876
Map: [<2 1 5 2 4|, <0 6 -1 10 8|]
EDOs: 22, 72e, 94e
Badness: 0.0578
Triton[edit]
Commas: 225/224, 1029/1000
POTE temperament: ~7/5 = 568.865
Map: [<1 0 6 7|, <0 3 -7 -8|]
Wedgie: <<3 -7 -8 -18 -21 1||
EDOs: 17d, 19, 78bd, 97bd
Badness: 0.0592
11-limit[edit]
Commas: 45/44, 56/55, 1029/1000
POTE generator: ~7/5 = 569.144
Map: [<1 0 6 7 4|, <0 3 -7 -8 -1|]
EDOs: 19, 59bde, 78bde, 97bde
Badness: 0.0457
Tritonic[edit]
Comnma: 553584375/536870912
POTE generator: ~45/32 = 580.219
Map: [<1 4 -3|, <0 -5 11|]
EDOs: 29, 31, 60, 91
Badness: 0.3785
7-limit[edit]
Commas: 225/224, 50421/50000
POTE temperament: 580.286
Map: [<1 4 -3 -3|, <0 -5 11 12|]
Wedgie: <<5 -11 -12 -29 -33 3||
EDOs: 29, 31, 60, 91, 122, 213bcd
11-limit[edit]
Commas: 121/120, 225/224, 441/440
POTE generator: ~7/5 = 580.267
Map: [<1 4 -3 -3 2|, <0 -5 11 12 3|]
EDOs: 29, 31, 60e
Badness: 0.0237
13-limit[edit]
Commas: 105/104, 121/120, 196/195, 275/273
POTE generator: ~7/5 = 580.108
Map: [<1 4 -3 -3 2 -5|, <0 -5 11 12 3 18|]
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230
Tritoni[edit]
Commas: 225/224, 385/384, 27783/27500
POTE generator: ~7/5 = 580.389
Map: [<1 4 -3 -3 17|, <0 -5 11 12 -28|]
EDOs: 31, 91, 122
Badness: 0.0455
Merman[edit]
Comma: 1121008359375/1099511627776
POTE generator: ~45/32 = 585.634
Map: [<1 5 -5|, <0 -7 15|]
EDOs: 41, 84, 125, 209, 334c, 543bc
Badness: 0.4538
7-limit[edit]
Commas: 225/224, 2500000/2470629
POTE generator: ~7/5 = 585.585
Map: [<1 5 -5 -5|, <0 -7 15 16|]
Wedgie: <<7 -15 -16 -40 -45 5||
EDOs: 41, 84, 125
Badness: 0.0551
11-limit[edit]
Commas: 225/224, 441/440, 1344/1331
POTE generator: ~7/5 = 585.606
Map: [<1 5 -5 -5 2|, <0 -7 15 16 3]]
EDOs: 41, 84, 125e
Badness: 0.0364
13-limit[edit]
Commas: 144/143, 225/224, 364/363, 441/440
POTE generator: ~7/5 = 585.657
Map: [<1 5 -5 -5 2 12|, <0 -7 15 16 3 -17|]
EDOs: 41, 84, 125e, 209ef, 293ef
Badness: 0.0275
Septimin[edit]
Comma: 35595703125/34359738368
POTE generator: ~75/64 = 263.658
Map: [<1 4 1|, <0 -11 6|]
EDOs: 9, 32, 41, 91, 132, 223c, 355bc
Badness: 0.6709
7-limit[edit]
Commas: 225/224, 84035/82944
POTE temperament: 263.632
Map: [<1 4 1 5|, <0 -11 6 -10|]
Wedgie: <<11 -6 10 -35 -15 40||
11-limit[edit]
Commas: 225/224, 385/384, 2401/2376
POTE temperament: 263.634
Map: [<1 4 1 5 5|, <0 -11 6 -10 -7|]
EDOs: 9, 32, 41, 91, 223cdef
13-limit[edit]
Commas: 105/104, 144/143, 196/195, 245/242
POTE temperament: 263.700
Map: [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|]
EDOs: 9, 41, 91
Slender[edit]
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||
EDOs: 31, 94, 125
Badness: 0.0569
11-limit[edit]
Commas: 225/224, 385/384, 1331/1323
POTE temperament: ~49/48 = 38.387
Map: [<1 2 2 3 4|, <0 -13 10 -6 -17|]
EDOs: 31, 63, 94, 125
Badness: 25.342
13-limit[edit]
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|]
EDOs: 31, 63, 94
Badness: 25.913
Marvo[edit]
Commas: 225/224, 78125000/78121827
POTE temperament: ~27/20 = 516.694
Map: [<1 5 12 29|, <0 -6 -17 -46|]
Wedgie: <<6, 17, 46, 13, 56 59||
EDOs: 65, 72
11-limit[edit]
Commas: 225/224, 243/242, 4000/3993
POTE temperament: ~27/20 = 516.699
Map: [<1 5 12 29 12|, <0 -6 -17 -46 -15|]
EDOs: 65, 72, 281, 497
13-limit[edit]
Commas: 243/242, 225/224, 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[edit]
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[edit]
Commas: 225/224, 441/440, 4000/3993
POTE generator: ~21/20 = 83.340
Map: [<1 2 1 1 2|, <0 -6 19 26 21|]
EDOs: 29, 43, 72
Badness: 0.0290
13-limit[edit]
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|]
EDOs: 29, 43, 72, 115f
Badness: 0.0215
Amavil[edit]
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||
EDOs: 9, 34d, 43, 77d
Badness: 0.1096
11-limit[edit]
Commas: 99/98, 176/175, 864/847
POTE generator: ~15/11 = 529.974
Map: [<1 6 1 9 7|, <0 -10 3 -14 -8|]
EDOs: 9, 34d, 43, 77de
Badness: 0.0426
13-limit[edit]
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|]
EDOs: 9, 34d, 43, 77de
Badness: 0.0258
Enneaportent[edit]
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||
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937
11-limit[edit]
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[edit]
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|]
EDOs: 9, 54, 63, 72, 279cf
Badness: 0.0223
Submajor[edit]
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[edit]
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||
EDOs: 10, 33, 43, 53
Badness: 0.0605
11-limit[edit]
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[edit]
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[edit]
Commas: 99/98, 176/175, 51200/50421
POTE generator: ~49/40 = 362.418
Map: [<1 4 -1 1 -5|, <0 -8 11 6 28|]
EDOs: 43, 53, 96, 149d
Badness: 0.0518
13-limit[edit]
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|]
EDOs: 43, 53, 96, 149d
Badness: 0.0297
Alphorn[edit]
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[edit]
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