Kleismic family
The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is |-6 -5 6>, and flipping that yields <<6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called Hanson, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.
valid range: [300.000, 327.273] (4 to 11b)
nice range: [315.641, 317.263]
strict range: [315.641, 317.263]
POTE generator: 317.007
Map: [<1 0 1|, <0 6 5|]
EDOs: 15, 19, 34, 53, 458, 882c
Music:
Analysis and diagrams:
11 note chain-of-minor-thirds scale, by David Keenan
Seven limit children[edit]
The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.
Keemun[edit]
Commas: 49/48, 126/125
valid range: [300.000, 327.273] (4 to 11b)
nice range: [308.744, 322.942]
strict range: [308.744, 322.942]
POTE generator: ~6/5 = 316.473
Map: [<1 0 1 2|, <0 6 5 3|]
Wedgie: <<6 5 3 -6 -12 -7||
Badness: 0.0274
11-limit[edit]
Commas: 49/48, 56/55, 100/99
valid range: [315.789, 320.000] (19 to 15)
nice range: [308.744, 324.341]
strict range: [315.789, 320.000]
POTE generator: ~6/5 = 317.576
Map: [<1 0 1 2 4|, <0 6 5 3 -2|]
Badness: 0.0274
13-limit[edit]
Commas: 49/48, 56/55, 78/77, 100/99
valid range: 315.789 (19)
nice range: [303.597, 324.341]
strict range: 315.789
POTE generator: ~6/5 = 316.611
Map: [<1 0 1 2 4 5|, <0 6 5 3 -2 -5|]
Badness: 0.0297
Kema[edit]
Commas: 49/48, 56/55, 91/90, 100/99
valid range: [315.789, 320.000] (19 to 15)
nice range: [308.744, 324.341]
strict range: [315.789, 320.000]
POTE generator: ~6/5 = 317.423
Map: [<1 0 1 2 4 0|, <0 6 5 3 -2 14|]
EDOs: 15, 19, 34, 87de
Badness: 0.0227
Kumbaya[edit]
Commas: 40/39, 49/48, 56/55, 66/65
POTE generator: ~6/5 = 318.595
Map: [<1 0 1 2 4 4|, <0 6 5 3 -2 -1|]
EDOs: 15, 34f
Badness: 0.0316
Qeema[edit]
Commas: 45/44, 49/48, 126/125
POTE generator: ~6/5 = 314.730
Map: [<1 0 1 2 -1|, <0 6 5 3 17|]
EDOs: 19, 42bcd, 61bcd
Badness: 0.0401
13-limit[edit]
Commas: 45/44, 49/48, 78/77, 126/125
POTE generator: ~6/5 = 315.044
Map: [<1 0 1 2 -1 0|, <0 6 5 3 17 14|]
EDOs: 19
Badness: 0.0294
Darjeeling[edit]
Commas: 49/48, 55/54, 77/75
POTE generator: ~6/5 = 317.656
Map: [<1 0 1 2 0|, <0 6 5 3 13|]
EDOs: 15, 19e, 34e
Badness: 0.0276
13-limit Darjeeling[edit]
Commas: 49/48, 55/54, 66/65, 77/75
POTE generator: ~6/5 = 317.298
Map: [<1 0 1 2 0 0|, <0 6 5 3 13 14|]
EDOs: 15, 19e, 34e, 53de
Badness: 0.02144
Catalan[edit]
Commas: 64/63, 15625/15552
POTE generator: ~6/5 = 318.267
Map: [<1 0 1 6|, <0 6 5 -12|]
Wedgie: <<6 5 -12 -6 -36 -42||
EDOs: 15, 34d, 49, 132bcd
Badness: 0.0949
11-limit[edit]
Commas: 64/63, 100/99, 1331/1323
POTE generator: ~6/5 = 318.282
Map: [<1 0 1 6 4|, <0 6 5 -12 -2|]
EDOs: 15, 34d, 49
Badness: 0.0369
Catakleismic[edit]
Commas: 225/224, 4375/4374
valid range: [315.789, 317.647] (19 to 34)
nice range: [315.641, 317.263]
strict range: [315.789, 317.263]
POTE generator: 316.732
Map: [<1 0 1 -3|, <0 6 5 22|]
Wedgie: <<6 5 22 -6 18 37||
Badness: 0.0215
11-limit[edit]
Commas: 225/224, 385/384, 4375/4374
valid range: [315.789, 316.981] (19 to 53)
nice range: [315.641, 317.263]
strict range: [315.789, 316.981]
POTE generator: 316.719
Map: [<1 0 1 -3 9|, <0 6 5 22 -21|]
EDOs: 19, 53, 72, 197e, 269ce, 341ce, 610bce
Badness: 0.0218
13-limit[edit]
Commas: 169/168, 225/224, 325/324, 540/539
valid range: [315.789, 316.981] (19 to 53)
nice range: [315.641, 318.309]
strict range: [315.789, 316.981]
POTE generator: 316.738
Map: [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|]
EDOs: 19, 53, 72, 125f, 197ef, 269cef
Badness: 0.0169
Cataclysmic[edit]
Commas: 99/98, 176/175, 2200/2187
POTE generator: ~6/5 = 317.042
Map: [<1 0 1 -3 -5|, <0 6 5 22 32|]
EDOs: 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de
Badness: 0.0400
13-limit[edit]
Commas: 99/98, 169/168, 176/175, 275/273
POTE generator: ~6/5 = 317.036
Map: [<1 0 1 -3 -5 0|, <0 6 5 22 32 14|]
EDOs: 53, 87d, 140d, 193de, 246de
Badness: 0.0226
Catalytic[edit]
Commas: 225/224, 441/440, 4375/4374
POTE generator: ~6/5 = 316.653
Map: [<1 0 1 -3 -10|, <0 6 5 22 51|]
EDOs: 53e, 72
Badness: 0.0304
13-limit[edit]
Commas: 169/168 225/224 325/324 1716/1715
POTE generator: ~6/5 = 316.639
Map: [<1 0 1 -3 -10 0|, <0 6 5 22 51 14|]
EDOs: 19e, 53e, 72
Badness: 0.0223
Cataleptic[edit]
Commas: 100/99, 225/224, 864/847
POTE generator: ~6/5 = 317.083
Map: [<1 0 1 -3 4|, <0 6 5 22 -2|]
EDOs: 19, 34d, 53e
Badness: 0.0443
13-limit[edit]
Commas: 78/77, 100/99, 144/143, 676/675
POTE generator: ~6/5 = 317.118
Map: [<1 0 1 -3 4 0|, <0 6 5 22 -2 14|]
EDOs: 19, 34d, 53e, 87de
Badness: 0.0273
Countercata[edit]
Commas: 15625/15552, 5120/5103
POTE generator: 317.121
Map: [<1 0 1 11|, <0 6 5 -31|]
Wedgie: <<6 5 -31 -6 -66 -86||
EDOs: 15, 19, 34, 53, 87, 140, 333, 473, 806b
Badness: 0.0521
11-limit[edit]
Commas: 385/384, 2200/2187, 3388/3375
POTE generator: ~6/5 = 317.162
Map: [<1 0 1 11 -5|, <0 6 5 -31 32|]
Badness: 0.0398
13-limit[edit]
Commas: 325/324, 352/351, 385/384, 625/624
POTE generator: ~6/5 = 317.162
Map: [<1 0 1 11 -5 0|, <0 6 5 -31 32 14|]
EDOs: 34, 53, 87, 140, 227, 367e, 507e
Badness: 0.0202
Metakleismic[edit]
Commas: 15625/15552, 179200/177147
POTE generator: ~6/5 = 317.314
Map: [<1 0 1 -12|, <0 6 5 56|]
Wedgie: <<6 5 56 -6 72 116||
EDOs: 15, 19, 34, 87, 121, 208
Badness: 0.1635
11-limit[edit]
Commas: 896/891, 2200/2187, 14700/14641
POTE generator: ~6/5 = 317.311
Map: [<1 0 1 -12 -5|, <0 6 5 56 32|]
EDOs: 15, 19, 34, 87, 121, 208
Badness: 0.0486
13-limit[edit]
Commas: 325/324, 352/351, 364/363, 625/624
POTE generator: ~6/5 = 317.311
Map: [<1 0 1 -12 -5 0|, <0 6 5 56 32 14|]
EDOs: 15, 19, 34, 87, 121, 208
Badness: 0.0244
Hemikleismic[edit]
Commas: 4000/3969, 6144/6125
POTE generator: 158.649
Map: [<1 0 1 4|, <0 12 10 -9|]
Badness: 0.0521
11-limit[edit]
Commas: 121/120, 176/175, 4000/3969
POTE generator: ~11/10 = 158.677
Map: [<1 0 1 4 2|, <0 12 10 -9 11|]
EDOs: 15, 38, 53, 68, 121e
Badness: 0.0380
13-limit[edit]
Commas: 121/120, 176/175, 275/273, 325/324
POTE generator: ~11/10 = 158.655
Map: [<1 0 1 4 2 0|, <0 12 10 -9 11 28|]
EDOs: 15, 53, 121e
Badness: 0.0260
Clyde[edit]
Commas: 245/243, 3136/3125
7 and 9 limit minimax
[|1 0 0 0>, |6/25 0 0 12/25>, |6/5 0 0 2/5>, |0 0 0 1>]
Eigenmonzos: 2, 7
POTE generator: ~9/7 = 441.335
Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.
Map: [<1 6 6 12|, <0 -12 -10 -25|]
Generators: 2, 9/7
Badness: 0.0473
11-limit[edit]
Commas: 245/243, 3136/3125, 385/384
POTE generator: ~9/7 = 441.355
Map: [<1 6 6 12 -5|, <0 -12 -10 -25 23|]
EDOs: 19, 68, 87, 329bd, 419bd, 503bd, 590bd
Badness: 0.0474
13-limit[edit]
Commas: 196/195, 245/243, 385/384, 625/624
POTE generator: ~9/7 = 441.363
Map: [<1 6 6 12 -5 14|, <0 -12 -10 -25 23 -28|]
EDOs: 19, 68, 87, 503bdf, 590bdf
Badness: 0.0268
Bikleismic[edit]
Commas: 225/224, 243/242, 4375/4356
POTE generator: ~6/5 = 316.721
Map: [<2 0 2 -6 -1|, <0 6 5 22 15|]
EDOs: 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc
Badness: 0.0293
13-limit[edit]
Commas: 169/168, 225/224, 243/242, 325/324
POTE generator: ~6/5 = 316.726
Map: [<2 0 2 -6 -1 0|, <0 6 5 22 15 14|]
EDOs: 72, 106, 322cff, 394cff, 466bcff, 538bcfff
Badness: 0.0218
Tritikleismic[edit]
Commas: 15625/15552, 1029/1024
POTE generator: 316.872
Map: [<3 0 3 10|, <0 6 5 -2|]
Wedgie: <<18 15 -6 -18 -60 -56||
Badness: 0.0563
11-limit[edit]
Commas: 385/384, 441/440, 4000/3993
POTE generator: 316.881
Map: [<3 0 3 10 8|, <0 6 5 -2 3|]
Badness: 0.0193
13-limit[edit]
Commas: 325/324, 364/363, 441/440, 625/624
POTE generator: 316.959
Map: [<3 0 3 10 8 0|, <0 6 5 -2 3 14|]
EDOs: 15, 72, 87, 159, 867, 1026
Badness: 0.0157
Quadritikleismic[edit]
Commas: 15625/15552, 2401/2400
POTE generator: 316.9999
Map: [<4 0 4 7|, <0 6 5 4|]
Wedgie: <<24 20 16 -24 -42 -19||
Badness: 0.0392
11-limit[edit]
Commas: 385/384, 1375/1372, 6250/6237
POTE generator: 316.925
Map: [<4 0 4 7 17|, <0 6 5 4 -3|]
EDOs: 68, 72, 140, 212, 284, 496, 780
Badness: 0.0234
13-limit[edit]
Commas: 325/324, 385/384, 625/624, 1573/1568
POTE generator: 316.989
Map: [<4 0 4 7 17 0|, <0 6 5 4 -3 14|]
Badness: 0.0187
Kleiboh[edit]
Commas: 1728/1715, 3125/3087
POTE generator: ~25/21 = 294.303
Map: [<1 6 6 6|, <0 -18 -15 -13|]
Wedgie: <<18 15 13 -18 -30 -12||
EDOs: 49, 53, 314d
Badness: 0.0765
11-limit[edit]
Commas: 176/175, 540/539, 3125/3087
POTE generator: ~25/21 = 294.181
Map: [<1 6 6 6 14|, <0 -18 -15 -13 -43|]
EDOs: 49, 53, 102d, 155d
Badness: 0.0528
13-limit[edit]
Commas: 176/175, 275/273, 325/324, 540/539
POTE generator: ~13/11 = 294.187
Map: [<1 6 6 6 14 14|, <0 -18 -15 -13 -43 -42|]
EDOs: 53, 102df, 155d
Badness: 0.0311
Novemkleismic[edit]
Commas: 15625/15552, 40353607/40310784
POTE generator: ~6/5 = 317.005
Map: [<9 0 9 11|, <0 6 5 6|]
Wedgie: <<54 45 54 -54 -66 -1||
EDOs: 72, 261, 333, 405, 477c, 882c
Badness: 0.1934
11-limit[edit]
Commas: 1375/1372, 4000/3993, 15625/15552
POTE generator: ~6/5 = 317.010
Map: [<9 0 9 11 24|, <0 6 5 6 3|]
EDOs: 72, 261, 333, 405, 882c
Badness: 0.05172
13-limit[edit]
Commas: 325/324, 625/624, 1375/1372, 4000/3993
POTE generator: ~6/5 = 317.086
Map: [<9 0 9 11 24 0|, <0 6 5 6 3 14|]
EDOs: 72, 261, 333, 738cf, 1071bcf
Badness: 0.0391
Sqrtphi[edit]
Commas: 15625/15552, 16875/16807
POTE generator: ~125/98 = 416.603 cents
Sqrt(phi) = 416.545 cents
Map: [<1 12 11 16|, <0 -30 -25 -38|]
Badness: 0.0704
11-limit[edit]
Commas: 540/539, 1375/1372, 4375/4356
POTE generator: ~14/11 = 416.604
Map: [<1 12 11 16 17|, <0 -30 -25 -38 -39|]
EDOs: 49, 72, 193, 265
Badness: 0.0255
13-limit[edit]
Commas: 325/324, 364/363, 625/624, 1375/1372
POTE generator: ~14/11 = 416.585
Map: [<1 12 11 16 17 28|, <0 -30 -25 -38 -39 -70|]
EDOs: 72, 121, 193
Badness: 0.0200
17-limit[edit]
Commas: 325/324, 364/363, 375/374, 540/539, 595/594
POTE generator: ~14/11 = 416.585
Map: [<1 12 11 16 17 28 27|, <0 -30 -25 -38 -39 -70 -66|]
EDOs: 72, 121, 193
Badness: 0.0130
19-limit[edit]
Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
POTE generator: ~14/11 = 416.580
Map: [<1 12 11 16 17 28 27 -2|, <0 -30 -25 -38 -39 -70 -66 18|]
EDOs: 72, 121, 193
Badness: 0.0147
Scales[edit]
Music[edit]
Prelude for Piano in Square root of Phi Tuning by Chris Vaisvil
A Fight for Phi by Vito Sicurella