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:

Hanson Improv by Petr Parizek

In Hanson11 by Chris Vaisvil

Analysis and diagrams:

11 note chain-of-minor-thirds scale, by David Keenan

Seven limit children

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

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||

EDOs: 15, 19, 53d, 72d, 91d

Badness: 0.0274

11-limit

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|]

EDOs: 4, 15, 19, 34, 102de

Badness: 0.0274

13-limit

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|]

EDOs: 4, 15f, 19, 53def, 72def

Badness: 0.0297

Kema

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

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

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

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

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

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

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

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

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||

EDOs: 19, 53, 72, 197, 269c

Badness: 0.0215

11-limit

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

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

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

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

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

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

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

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

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: 19d, 34, 53, 87, 140, 333, 473, 806b

Badness: 0.0521

11-limit

Commas: 385/384, 2200/2187, 3388/3375

POTE generator: ~6/5 = 317.162

Map: [<1 0 1 11 -5|, <0 6 5 -31 32|]

EDOs: 34, 53, 87, 140, 227

Badness: 0.0398

13-limit

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

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: 34d, 53d, 87, 121, 208

Badness: 0.1635

11-limit

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: 34d, 53d, 87, 121, 208

Badness: 0.0486

13-limit

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: 34d, 53d, 87, 121, 208

Badness: 0.0244

Hemikleismic

Commas: 4000/3969, 6144/6125

POTE generator: 158.649

Map: [<1 0 1 4|, <0 12 10 -9|]

EDOs: 15, 53, 121

Badness: 0.0521

11-limit

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

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

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

Edos: 19, 49, 68, 87, 155

Badness: 0.0473

11-limit

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

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

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

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

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||

EDOs: 15, 57, 72, 159, 231

Badness: 0.0563

11-limit

Commas: 385/384, 441/440, 4000/3993

POTE generator: 316.881

Map: [<3 0 3 10 8|, <0 6 5 -2 3|]

EDOs: 15, 57, 72, 159, 231

Badness: 0.0193

13-limit

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

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||

EDOs: 68, 72, 140, 212, 1200

Badness: 0.0392

11-limit

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

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|]

EDOs: 68, 72, 140, 212

Badness: 0.0187

Kleiboh

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

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

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

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

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

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

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|]

EDOs: 49, 72, 193, 265

Badness: 0.0704

11-limit

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

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

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

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

sqrtphi17

sqrtphi23

sqrtphi49

Music

Prelude for Piano in Square root of Phi Tuning by Chris Vaisvil

A Fight for Phi by Vito Sicurella