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.

Hanson

Comma list: 15625/15552

POTE generator: ~6/5 = 317.007

Mapping: [⟨1 0 1], ⟨0 6 5]]

Tuning ranges:

• valid range: [300.000, 327.273] (4 to 11b)
• nice range: [315.641, 317.263]
• strict range: [315.641, 317.263]

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 extensions

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

Comma list: 49/48, 126/125

POTE generator: ~6/5 = 316.473

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Wedgie: ⟨⟨ 6 5 3 -6 -12 -7 ]]

Tuning ranges:

• valid range: [300.000, 327.273] (4 to 11b)
• nice range: [308.744, 322.942]
• strict range: [308.744, 322.942]

Badness: 0.0274

11-limit

Comma list: 49/48, 56/55, 100/99

POTE generator: ~6/5 = 317.576

Mapping: [⟨1 0 1 2 4], ⟨0 6 5 3 -2]]

Tuning ranges:

• valid range: [315.789, 320.000] (19 to 15)
• nice range: [308.744, 324.341]
• strict range: [315.789, 320.000]

Badness: 0.0274

13-limit

Comma list: 49/48, 56/55, 78/77, 100/99

POTE generator: ~6/5 = 316.611

Mapping: [⟨1 0 1 2 4 5], ⟨0 6 5 3 -2 -5]]

Tuning ranges:

• valid range: 315.789 (19)
• nice range: [303.597, 324.341]
• strict range: 315.789

Badness: 0.0297

Kema

Comma list: 49/48, 56/55, 91/90, 100/99

POTE generator: ~6/5 = 317.423

Mapping: [⟨1 0 1 2 4 0], ⟨0 6 5 3 -2 14]]

Tuning ranges:

• valid range: [315.789, 320.000] (19 to 15)
• nice range: [308.744, 324.341]
• strict range: [315.789, 320.000]

Badness: 0.0227

Kumbaya

Comma list: 40/39, 49/48, 56/55, 66/65

POTE generator: ~6/5 = 318.595

Mapping: [⟨1 0 1 2 4 4], ⟨0 6 5 3 -2 -1]]

Badness: 0.0316

Qeema

Comma list: 45/44, 49/48, 126/125

POTE generator: ~6/5 = 314.730

Mapping: [⟨1 0 1 2 -1], ⟨0 6 5 3 17]]

Badness: 0.0401

13-limit

Comma list: 45/44, 49/48, 78/77, 126/125

POTE generator: ~6/5 = 315.044

Mapping: [⟨1 0 1 2 -1 0], ⟨0 6 5 3 17 14]]

Badness: 0.0294

Darjeeling

Comma list: 49/48, 55/54, 77/75

POTE generator: ~6/5 = 317.656

Mapping: [⟨1 0 1 2 0], ⟨0 6 5 3 13]]

Badness: 0.0276

13-limit

Comma list: 49/48, 55/54, 66/65, 77/75

POTE generator: ~6/5 = 317.298

Mapping: [⟨1 0 1 2 0 0], ⟨0 6 5 3 13 14]]

Badness: 0.02144

Catalan

Comma list: 64/63, 15625/15552

POTE generator: ~6/5 = 318.267

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Wedgie: ⟨⟨ 6 5 -12 -6 -36 -42 ]]

Badness: 0.0949

11-limit

Comma list: 64/63, 100/99, 1331/1323

POTE generator: ~6/5 = 318.282

Mapping: [⟨1 0 1 6 4], ⟨0 6 5 -12 -2]]

Badness: 0.0369

Catakleismic

Comma list: 225/224, 4375/4374

POTE generator: ~6/5 = 316.732

Mapping: [⟨1 0 1 -3], ⟨0 6 5 22]]

Wedgie: ⟨⟨ 6 5 22 -6 18 37 ]]

Tuning ranges:

• valid range: [315.789, 317.647] (19 to 34)
• nice range: [315.641, 317.263]
• strict range: [315.789, 317.263]

Badness: 0.0215

11-limit

Comma list: 225/224, 385/384, 4375/4374

POTE generator: ~6/5 = 316.719

Mapping: [⟨1 0 1 -3 9], ⟨0 6 5 22 -21]]

Tuning ranges:

• valid range: [315.789, 316.981] (19 to 53)
• nice range: [315.641, 317.263]
• strict range: [315.789, 316.981]

Badness: 0.0218

13-limit

Comma list: 169/168, 225/224, 325/324, 540/539

POTE generator: ~6/5 = 316.738

Mapping: [⟨1 0 1 -3 9 0], ⟨0 6 5 22 -21 14]]

Tuning ranges:

• valid range: [315.789, 316.981] (19 to 53)
• nice range: [315.641, 318.309]
• strict range: [315.789, 316.981]

Badness: 0.0169

Cataclysmic

Comma list: 99/98, 176/175, 2200/2187

POTE generator: ~6/5 = 317.042

Mapping: [⟨1 0 1 -3 -5], ⟨0 6 5 22 32]]

Badness: 0.0400

13-limit

Comma list: 99/98, 169/168, 176/175, 275/273

POTE generator: ~6/5 = 317.036

Mapping: [⟨1 0 1 -3 -5 0], ⟨0 6 5 22 32 14]]

Badness: 0.0226

Catalytic

Comma list: 225/224, 441/440, 4375/4374

POTE generator: ~6/5 = 316.653

Mapping: [⟨1 0 1 -3 -10], ⟨0 6 5 22 51]]

Badness: 0.0304

13-limit

Comma list: 169/168 225/224 325/324 1716/1715

POTE generator: ~6/5 = 316.639

Mapping: [⟨1 0 1 -3 -10 0], ⟨0 6 5 22 51 14]]

Badness: 0.0223

Cataleptic

Comma list: 100/99, 225/224, 864/847

POTE generator: ~6/5 = 317.083

Mapping: [⟨1 0 1 -3 4], ⟨0 6 5 22 -2]]

Badness: 0.0443

13-limit

Comma list: 78/77, 100/99, 144/143, 676/675

POTE generator: ~6/5 = 317.118

Mapping: [⟨1 0 1 -3 4 0], ⟨0 6 5 22 -2 14]]

Badness: 0.0273

Bikleismic

Comma list: 225/224, 243/242, 4375/4356

POTE generator: ~6/5 = 316.721

Mapping: [⟨2 0 2 -6 -1], ⟨0 6 5 22 15]]

Badness: 0.0293

13-limit

Comma list: 169/168, 225/224, 243/242, 325/324

POTE generator: ~6/5 = 316.726

Mapping: [⟨2 0 2 -6 -1 0], ⟨0 6 5 22 15 14]]

Badness: 0.0218

Countercata

Comma list: 5120/5103, 15625/15552

POTE generator: ~6/5 = 317.121

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Wedgie: ⟨⟨ 6 5 -31 -6 -66 -86 ]]

Badness: 0.0521

11-limit

Comma list: 385/384, 2200/2187, 3388/3375

POTE generator: ~6/5 = 317.162

Mapping: [⟨1 0 1 11 -5], ⟨0 6 5 -31 32]]

Badness: 0.0398

13-limit

Comma list: 325/324, 352/351, 385/384, 625/624

POTE generator: ~6/5 = 317.162

Mapping: [⟨1 0 1 11 -5 0], ⟨0 6 5 -31 32 14]]

Badness: 0.0202

Metakleismic

Comma list: 15625/15552, 179200/177147

POTE generator: ~6/5 = 317.314

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Wedgie: ⟨⟨ 6 5 56 -6 72 116 ]]

Badness: 0.1635

11-limit

Comma list: 896/891, 2200/2187, 14700/14641

POTE generator: ~6/5 = 317.311

Mapping: [⟨1 0 1 -12 -5], ⟨0 6 5 56 32]]

Badness: 0.0486

13-limit

Comma list: 325/324, 352/351, 364/363, 625/624

POTE generator: ~6/5 = 317.311

Mapping: [⟨1 0 1 -12 -5 0], ⟨0 6 5 56 32 14]]

Badness: 0.0244

Hemikleismic

Comma list: 4000/3969, 6144/6125

POTE generator: 158.649

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

Badness: 0.0521

11-limit

Comma list: 121/120, 176/175, 4000/3969

POTE generator: ~11/10 = 158.677

Mapping: [⟨1 0 1 4 2], ⟨0 12 10 -9 11]]

Badness: 0.0380

13-limit

Comma list: 121/120, 176/175, 275/273, 325/324

POTE generator: ~11/10 = 158.655

Mapping: [⟨1 0 1 4 2 0], ⟨0 12 10 -9 11 28]]

Badness: 0.0260

Clyde

Comma list: 245/243, 3136/3125

POTE generator: ~9/7 = 441.335

Mapping: [⟨1 6 6 12], ⟨0 -12 -10 -25]]

Mapping generators: ~2, ~9/7

Minimax tuning:

• 7- and 9-odd-limit:

[[1 0 0 0⟩, [6/25 0 0 12/25⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]

Eigenmonzos: 2, 7

Algebraic generator: real root of 5x³ - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

Badness: 0.0473

11-limit

Comma list: 245/243, 3136/3125, 385/384

POTE generator: ~9/7 = 441.355

Mapping: [⟨1 6 6 12 -5], ⟨0 -12 -10 -25 23]]

Badness: 0.0474

13-limit

Comma list: 196/195, 245/243, 385/384, 625/624

POTE generator: ~9/7 = 441.363

Mapping: [⟨1 6 6 12 -5 14], ⟨0 -12 -10 -25 23 -28]]

Badness: 0.0268

Tritikleismic

See also: Gamelismic clan § Tritikleismic

Comma list: 1029/1024, 15625/15552

POTE generator: 316.872

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

Wedgie: ⟨⟨ 18 15 -6 -18 -60 -56 ]]

Badness: 0.0563

11-limit

Comma list: 385/384, 441/440, 4000/3993

POTE generator: ~6/5 = 316.881

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Badness: 0.0193

13-limit

Comma list: 325/324, 364/363, 441/440, 625/624

POTE generator: ~6/5 = 316.959

Mapping: [⟨3 0 3 10 8 0], ⟨0 6 5 -2 3 14]]

Badness: 0.0157

Quadritikleismic

Comma list: 2401/2400, 15625/15552

POTE generator: ~6/5 = 316.9999

Mapping: [⟨4 0 4 7], ⟨0 6 5 4]]

Wedgie: ⟨⟨ 24 20 16 -24 -42 -19 ]]

Badness: 0.0392

11-limit

Comma list: 385/384, 1375/1372, 6250/6237

POTE generator: ~6/5 = 316.925

Mapping: [⟨4 0 4 7 17], ⟨0 6 5 4 -3]]

Badness: 0.0234

13-limit

Comma list: 325/324, 385/384, 625/624, 1573/1568

POTE generator: ~6/5 = 316.989

Mapping: [⟨4 0 4 7 17 0], ⟨0 6 5 4 -3 14]]

Badness: 0.0187

Kleiboh

Comma list: 1728/1715, 3125/3087

POTE generator: ~25/21 = 294.303

Mapping: [⟨1 6 6 6], ⟨0 -18 -15 -13]]

Wedgie: ⟨⟨ 18 15 13 -18 -30 -12 ]]

Badness: 0.0765

11-limit

Comma list: 176/175, 540/539, 3125/3087

POTE generator: ~25/21 = 294.181

Mapping: [⟨1 6 6 6 14], ⟨0 -18 -15 -13 -43]]

Badness: 0.0528

13-limit

Comma list: 176/175, 275/273, 325/324, 540/539

POTE generator: ~13/11 = 294.187

Mapping: [⟨1 6 6 6 14 14], ⟨0 -18 -15 -13 -43 -42]]

Badness: 0.0311

Novemkleismic

Comma list: 15625/15552, 40353607/40310784

POTE generator: ~6/5 = 317.005

Mapping: [⟨9 0 9 11], ⟨0 6 5 6]]

Wedgie: ⟨⟨ 54 45 54 -54 -66 -1 ]]

Badness: 0.1934

11-limit

Comma list: 1375/1372, 4000/3993, 15625/15552

POTE generator: ~6/5 = 317.010

Mapping: [⟨9 0 9 11 24], ⟨0 6 5 6 3]]

Badness: 0.05172

13-limit

Comma list: 325/324, 625/624, 1375/1372, 4000/3993

POTE generator: ~6/5 = 317.086

Mapping: [⟨9 0 9 11 24 0], ⟨0 6 5 6 3 14]]

Badness: 0.0391

Sqrtphi

The just value of sqrt (φ) is 416.545 cents.

Comma list: 15625/15552, 16875/16807

POTE generator: ~125/98 = 416.603 cents

Mapping: [⟨1 12 11 16], ⟨0 -30 -25 -38]]

Badness: 0.0704

11-limit

Comma list: 540/539, 1375/1372, 4375/4356

POTE generator: ~14/11 = 416.604

Mapping: [⟨1 12 11 16 17], ⟨0 -30 -25 -38 -39]]

Badness: 0.0255

13-limit

Comma list: 325/324, 364/363, 625/624, 1375/1372

POTE generator: ~14/11 = 416.585

Mapping: [⟨1 12 11 16 17 28], ⟨0 -30 -25 -38 -39 -70]]

Badness: 0.0200

17-limit

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

POTE generator: ~14/11 = 416.585

Mapping: [⟨1 12 11 16 17 28 27], ⟨0 -30 -25 -38 -39 -70 -66]]

Badness: 0.0130

19-limit

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

POTE generator: ~14/11 = 416.580

Mapping: [⟨1 12 11 16 17 28 27 -2], ⟨0 -30 -25 -38 -39 -70 -66 18]]

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