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 classical minor third (6/5), 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.

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. 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.

Hanson

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.007

Tuning ranges:

• 5-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
• 5-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)
• 5-odd-limit diamond monotone and tradeoff: ~6/5 = [315.641, 317.263]

Optimal ET sequence: 15, 19, 34, 53, 458, 511c, …, 882c

Badness: 0.013234

Music

• Hanson Improv by Petr Parizek
• In Hanson11 by Chris Vaisvil

Analysis and diagrams

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

Cata

Main article: Catakleismic

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Sval mapping: [⟨1 0 1 0], ⟨0 6 5 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.0756

Optimal ET sequence: 15, 19, 34, 53, 140, 193, 246

Badness: 0.394

Keemun

Main article: Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Wedgie: ⟨⟨ 6 5 3 -6 -12 -7 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.473

Tuning ranges:

• 7-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
• 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [308.744, 322.942]
• 7-odd-limit diamond monotone and tradeoff: ~6/5 = [308.744, 322.942]
• 9-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]

Optimal ET sequence: 15, 19, 53d, 72dd, 91dd

Badness: 0.027408

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 1 2 4], ⟨0 6 5 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.576

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]

Optimal ET sequence: 4, 15, 19, 34

Badness: 0.027410

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 65/64, 100/99

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.611

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~6/5 = 315.789

Optimal ET sequence: 4, 15f, 19, 53def, 72def

Badness: 0.029749

Kema

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.423

Tuning ranges:

• 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
• 13-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]
• 15-odd-limit diamond monotone and tradeoff: ~6/5 = 315.789

Optimal ET sequence: 15, 19, 34, 87ddee

Badness: 0.022749

Kumbaya

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.595

Optimal ET sequence: 4, 15, 19f, 34ff

Badness: 0.031633

Qeema

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 1 2 -1], ⟨0 6 5 3 17]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 314.730

Optimal ET sequence: 4e, 19, 42bcd, 61bcdd

Badness: 0.040056

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.044

Optimal ET sequence: 4ef, 19

Badness: 0.029419

Darjeeling

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 1 2 0], ⟨0 6 5 3 13]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.656

Optimal ET sequence: 15, 19e, 34e

Badness: 0.027648

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.298

Optimal ET sequence: 15, 19e, 34e, 53dee

Badness: 0.021445

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Wedgie: ⟨⟨ 6 5 -12 -6 -36 -42 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.267

Tuning ranges:

• 7- and 9-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [315.641, 319.265]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [317.647, 319.265]

Optimal ET sequence: 15, 34d, 49, 132bcd

Badness: 0.094872

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.282

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
• 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [317.647, 320.000]

Optimal ET sequence: 15, 34d, 49

Badness: 0.036894

Catakleismic

Main article: Catakleismic

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3], ⟨0 6 5 22]]

Wedgie: ⟨⟨ 6 5 22 -6 18 37 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.732

Tuning ranges:

• 7- and 9-odd-limit diamond monotone: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.263]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 317.263]

Optimal ET sequence: 19, 34d, 53, 72, 197, 269c

Badness: 0.021501

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

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

Sval mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 14]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8865

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197f

Badness: 0.0118

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.719

Tuning ranges:

• 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
• 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 316.981]

Optimal ET sequence: 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee

Badness: 0.021849

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 385/384

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.738

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 316.981]

Optimal ET sequence: 19, 34de, 53, 72, 125f, 197ef, 269cef

Badness: 0.016883

Cataclysmic

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.042

Optimal ET sequence: 19e, 34d, 53

Badness: 0.039954

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.036

Optimal ET sequence: 19e, 34d, 53

Badness: 0.022555

Catalytic

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.653

Optimal ET sequence: 19e, 53e, 72

Badness: 0.030422

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.639

Optimal ET sequence: 19e, 53e, 72

Badness: 0.022337

Cataleptic

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.083

Optimal ET sequence: 19, 34d, 53e

Badness: 0.044335

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.118

Optimal ET sequence: 19, 34d, 53e, 87dee

Badness: 0.027343

Bikleismic

Subgroup: 2.3.5.7.11

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

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

mapping generators: ~99/70, ~6/5

Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 316.721

Optimal ET sequence: 34d, 72, 322c, …, 610bcc

Badness: 0.029319

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 316.726

Optimal ET sequence: 34d, 72

Badness: 0.021814

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 325/324

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

Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726

Optimal ET sequence: 34d, 38df, 72

Badness: 0.015656

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0 5 -1], ⟨0 6 5 22 15 14 6 18]]

Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726

Optimal ET sequence: 34dh, 38df, 72

Badness: 0.015771

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Wedgie: ⟨⟨ 6 5 -31 -6 -66 -86 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.121

Tuning ranges:

• 7- and 9-odd-limit diamond monotone: ~6/5 = [316.667, 317.647] (19\72 to 9\34)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.263]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [316.667, 317.263]

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

Badness: 0.052129

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
• 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.370]

Optimal ET sequence: 34, 53, 87, 140, 227, 367e, 507e

Badness: 0.039770

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162

Tuning ranges:

• 13-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
• 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
• 13-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.647]
• 15-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.241]

Optimal ET sequence: 34, 53, 87, 140, 367e, 507e

Badness: 0.020156

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Wedgie: ⟨⟨ 6 5 56 -6 72 116 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.314

Optimal ET sequence: 34d, 87, 121, 208

Badness: 0.163519

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311

Optimal ET sequence: 34d, 53d, 87, 121, 208

Badness: 0.048570

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311

Optimal ET sequence: 34d, 53d, 87, 121, 208

Badness: 0.024371

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

Wedgie: ⟨⟨ 12 10 -9 -12 -48 -49 ]]

Optimal tuning (POTE): ~2 = 1\1, ~35/32 = 158.649

Optimal ET sequence: 15, 38, 53, 121

Badness: 0.052054

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.677

Optimal ET sequence: 15, 38, 53, 68, 121e

Badness: 0.038023

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.655

Optimal ET sequence: 15, 38f, 53, 121e

Badness: 0.026005

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

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

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 12 10 25 -12 6 30 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.335

Minimax tuning:

• 7- and 9-odd-limit: ~9/7 = [12/25 0 0 -1/25⟩

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

eigenmonzo (unchanged-interval) basis: 2.7

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

Optimal ET sequence: 19, 49, 68, 87, 155

Badness: 0.047261

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.355

Optimal ET sequence: 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd

Badness: 0.047417

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.363

Optimal ET sequence: 19, 49ef, 68, 87, 503bdf, 590bdf

Badness: 0.026842

Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

mapping generators: ~63/50, ~6/5

Wedgie: ⟨⟨ 18 15 -6 -18 -60 -56 ]]

Optimal tuning (POTE): ~63/50 = 1\3, ~6/5 = 316.872

Minimax tuning:

• 7-odd-limit: ~6/5 = [1/3 0 1/7 -1/7⟩

[[1 0 0 0⟩, [2 0 6/7 -6/7⟩, [8/3 0 5/7 -5/7⟩, [8/3 0 -2/7 2/7⟩]

eigenmonzo (unchanged-interval) basis: 2.7/5

• 9-odd-limit: ~6/5 = [5/21 1/7 0 -1/14⟩

[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [46/21 5/7 0 -5/14⟩, [20/7 -2/7 0 1/7⟩]

eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 15, 42bc, 57, 72, 159, 231

Badness: 0.056337

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.881

Minimax tuning:

• 11-odd-limit: ~6/5 = [5/21 1/7 0 -1/14⟩

[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [46/21 5/7 0 -5/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [71/21 3/7 0 -3/14 0⟩]

eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 15, 42bc, 57, 72, 159, 231

Badness: 0.019333

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

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

Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.9585

Optimal ET sequence: 72, 87, 159

Badness: 0.015652

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [⟨3 0 3 10 8 0 -2], ⟨0 6 5 -2 3 14 18]]

Optimal tuning (POTE): ~34/27 = 1\3, ~6/5 = 316.9082

Optimal ET sequence: 72, 159, 231f

Badness: 0.013551

Quadritikleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping: [⟨4 0 4 7], ⟨0 6 5 4]]

mapping generators: ~25/21, 6/5

Wedgie: ⟨⟨ 24 20 16 -24 -42 -19 ]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9999

Optimal ET sequence: 68, 72, 140, 212, 776cd, 988ccd, 1200ccd

Badness: 0.039231

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 0 4 7 17], ⟨0 6 5 4 -3]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9247

Optimal ET sequence: 68, 72, 140, 212, 284, 496ce, 780ccdee

Badness: 0.023406

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 1375/1372

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

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9887

Optimal ET sequence: 68, 72, 140, 212

Badness: 0.018731

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 625/624

Mapping: [⟨4 0 4 7 17 0 10], ⟨0 6 5 4 -3 14 6]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9846

Optimal ET sequence: 68, 72, 140, 212g

Badness: 0.012784

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [⟨1 6 6 6], ⟨0 -18 -15 -13]]

mapping generators: ~2, ~25/21

Wedgie: ⟨⟨ 18 15 13 -18 -30 -12 ]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.303

Optimal ET sequence: 49, 53, 314d

Badness: 0.076460

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.181

Optimal ET sequence: 49, 53, 102d, 155d

Badness: 0.052805

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 294.187

Optimal ET sequence: 49f, 53, 102df, 155d

Badness: 0.031074

Marfifths

The marfifths temperament (19&140) tempers out the hemimage comma, 10976/10935. It splits the interval of major tenth (~10/3) into three marvelous fifth (112/75) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 15625/15552

Mapping: [⟨1 -6 -4 -17], ⟨0 18 15 47]]

mapping generators: ~2, ~75/56

Wedgie: ⟨⟨ 18 15 47 -18 24 67 ]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.705

Optimal ET sequence: 19, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd

Badness: 0.063448

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 10976/10935

Mapping: [⟨1 -6 -4 -17 22], ⟨0 18 15 47 -44]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.684

Optimal ET sequence: 19, 140, 159, 299

Badness: 0.058902

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 10976/10935

Mapping: [⟨1 -6 -4 -17 22 -14], ⟨0 18 15 47 -44 42]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.686

Optimal ET sequence: 19, 140, 159, 299

Badness: 0.030082

Diatessic

The diatessic temperament (121 & 140) is closely related to the diatess tuning (generator: 505.727281 cents).

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2200/2187, 5632/5625

Mapping: [⟨1 -6 -4 -17 -37], ⟨0 18 15 47 96]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740

Optimal ET sequence: 19e, 121, 140, 261, 401, 662bd

Badness: 0.061172

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 625/624, 1375/1372

Mapping: [⟨1 -6 -4 -17 -37 -14], ⟨0 18 15 47 96 42]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740

Optimal ET sequence: 19e, 121, 140, 261, 401, 662bd

Badness: 0.028671

Marf

The marf temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 15625/15552

Mapping: [⟨1 -6 -4 -17 14], ⟨0 18 15 47 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.769

Optimal ET sequence: 19, 102d, 121

Badness: 0.075112

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539, 625/624, 896/891

Mapping: [⟨1 -6 -4 -17 14 -14], ⟨0 18 15 47 -25 42]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.771

Optimal ET sequence: 19, 121

Badness: 0.038317

Marthirds

The marthirds temperament (19 & 193) tempers out the sesquiquartisma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third (56/45) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 2460375/2458624

Mapping: [⟨1 -6 -4 -19], ⟨0 24 20 69]]

Wedgie: ⟨⟨ 24 20 69 -24 42 104 ]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.252

Optimal ET sequence: 19, 193, 212, 617c, 829c

Badness: 0.104253

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 15625/15552, 19712/19683

Mapping: [⟨1 -6 -4 -19 -43], ⟨0 24 20 69 147]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.257

Optimal ET sequence: 19e, 193, 212, 405, 617c, 1022cce

Badness: 0.075624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 19712/19683

Mapping: [⟨1 -6 -4 -19 -43 -14], ⟨0 24 20 69 147 56]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.256

Optimal ET sequence: 19e, 193, 212, 405f, 617cff

Badness: 0.043728

Quartkeenlig

This temperament uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the quartisma by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of stretched 23edo.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 117649/116640

Mapping: [⟨1 0 1 1], ⟨0 36 30 41]]

mapping generator: ~2, ~36/35

Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 52.8562

Optimal ET sequence: 68, 91, 159, 386d, 545dd

Badness: 0.146

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 67228/66825

Mapping: [⟨1 0 1 1 5], ⟨0 36 30 41 -35]]

Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8524

Optimal ET sequence: 68, 91, 159, 386d, 545dd

Badness: 0.0865

Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping: [⟨9 0 9 11], ⟨0 6 5 6]]

mapping generators: ~2592/2401, ~6/5

Wedgie: ⟨⟨ 54 45 54 -54 -66 -1 ]]

Optimal tuning (POTE): ~2592/2401 = 1\9, ~6/5 = 317.005

Optimal ET sequence: 72, 261, 333, 405, 477c, 882c

Badness: 0.193429

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨9 0 9 11 24], ⟨0 6 5 6 3]]

Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.010

Optimal ET sequence: 72, 261, 333, 405, 882c

Badness: 0.051730

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.086

Optimal ET sequence: 72, 261, 333, 738cf, 1071bcff

Badness: 0.039072

Sqrtphi

Main article: Sqrtphi

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

Subgroup: 2.3.5.7

Comma list: 15625/15552, 16875/16807

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

mapping generators: ~2, 125/98

Optimal tuning (POTE): ~2 = 1\1, ~125/98 = 416.603

Optimal ET sequence: 49, 72, 193, 265

Badness: 0.070378

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.604

Optimal ET sequence: 49, 72, 193, 265

Badness: 0.025515

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585

Optimal ET sequence: 72, 121, 193

Badness: 0.020040

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585

Optimal ET sequence: 72, 121, 193

Badness: 0.013028

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.580

Optimal ET sequence: 72, 121, 193

Badness: 0.014748

Scales

• Sqrtphi17
• Sqrtphi23
• Sqrtphi49

Music

• Prelude for Piano in Square root of Phi Tuning by Chris Vaisvil
• A Fight for Phi by Vito Sicurella