Kleismic family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma, which is the amount by which a stack of six classical minor thirds falls short of the 3rd harmonic.

Kleismic a.k.a. hanson

Main article: Kleismic

The generator of kleismic is a classical minor third, and to get to the interval class of major thirds requires five of these, and so to get to fifths requires six. In fact, (6/5)⁵ = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a microtemperament) is sometimes called hanson, and 14\53 is about perfect as a generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

mapping generators: ~2, ~6/5

Optimal tunings:

CTE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢
POTE: ~2 = 1200.000 ¢, ~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)

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

Badness (Smith): 0.013234

Overview to 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. 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.

6144/6125, the porwell comma, gives hemikleismic. 245/243, sensamagic, gives clyde. 1029/1024, the gamelisma, gives tritikleismic. 2401/2400 the breedsma, gives quadritikleismic. 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

Main article: Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~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]

Optimal ET sequence: 4, 15, 19, 34

Badness (Smith): 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 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 318.595 ¢

Optimal ET sequence: 4, 15, 19f, 34ff

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 314.730 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 315.044 ¢

Optimal ET sequence: 4ef, 19

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.656 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.298 ¢

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

Badness (Smith): 0.021445

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~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]

Optimal ET sequence: 15, 34d, 49, 132bcdd, 181bbcddd

Badness (Smith): 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 = 1200.000 ¢, ~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]

Optimal ET sequence: 15, 34d, 49, 181bbcdddeee

Badness (Smith): 0.036894

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 144/143, 275/273

Mapping: [⟨1 0 1 6 4 0], ⟨0 6 5 -12 -2 14]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 317.9159 ¢

Optimal ET sequence: 15, 34d, 49f, 83def, 132bcddeefff

Badness (Smith): 0.0263

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]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~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]

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

Badness (Smith): 0.021501

2.3.5.7.13 subgroup

The S-expression-based comma list of this temperament is {S13, S15 = S25*S26*S27, S10/S12 = S25*S26, (S25, S26 = S13/S15, S27)}.

Subgroup: 2.3.5.7.13

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

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 316.8865 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.042 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.036 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 316.653 ¢

Optimal ET sequence: 19e, 53e, 72

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 316.639 ¢

Optimal ET sequence: 19e, 53e, 72

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.083 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.118 ¢

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

Badness (Smith): 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 = 600.000 ¢, ~6/5 = 316.721 ¢

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

Badness (Smith): 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 = 600.000 ¢, ~6/5 = 316.726 ¢

Optimal ET sequence: 34d, 72

Badness (Smith): 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 = 600.000 ¢, ~6/5 = 316.726 ¢

Optimal ET sequence: 34d, 38df, 72

Badness (Smith): 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 = 600.000 ¢, ~6/5 = 316.726 ¢

Optimal ET sequence: 34dh, 38df, 72

Badness (Smith): 0.015771

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~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]

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

Badness (Smith): 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 = 1200.000 ¢, ~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]

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

Badness (Smith): 0.020156

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 317.314 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.311 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~6/5 = 317.311 ¢

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

Badness (Smith): 0.024371

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/32 = 158.649 ¢

Optimal ET sequence: 15, 38, 53, 121

Badness (Smith): 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 = 1200.000 ¢, ~11/10 = 158.677 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~11/10 = 158.655 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~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⟩]

unchanged-interval (eigenmonzo) 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 (Smith): 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 = 1200.000 ¢, ~9/7 = 441.355 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~9/7 = 441.363 ¢

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

Badness (Smith): 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

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~6/5 = 316.872 ¢ (~21/20 = 83.128 ¢)

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⟩]

unchanged-interval (eigenmonzo) 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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/7

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

Badness (Smith): 0.056337

Music:

Four Short Experiments in Octave Stretched 42edo (Dec 2024) by Budjarn Lambeth

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 = 400.000 ¢, ~6/5 = 316.881 ¢ (~21/20 = 83.119 ¢)

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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/7

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

Badness (Smith): 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 = 400.0000 ¢, ~6/5 = 316.9585 ¢ (~21/20 = 83.0415 ¢)

Optimal ET sequence: 72, 87, 159

Badness (Smith): 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 = 400.0000 ¢, ~6/5 = 316.9082 ¢ (~21/20 = 83.0918 ¢)

Optimal ET sequence: 72, 159, 231f

Badness (Smith): 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

Optimal tuning (POTE): ~25/21 = 300.0000 ¢, ~6/5 = 316.9999 ¢ (~126/125 = 16.9999 ¢)

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

Badness (Smith): 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 = 300.0000 ¢, ~6/5 = 316.9247 ¢ (~100/99 = 16.9247 ¢)

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

Badness (Smith): 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 = 300.0000 ¢, ~6/5 = 316.9887 ¢ (~100/99 = 16.9887 ¢)

Optimal ET sequence: 68, 72, 140, 212

Badness (Smith): 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 = 300.0000 ¢, ~6/5 = 316.9846 ¢ (~100/99 = 16.9846 ¢)

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

Badness (Smith): 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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 294.303 ¢

Optimal ET sequence: 49, 53, 314d

Badness (Smith): 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 = 1200.000 ¢, ~25/21 = 294.181 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~13/11 = 294.187 ¢

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

Badness (Smith): 0.031074

Marfifths

The marfifths temperament (19 & 140) tempers out the hemimage comma, 10976/10935. It splits the interval of a major thirteenth (~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, ~75/5

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~75/56 = 505.705 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~75/56 = 505.684 ¢

Optimal ET sequence: 19, 121e, 140, 159, 299

Badness (Smith): 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 = 1200.000 ¢, ~75/56 = 505.686 ¢

Optimal ET sequence: 19, 121e, 140, 159, 299

Badness (Smith): 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 = 1200.000 ¢, ~75/56 = 505.740 ¢

Optimal ET sequence: 19e, …, 121, 140, 261, 401

Badness (Smith): 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 = 1200.000 ¢, ~75/56 = 505.740 ¢

Optimal ET sequence: 19e, …, 121, 140, 261, 401

Badness (Smith): 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 = 1200.000 ¢, ~75/56 = 505.769 ¢

Optimal ET sequence: 19, 102d, 121

Badness (Smith): 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 = 1200.000 ¢, ~75/56 = 505.771 ¢

Optimal ET sequence: 19, 102df, 121

Badness (Smith): 0.038317

Marthirds

The marthirds temperament (19 & 193) tempers out the breeze comma (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]]

mapping generators: ~2, ~56/45

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~56/45 = 379.252 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~56/45 = 379.257 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~56/45 = 379.256 ¢

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

Badness (Smith): 0.043728

Quartkeenlig

Quartkeenlig 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 = 1200.0000 ¢, ~36/35 = 52.8562 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~33/32 = 52.8524 ¢

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

Badness (Smith): 0.0865

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 16807/16731

Mapping: [⟨1 0 1 1 5 0], ⟨0 36 30 41 -35 84]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~33/32 = 52.8562 ¢

Optimal ET sequence: 68, 159, 386d, 545ddf

Badness (Smith): 0.0477

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

Optimal tuning (POTE): ~2592/2401 = 133.333 ¢, ~6/5 = 317.005 ¢ (~36/35 = 50.338 ¢)

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

Badness (Smith): 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 = 133.333 ¢, ~6/5 = 317.010 ¢ (~36/35 = 50.343 ¢)

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

Badness (Smith): 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 = 133.333 ¢, ~6/5 = 317.086 ¢ (~36/35 = 50.419 ¢)

Optimal ET sequence: 72, 189f, 261, 333, 738cf

Badness (Smith): 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 = 1200.000 ¢, ~125/98 = 416.603 ¢

Optimal ET sequence: 49, 72, 193, 265

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 416.604 ¢

Optimal ET sequence: 49, 72, 193, 265

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 416.585 ¢

Optimal ET sequence: 49f, 72, 121, 193

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 416.585 ¢

Optimal ET sequence: 49fg, 72, 121, 193

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 416.580 ¢

Optimal ET sequence: 49fg, 72, 121, 193

Badness (Smith): 0.014748

Scales

Music

Subgroup extensions

Kleismic (2.3.5.13) a.k.a. cata

Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as 26/15. Notice 15625/15552 = (325/324)(625/624) and 325/324 = (625/624)(676/675). The S-expression-based comma list of the temperament is {S10/S12 = S25*S26, (S25,) S13/S15 = S26}. For the high-limit version of cata with a 1\5 period, see thunderclysmic.

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Subgroup-val mapping: [⟨1 0 1 0], ⟨0 6 5 14]]

Optimal tunings:

CTE: ~2 = 1200.0000 ¢, ~6/5 = 317.1110 ¢
POTE: ~2 = 1200.0000 ¢, ~6/5 = 317.0756 ¢

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

Badness (Sintel): 0.131

2.3.5.13.37.41 subgroup

Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.

Subgroup: 2.3.5.13.37.41

Comma list: 325/324, 625/624, 481/480, 1600/1599

Subgroup-val mapping: [⟨1 0 1 0 6 8], ⟨0 6 5 14 -3 -10]]

Optimal tunings:

WE: ~2 = 1200.165 ¢, ~6/5 = 317.113 ¢
CWE: ~2 = 1200.000 ¢, ~6/5 = 317.075 ¢

Badness (Sintel): 0.223