Xenharmonic Wiki

Kleismic family

Revision as of 10:20, 6 January 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings (2/))
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:

  • WE: ~2 = 1200.1659 ¢, ~6/5 = 317.0504 ¢
error map: ⟨+0.166 +0.347 -0.896]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢
error map: ⟨0.000 +0.230 -1.160]

Tuning ranges:

Optimal ET sequence: 15, 19, 34, 53, 458, 511c, …, 829c, 882c

Badness (Sintel): 0.310

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.

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 tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126]

Tuning ranges:

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

Badness (Sintel): 0.544

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 tunings:

  • WE: ~2 = 1200.7838 ¢, ~6/5 = 316.9478 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢

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

Badness (Sintel): 0.410

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 tunings:

  • WE: ~2 = 1200.6524 ¢, ~6/5 = 316.8911 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7267 ¢

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, 53, 72, 197e, 269ce, 341ce

Badness (Sintel): 0.722

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 tunings:

  • WE: ~2 = 1200.7982 ¢, ~6/5 = 316.9482 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7491 ¢

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, 53, 72, 125f, 197ef

Badness (Sintel): 0.698

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 tunings:

  • WE: ~2 = 1199.9590 ¢, ~6/5 = 317.0315 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0403 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Sintel): 1.32

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 tunings:

  • WE: ~2 = 1200.0797 ¢, ~6/5 = 317.0571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0400 ¢

Optimal ET sequence: 19e, 34d, 53

Badness (Sintel): 0.932

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 tunings:

  • WE: ~2 = 1200.8102 ¢, ~6/5 = 316.8669 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6768 ¢

Optimal ET sequence: 19e, 53e, 72

Badness (Sintel): 1.01

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 tunings:

  • WE: ~2 = 1201.0807 ¢, ~6/5 = 316.9246 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6700 ¢

Optimal ET sequence: 19e, 53e, 72, 307bcdeeffff

Badness (Sintel): 0.923

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 tunings:

  • WE: ~2 = 1198.6575 ¢, ~6/5 = 316.7282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0257 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Sintel): 1.47

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 tunings:

  • WE: ~2 = 1198.8403 ¢, ~6/5 = 316.8111 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0652 ¢

Optimal ET sequence: 19, 34d, 53e

Badness (Sintel): 1.13

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 tunings:

  • WE: ~99/70 = 600.2674 ¢, ~6/5 = 316.8624 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~6/5 = 316.7575 ¢

Optimal ET sequence: 34d, 72, 322c, 394c

Badness (Sintel): 0.969

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 tunings:

  • WE: ~55/39 = 600.3582 ¢, ~6/5 = 316.9152 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~6/5 = 316.7759 ¢

Optimal ET sequence: 34d, 72

Badness (Sintel): 0.901

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 tunings:

  • WE: ~17/12 = 600.4210 ¢, ~6/5 = 316.9282 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~6/5 = 316.7578 ¢

Optimal ET sequence: 34d, 38df, 72

Badness (Sintel): 0.798

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 tunings:

  • WE: ~17/12 = 600.3763 ¢, ~6/5 = 316.8720 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~6/5 = 316.7205 ¢

Optimal ET sequence: 34dh, 38df, 72

Badness (Sintel): 0.959

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 tunings:

  • WE: ~2 = 1202.6235 ¢, ~6/5 = 317.1646 ¢
error map: ⟨+2.624 +1.033 +2.133 -12.085]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.8293 ¢
error map: ⟨0.000 -0.979 -2.167 -18.388]

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

Badness (Sintel): 0.694

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 tunings:

  • WE: ~2 = 1199.7353 ¢, ~6/5 = 317.5055 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.5546 ¢

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: 15, 19, 34

Badness (Sintel): 0.906

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 tunings:

  • WE: ~2 = 1201.8360 ¢, ~6/5 = 317.0958 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.6829 ¢

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

Badness (Sintel): 1.23

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 tunings:

  • WE: ~2 = 1199.7816 ¢, ~6/5 = 317.3653 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.4070 ¢

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

Badness (Sintel): 0.940

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 tunings:

  • WE: ~2 = 1196.7615 ¢, ~6/5 = 317.7353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.4059 ¢

Optimal ET sequence: 4, 11b, 15

Badness (Sintel): 1.31

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 tunings:

  • WE: ~2 = 1204.5534 ¢, ~6/5 = 315.9247 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 315.1686 ¢

Optimal ET sequence: 4e, 19, 42bcd

Badness (Sintel): 1.32

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 tunings:

  • WE: ~2 = 1204.4937 ¢, ~6/5 = 316.2241 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 315.4748 ¢

Optimal ET sequence: 4ef, 19

Badness (Sintel): 1.22

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 tunings:

  • WE: ~2 = 1201.6569 ¢, ~6/5 = 318.0942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.8547 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Sintel): 0.914

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 tunings:

  • WE: ~2 = 1201.9324 ¢, ~6/5 = 317.8090 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.5170 ¢

Optimal ET sequence: 15, 19e, 34e

Badness (Sintel): 0.886

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Optimal tunings:

  • WE: ~2 = 1197.1789 ¢, ~6/5 = 317.5185 ¢
error map: ⟨-2.821 +3.156 -1.542 +4.025]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.2411 ¢
error map: ⟨0.000 +7.492 +4.892 +12.281]

Tuning ranges:

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

Badness (Sintel): 2.40

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 tunings:

  • WE: ~2 = 1197.0368 ¢, ~6/5 = 317.4956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.2672 ¢

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 (Sintel): 1.22

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 tunings:

  • WE: ~2 = 1196.8961 ¢, ~6/5 = 317.3837 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 318.1621 ¢

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

Badness (Sintel): 1.09

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Optimal tunings:

  • WE: ~2 = 1199.9172 ¢, ~6/5 = 317.0995 ¢
error map: ⟨-0.083 +0.642 -0.899 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1220 ¢
error map: ⟨0.000 +0.777 -0.704 +0.391]

Tuning ranges:

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

Badness (Sintel): 1.32

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 tunings:

  • WE: ~2 = 1200.0980 ¢, ~6/5 = 317.1879 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1623 ¢

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

Badness (Sintel): 1.31

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 tunings:

  • WE: ~2 = 1200.0936 ¢, ~6/5 = 317.1864 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.1622 ¢

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 (Sintel): 0.833

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Optimal tunings:

  • WE: ~2 = 1199.5969 ¢, ~6/5 = 317.2079 ¢
error map: ⟨-0.403 +1.292 -0.678 -0.349]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3071 ¢
error map: ⟨0.000 +1.887 +0.222 +0.370]

Optimal ET sequence: 34d, 87, 121, 208, 537b

Badness (Sintel): 4.14

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 tunings:

  • WE: ~2 = 1199.5425 ¢, ~6/5 = 317.1901 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3020 ¢

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

Badness (Sintel): 1.61

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 tunings:

  • WE: ~2 = 1199.5339 ¢, ~6/5 = 317.1882 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.3028 ¢

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

Badness (Sintel): 1.01

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

mapping generators: ~2, ~35/32

Optimal tunings:

  • WE: ~2 = 1199.3950 ¢, ~35/32 = 158.5686 ¢
error map: ⟨-0.605 +0.868 -1.233 +1.637]
  • CWE: ~2 = 1200.0000 ¢, ~35/32 = 158.6338 ¢
error map: ⟨0.000 +1.651 +0.024 +3.470]

Optimal ET sequence: 15, 38, 53, 121, 174d, 295d

Badness (Sintel): 1.32

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 tunings:

  • WE: ~2 = 1199.8009 ¢, ~11/10 = 158.6508 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 158.6717 ¢

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

Badness (Sintel): 1.26

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 tunings:

  • WE: ~2 = 1199.7952 ¢, ~11/10 = 158.6279 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 158.6493 ¢

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

Badness (Sintel): 1.07

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping: [⟨1 -6 -4 -13], ⟨0 12 10 25]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.8369 ¢, ~14/9 = 758.5621 ¢
error map: ⟨-0.163 +1.769 -0.040 -2.652]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 758.6554 ¢
error map: ⟨0.000 +1.910 +0.240 -2.441]

Minimax tuning:

[[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, 242

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -6 -4 -13 18], ⟨0 12 10 25 -23]]

Optimal tunings:

  • WE: ~2 = 1199.9620 ¢, ~14/9 = 758.6210 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 758.6445 ¢

Optimal ET sequence: 19, 49e, 68, 87

Badness (Sintel): 1.57

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -6 -4 -13 18 -14], ⟨0 12 10 25 -23 28]]

Optimal tunings:

  • WE: ~2 = 1199.9292 ¢, ~14/9 = 758.5919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 758.6355 ¢

Optimal ET sequence: 19, 68, 87

Badness (Sintel): 1.11

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 tunings:

  • WE: ~63/50 = 400.1845 ¢, ~6/5 = 317.0178 ¢ (~21/20 = 83.1667 ¢)
error map: ⟨+0.553 +0.152 -0.671 -1.017]
  • CWE: ~63/50 = 400.0000 ¢, ~6/5 = 316.9129 ¢ (~21/20 = 83.0871 ¢)
error map: ⟨0.000 -0.478 -1.749 -2.652]

Minimax tuning:

[[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
[[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, 765ccddd

Badness (Sintel): 1.43

Music:

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 tunings:

  • WE: ~44/35 = 400.1571 ¢, ~6/5 = 317.0058 ¢ (~21/20 = 83.1514 ¢)
  • CWE: ~44/35 = 400.0000 ¢, ~6/5 = 316.9154 ¢ (~21/20 = 83.0846 ¢)

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 (Sintel): 0.639

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 tunings:

  • WE: ~44/35 = 400.1514 ¢, ~6/5 = 317.0785 ¢ (~21/20 = 83.0729 ¢)
  • CWE: ~44/35 = 400.0000 ¢, ~6/5 = 316.9896 ¢ (~21/20 = 83.0104 ¢)

Optimal ET sequence: 15, 57f, 72, 87, 159

Badness (Sintel): 0.647

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 tunings:

  • WE: ~34/27 = 400.1604 ¢, ~6/5 = 317.0353 ¢ (~21/20 = 83.1251 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~6/5 = 316.9384 ¢ (~21/20 = 83.0616 ¢)

Optimal ET sequence: 15g, 57fg, 72, 159, 231f

Badness (Sintel): 0.690

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 tunings:

  • WE: ~25/21 = 300.0520 ¢, ~6/5 = 317.0548 ¢ (~126/125 = 17.0029 ¢)
error map: ⟨+0.208 +0.374 -0.832 -0.243]
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 317.0301 ¢ (~126/125 = 17.0301 ¢)
error map: ⟨0.000 +0.225 -1.163 -0.706]

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

Badness (Sintel): 0.993

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 tunings:

  • WE: ~25/21 = 300.0995 ¢, ~6/5 = 317.0298 ¢ (~100/99 = 16.9303 ¢)
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 316.9540 ¢ (~100/99 = 16.9540 ¢)

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

Badness (Sintel): 0.774

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 tunings:

  • WE: ~25/21 = 300.0985 ¢, ~6/5 = 317.0899 ¢ (~100/99 = 16.9941 ¢)
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 317.0155 ¢ (~100/99 = 17.0155 ¢)

Optimal ET sequence: 68, 72, 140, 212

Badness (Sintel): 0.774

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 tunings:

  • WE: ~25/21 = 300.1102 ¢, ~6/5 = 317.1011 ¢ (~100/99 = 16.9909 ¢)
  • CWE: ~25/21 = 300.0000 ¢, ~6/5 = 317.0155 ¢ (~100/99 = 17.0155 ¢)

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

Badness (Sintel): 0.651

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [⟨1 -12 -9 -7], ⟨0 18 15 13]]

mapping generators: ~2, ~42/25

Optimal tunings:

  • WE: ~2 = 1199.5290 ¢, ~42/25 = 905.3417 ¢
error map: ⟨-0.471 -0.152 -1.949 +3.914]
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 905.6741 ¢
error map: ⟨0.000 +0.178 -1.203 +4.937]

Optimal ET sequence: 49, 53

Badness (Sintel): 1.93

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -12 -9 -7 -29], ⟨0 18 15 13 43]]

Optimal tunings:

  • WE: ~2 = 1199.1389 ¢, ~42/25 = 905.1688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 905.7840 ¢

Optimal ET sequence: 49, 53, 102d

Badness (Sintel): 1.75

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -12 -9 -7 -29 -28], ⟨0 18 15 13 43 42]]

Optimal tunings:

  • WE: ~2 = 1199.1517 ¢, ~22/13 = 905.1727 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 905.7801 ¢

Optimal ET sequence: 49f, 53, 102df

Badness (Sintel): 1.28

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, ~75/56

Optimal tunings:

  • WE: ~2 = 1200.0223 ¢, ~75/56 = 505.7147 ¢
error map: ⟨+0.022 +0.775 -0.683 -0.615]
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7060 ¢
error map: ⟨0.000 +0.753 -0.724 -0.643]

Optimal ET sequence: 19, …, 121, 140, 579, 719

Badness (Sintel): 1.61

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 tunings:

  • WE: ~2 = 1200.2484 ¢, ~75/56 = 505.7882 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.6853 ¢

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

Badness (Sintel): 1.95

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 tunings:

  • WE: ~2 = 1200.2747 ¢, ~75/56 = 505.8019 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.6883 ¢

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

Badness (Sintel): 1.24

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 tunings:

  • WE: ~2 = 1199.7886 ¢, ~75/56 = 505.6513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7366 ¢

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

Badness (Sintel): 2.02

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 tunings:

  • WE: ~2 = 1199.7996 ¢, ~75/56 = 505.6558 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7366 ¢

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

Badness (Sintel): 1.18

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 tunings:

  • WE: ~2 = 1199.3198 ¢, ~75/56 = 505.4822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7607 ¢

Optimal ET sequence: 19, 102d, 121

Badness (Sintel): 2.48

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 tunings:

  • WE: ~2 = 1199.3368 ¢, ~75/56 = 505.4919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/56 = 505.7627 ¢

Optimal ET sequence: 19, 102df, 121

Badness (Sintel): 1.58

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 tunings:

  • WE: ~2 = 1200.1662 ¢, ~56/45 = 379.3041 ¢
error map: ⟨+0.166 +0.347 -0.896 +0.000]
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 379.2552 ¢
error map: ⟨0.000 +0.171 -1.209 -0.214]

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

Badness (Sintel): 2.64

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 tunings:

  • WE: ~2 = 1200.1189 ¢, ~56/45 = 379.2942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 379.2580 ¢

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

Badness (Sintel): 2.50

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 tunings:

  • WE: ~2 = 1200.2154 ¢, ~56/45 = 379.3236 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 379.2580 ¢

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

Badness (Sintel): 1.81

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 tunings:

  • WE: ~2 = 1200.2825 ¢, ~36/35 = 52.8528 ¢
error map: ⟨+0.282 +0.745 -0.448 -1.579]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 52.8476 ¢
error map: ⟨0.000 +0.558 -0.886 -2.074]

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

Badness (Sintel): 3.69

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 tunings:

  • WE: ~2 = 1200.2526 ¢, ~36/35 = 52.8534 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 52.8446 ¢

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

Badness (Sintel): 2.86

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 tunings:

  • WE: ~2 = 1200.2564 ¢, ~36/35 = 52.8568 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 52.8479 ¢

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

Badness (Sintel): 1.97

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 tunings:

  • WE: ~2592/2401 = 133.3488 ¢, ~6/5 = 317.0413 ¢ (~36/35 = 50.3437 ¢)
error map: ⟨+0.139 +0.293 -0.968 +0.259]
  • CWE: ~2592/2401 = 133.3333 ¢, ~6/5 = 317.0260 ¢ (~36/35 = 50.3593 ¢)
error map: ⟨0.000 +0.201 -1.184 -0.003]

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

Badness (Sintel): 4.90

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 tunings:

  • WE: ~250/231 = 133.3465 ¢, ~6/5 = 317.0416 ¢ (~36/35 = 50.3486 ¢)
  • CWE: ~250/231 = 133.3333 ¢, ~6/5 = 317.0264 ¢ (~36/35 = 50.3597 ¢)

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

Badness (Sintel): 1.71

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 tunings:

  • WE: ~250/231 = 133.3385 ¢, ~6/5 = 317.0978 ¢ (~36/35 = 50.4208 ¢)
  • CWE: ~250/231 = 133.3333 ¢, ~6/5 = 317.0910 ¢ (~36/35 = 50.4243 ¢)

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

Badness (Sintel): 1.61

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 -18 -14 -22], ⟨0 30 25 38]]

mapping generators: ~2, 196/125

Optimal tunings:

  • WE: ~2 = 1200.1357 ¢, ~196/125 = 783.4853 ¢
error map: ⟨+0.136 +0.163 -1.080 +0.632]
  • CWE: ~2 = 1200.0000 ¢, ~196/125 = 783.4009 ¢
error map: ⟨0.000 +0.072 -1.291 +0.408]

Optimal ET sequence: 23d, 49, 72, 193, 265

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -18 -14 -22 -22], ⟨0 30 25 38 39]]

Optimal tunings:

  • WE: ~2 = 1200.0514 ¢, ~11/7 = 783.4294 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.3975 ¢

Optimal ET sequence: 23de, 49, 72, 193, 265

Badness (Sintel): 0.844

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -18 -14 -22 -22 -42], ⟨0 30 25 38 39 70]]

Optimal tunings:

  • WE: ~2 = 1199.9314 ¢, ~11/7 = 783.3705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.4134 ¢

Optimal ET sequence: 23deff, 49f, 72, 121, 193

Badness (Sintel): 0.828

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -18 -14 -22 -22 -42 -39], ⟨0 30 25 38 39 70 66]]

Optimal tunings:

  • WE: ~2 = 1199.9324 ¢, ~11/7 = 783.3706 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.4129 ¢

Optimal ET sequence: 23deffgg, 49fg, 72, 121, 193

Badness (Sintel): 0.664

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 -18 -14 -22 -22 -42 -39 16], ⟨0 30 25 38 39 70 66 -18]]

Optimal tunings:

  • WE: ~2 = 1199.8567 ¢, ~11/7 = 783.3262 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 783.4176 ¢

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

Badness (Sintel): 0.897

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

Retrieved from "https://en.xen.wiki/index.php?title=Kleismic_family&oldid=221002"