Parakleismic family

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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 parakleismic family of temperaments tempers out the parakleisma (monzo: [8 14 -13⟩, ratio: 1 224 440 064 / 1 220 703 125).

Parakleismic

Main article: Parakleismic

Parakleismic has a generator a very slightly flat classical minor third. Note that in the data that follow, the generator is its octave complement, ~5/3, thirteen of which give the interval class of 3, and fourteen give the interval class of 5. It is a member of the syntonic–kleismic equivalence continuum, with equivalence number n = 13/2. It is an undoubted microtemperament in the 5-limit, with the 118edo tuning giving errors well under a cent.

Subgroup: 2.3.5

Comma list: 1224440064/1220703125

Mapping: [⟨1 -8 -8], ⟨0 13 14]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.971 ¢, ~5/3 = 884.7383 ¢
error map: ⟨-0.029 -0.127 +0.253]
  • CWE: ~2 = 1200.000 ¢, ~5/3 = 884.8576 ¢
error map: ⟨0.000 -0.106 +0.293]

Optimal ET sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496

Badness (Sintel): 1.02

Septimal parakleismic

Main article: Parakleismic

While 118edo no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, for which 99edo, 118edo, and especially 217edo are accurate tunings.

Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23], ⟨0 13 14 35]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.7820 ¢, ~5/3 = 884.6581 ¢
error map: ⟨-0.218 +0.344 +0.644 -0.779]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8088 ¢
error map: ⟨0.000 +0.560 +1.010 -0.516]

Optimal ET sequence: 19, 61d, 80, 99, 217, 316, 415

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 30], ⟨0 13 14 35 -36]]

Optimal tunings:

  • WE: ~2 = 1200.3296 ¢, ~5/3 = 884.9921 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7519 ¢

Optimal ET sequence: 19, 99, 118

Badness (Sintel): 1.64

Paralytic

Paralytic (99e & 118) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57], ⟨0 13 14 35 82]]

Optimal tunings:

  • WE: ~2 = 1199.9940 ¢, ~5/3 = 884.7757 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7800 ¢

Optimal ET sequence: 19e, …, 99e, 118, 217, 335

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57 59], ⟨0 13 14 35 82 -75]]

Optimal tunings:

  • WE: ~2 = 1199.9218 ¢, ~5/3 = 884.7285 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7858 ¢

Optimal ET sequence: 99e, 118, 217

Badness (Sintel): 1.85

Paraklein

Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates 13/11 with 32/27, 14/13 with 15/14, 25/24 with 26/25, and 27/26 with 28/27.

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: [⟨1 -8 -8 -23 -57 -28], ⟨0 13 14 35 82 43]]

Optimal tunings:

  • WE: ~2 = 1199.8239 ¢, ~5/3 = 884.6449 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7709 ¢

Optimal ET sequence: 19e, …, 99ef, 118

Badness (Sintel): 1.55

Parkleismic

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: [⟨1 -8 -8 -23 -43], ⟨0 13 14 35 63]]

Optimal tunings:

  • WE: ~2 = 1199.1848 ¢, ~5/3 = 884.3386 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9158 ¢

Optimal ET sequence: 19e, 61de, 80, 179, 259cd

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: [⟨1 -8 -8 -23 -43 -14], ⟨0 13 14 35 63 24]]

Optimal tunings:

  • WE: ~2 = 1199.5318 ¢, ~5/3 = 884.5800 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9118 ¢

Optimal ET sequence: 19e, 61de, 80, 179

Badness (Sintel): 1.51

Paradigmic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: [⟨1 -8 -8 -23 16], ⟨0 13 14 35 -17]]

Optimal tunings:

  • WE: ~2 = 1199.0616 ¢, ~5/3 = 884.2124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8877 ¢

Optimal ET sequence: 19, 61d, 80, 99e, 179e, 457bcddeeee

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: [⟨1 -8 -8 -23 16 -14], ⟨0 13 14 35 -17 24]]

Optimal tunings:

  • WE: ~2 = 1199.2683 ¢, ~5/3 = 884.3805 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9061 ¢

Optimal ET sequence: 19, 61d, 80, 99e

Badness (Sintel): 1.48

Semiparakleismic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4], ⟨0 13 14 35 23]]

mapping generators: ~99/70, ~33/28

Optimal tunings:

  • WE: ~99/70 = 599.9270 ¢, ~33/28 = 284.7841 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8119 ¢

Optimal ET sequence: 80, 118, 198, 316, 514c

Badness (Sintel): 1.13

Semiparamint

This extension was named semiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4 15], ⟨0 13 14 35 23 -16]]

Optimal tunings:

  • WE: ~99/70 = 599.8253 ¢, ~33/28 = 284.7608 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8366 ¢

Optimal ET sequence: 80, 118, 198

Badness (Sintel): 1.40

Semiparawolf

This extension was named gentsemiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: [⟨2 -3 -2 -11 -4 -4], ⟨0 13 14 35 23 24]]

Optimal tunings:

  • WE: ~99/70 = 600.0569 ¢, ~13/11 = 284.8431 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~13/11 = 284.8216 ¢

Optimal ET sequence: 80, 118f, 198f

Badness (Sintel): 1.67

Paraguay

Named by Xenllium in 2026, paraguay tempers out 12005/11664 and may be described as the 19 & 61 temperament. It is a variant of septimal parakleismic, mapping 7th harmonic to 16 generators.

Subgroup: 2.3.5.7

Comma list: 126/125, 12005/11664

Mapping: [⟨1 -8 -8 -9], ⟨0 13 14 16]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.6421 ¢, ~5/3 = 885.3232 ¢
error map: ⟨+0.642 +2.110 +3.074 -9.434]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8949 ¢
error map: ⟨0.000 +1.678 +2.214 -10.508]

Optimal ET sequence: 19, 61, 80d, 99d

Badness (Sintel): 2.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 12005/11664

Mapping: [⟨1 -8 -8 -9 2], ⟨0 13 14 16 2]]

Optimal tunings:

  • WE: ~2 = 1197.7783 ¢, ~5/3 = 883.6140 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.1383 ¢

Optimal ET sequence: 19, 42e, 61e

Badness (Sintel): 2.49

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 343/338

Mapping: [⟨1 -8 -8 -9 2 -14], ⟨0 13 14 16 2 24]]

Optimal tunings:

  • WE: ~2 = 1197.7848 ¢, ~5/3 = 883.6431 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.1623 ¢

Optimal ET sequence: 19, 42ef, 61e

Badness (Sintel): 1.86

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