Ditonmic 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 ditonmic family of temperaments tempers out the ditonma (monzo: [-27 -2 13⟩, ratio: 1 220 703 125 / 1 207 959 552).

Ditonic

Named by Petr Pařízek in 2011, ditonic splits ~5/2 in two for a generator, which happens to be an interval very close in size to the octave complement of the ditone (i.e. the Pythagorean major third, 81/64). It can be described as 3 & 53, and is part of the schismic–Mercator equivalence continuum with equivalence number n = 13/2. Its ploidacot is eta-13-cot. Note that the ditone itself is 52 generator steps away.

Subgroup: 2.3.5

Comma list: 1220703125/1207959552

Mapping: [⟨1 -7 1], ⟨0 13 2]]

mapping generators: ~2, ~24576/15625

Optimal tunings:

• WE: ~2 = 1200.2971 ¢, ~24576/15625 = 792.6223 ¢

error map: ⟨+0.297 +0.055 -0.772]

• CWE: ~2 = 1200.0000 ¢, ~24576/15625 = 792.4403 ¢

error map: ⟨0.000 -0.231 -1.433]

Optimal ET sequence: 3, …, 47, 50, 53, 474c, 527c, 580c, 633c, 686c, 739c, 792c, 845cc

Badness (Sintel): 3.92

Coditone

Subgroup: 2.3.5.7

Comma list: 225/224, 2125764/2100875

Mapping: [⟨1 -7 1 -17], ⟨0 13 2 30]]

Optimal tunings:

• WE: ~2 = 1200.4411 ¢, ~1944/1225 = 792.6016 ¢

error map: ⟨+0.441 -1.223 -0.669 +1.722]

• CWE: ~2 = 1200.0000 ¢, ~1944/1225 = 792.3270 ¢

error map: ⟨0.000 -1.704 -1.660 +0.985]

Optimal ET sequence: 50, 53, 103, 156

Badness (Sintel): 1.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 78408/78125

Mapping: [⟨1 -7 1 -17 16], ⟨0 13 2 30 -19]]

Optimal tunings:

• WE: ~2 = 1200.8016 ¢, ~198/125 = 792.7878 ¢
• CWE: ~2 = 1200.0000 ¢, ~198/125 = 792.2692 ¢

Optimal ET sequence: 50, 53, 103, 259be, 362bcee

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 385/384, 847/845

Mapping: [⟨1 -7 1 -17 16 7], ⟨0 13 2 30 -19 -5]]

Optimal tunings:

• WE: ~2 = 1200.7372 ¢, ~198/125 = 792.7511 ¢
• CWE: ~2 = 1200.0000 ¢, ~198/125 = 792.2715 ¢

Optimal ET sequence: 50, 53, 103, 259be, 362bceef

Badness (Sintel): 1.01

Coditonic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 6655/6561

Mapping: [⟨1 -7 1 -17 -19], ⟨0 13 2 30 34]]

Optimal tunings:

• WE: ~2 = 1200.3578 ¢, ~189/121 = 792.6693 ¢
• CWE: ~2 = 1200.0000 ¢, ~189/121 = 792.4463 ¢

Optimal ET sequence: 3de, 50e, 53

Badness (Sintel): 2.11

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 325/324, 847/845

Mapping: [⟨1 -7 1 -17 -19 -28], ⟨0 13 2 30 34 48]]

Optimal tunings:

• WE: ~2 = 1200.2797 ¢, ~52/33 = 792.6439 ¢
• CWE: ~2 = 1200.0000 ¢, ~52/33 = 792.4682 ¢

Optimal ET sequence: 3def, 50eff, 53

Badness (Sintel): 1.82

Diton

This low-accuracy temperament was considered the canonical extension of ditonic, and catalogued as so in Graham Breed's Temperament Finder.

Subgroup: 2.3.5.7

Comma list: 126/125, 8751645/8388608

Mapping: [⟨1 -7 1 16], ⟨0 13 2 -20]]

Optimal tunings:

• WE: ~2 = 1201.9455 ¢, ~2048/1323 = 793.3304 ¢

error map: ⟨+1.946 -2.278 +2.293 -4.305]

• CWE: ~2 = 1200.000 ¢, ~2048/1323 = 792.0527 ¢

error map: ⟨0.000 -5.270 -2.208 -9.879]

Optimal ET sequence: 3, 47, 50

Badness (Sintel): 6.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 2079/2048

Mapping: [⟨1 -7 1 16 16], ⟨0 13 2 -20 -19]]

Optimal tunings:

• WE: ~2 = 1201.6076 ¢, ~11/7 = 793.1692 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/7 = 792.1016 ¢

Optimal ET sequence: 3, 47, 50

Badness (Sintel): 3.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 1287/1280

Mapping: [⟨1 -7 1 16 16 7], ⟨0 13 2 -20 -19 -5]]

Optimal tunings:

• WE: ~2 = 1201.3885 ¢, ~11/7 = 793.0294 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/7 = 792.1096 ¢

Optimal ET sequence: 3, 47, 50

Badness (Sintel): 2.27