Triwellismic temperaments

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.

This page collects miscellaneous rank-2 triwellismic temperaments, which temper out the triwellisma (monzo: [1 -1 -7 6⟩, ratio: 235298/234375), the difference between seven 7/5's octave reduced and a septimal subfourth (21/16).

Temperaments discussed elsewhere include ammonite, chromat, decile, fourfives, hemiwürschmidt, nusecond, semidimfourth, sextilischis, tritonic, triwell and undecental.

Interaufo

For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.

The name interaufo means an augmented fourth interval between 45/32 (classic diatonic tritone) and 729/512 (Pythagorean tritone). The interaufo temperament (159 & 161) tempers out the same 5-limit comma as the untriton, but has a generator as 24/17-wide tritone, three generators makes 45/32 and five of them makes 7/5 with octave reduction.

Subgroup: 2.3.5.7

Comma list: 235298/234375, 33756345/33554432

Mapping: [⟨1 15 -26 -28], ⟨0 -27 57 62]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~10125/7168 = 596.247 ¢

Optimal ET sequence: 2, …, 157cd, 159, 320, 799dd, 1119bdd

Badness (Smith): 0.342890

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 102487/102400, 235298/234375

Mapping: [⟨1 15 -26 -28 -3], ⟨0 -27 57 62 13]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~512/363 = 596.248 ¢

Optimal ET sequence: 2, …, 157cd, 159, 320

Badness (Smith): 0.080833

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364

Mapping: [⟨1 15 -26 -28 -3 -44], ⟨0 -27 57 62 13 96]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/39 = 596.248 ¢

Optimal ET sequence: 2f, …, 157cdf, 159, 320

Badness (Smith): 0.041506

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 833/832, 1001/1000, 1089/1088, 8624/8619

Mapping: [⟨1 15 -26 -28 -3 -44 18], ⟨0 -27 57 62 13 96 -28]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.250 ¢

Optimal ET sequence: 2f, …, 157cdf, 159, 320

Badness (Smith): 0.028653

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 441/440, 513/512, 833/832, 969/968, 1001/1000, 1521/1520

Mapping: [⟨1 15 -26 -28 -3 -44 18 -36], ⟨0 -27 57 62 13 96 -28 81]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.250 ¢

Optimal ET sequence: 2fh, …, 157cdfh, 159, 161, 320

Badness (Smith): 0.020593

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 441/440, 513/512, 529/528, 833/832, 897/896, 969/968, 1001/1000

Mapping: [⟨1 15 -26 -28 -3 -44 18 -36 8], ⟨0 -27 57 62 13 96 -28 81 -7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.249 ¢

Optimal ET sequence: 2fh, …, 157cdfh, 159, 320i

Badness (Smith): 0.017242

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 441/440, 513/512, 529/528, 551/550, 609/608, 783/782, 833/832, 969/968

Mapping: [⟨1 15 -26 -28 -3 -44 18 -36 8 -18], ⟨0 -27 57 62 13 96 -28 81 -7 46]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.249 ¢

Optimal ET sequence: 2fh, …, 157cdfhj, 159, 320ij

Badness (Smith): 0.014267