Sengic 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 is a collection of rank-2 sengic temperaments, which temper out the senga (monzo: [1 -3 -2 3⟩, ratio: 686/675), with S-expression S13⋅S14².

Temperaments discussed elsewhere are:

• Fluorine (+21/20) → Septisemi temperaments
• Octokaidecal (+28/27) → Trienstonic clan
• Beatles (+64/63) → Archytas clan
• Liese (+81/80) → Meantone family
• Squirrel (+32805/32768) → Schismatic family
• Negri (+49/48) → Semaphoresmic clan
• Progression (+36/35) → Mint temperaments
• Echidnic (+1029/1024) → Diaschismic family
• Sensi (+126/125) → Sensipent family
• Niner (+128/125) → Augmented family
• Ammonite (+250/243) → Porcupine family
• Sycamore (+875/864) → Sycamore family

Considered below are leapday and twothirdtonic, in the order of increasing badness.

Leapday

Main article: Leapday

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

Leapday tempers out 5120/5103, the aberschisma, in addition to the senga, and may be described as the 29 & 46 temperament. It extends leapfrog, such that 7/4 is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). 46edo itself is an excellent tuning for this.

Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying 14/11 by a major third and 13/11 by a minor third, tempering out not only 352/351 and 364/363 but 91/90, 121/120, 169/168 and 196/195. It can be further extended to include the 17th and 23rd harmonics. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.

Leapday has an alternative extension called polypyth, which tempers out the same 5-limit comma as leapday, but with the porwell comma (6144/6125) rather than the aberschisma tempered out.

Subgroup: 2.3.5.7

Comma list: 686/675, 5120/5103

Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]

mapping generators: ~2, ~3

Optimal tunings:

• WE: ~2 = 1199.7167 ¢, ~3/2 = 704.0971 ¢

error map: ⟨-0.283 +1.859 +2.559 -5.669]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2504 ¢

error map: ⟨0.000 +2.295 +2.945 -5.070]

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 2.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

Optimal tunings:

• WE: ~2 = 1200.0731 ¢, ~3/2 = 704.2933 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2538 ¢

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 1.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

Optimal tunings:

• WE: ~2 = 1200.4758 ¢, ~3/2 = 704.4930 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2346 ¢

Optimal ET sequence: 17c, 29, 46, 121def

Badness (Sintel): 1.02

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

Optimal tunings:

• WE: ~2 = 1200.4818 ¢, ~3/2 = 704.5121 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2507 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.910

2.3.5.7.11.13.17.23 subgroup

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -5], ⟨0 1 21 15 11 8 24 6]]

Optimal tunings:

• WE: ~2 = 1200.5169 ¢, ~3/2 = 704.5279 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2450 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.872

Twothirdtonic

Twothirdtonic tempers out 6144/6125, the porwell comma, in addition to the senga, and may be described as the 37 & 46 temperament, generated by one third of a classical major third that represents 15/14, 14/13, and 13/12 in the 13-limit interpretation. Note that in the data below, the generator is taken to be its octave complement, thirteen of which octave reduced make the perfect fifth; it follows that the ploidacot for this temperament is 11-sheared 13-cot. 46edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 -10 5 -7], ⟨0 13 -3 11]]

mapping generators: ~2, ~28/15

Optimal tunings:

• WE: ~2 = 1199.3074 ¢, ~28/15 = 1068.9820 ¢

error map: ⟨-0.693 +1.736 +3.278 -5.176]

• CWE: ~2 = 1200.0000 ¢, ~28/15 = 1069.5746 ¢

error map: ⟨0.000 +2.515 +4.962 -3.505]

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 2.52

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 686/675

Mapping: [⟨1 -10 5 -7 -1], ⟨0 13 -3 11 5]]

Optimal tunings:

• WE: ~2 = 1199.7068 ¢, ~28/15 = 1069.3084 ¢
• CWE: ~2 = 1200.0000 ¢, ~28/15 = 1069.5600 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

Mapping: [⟨1 -10 5 -7 -1 -7], ⟨0 13 -3 11 5 12]]

Optimal tunings:

• WE: ~2 = 1199.9531 ¢, ~13/7 = 1069.5492 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/7 = 1069.5893 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 1.07