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:
- Undecental (+5120/5103) → Aberschismic temperaments
- Tritonic (+225/224) → Marvel temperaments
- Sextilifourths (+32805/32768) → Schismatic family
- Ammonite (+250/243) → Porcupine family
- Decile (+321489/320000) → Quintile family
- Nusecond (+126/125) → Starling temperaments
- Chromat (+10976/10935) → Amity family
- Hemiwürschmidt (+2401/2400) → Hemimean clan
- Fourfives (+245/243) → Fifive family
- Triwell (+1029/1024) → Semicomma family
- Semidimfourth (+4375/4374) → Ragismic microtemperaments
Interaufo
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.
Interaufo may be described as the 159 & 161 temperament. It tempers out the same 5-limit comma as the untriton, but has a generator as a ~17/12 tritone, three generators make 64/45 and five make 10/7 with octave reduction. It was named by Xenllium in 2022, referring to the augmented fourth between 45/32 (ptolemaic augmented fourth) and 729/512 (Pythagorean augmented fourth).
Subgroup: 2.3.5.7
Comma list: 235298/234375, 33756345/33554432
Mapping: [⟨1 -12 31 34], ⟨0 27 -57 -62]]
- mapping generators: ~2, ~14336/10125
- WE: ~2 = 1200.2580 ¢, ~14336/10125 = 603.8868 ¢
- error map: {[val| +0.258 -0.216 +0.366 -0.787 }}
- CWE: ~2 = 1200.0000 ¢, ~14336/10125 = 603.7517 ¢
- error map: {[val| 0.000 -0.659 -0.161 -1.432 }}
Optimal ET sequence: 159, 320, 799dd, 1119bdd
Badness (Sintel): 8.68
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 102487/102400, 235298/234375
Mapping: [⟨1 -12 31 34 10], ⟨0 27 -57 -62 -13]]
Optimal tunings:
- WE: ~2 = 1200.2088 ¢, ~363/256 = 603.8574 ¢
- CWE: ~2 = 1200.0000 ¢, ~363/256 = 603.7515 ¢
Badness (Sintel): 2.67
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364
Mapping: [⟨1 -12 31 34 10 52], ⟨0 27 -57 -62 -13 -96]]
Optimal tunings:
- WE: ~2 = 1200.2085 ¢, ~78/55 = 603.8574 ¢
- CWE: ~2 = 1200.0000 ¢, ~78/55 = 603.7513 ¢
Badness (Sintel): 1.72
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 833/832, 1001/1000, 1089/1088, 8624/8619
Mapping: [⟨1 -12 31 34 10 52 -10], ⟨0 27 -57 -62 -13 -96 28]]
Optimal tunings:
- WE: ~2 = 1200.1599 ¢, ~17/12 = 603.8310 ¢
- CWE: ~2 = 1200.0000 ¢, ~17/12 = 603.7499 ¢
Badness (Sintel): 1.46
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 -12 31 34 10 52 -10 45], ⟨0 27 -57 -62 -13 -96 28 -81]]
Optimal tunings:
- WE: ~2 = 1200.1710 ¢, ~17/12 = 603.8357 ¢
- CWE: ~2 = 1200.0000 ¢, ~17/12 = 603.7490 ¢
Optimal ET sequence: 159, 161, 320
Badness (Sintel): 1.25
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 -12 31 34 10 52 -10 45 1], ⟨0 27 -57 -62 -13 -96 28 -81 7]]
Optimal tunings:
- WE: ~2 = 1200.2121 ¢, ~17/12 = 603.8577 ¢
- CWE: ~2 = 1200.0000 ¢, ~17/12 = 603.7503 ¢
Optimal ET sequence: 159, 320i
Badness (Sintel): 1.23
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 -12 31 34 10 52 -10 45 1 28], ⟨0 27 -57 -62 -13 -96 28 -81 7 -46]]
Optimal tunings:
- WE: ~2 = 1200.2331 ¢, ~17/12 = 603.8681 ¢
- CWE: ~2 = 1200.0000 ¢, ~17/12 = 603.7500 ¢
Optimal ET sequence: 159, 320ij
Badness (Sintel): 1.19