16th-octave temperaments
16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.
Temperaments discussed elsewhere include hexadecoid and sedecic.
Sulfur
Subgroup: 2.3.5
Comma list: [-115 96 -16⟩
Mapping: [⟨16 0 -115], ⟨0 1 6]]
- Mapping generators: ~214748364800000/205891132094649 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.8895
Supporting ETs: 48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472
7-limit
Subgroup: 2.3.5.7
Comma list: 14348907/14336000, 2147483648/2144153025
Mapping: [⟨16 0 -115 121], ⟨0 1 6 -3]]
- Mapping generators: ~256/245 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9129
Optimal ET sequence: 48, 128c, 176, 224, 400, 624
Badness: 0.166051
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 46656/46585, 131072/130977
Mapping: [⟨16 0 -115 121 30], ⟨0 1 6 -3 1]]
- Mapping generators: ~256/245 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9070
Optimal ET sequence: 48, 128c, 176, 224, 400, 624
Badness: 0.041764
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095, 39366/39325
Mapping: [⟨16 0 -115 121 30 186], ⟨0 1 6 -3 1 -5]]
- Mapping generators: ~117/112 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9047
Optimal ET sequence: 48, 128cf, 176, 224, 400, 624, 848
Badness: 0.025967
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1701/1700, 1716/1715, 4096/4095, 11016/11011
Mapping: [⟨16 0 -115 121 30 186 319], ⟨0 1 6 -3 1 -5 -10]]
- Mapping generators: ~117/112 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9418
Optimal ET sequence: 176, 224, 400, 624
Badness: 0.023704
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1521/1520, 1701/1700, 1716/1715, 4096/4095, 11016/11011
Mapping: [⟨16 0 -115 121 30 186 319 423], ⟨0 1 6 -3 1 -5 -10 -14]]
- Mapping generators: ~117/112 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9505
Optimal ET sequence: 176h, 224, 400, 624
Badness: 0.020421
Ntiscifer
Ntiscifer tempers out the Pythagorean double-augmented second, and is equivalent to the 16edo circle of fifths with an added dimension for 5/4. In 16edo, this maps 5/4 to 375 cents, as in mavila temperament. Tunings with a separate, more accurate third include 64edo, 80edo, and 96edo; 96edo is a particularly accurate tuning, though 64edo might be considered more practical.
Subgroup: 2.3.5
Mapping: [⟨16 25 0], ⟨0 0 1]]
- mapping generators: ~2048/2187, ~5
- CTE: ~2048/2187 = 1\16, ~5/4 = 386.3137 (~135/128 = 11.3137)
- CWE: ~2048/2187 = 1\16, ~5/4 = 373.1508 (~128/135 = 1.8492)
Badness: 3.05
Hilarity
Hilarity, the 80 & 128 temperament, tempers 40/27 to 9\16.
Subgroup: 2.3.5
Comma list: [39 -48 16⟩
Mapping: [⟨16 0 -39], ⟨0 1 3]]
- mapping generators: ~10460353203/10240000000, ~3
- CTE: ~10460353203/1024000000 = 1\16, ~3/2 = 703.422
- CWE: ~10460353203/1024000000 = 1\16, ~3/2 = 703.621
Optimal ET sequence: 16cc, 32c, 48, 80, 128, 208
Badness: 4.22
7-limit
Subgroup: 2.3.5.7
Comma list: 179200/177147, 390625/388962
Mapping: [⟨16 0 -39 -82], ⟨0 1 3 5]]
- mapping generators: ~392/375, ~3
Optimal ET sequence: 32cdd, 48d, 80, 128, 208, 496bd
Badness: 0.302
11-limit
Subgroup: 2.3.5.7.11
Comma list: 896/891, 2200/2187, 125000/124509
Mapping: [⟨16 0 -39 -82 30], ⟨0 1 3 5 1]]
- mapping generators: ~392/375, ~3
Optimal ET sequence: 48d, 80, 128, 208, 288be, 496bdee
Badness: 0.0886
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 896/891, 540000/539539
Mapping: [⟨16 0 -39 -82 30 110], ⟨0 1 3 5 1 -2]]
- mapping generators: ~392/375, ~3
Optimal ET sequence: 80, 128, 208, 496bdeef
Badness: 0.0611