16th-octave temperaments

Fractional-octave temperaments

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← 15th-octave • 16th-octave • 17th-octave →

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

Main article: 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

Comma list: 43046721/33554432

Mapping: [⟨16 25 0], ⟨0 0 1]]

mapping generators: ~2048/2187, ~5

Optimal tunings:

• 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)

Optimal ET sequence: 16

Badness: 3.05