# 16th-octave temperaments

**Fractional-octave temperaments**

← 15th-octave temperaments 16th-octave temperaments 17th-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.

A temperament discussed elsewhere is hexadecoid, a weak extension of octopus / octoid to the 19-limit with slightly different mappings.

## 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 = 701.980

Supporting ETs: 48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 936/935, 1716/1715, 2080/2079, 2376/2375, 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 = 701.950

Supporting ETs: 48ghh, 176h, 224, 400, 624, 848gh, 1024e

## 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