# 5th-octave temperaments

5edo is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of 12edo.

The most notable 5th-octave family is limmic temperamentstempering out 256/243 and associates 3\5 to 3/2 as well as 1\5 to 9/8, producing temperaments like blackwood. Equally notable among small equal divisions are the cloudy temperaments – identifying 8/7 with one step of 5edo.

Other families of 5-limit 5th-octave commas are:

## Quint

Quint preserves the 5-limit mapping of 5edo, and the harmonic 7 is mapped to an independent generator. In what way is this useful is unexplained.

Subgroup: 2.3.5.7

Comma list: 16/15, 27/25

Mapping[5 8 12 0], 0 0 0 1]]

Mapping generators: ~9/8, ~7

Wedgie⟨⟨0 0 5 0 8 12]]

Optimal tuning (POTE): ~9/8 = 1＼5, ~7/4 = 1017.903

## Slendrismic

Subgroup: 2.3.7

Comma list: 68719476736/68641485507

Mapping[5 0 18], 0 2 -1]]

Mapping generators: ~147/128 = 1＼5, ~262144/151263

Optimal tuning (CTE): ~8/7 = 230.9930 (or ~1029/1024 = 9.0080)