# 5th-octave temperaments

**Fractional-octave temperaments**

← 4th-octave temperaments 5th-octave temperaments 6th-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 temperaments – tempering 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:

- Pental temperaments - tempers out the [-28 25 -5⟩ comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
- Quintosec temperaments
- Trisedodge temperaments

## 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

Badness: 0.048312

## Slendrismic

*See also: No-fives subgroup temperaments #Slendrismic and Slendrisma*

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)

Optimal ET sequence: 130, 135, 265, 400, 1065, 1465, 1865

Badness: 0.013309

## Pentonismic (rank-5)

Subgroup: 2.3.5.7.11.13

Comma list: 281974669312/281950621875

Mapping: [⟨5 0 0 0 0 24], ⟨0 1 0 0 0 -1], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 0 1], ⟨0 0 0 0 1 0]]

- Mapping generators: ~224/195 = 1＼5, ~3, ~5, ~7, ~11

Supporting ETs: 10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585