5th-octave temperaments
Template:Fractional-octave navigation 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.
Considered below is a temperament called quint, which uses exactly the same 5-limit as 5et, but the harmonic 7 is mapped to an independent generator. In what way is this useful is unexplained.
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
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
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]]
- mapping generators: ~224/195 = 1\5, ~3, ~5, ~7, ~11
Supporting ETs: 10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585