5th-octave temperaments

From Xenharmonic Wiki
Jump to navigation Jump to search

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

Optimal ET sequence5, 15ccd

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 sequence130, 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