2.5 subgroup

The 2.5 subgroup is just intonation subgroup consisting of rational intervals where 2 and 5 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 2 and 5. This is an infinite set. Some examples of intervals in this subgroup are 5/4, 8/5, 25/16, 32/25, and so on.

The 2.5 subgroup is a retraction of the 5-limit, obtained by removing prime 3. Its simplest expansion is the 2.5.7 subgroup, which adds prime 7.

MOS scales

As it is a rank-2 system, stacking 5/4 with octave-reduction can produce MOS scales, just like how Pythagorean tuning famously produces a version of the pentatonic, diatonic and chromatic scales. An important structural feature of the 2.5 subgroup is that 5/4³ = 125/64, which is only a diesis (128/125) short of an octave. This means that the cube root of 2, or the familiar 3edo/12edo major third, is a decent approximation of 5/4. It also means that most small MOSes generated by the 2.5 subgroup are of the form 3L xs, and are extremely hard (high L:s ratio) which makes them difficult to use. It takes all the way up to a 31-tone MOS (28L 3s) to break out of the 3L xs chain, which corresponds to how 31edo is the smallest EDO to have an approximation of 5/4 in-between just 5/4 and the cube root of 3.

The non-trivial MOS scales that are produced are: 1L 2s, 3L 1s, 3L 4s, 3L 7s, 3L 10s, 3L 13s, 3L 16s, 3L 19s, 3L 22s, 3L 25s, 28L 3s, 28L 31s, 59L 28s, 59L 87s, etc.

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