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 2.

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.

Equating 5/4 to the cube root of 2, or tempering out 128/125, results in the augmented temperament, which has a period of ~5/4 and generator of ~3/2. This temperament may be familiar because 12edo supports it, and 12edo's tuning of the Augmented[6] 3L 3s scale is the augmented scale of standard music theory. There are regular temperaments generated by ~5/4, which have a one-to-one correspondence to the intervals of 2.5 subgroup the same way meantone does to Pythagorean tuning. These temperaments often exploit the 128/125 diesis, even though they don't temper it out: for example, magic temperament has a flat 5/4 of about 380 cents to equate (5/4)⁵ octave-reduced to 3/2, which is the same thing as equating (25/16)/(128/125) to 3/2. Wurschmidt has a more accurate 5/4 and equates (5/4)⁸ octave-reduced to 3/2, which is the same thing as equating (25/16)/((128/125)²) to 3/2. These can be seen as cluster temperaments using 128/125 as a versatile chroma.

See also

• 3-limit (2.3)
• 3.5 subgroup

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