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.

EDO approximation

EDOs which do relatively well at approximating 2.5 subgroup intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm base 2 of 5. The sequence is equivalent to the numbers of notes per octave in the mos scales generated by a pure 5/4 generator: 2, 3, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 59, 87, 146, 205, 351, 497, 643, 789, 1432 …

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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