Half-prime subgroup

Half-prime subgroups are a family of nonoctave just intonation subgroups where the basis elements are the halves of primes (3/2, 5/2, 7/2, 11/2 and etc.), rather than the primes themselves. Similar to howno-twos subgroups are usually considered with a period of 3/1, half-prime subgroups can be considered with a period of 3/2 or more complexly 5/2, so present a possible JI interpretation of EDFs and Ed5/2s. They were first considered by CompactStar in 2023.

There are rank-1 and rank-2 regular temperaments that can be built on this system. 11edf and 12edf (which have Ed5/2 counterparts as 25ed5/2 and 27ed5/2) are the smallest EDFs which offer a plausible rendition of 3/2.5/2.7/2 subgroup. Notable commas are the hemimage comma, which if tempered results in a chain of 28/27s that is similar to the previously-mentioned 11edf and 12edf, the Sirius comma 3125/3087 that is also a member of the no-twos 3.5.7 subgroup, and 20480/19683.

Intervals and chords

These subgroups offer a wide diversity of intervals but very few are simple or of low odd limit, at least if 3/2 is used as the interval of equivalence. The simplest interval in any half-prime subgroup that is below 3/2 is 7/5, arising from the 3/2.5/2.7/2 subgroup. This is followed by 10/9 (the fifth-reduced form of 5/2), 15/14, 25/21, 27/20, and 28/27 (the fifth-reduced form of 7/2). 11/2 reduces to 88/81 and higher half-primes are even more complex. There is a similar situation for chords with multiple intervals–the simplest that can fit inside 3/2 would be 27:28:30, A dense tone cluster. For a non-tone cluster, the simplest would be 45:50:63, a sort of diminished triad, but using 10/9 instead of a minor third above the root. So it appears that harmony in this system would be largely built on dyads if it is based on simple just intervals. Although if the interval of equivalence is chosen as wider, like 5/2 or 7/2, simpler chords and intervals become available like 14/9 and thus 9:10:14.