Half-prime subgroup

Half-prime subgroups^{[idiosyncratic term]} 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 how no-twos subgroups are usually considered with a period of 3/1, half-prime subgroups can be considered with a period of 3/2, so present a possible JI interpretation of EDFs. 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 are the smallest EDFs which offer a plausible rendition of 3/2.5/2.7/2 subgroup. Notable commas that could be tempered 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, 20480/19683, and 99/98.

Generalizations

The concepts of half-prime subgroups and no-twos subgroups can be combined to create "no-3/2s half-prime subgroups" (5/2.7/2.11/2.13/2....) which are suitable for Ed5/2 systems. Additionally, half-prime subgroups can be generalized for other denominators, such as to "third-prime subgroups" (5/3.7/3.11/3.13/3...), or "quarter-prime subgroups" (5/4.7/4.11/4.13/4...).

Harmony

If a low-complexity JI-based perspective is used, then harmony would be largely established using 2 notes at a time rather than three, with an absence of low-complexity triads that can be practically used. The chord 3:5:7, which is shared with Bohlen-Pierce and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use dyads of 10/9, 27/20 or 7/5, or the 12edo-like minor and major thirds 25/21 and 63/50. Note that in a fifth-repeating system, the dyad of 25/21 is equivalent to the minor triad 1-25/21-3/2, the minor seventh chord 1-25/21-3/2-25/14, and so on.