# 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 3/1 as the equivalence interval, half-prime subgroups can be considered with 3/2 as the equivalence interval, presenting a possible JI interpretation of EDFs. They were first considered by CompactStar in 2023.

They correspond to EDFs if used as a rank-1 tempered systems.

## Generalizations

Half-prime subgroups can be generalized for other denominators, such as to third-prime subgroups (5/3.7/3.11/3.13/3..., which are suitable for 5/3 as the equave), or "quarter-prime subgroups" (5/4.7/4.11/4.13/4..., which are suitable for 5/4 as the equave). They can also be restricted to remove 3/2 for usage in Ed5/2 systems.

## Harmony

If a low-complexity JI-based perspective is used, there is an absence of low-complexity chords with 3 or more notes 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 3/2-repeating system, spanning more than twice the equivalence interval of 3/2. Thus, harmony would be largely established using two notes at a time rather than three, using dyads with intervals of 10/9, 25/21, 27/20 or 7/5, as well as 28/27 or 15/14 if extreme tension is permitted. This can be compared to 2edo, 3edo and 4edo, but with far more sophisticated types of harmonic progression. Note that in a 3/2-repeating system, tertian chords are considered voicings of a dyad–for example, the minor dyad with the interval 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.

There is however infiitely many high-complexity JI chords contained within half-prime subgroups, as with any just intonation system, with the diminished triad 125:147:175 (1-25/21-7/5) being of interest.