Half-prime subgroup

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Half-prime subgroups are a family of nonoctave just intonation subgroup 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

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.