Half-prime subgroup: Difference between revisions

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 (notably, the dyad of 1-[[25/21]] is considered fifth-equivalent to a standard minor triad of 1-[[25/21]]-[[3/2]]). 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.