Half-prime subgroup: Difference between revisions

Line 1:

'''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 hown[[o-twos subgroup]]s 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 [[EDF]]s and [[Ed5/2]]s. They were first considered by [[User:CompactStar|CompactStar]] in 2023

'''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 hown[[o-twos subgroup]]s 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 [[EDF]]s and [[Ed5/2]]s. They were first considered by [[User:CompactStar|CompactStar]] in 2023.

There are rank-1 and rank-2 [[regular temperament]]s 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 [[28/27]]s that is similar to the previously-mentioned 11edf and 12edf,

== 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.

[[Category:Subgroup]]

[[Category:Just intonation]]

@@ Line 1: / Line 1: @@
 {{Stub}}
-'''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 hown[[o-twos subgroup]]s 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 [[EDF]]s and [[Ed5/2]]s. They were first considered by [[User:CompactStar|CompactStar]] in 2023
+'''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 hown[[o-twos subgroup]]s 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 [[EDF]]s and [[Ed5/2]]s. They were first considered by [[User:CompactStar|CompactStar]] in 2023.
+There are rank-1 and rank-2 [[regular temperament]]s 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 [[28/27]]s that is similar to the previously-mentioned 11edf and 12edf,
 == 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.
 [[Category:Subgroup]]
 [[Category:Just intonation]]