Half-prime subgroup: Difference between revisions

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They correspond to [[EDF]]s 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.

If numerators are allowed to be composite numbers as well as primes in a subgroup, then it could be called half-basis subgroups{{idiosyncratic}}, third basis subgroups{{idiosyncratic}}, quarter basis subgroups{{idiosyncratic}}, etc. Because "[[basis element]]s" is the generalized form of "primes" in a subgroup.

== Harmony ==

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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.

== See also ==

* [[Subgroup temperament]]

* [[Carlos Alpha]], [[Carlos Beta]], [[Carlos Gamma]]

* [[Basal subgroup]]

[[Category:Subgroup]]

[[Category:Just intonation]]

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 They correspond to [[EDF]]s 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.
+If numerators are allowed to be composite numbers as well as primes in a subgroup, then it could be called half-basis subgroups{{idiosyncratic}}, third basis subgroups{{idiosyncratic}}, quarter basis subgroups{{idiosyncratic}}, etc. Because "[[basis element]]s" is the generalized form of "primes" in a subgroup.
 == Harmony ==
@@ Line 10: / Line 12: @@
 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.
+== See also ==
+* [[Subgroup temperament]]
+* [[Carlos Alpha]], [[Carlos Beta]], [[Carlos Gamma]]
+* [[Basal subgroup]]
 [[Category:Subgroup]]
 [[Category:Just intonation]]