Half-prime subgroup: Difference between revisions

Revision as of 03:58, 2 April 2024

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 a period of 3/1, half-prime subgroups can be considered with a period of 3/2, so present a possible JI interpretation of EDFs. They were first considered by CompactStar in 2023.

There are rank-1 and rank-2 regular temperaments that can be built on this system. 11edf and 12edf are the smallest EDFs which offer a plausible rendition of 3/2.5/2.7/2 subgroup. Notable commas that could be tempered are the hemimage comma, which if tempered results in a chain of 28/27s that is similar to the previously-mentioned 11edf and 12edf, the Sirius comma 3125/3087, 20480/19683, and 99/98.

Generalizations

The concepts of half-prime subgroups and no-twos subgroups can be combined to create "no-3/2s half-prime subgroups" (5/2.7/2.11/2.13/2....) which are suitable for Ed5/2 systems. Additionally, half-prime subgroups can be generalized for other denominators, such as to "third-prime subgroups" (5/3.7/3.11/3.13/3...), or "quarter-prime subgroups" (5/4.7/4.11/4.13/4...).

Harmony

If a low-complexity JI-based perspective is used, then harmony would be largely established using 2 notes at a time rather than three, with an absence of low-complexity triads that can be practically used. The chord 3:5:7, which is shared with Bohlen-Pierce systems, is available but it is unwieldy to manage \

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 The concepts of half-prime subgroups and no-twos subgroups can be combined to create "no-3/2s half-prime subgroups" (5/2.7/2.11/2.13/2....) which are suitable for [[Ed5/2]] systems. Additionally, half-prime subgroups can be generalized for other denominators, such as to "third-prime subgroups" (5/3.7/3.11/3.13/3...), or "quarter-prime subgroups" (5/4.7/4.11/4.13/4...).
-== Intervals and chords ==
+== Harmony ==
-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 (which is in fact fifth-equivalent to 27:28:30).
+If a [[low-complexity JI]]-based perspective is used, then harmony would be largely established using 2 notes at a time rather than three, with an absence of low-complexity triads that can be practically used. The chord 3:5:7, which is shared with Bohlen-Pierce systems, is available but it is unwieldy to manage \
 [[Category:Subgroup]]
 [[Category:Just intonation]]

Half-prime subgroup: Difference between revisions

Revision as of 03:58, 2 April 2024

Generalizations

Harmony

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