Half-prime subgroup: Difference between revisions

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'''Half-prime subgroups'''{{idiosyncratic}} are a family of [[nonoctave]] [[just intonation subgroup]]s 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 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'''{{idiosyncratic}} are a family of [[nonoctave]] [[just intonation subgroup]]s 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 subgroup]]s are usually considered with [[3/1]] as the [[equivalence interval]], half-prime subgroups can be considered with [[3/2]] as the [[equivalence interval]], presenting a possible JI interpretation of [[EDF]]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]] are the smallest [[EDF]]s 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 [[28/27]]s that is similar to the previously-mentioned 11edf and 12edf, the Sirius comma [[3125/3087]], [[20480/19683]], and [[99/98]].
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.


== Intervals and chords ==
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.
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).


== Harmony ==
If a [[low-complexity JI]]-based perspective is used, there is an absence of low-complexity chords with 3 or more notes that can be practically used. The chord 3:5:7, which is shared with [[Bohlen-Pierce]] and other no-twos systems, is available but it is unwieldy to manage in a 3/2-repeating system, spanning more than twice the equivalence interval of 3/2. Thus, harmony would be largely established using two notes at a time rather than three, using dyads with intervals of [[10/9]], [[25/21]], [[27/20]] or [[7/5]], as well as [[28/27]] or [[15/14]] if extreme tension is permitted. This can be compared to [[2edo]], [[3edo]] and [[4edo]], but with far more sophisticated types of harmonic progression. Note that in a 3/2-repeating system, tertian chords are considered voicings of a dyad–for example, the minor dyad with the interval of 25/21 is equivalent to the minor triad 1-25/21-3/2, the minor seventh chord 1-25/21-3/2-25/14, and so on.
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:Subgroup]]
[[Category:Just intonation]]
[[Category:Just intonation]]

Latest revision as of 07:03, 22 December 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 3/1 as the equivalence interval, half-prime subgroups can be considered with 3/2 as the equivalence interval, presenting a possible JI interpretation of EDFs. They were first considered by CompactStar in 2023.

They correspond to EDFs 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 term], third basis subgroups[idiosyncratic term], quarter basis subgroups[idiosyncratic term], etc. Because "basis elements" is the generalized form of "primes" in a subgroup.

Harmony

If a low-complexity JI-based perspective is used, there is an absence of low-complexity chords with 3 or more notes that can be practically used. The chord 3:5:7, which is shared with Bohlen-Pierce and other no-twos systems, is available but it is unwieldy to manage in a 3/2-repeating system, spanning more than twice the equivalence interval of 3/2. Thus, harmony would be largely established using two notes at a time rather than three, using dyads with intervals of 10/9, 25/21, 27/20 or 7/5, as well as 28/27 or 15/14 if extreme tension is permitted. This can be compared to 2edo, 3edo and 4edo, but with far more sophisticated types of harmonic progression. Note that in a 3/2-repeating system, tertian chords are considered voicings of a dyad–for example, the minor dyad with the interval of 25/21 is equivalent to the minor triad 1-25/21-3/2, the minor seventh chord 1-25/21-3/2-25/14, and so on.

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

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