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 [[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.

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 ==

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~~. One possible approach would to be use minor and major~~ dyads with ~~the~~ intervals of [[25/21]] ~~and~~ [[63/50]], ~~in a similar fashion to~~ [[~~4edo~~]] ~~and~~ [[~~3edo~~]] ~~respectively, although with much more sophisticated types of harmonic progression~~. ~~More xenharmonic options include using dyads based on~~ [[~~10/9~~]], [[~~27/20~~]] or [[~~7/5~~]]. Note that in a 3/2-repeating system, the minor dyad 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~~, and the same applies to the major dyad~~.

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 ~~an abundance of~~ high-complexity JI chords contained within half-prime subgroups, as with any just intonation system, ~~like 45:50:63 (1-10/9-7/5), or~~ the diminished triad ~~105:~~125:147 (1-25/21~~-7/5), which is made symmetric in temperaments that [[temper out~~]] [[~~3125~~/~~3087~~]].

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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 '''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.
+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 ==
-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. One possible approach would to be use minor and major dyads with the intervals of [[25/21]] and [[63/50]], in a similar fashion to [[4edo]] and [[3edo]] respectively, although with much more sophisticated types of harmonic progression. More xenharmonic options include using dyads based on [[10/9]], [[27/20]] or [[7/5]]. Note that in a 3/2-repeating system, the minor dyad 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, and the same applies to the major dyad.
+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 an abundance of high-complexity JI chords contained within half-prime subgroups, as with any just intonation system, like 45:50:63 (1-10/9-7/5), or the diminished triad 105:125:147 (1-25/21-7/5), which is made symmetric in temperaments that [[temper out]] [[3125/3087]].
+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]]