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. | '''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 == | == 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. | 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. | ||
== Harmony == | == 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 | 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 | 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. | ||
[[Category:Subgroup]] | [[Category:Subgroup]] | ||
[[Category:Just intonation]] | [[Category:Just intonation]] | ||
Revision as of 00:27, 11 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 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.
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.