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 a period of [[3/1]], half-prime subgroups can be considered with a period of [[3/2]]], so present 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]].

== Generalizations ==

The concepts of half-prime subgroups and no-twos subgroups can be combined to create "no-3/2 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 ==

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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 a period of [[3/1]], half-prime subgroups can be considered with a period of [[3/2]]], so present 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]].
+== Generalizations ==
+The concepts of half-prime subgroups and no-twos subgroups can be combined to create "no-3/2 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 ==