Half-prime subgroup - Revision history

BudjarnLambeth: /* Generalizations */ Add half-basis, third-basis, etc.

2024-12-22T07:03:55Z

Generalizations: Add half-basis, third-basis, etc.

← Older revision		Revision as of 07:03, 22 December 2024
Line 3:		Line 3:
	They correspond to [[EDF]]s if used as a rank-1 tempered systems.		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.

			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 ==		== Harmony ==

BudjarnLambeth: /* See also */

2024-11-05T13:14:37Z

BudjarnLambeth: Add see also section for ease of navigation

2024-11-05T02:02:16Z

Add see also section for ease of navigation

← Older revision		Revision as of 02:02, 5 November 2024
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	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.		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]]

	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

CompactStar at 00:28, 11 April 2024

2024-04-11T00:28:23Z

← Older revision		Revision as of 00:28, 11 April 2024
Line 6:		Line 6:

	== 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, 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.		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.		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.

CompactStar: I will rewrite the deleted parts soon

2024-04-11T00:27:51Z

I will rewrite the deleted parts soon

← Older revision		Revision as of 00:27, 11 April 2024
Line 1:		Line 1:
	'''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~~. 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.


	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

CompactStar at 00:17, 11 April 2024

2024-04-11T00:17:45Z

← Older revision		Revision as of 00:17, 11 April 2024
Line 1:		Line 1:
	'''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.		'''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]].

	== Generalizations ==		== 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...).		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, ~~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]] and other no-twos systems, is available but it is unwieldy to manage in a ~~fifth~~-repeating system, spanning more than 2 ~~fifths~~. ~~It may~~ be ~~considered to use~~ to use [[~~dyad~~]]~~s of~~ [[10/9]], [[~~27/20~~]], [[7/5]], ~~or the 12edo-like minor and major thirds~~ [[25/21]] ~~and~~ [[63/50]]. Note that in a ~~fifth~~-repeating system, ~~fifth doublings are fungible, meaning~~ the 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.		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.

			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 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 EDF systems that temper out the hemimage comma.
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

CompactStar at 04:48, 3 April 2024

2024-04-03T04:48:49Z

← Older revision		Revision as of 04:48, 3 April 2024
Line 7:		Line 7:

	== Harmony ==		== 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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, fifth doublings are fungible, meaning the 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,.		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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, fifth doublings are fungible, meaning the 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.

	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 EDF systems that temper out the hemimage comma.		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 EDF systems that temper out the hemimage comma.
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

CompactStar at 04:48, 3 April 2024

2024-04-03T04:48:33Z

← Older revision		Revision as of 04:48, 3 April 2024
Line 7:		Line 7:

	== Harmony ==		== 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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, the 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.		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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, fifth doublings are fungible, meaning the 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,.

	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 EDF systems that temper out the hemimage comma.		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 EDF systems that temper out the hemimage comma.
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

CompactStar at 04:47, 3 April 2024

2024-04-03T04:47:54Z

← Older revision		Revision as of 04:47, 3 April 2024
Line 9:		Line 9:
	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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, the 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.		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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, the 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.

	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 EDF systems that temper out the hemimage comma.		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 EDF systems that temper out the hemimage comma.
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

CompactStar at 04:47, 3 April 2024

2024-04-03T04:47:38Z

← Older revision		Revision as of 04:47, 3 April 2024
Line 9:		Line 9:
	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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, the 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.		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]] and other no-twos systems, is available but it is unwieldy to manage in a fifth-repeating system, spanning more than 2 fifths. It may be considered to use to use [[dyad]]s of [[10/9]], [[27/20]], [[7/5]], or the 12edo-like minor and major thirds [[25/21]] and [[63/50]]. Note that in a fifth-repeating system, the 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.

	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 EDF systems ~~where~~ the hemimage comma ~~is tempered out~~.		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 EDF systems that temper out the hemimage comma.
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]

← Older revision		Revision as of 13:14, 5 November 2024
Line 13:		Line 13:
	* [[Subgroup temperament]]		* [[Subgroup temperament]]
	* [[Carlos Alpha]], [[Carlos Beta]], [[Carlos Gamma]]		* [[Carlos Alpha]], [[Carlos Beta]], [[Carlos Gamma]]
			* [[Basal subgroup]]

	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Just intonation]]		[[Category:Just intonation]]