3.5 subgroup - Revision history

BudjarnLambeth: categories

2026-09-19T06:58:24Z

BudjarnLambeth: I think this was intended to say 3/2

2026-09-19T06:57:10Z

I think this was intended to say 3/2

← Older revision		Revision as of 06:57, 19 September 2026
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	The 3.5 subgroup is a [[retraction]] of the [[5-limit]], obtained by removing prime 2. Its simplest [[expansion]] is the [[3.5.7 subgroup]], which adds prime [[7/1\|7]].		The 3.5 subgroup is a [[retraction]] of the [[5-limit]], obtained by removing prime 2. Its simplest [[expansion]] is the [[3.5.7 subgroup]], which adds prime [[7/1\|7]].

	If used with [[3/1\|tritave]]-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] (also known as the 2.3 subgroup or 3-limit) involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the Pythagorean principle (stacking the smallest prime harmonic larger than the [[equave]]) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent [[rank-2 temperament]]s generated by ~[[5/3]], such as 3.5.7 [[Arcturus]] and 3.5.11 [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]].		If used with [[3/1\|tritave]]-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] (also known as the 2.3 subgroup or 3-limit) involves stacking [[3/2]] with octave-reduction. It can be regarded as an application of the Pythagorean principle (stacking the smallest prime harmonic larger than the [[equave]]) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent [[rank-2 temperament]]s generated by ~[[5/3]], such as 3.5.7 [[Arcturus]] and 3.5.11 [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]].

	== MOS scales ==		== MOS scales ==

CompactStar at 05:07, 19 September 2026

2026-09-19T05:07:26Z

← Older revision		Revision as of 05:07, 19 September 2026
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	== Generator chain ==		== Generator chain ==
			{\|class="wikitable"
			\|-
			!Number of generators
			!Interval
			!Cents
			\|-
			\|0
			\|[[1/1]]
			\|0.000
			\|-
			\|1
			\|[[5/3]]
			\|884.359
			\|-
			\|2
			\|[[25/9]]
			\|1768.717
			\|-
			\|3
			\|[[125/81]]
			\|751.121
			\|-
			\|4
			\|[[625/343]]
			\|1635.480
			\|}

CompactStar at 05:05, 19 September 2026

2026-09-19T05:05:29Z

← Older revision		Revision as of 05:05, 19 September 2026
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	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament, and removing octaves from bug results in [[2edt]] as a 3.5 equal temperament).		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament, and removing octaves from bug results in [[2edt]] as a 3.5 equal temperament).

			== Generator chain ==

CompactStar: /* MOS scales */

2026-09-19T04:52:47Z

MOS scales

← Older revision		Revision as of 04:52, 19 September 2026
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	As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.		As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament).		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament, and removing octaves from bug results in [[2edt]] as a 3.5 equal temperament).

CompactStar at 04:52, 19 September 2026

2026-09-19T04:52:03Z

← Older revision		Revision as of 04:52, 19 September 2026
Line 8:		Line 8:
	As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.		As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales~~/"diatonic-like'~~ are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament).		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament).

CompactStar at 04:51, 19 September 2026

2026-09-19T04:51:50Z

← Older revision		Revision as of 04:51, 19 September 2026
Line 8:		Line 8:
	As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.		As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament).		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales/"diatonic-like' are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament).

CompactStar at 04:51, 19 September 2026

2026-09-19T04:51:13Z

← Older revision		Revision as of 04:51, 19 September 2026
Line 8:		Line 8:
	As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.		As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (this fact is ~~incidentally~~ realized by the [[bug]] temperament).		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the [[bug]] temperament).

CompactStar at 04:51, 19 September 2026

2026-09-19T04:51:05Z

← Older revision		Revision as of 04:51, 19 September 2026
Line 8:		Line 8:
	As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.		As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents)~~, a~~ fact ~~which~~ is ~~actually connected to~~ the [[bug]] temperament.		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (this fact is incidentally realized by the [[bug]] temperament).

CompactStar at 04:49, 19 September 2026

2026-09-19T04:49:17Z

← Older revision		Revision as of 04:49, 19 September 2026
Line 8:		Line 8:
	As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.		As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

	The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents), a fact which is ~~realized by~~ the [[bug]] temperament.		The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)\|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)\|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))\|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)\|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)\|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)\|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)\|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)\|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)\|28L 15s<3/1>]]. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents), a fact which is actually connected to the [[bug]] temperament.

← Older revision		Revision as of 06:58, 19 September 2026
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	\|1635.480		\|1635.480
	\|}		\|}

			[[Category:Just intonation subgroups\|#]]
			[[Category:5-limit\|#]]