4L 5s (3/1-equivalent) - Revision history

FloraC: Misc. cleanup and style improvements

2025-05-27T09:00:38Z

Misc. cleanup and style improvements

← Older revision		Revision as of 09:00, 27 May 2025
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	: ''For the octave equivalent 4L 5s pattern, see [[4L 5s]].''		: ''For the octave equivalent 4L 5s pattern, see [[4L 5s]].''

	{{Infobox MOS		{{Infobox MOS\|Other names=Lambda}}
	\| Other names = Lambda		{{MOS intro\|Other Names=Lambda}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].
	}}
	{{MOS intro
	\| Other Names = Lambda
	}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].

	{{MOS scalesig\|4L 5s <3/1>}} can be thought of as a ~~MOS~~ generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is ~~supported~~ by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].		{{MOS scalesig\|4L 5s <3/1>}} can be thought of as a mos generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a [[3.5.7 subgroup\|3.5.7-subgroup]] [[rank-2 temperament]] that [[tempering out\|tempers out]] [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is [[support]]ed by many [[edt]]s (and even [[edo]]s) besides [[13edt]].

	Some low-numbered ~~EDOs~~ that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered ~~EDTs~~ that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and ~~EDOs~~ approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.		Some low-numbered edos that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered edts that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a [[detempering\|detempered]] version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how [[meantone]] temperaments such as {{EDOs\| 19, 31, and 43 }} and edos approximating [[Pythagorean tuning]] ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

	When playing this scale in some ~~EDO~~, it may be desired to [[stretched and compressed tuning\|stretch or compress the octaves]] to make 3/1 just (or closer to just), rather than the octave being pure—or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.		When playing this scale in some edo, it may be desired to [[stretched and compressed tuning\|stretch or compress the octaves]] to make 3/1 just (or closer to just), rather than the octave being pure—or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.

	One can add the octave to BPS by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup" ("add-octave") extension of BPS.		One can add the octave to BPS by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup" ("add-octave") extension of BPS.
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	=== Proposed mode names ===		=== Proposed mode names ===
	[[User:Lériendil\|Lériendil]] proposes mode names derived from the constellations of the northern sky.		[[User:Lériendil\|Lériendil]] proposes mode names derived from the constellations of the northern sky.
	{{MOS modes		{{MOS modes\| Mode Names=
	\| Mode Names=
	Lyncian $		Lyncian $
	Aurigan $		Aurigan $
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	== Notation ==		== Notation ==
	Bohlen–Pierce theory possesses a well-established [[non-octave]] notation system for [[~~EDT~~]]s and no-~~twos~~ music, which is based on this ~~MOS~~ scale as generated by approximately [[7/3]], relating it to BPS. The preferred generator for any edt is its patent val approximation of 7/3.		Bohlen–Pierce theory possesses a well-established [[non-octave]] notation system for [[edt]]s and no-2's music, which is based on this mos scale as generated by approximately [[7/3]], relating it to BPS. The preferred generator for any edt is its patent val approximation of 7/3.

	This notation uses 9 nominals: for compatibility with [[diamond-~~MOS~~ notation]], the current recommendation is to use the notes {{nowrap\|J K L M N O P Q R}} as presented in the J~~ ~~Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: {{dash\|~~...Q♯~~, ~~O♯~~, ~~M♯~~, ~~K♯~~, R, P, N, L, J, Q, O, M, K, ~~R♭~~, ~~P♭~~, ~~N♭~~, ~~L♭...~~\|hair\|med}} However, an alternative convention ({{w\|Bohlen–Pierce scale#Intervals and scale diagrams\|as seen on Wikipedia}} and some other articles of this wiki) labels them {{nowrap\|C D E F G H J A B}} in the C~~ ~~Andromedan (LssLsLsLs) mode, which rotates to the E symmetric mode.		This notation uses 9 nominals: for compatibility with [[diamond-mos notation]], the current recommendation is to use the notes {{nowrap\| J K L M N O P Q R }} as presented in the J Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: {{dash\|…, Q♯, O♯, M♯, K♯, R, P, N, L, J, Q, O, M, K, R♭, P♭, N♭, L♭, …\|hair\|med}} However, an alternative convention ({{w\|Bohlen–Pierce scale #Intervals and scale diagrams\|as seen on Wikipedia}} and some other articles of this wiki) labels them {{nowrap\| C D E F G H J A B }} in the C Andromedan (LssLsLsLs) mode, which rotates to the E symmetric mode.

	An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]].		An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]].
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	\| P8		\| P8
	\| P9		\| P9
	\|}<br />		\|}<br>

	{\| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"		{\| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
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	\| M8		\| M8
	\| P9		\| P9
	\|}<br />		\|}<br>

	{\| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"		{\| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
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	\|}		\|}

	== ~~List of edts supporting the Lambda scale~~ ==		== Scale tree ==
	Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5{{c}}.		Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5{{c}}.
	{{MOS tuning spectrum		{{MOS tuning spectrum
	\| Depth = 7		\| Depth = 7
	\| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]		\| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]
	\| 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) ~~is in this~~ region		\| 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) region
	\| 22/13 = Essentially just 7/3		\| 22/13 = Essentially just 7/3
	}}		}}

	Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]].		Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by [[48edt\|37\48edt]] and extremely closely approximated by [[153edt\|118\153edt]].

	[[Category:Bohlen–Pierce]]		[[Category:Bohlen–Pierce]]

ArrowHead294: Can't believe this was on this page

2025-05-05T14:19:20Z

Can't believe this was on this page

← Older revision		Revision as of 14:19, 5 May 2025
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	== Scale properties ==		== Scale properties ==
	{{TAMNAMS use}}		{{TAMNAMS use}}
	{{MOS ~~data~~}}
			=== Intervals ===
			{{MOS intervals}}

			=== Generator chain ===
			{{MOS genchain}}

			=== Modes ===
			{{MOS mode degrees}}

	=== Proposed mode names ===		=== Proposed mode names ===

ArrowHead294 at 13:01, 10 March 2025

2025-03-10T13:01:29Z

← Older revision		Revision as of 13:01, 10 March 2025
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	\| Depth = 7		\| Depth = 7
	\| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]		\| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]
	\| 13/6 = [[~~Bohlen–Pierce–Stearns~~]] is in this region		\| 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) is in this region
	\| 22/13 = Essentially just 7/3		\| 22/13 = Essentially just 7/3
	}}		}}
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	Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]].		Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]].

	[[Category:~~Bohlen-Pierce~~]]		[[Category:Bohlen–Pierce]]

ArrowHead294 at 00:53, 8 March 2025

2025-03-08T00:53:06Z

← Older revision		Revision as of 00:53, 8 March 2025
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	== List of edts supporting the Lambda scale ==		== List of edts supporting the Lambda scale ==
	Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5¢.		Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5{{c}}.
	{{MOS tuning spectrum		{{MOS tuning spectrum
	\| Depth = 7		\| Depth = 7

Ganaram inukshuk at 08:53, 7 March 2025

2025-03-07T08:53:36Z

← Older revision		Revision as of 08:53, 7 March 2025
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	}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].		}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].

	{{MOS scalesig}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].		{{MOS scalesig\|4L 5s <3/1>}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].

	Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.		Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

ArrowHead294: Replace MOSPAGENAME with MOS scalesig

2025-03-06T19:24:39Z

Replace MOSPAGENAME with MOS scalesig

← Older revision		Revision as of 19:24, 6 March 2025
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	}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].		}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].

	{{~~MOSPAGENAME~~}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].		{{MOS scalesig}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].

	Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.		Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

ArrowHead294 at 18:52, 6 March 2025

2025-03-06T18:52:12Z

← Older revision		Revision as of 18:52, 6 March 2025
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	}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].		}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].

	~~4L 5s~~{{~~angbr\|3/1~~}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].		{{MOSPAGENAME}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].

	Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.		Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

ArrowHead294 at 16:35, 5 March 2025

2025-03-05T16:35:19Z

← Older revision		Revision as of 16:35, 5 March 2025
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	}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].		}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].

	4L 5s{{angbr\|3/1}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[~~Bohlen–Pierce–Stearns~~]] (~~BPS~~), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].		4L 5s{{angbr\|3/1}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].

	Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.		Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

ArrowHead294: /* List of edts supporting the Lambda scale */

2025-03-03T16:40:16Z

List of edts supporting the Lambda scale

← Older revision		Revision as of 16:40, 3 March 2025
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	Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5¢.		Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5¢.
	{{MOS tuning spectrum		{{MOS tuning spectrum
			\| Depth = 7
			\| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]
	\| 13/6 = [[Bohlen–Pierce–Stearns]] is in this region		\| 13/6 = [[Bohlen–Pierce–Stearns]] is in this region
	\| 22/13 = Essentially just 7/3		\| 22/13 = Essentially just 7/3

ArrowHead294 at 16:39, 3 March 2025

2025-03-03T16:39:10Z

← Older revision		Revision as of 16:39, 3 March 2025
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	== List of edts supporting the Lambda scale ==		== List of edts supporting the Lambda scale ==
	Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5¢.		Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5¢.
	{{~~Scale tree~~\|~~depth=7\|Comments=~~13/6: [[Bohlen–Pierce–Stearns]] is in this region; 22/13: Essentially just 7/3}}		{{MOS tuning spectrum
			\| 13/6 = [[Bohlen–Pierce–Stearns]] is in this region
			\| 22/13 = Essentially just 7/3
			}}

	Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]].		Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]].

	~~== Intervals ==~~
	~~{{MOS intervals}}~~

	[[Category:Bohlen-Pierce]]		[[Category:Bohlen-Pierce]]

← Older revision		Revision as of 16:35, 5 March 2025
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	}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].		}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s\|diatonic scale]].

	4L 5s{{angbr\|3/1}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[~~Bohlen–Pierce–Stearns~~]] (~~BPS~~), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].		4L 5s{{angbr\|3/1}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is supported by many [[edt]]'s (and even [[edo]]s) besides [[13edt]].

	Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.		Some low-numbered EDOs that support BPS are {{EDOs\| 19, 22, 27, 41, and 46 }}, and some low-numbered EDTs that support it are {{EDTs\| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how meantone temperaments such as {{EDOs\| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs\| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.