4L 5s (3/1-equivalent): Difference between revisions

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{{Infobox MOS

: ''For the octave equivalent 4L 5s pattern, see [[4L 5s]].''

| Other names = Lambda

}}

{{MOS intro

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

| Other Names = Lambda

}}

{{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]].

~~Suggested for use as the analog~~ of the [[5L 2s|diatonic scale]] ~~when playing [[Bohlen–Pierce]] is this 9-note Lambda scale, which is the~~ 4L 5s ~~mos with [[equave]]~~ 3/1~~. This~~ can be thought of as a mos generated by a 3.5.7-[[subgroup]] [[rank-2 temperament]] ~~called~~ [[~~Bohlen–Pierce–Stearns~~]] ~~that tempers out only the comma~~ [[245/243]]~~, so that (9/7)<sup>2</sup> is equated with 5/3~~. ~~This~~ is a very good temperament on the 3.5.7 subgroup, and ~~additionally~~ 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 [[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.

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.

~~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 BP music to some reasonable extent. These equal temperaments contain not only the Lambda "BP diatonic" scale, but, with the exception of 9edt, also 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 19edo and 31edo do not contain 12edo as a subset, but they do contain the meantone[12] chromatic scale.

== Scale properties ==

When playing this temperament in some edo, it may be desired to [[stretched and compressed tuning|stretch/compress the tuning]] so that the tritave is pure, 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.

=== Intervals ===

One can add the octave to BPS temperament 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 extension" of the original 3.5.7-subgroup BPS temperament.

=== Generator chain ===

== Modes ==

=== Modes ===

=== Proposed names ===

=== Proposed mode names ===

[[User:Lériendil|Lériendil]] proposes mode names derived from the constellations of the northern sky.

{{MOS modes~~|Scale Signature=4L 5s~~|Mode Names=Lyncian; Aurigan; Persean; Andromedan; Cassiopeian; Lacertian; Cygnian; Draconian; Herculean}}

{{MOS modes| Mode Names=

Lyncian $

Aurigan $

Persean $

Andromedan $

Cassiopeian $

Lacertian $

Cygnian $

Draconian $

Herculean $

}}

== Notation ==

~~Bohlen–Pierce~~ theory possesses a well-established [[~~nonoctave~~]] 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 ~~[[Bohlen–Pierce–Stearns]] temperament, where two 7/3 generators are equated to 27/5~~. 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 J}} 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 used on Wikipedia}} and certain articles of this wiki) labels them {{nowrap|C D E F G H J A B C}} 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]].

=== Examples ===

{| class="wikitable article-table"

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

|+ style="font-size: 105%;" | 4L 5s in [[9edt]] (equalized)

|+ style="font-size: 105%;" | 4L 5s in [[9edt]] (equalized)

|-

! 0

Line 62:

Line 81:

| P8

| P9

|}

|}<br>

{| class="wikitable article-table"

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

|+ style="font-size: 105%;" | 4L 5s in [[13edt]] (~~Bohlen–Pierce~~)

|+ style="font-size: 105%;" | 4L 5s in [[13edt]] (Bohlen–Pierce)

|-

! 0

Line 84:

Line 103:

| J

| K

| K#, Lb

| K♯, L♭

| L

| M

| M#, Nb

| M♯, N♭

| N

| O

| O#, Pb

| O♯, P♭

| P

| Q

| Rb, Q#

| Q♯, R♭

| R

| J

Line 111:

Line 130:

| M8

| P9

|}

|}<br>

{| class="wikitable article-table"

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

|+ style="font-size: 105%;" | 4L 5s in [[30edt]]

|+ style="font-size: 105%;" | 4L 5s in [[30edt]]

|-

! 0

Line 149:

Line 168:

|-

| J

| R#, ~~Lbb~~

| R♯, L𝄫

| K

| J#

| J♯

| Lb

| L♭

| K#

| K♯

| Mb

| M♭

| L

| Kx, ~~Nbb~~

| K𝄪, N𝄫

| M

| L#

| L♯

| Nb

| N♭

| M#

| M♯

| Ob

| O♭

| N

| Mx, ~~Pbb~~

| M𝄪, P𝄫

| O

| N#

| N♯

| Pb

| P♭

| O#

| O♯

| Qb

| Q♭

| P

| Ox, ~~Rbb~~

| O𝄪, R𝄫

| Q

| P#

| P♯

| Rb

| R♭

| Q#

| Q♯

| Jb

| J♭

| R

| Kb, Qx

| K♭, Q𝄪

| J

|-

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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 ~~cents~~ and 475.5 ~~cents~~.

Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5{{c}}.

{{~~Scale tree~~|~~depth~~=7|~~Comments~~=13/6: [[~~Bohlen–Pierce–Stearns~~]] ~~is in this~~ region; 22/13: Essentially just 7/3}}

{{MOS tuning spectrum

| Depth = 7

| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]

| 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) 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 [[48edt|37\48edt]] and extremely closely approximated by [[153edt|118\153edt]].

~~== Intervals ==~~

~~{{MOS intervals}}~~

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

[[Category:Bohlen–Pierce]]

@@ Line 1: / Line 1: @@
-{{Infobox MOS
+: ''For the octave equivalent 4L&nbsp;5s pattern, see [[4L&nbsp;5s]].''
-| Other names = Lambda
-}}
+{{Infobox MOS|Other names=Lambda}}
-{{MOS intro
+{{MOS intro|Other Names=Lambda}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s|diatonic scale]].
-| Other Names = Lambda
-}}
+{{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]].
-Suggested for use as the analog of the [[5L 2s|diatonic scale]] when playing [[Bohlen&ndash;Pierce]] is this 9-note Lambda scale, which is the 4L 5s mos with [[equave]] 3/1. This can be thought of as a mos generated by a 3.5.7-[[subgroup]] [[rank-2 temperament]] called [[Bohlen&ndash;Pierce&ndash;Stearns]] that tempers out only the comma [[245/243]], so that (9/7)<sup>2</sup> is equated with 5/3. This is a very good temperament on the 3.5.7 subgroup, and additionally 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 [[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.
+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.
-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 BP music to some reasonable extent. These equal temperaments contain not only the Lambda "BP diatonic" scale, but, with the exception of 9edt, also 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 19edo and 31edo do not contain 12edo as a subset, but they do contain the meantone[12] chromatic scale.
+== Scale properties ==
+{{TAMNAMS use}}
-When playing this temperament in some edo, it may be desired to [[stretched and compressed tuning|stretch/compress the tuning]] so that the tritave is pure, 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.
+=== Intervals ===
+{{MOS intervals}}
-One can add the octave to BPS temperament 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 extension" of the original 3.5.7-subgroup BPS temperament.
+=== Generator chain ===
+{{MOS genchain}}
-== Modes ==
+=== Modes ===
 {{MOS mode degrees}}
-=== Proposed names ===
+=== Proposed mode names ===
 [[User:Lériendil|Lériendil]] proposes mode names derived from the constellations of the northern sky.
-{{MOS modes|Scale Signature=4L 5s|Mode Names=Lyncian; Aurigan; Persean; Andromedan; Cassiopeian; Lacertian; Cygnian; Draconian; Herculean}}
+{{MOS modes| Mode Names=
+Lyncian $
+Aurigan $
+Persean $
+Andromedan $
+Cassiopeian $
+Lacertian $
+Cygnian $
+Draconian $
+Herculean $
+}}
 == Notation ==
-Bohlen&ndash;Pierce theory possesses a well-established [[nonoctave]] 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 [[Bohlen&ndash;Pierce&ndash;Stearns]] temperament, where two 7/3 generators are equated to 27/5. 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 J}} as presented in the J&nbsp;Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: {{dash|...Q&#x266F;, O&#x266F;, M&#x266F;, K&#x266F;, R, P, N, L, J, Q, O, M, K, R&#x266D;, P&#x266D;, N&#x266D;, L&#x266D;...|hair|med}} However, an alternative convention ({{w|Bohlen&ndash;Pierce scale#Intervals and scale diagrams|as used on Wikipedia}} and certain articles of this wiki) labels them {{nowrap|C D E F G H J A B C}} in the C&nbsp;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]].
 === Examples ===
-{| class="wikitable article-table"
+{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
-|+ style="font-size: 105%;" | 4L 5s in [[9edt]] (equalized)
+|+ style="font-size: 105%;" | 4L&nbsp;5s in [[9edt]] (equalized)
 |-
 ! 0
@@ Line 62: / Line 81: @@
 | P8
 | P9
-|}
+|}<br>
-{| class="wikitable article-table"
+{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
-|+ style="font-size: 105%;" | 4L 5s in [[13edt]] (Bohlen&ndash;Pierce)
+|+ style="font-size: 105%;" | 4L&nbsp;5s in [[13edt]] (Bohlen–Pierce)
 |-
 ! 0
@@ Line 84: / Line 103: @@
 | J
 | K
-| K#, Lb
+| K♯, L♭
 | L
 | M
-| M#, Nb
+| M♯, N♭
 | N
 | O
-| O#, Pb
+| O♯, P♭
 | P
 | Q
-| Rb, Q#
+| Q♯, R♭
 | R
 | J
@@ Line 111: / Line 130: @@
 | M8
 | P9
-|}
+|}<br>
-{| class="wikitable article-table"
+{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
-|+ style="font-size: 105%;" | 4L 5s in [[30edt]]
+|+ style="font-size: 105%;" | 4L&nbsp;5s in [[30edt]]
 |-
 ! 0
@@ Line 149: / Line 168: @@
 |-
 | J
-| R#, Lbb
+| R♯, L𝄫
 | K
-| J#
+| J♯
-| Lb
+| L♭
-| K#
+| K♯
-| Mb
+| M♭
 | L
-| Kx, Nbb
+| K𝄪, N𝄫
 | M
-| L#
+| L♯
-| Nb
+| N♭
-| M#
+| M♯
-| Ob
+| O♭
 | N
-| Mx, Pbb
+| M𝄪, P𝄫
 | O
-| N#
+| N♯
-| Pb
+| P♭
-| O#
+| O♯
-| Qb
+| Q♭
 | P
-| Ox, Rbb
+| O𝄪, R𝄫
 | Q
-| P#
+| P♯
-| Rb
+| R♭
-| Q#
+| Q♯
-| Jb
+| J♭
 | R
-| Kb, Qx
+| K♭, Q𝄪
 | J
 |-
@@ Line 213: / Line 232: @@
 |}
-== 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 cents and 475.5 cents.
+Below is a list of equal temperaments which contain a 4L&nbsp;5s scale using generators between 422.7 and 475.5{{c}}.
-{{Scale tree|depth=7|Comments=13/6: [[Bohlen&ndash;Pierce&ndash;Stearns]] is in this region; 22/13: Essentially just 7/3}}
+{{MOS tuning spectrum
+| Depth = 7
+| 2/1 = Equally-tempered [[Bohlen–Pierce scale]]
+| 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) 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 [[48edt|37\48edt]] and extremely closely approximated by [[153edt|118\153edt]].
-== Intervals ==
-{{MOS intervals}}
-[[Category:Bohlen-Pierce]]
+[[Category:Bohlen–Pierce]]