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

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{{MOS intro

| Other Names = Lambda

}}

}} This scale is oftsn suggested for use as [[Bohlen–Pierce]]'s analog of the [[5L 2s|diatonic scale]], which is the 4L 5s mos with tritave equivalence. 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)2 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]].

~~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)2 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 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.

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 this 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.

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.

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.

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~~.

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 BPS.

== Modes ==

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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.

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.

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|seen on Wikipedia}} and certain 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]].

=== 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)

|-

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| P8

| P9

|}

|}

{| 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)

|-

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| 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

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| M8

| P9

|}

|}

{| 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]]

|-

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|-

| 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 4: / Line 4: @@
 {{MOS intro
 | Other Names = Lambda
-}}
+}} This scale is oftsn suggested for use as [[Bohlen–Pierce]]'s analog of the [[5L 2s|diatonic scale]], which is the 4L&nbsp;5s mos with tritave equivalence. 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]].
-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 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.
+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 this 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.
-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.
+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.
-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.
+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 BPS.
 == Modes ==
@@ Line 23: / Line 22: @@
 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.
-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–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.
+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&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–Pierce scale#Intervals and scale diagrams|seen on Wikipedia}} and certain articles of this wiki) labels them {{nowrap|C D E F G H J A B}} 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)
 |-
@@ Line 62: / Line 62: @@
 | 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)
 |-
@@ Line 84: / Line 84: @@
 | 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 111: @@
 | 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]]
 |-
@@ Line 149: / Line 149: @@
 |-
 | 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
 |-