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

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{{Infobox MOS
: ''For the octave equivalent 4L 5s pattern, see [[4L 5s]].''
| Name =  
 
| Equave = 3/1
{{Infobox MOS|Other names=Lambda}}
| nLargeSteps = 4
{{MOS intro|Other Names=Lambda}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s|diatonic scale]].
| nSmallSteps = 5
 
| Equalized = 2
{{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]].
| Collapsed = 1
 
| Pattern = LsLsLsLss
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.
}}
 
{{MOS intro}}
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.
 
== Scale properties ==
{{TAMNAMS use}}


Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS with [[equave]] 3/1. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament called BPS (Bohlen-Pierce-Stearns) that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.
=== Intervals ===
{{MOS intervals}}


This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many EDT's (and even EDOs!) besides 13-EDT.
=== Generator chain ===
{{MOS genchain}}


Some low-numbered EDOs that support Lambda are 19, 22, 27, 41, and 46, all of which make it possible to play BP music to some reasonable extent. These EDOs contain not only the Lambda BP diatonic scale, but also the 13-note "Lambda chromatic" MOS scale, or Lambda[13], which can be thought of as a "detempered" version of the 13-EDT Bohlen Pierce scale. This scale may be a suitable melodic substitute for the BP chromatic scale, and is basically the same as how 19-EDO and 31-EDO do not contain 12-EDO as a subset, but they do contain the meantone[12] chromatic scale.
=== Modes ===
{{MOS mode degrees}}


When playing this temperament in some EDO, it may be desired to 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.
=== Proposed mode names ===
[[User:Lériendil|Lériendil]] proposes mode names derived from the constellations of the northern sky.
{{MOS modes| Mode Names=
Lyncian $
Aurigan $
Persean $
Andromedan $
Cassiopeian $
Lacertian $
Cygnian $
Draconian $
Herculean $
}}


One can "add" the octave to Lambda 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 is [[Sensi]] temperament, in essence treating it as a "3.5.7.2 extension" of the original 3.5.7 Lambda temperament.
== Notation ==
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.


== Modes ==
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.
{{MOS modes}}


== List of EDT's supporting Lambda Temperament ==
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" style="text-align: center; margin: auto auto auto auto;"
|+ style="font-size: 105%;" | 4L&nbsp;5s in [[9edt]] (equalized)
|-
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
| J
| K
| L
| M
| N
| O
| P
| Q
| R
| J
|-
| P0
| P1
| P2
| P3
| P4
| P5
| P6
| P7
| P8
| P9
|}<br>


Below is a list of the equal-temperaments which contain a 4L+5s scale using generators between 422.7 cents and 475.5 cents.
{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
|+ style="font-size: 105%;" | 4L&nbsp;5s in [[13edt]] (Bohlen–Pierce)
|-
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
|-
| J
| K
| K♯, L♭
| L
| M
| M♯, N♭
| N
| O
| O♯, P♭
| P
| Q
| Q♯, R♭
| R
| J
|-
| P0
| m1
| M1, d2
| P2
| m3
| M3, m4
| M4
| m5
| M5, m6
| M6
| P7
| m8, A7
| M8
| P9
|}<br>


{{Scale tree|depth=7}}
{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;"
|+ style="font-size: 105%;" | 4L&nbsp;5s in [[30edt]]
|-
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
|-
| J
| R♯, L𝄫
| K
| J♯
| L♭
| K♯
| M♭
| L
| K𝄪, N𝄫
| M
| L♯
| N♭
| M♯
| O♭
| N
| M𝄪, P𝄫
| O
| N♯
| P♭
| O♯
| Q♭
| P
| O𝄪, R𝄫
| Q
| P♯
| R♭
| Q♯
| J♭
| R
| K♭, Q𝄪
| J
|-
| P0
| MM8, dd2
| m1
| A0
| d2
| M1
| mm3
| P2
| MM1, mm4
| m3
| A2
| m4
| M3
| mm5
| M4
| MM3, mm6
| m5
| MM4
| m6
| M5
| d7
| M6
| MM5, mm8
| P7
| MM6
| m8
| A7
| d9
| M8
| mm1, AA7
| P9
|}


*Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS.
== Scale tree ==
Below is a list of equal temperaments which contain a 4L&nbsp;5s scale using generators between 422.7 and 475.5{{c}}.
{{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
}}


[Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.]
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:Nonoctave]]
[[Category:Bohlen–Pierce]]
[[Category:9-tone scales]]
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