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{{Infobox MOS
| Other names = Lambda
}}
{{MOS intro
| Other Names = Lambda
}}
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 [[BPS|BPS (Bohlen-Pierce-Stearns)]] that eliminates 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 [[9edt|9]], [[13edt|13]], [[17edt|17]], and [[30edt|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 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.
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 is [[sensi]] temperament, in essence treating it as a "3.5.7.2-subgroup extension" of the original 3.5.7-subgroup BPS temperament.
== Modes ==
{{MOS modes}}
=== Proposed 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}}
== Notation ==
Bohlen-Pierce theory possesses a [[nonoctave]] notation system for [[EDT]]s and no-twos music, which is well-established<ref>[https://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale#Intervals_and_scale_diagrams|Wikipedia article]</ref>. It is based on the [[2L 9s (3/1-equivalent)|2L 9s⟨3/1⟩]] MOS scale generated by approximately [[5/3]], relating it to [[Arcturus]] temperament. The preferred generator for a given EDT is the patent val 5/3.
The 9|1 mode of 2L 9s⟨3/1⟩ is represented by 11 naturals CDEFGHJKLAB, and #/b accidentals are used to raise and lower by a chroma (L - s). For relative notation, the [[TAMNAMS]] convention of zero-based indices is used.  Assuming Arcturus temperament where [[7/3]] is mapped to the major 8-step, the chord 3:5:7 is notated as C - H - L (P0 - P5 - M8). To notate larger EDTs, Arcturus hendecatonic notation can be combined with [[ups and downs notation]], with ^/v accidentals raising and lowering by 1 EDTstep.
== Examples ==
[[11edt]] (equalized):
{|class="wikitable article-table"
|-
!0
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
|-
|C
|D
|E
|F
|G
|H
|J
|K
|L
|A
|B
|C
|-
|P0
|P1
|P2
|P3
|P4
|P5
|P6
|P7
|P8
|P9
|P10
|P11
|}
[[13edt]] (Bohlen-Pierce):
{|class="wikitable article-table"
|-
!0
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|C
|Db
|D, Eb
|E, Fb
|F, Gb
|G, Hb
|H
|J
|J#, Kb
|K
|L, Ab
|A, Bb
|B
|C
|-
|P0
|m1
|M1, m2
|M2, m3
|M3, m4
|M4, d5
|P5
|P6
|A6, m7
|M7
|M8, m9
|M9, m10
|M10
|P11
|}
[[24edt]]:
{|class="wikitable article-table"
|-
!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
|-
|C
|C#, Dbb
|Db
|D
|Eb
|E
|Fb
|F
|Gb
|G
|G#, Hb
|H
|H#, Jb
|J
|J#, Kbb
|Kb
|K
|Lb
|L
|Ab
|A
|Bb
|B
|B#, Cb
|C
|-
|P0
|A0, d1
|m1
|M1
|m2
|M2
|m3
|M3
|m4
|M4
|A4, d5
|P5
|A5, d6
|P6
|A6, d7
|m7
|M7
|m8
|M8
|m9
|M9
|m10
|M10
|A10, d11
|P11
|}
== 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 cents and 475.5 cents.
{{Scale tree|depth=7|Comments=13/6:[[Bohlen-Pierce-Stearns|BPS]] 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]].
[[Category:Notation]]
[[Category:Nonoctave]]

Latest revision as of 18:39, 22 June 2025

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