4L 5s (3/1-equivalent)

↖ 3L 4s⟨3/1⟩	↑ 4L 4s⟨3/1⟩	5L 4s⟨3/1⟩ ↗
← 3L 5s⟨3/1⟩	4L 5s (3/1-equivalent)	5L 5s⟨3/1⟩ →
↙ 3L 6s⟨3/1⟩	↓ 4L 6s⟨3/1⟩	5L 6s⟨3/1⟩ ↘

┌╥┬╥┬╥┬╥┬┬┐ │║│║│║│║│││ │││││││││││ └┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LsLsLsLss ssLsLsLsL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	2\9 to 1\4 (422.7 ¢ to 475.5 ¢)
Dark	3\4 to 7\9 (1426.5 ¢ to 1479.3 ¢)
Related MOS scales
Parent	4L 1s⟨3/1⟩
Sister	5L 4s⟨3/1⟩
Daughters	9L 4s⟨3/1⟩, 4L 9s⟨3/1⟩
Neutralized	8L 1s⟨3/1⟩
2-Flought	13L 5s⟨3/1⟩, 4L 14s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	2\9 (422.7 ¢)
Supersoft (L:s = 4:3)	7\31 (429.5 ¢)
Soft (L:s = 3:2)	5\22 (432.3 ¢)
Semisoft (L:s = 5:3)	8\35 (434.7 ¢)
Basic (L:s = 2:1)	3\13 (438.9 ¢)
Semihard (L:s = 5:2)	7\30 (443.8 ¢)
Hard (L:s = 3:1)	4\17 (447.5 ¢)
Superhard (L:s = 4:1)	5\21 (452.8 ¢)
Collapsed (L:s = 1:0)	1\4 (475.5 ¢)

4L 5s⟨3/1⟩, also called Lambda, is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 5 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 422.7 ¢ to 475.5 ¢, or from 1426.5 ¢ to 1479.3 ¢. Suggested for use as the analog of the 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 (Bohlen-Pierce-Stearns) that eliminates 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 edos!) besides 13edt.

Some low-numbered edos that support BPS are 19, 22, 27, 41, and 46, and some low-numbered edts that support it are 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 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 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

Scale degrees of the modes of 4L 5s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9
8|0	1	LsLsLsLss	Perf.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Aug.	Maj.	Perf.
7|1	3	LsLsLssLs	Perf.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
6|2	5	LsLssLsLs	Perf.	Maj.	Perf.	Maj.	Maj.	Min.	Maj.	Perf.	Maj.	Perf.
5|3	7	LssLsLsLs	Perf.	Maj.	Perf.	Min.	Maj.	Min.	Maj.	Perf.	Maj.	Perf.
4|4	9	sLsLsLsLs	Perf.	Min.	Perf.	Min.	Maj.	Min.	Maj.	Perf.	Maj.	Perf.
3|5	2	sLsLsLssL	Perf.	Min.	Perf.	Min.	Maj.	Min.	Maj.	Perf.	Min.	Perf.
2|6	4	sLsLssLsL	Perf.	Min.	Perf.	Min.	Maj.	Min.	Min.	Perf.	Min.	Perf.
1|7	6	sLssLsLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Perf.	Min.	Perf.
0|8	8	ssLsLsLsL	Perf.	Min.	Dim.	Min.	Min.	Min.	Min.	Perf.	Min.	Perf.

Proposed names

Lériendil proposes mode names derived from the constellations of the northern sky.

Modes of 4L 5s

UDP	Cyclic order	Step pattern
8|0	1	LsLsLsLss
7|1	3	LsLsLssLs
6|2	5	LsLssLsLs
5|3	7	LssLsLsLs
4|4	9	sLsLsLsLs
3|5	2	sLsLsLssL
2|6	4	sLsLssLsL
1|7	6	sLssLsLsL
0|8	8	ssLsLsLsL

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.

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.