4L 5s (3/1-equivalent)

For the octave equivalent 4L 5s pattern, see 4L 5s.

↖ 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 ¢. It is often considered to be Bohlen–Pierce's equivalent of the ubiquitous diatonic scale.

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 Bohlen–Pierce–Stearns (BPS), a 3.5.7-subgroup rank-2 temperament that tempers out 245/243. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and 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 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 "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 19, 31, and 43 and EDOs approximating Pythagorean tuning 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 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.

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

Notation

Bohlen–Pierce theory possesses a well-established non-octave notation system for EDTs and no-twos 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 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: ...Q♯ – O♯ – M♯ – K♯ – R – P – N – L – J – Q – O – M – K – R♭ – P♭ – N♭ – L♭... However, an alternative convention (as seen on Wikipedia and some other articles of this wiki) labels them 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

4L 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

4L 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

4L 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

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 and 475.5¢. Template:Scale tree

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.

Intervals

Intervals of 4L 5s⟨3/1⟩

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Minor 1-mosstep	m1ms	s	0.0 ¢ to 211.3 ¢
	Major 1-mosstep	M1ms	L	211.3 ¢ to 475.5 ¢
2-mosstep	Diminished 2-mosstep	d2ms	2s	0.0 ¢ to 422.7 ¢
	Perfect 2-mosstep	P2ms	L + s	422.7 ¢ to 475.5 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	475.5 ¢ to 634.0 ¢
	Major 3-mosstep	M3ms	2L + s	634.0 ¢ to 951.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	L + 3s	475.5 ¢ to 845.3 ¢
	Major 4-mosstep	M4ms	2L + 2s	845.3 ¢ to 951.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	951.0 ¢ to 1056.6 ¢
	Major 5-mosstep	M5ms	3L + 2s	1056.6 ¢ to 1426.5 ¢
6-mosstep	Minor 6-mosstep	m6ms	2L + 4s	951.0 ¢ to 1268.0 ¢
	Major 6-mosstep	M6ms	3L + 3s	1268.0 ¢ to 1426.5 ¢
7-mosstep	Perfect 7-mosstep	P7ms	3L + 4s	1426.5 ¢ to 1479.3 ¢
	Augmented 7-mosstep	A7ms	4L + 3s	1479.3 ¢ to 1902.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	3L + 5s	1426.5 ¢ to 1690.6 ¢
	Major 8-mosstep	M8ms	4L + 4s	1690.6 ¢ to 1902.0 ¢
9-mosstep	Perfect 9-mosstep	P9ms	4L + 5s	1902.0 ¢