4L 5s (3/1-equivalent)

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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¢)
Other names
Name(s) Lambda
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		 Mode names
8|0 1 LsLsLsLss Lyncian
7|1 3 LsLsLssLs Aurigan
6|2 5 LsLssLsLs Persean
5|3 7 LssLsLsLs Andromedan
4|4 9 sLsLsLsLs Cassiopeian
3|5 2 sLsLsLssL Lacertian
2|6 4 sLsLssLsL Cygnian
1|7 6 sLssLsLsL Draconian
0|8 8 ssLsLsLsL Herculean

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¢.

Scale Tree and Tuning Spectrum of 4L 5s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
2\9 422.657 1479.298 1:1 1.000 Equalized 4L 5s⟨3/1⟩
15\67 425.811 1476.144 8:7 1.143
13\58 426.300 1475.655 7:6 1.167
24\107 426.607 1475.348 13:11 1.182
11\49 426.969 1474.986 6:5 1.200
31\138 427.251 1474.704 17:14 1.214
20\89 427.406 1474.549 11:9 1.222
29\129 427.571 1474.384 16:13 1.231
9\40 427.940 1474.015 5:4 1.250
34\151 428.255 1473.700 19:15 1.267
25\111 428.368 1473.587 14:11 1.273
41\182 428.462 1473.493 23:18 1.278
16\71 428.610 1473.345 9:7 1.286
39\173 428.764 1473.191 22:17 1.294
23\102 428.872 1473.083 13:10 1.300
30\133 429.012 1472.943 17:13 1.308
7\31 429.474 1472.481 4:3 1.333 Supersoft 4L 5s⟨3/1⟩
33\146 429.894 1472.061 19:14 1.357
26\115 430.007 1471.948 15:11 1.364
45\199 430.090 1471.865 26:19 1.368
19\84 430.204 1471.751 11:8 1.375
50\221 430.307 1471.648 29:21 1.381
31\137 430.369 1471.586 18:13 1.385
43\190 430.442 1471.513 25:18 1.389
12\53 430.631 1471.324 7:5 1.400
41\181 430.830 1471.125 24:17 1.412
29\128 430.912 1471.043 17:12 1.417
46\203 430.985 1470.970 27:19 1.421
17\75 431.110 1470.845 10:7 1.429
39\172 431.257 1470.698 23:16 1.438
22\97 431.371 1470.584 13:9 1.444
27\119 431.536 1470.419 16:11 1.455
5\22 432.263 1469.693 3:2 1.500 Soft 4L 5s⟨3/1⟩
28\123 432.965 1468.990 17:11 1.545
23\101 433.118 1468.837 14:9 1.556
41\180 433.223 1468.732 25:16 1.562
18\79 433.357 1468.598 11:7 1.571
49\215 433.469 1468.486 30:19 1.579
31\136 433.534 1468.421 19:12 1.583
44\193 433.606 1468.349 27:17 1.588
13\57 433.779 1468.176 8:5 1.600
47\206 433.941 1468.014 29:18 1.611
34\149 434.003 1467.952 21:13 1.615
55\241 434.056 1467.899 34:21 1.619
21\92 434.142 1467.813 13:8 1.625
50\219 434.236 1467.719 31:19 1.632
29\127 434.305 1467.650 18:11 1.636
37\162 434.397 1467.558 23:14 1.643
8\35 434.733 1467.222 5:3 1.667 Semisoft 4L 5s⟨3/1⟩
35\153 435.088 1466.867 22:13 1.692 Essentially just 7/3
27\118 435.193 1466.762 17:10 1.700
46\201 435.273 1466.682 29:17 1.706
19\83 435.387 1466.568 12:7 1.714
49\214 435.494 1466.461 31:18 1.722
30\131 435.562 1466.393 19:11 1.727
41\179 435.643 1466.312 26:15 1.733
11\48 435.865 1466.090 7:4 1.750
36\157 436.117 1465.838 23:13 1.769
25\109 436.228 1465.727 16:9 1.778
39\170 436.331 1465.624 25:14 1.786
14\61 436.514 1465.441 9:5 1.800
31\135 436.745 1465.210 20:11 1.818
17\74 436.936 1465.019 11:6 1.833
20\87 437.231 1464.724 13:7 1.857
3\13 438.913 1463.042 2:1 2.000 Basic 4L 5s⟨3/1⟩
Scales with tunings softer than this are proper
19\82 440.697 1461.258 13:6 2.167 Bohlen–Pierce–Stearns is in this region
16\69 441.033 1460.922 11:5 2.200
29\125 441.254 1460.701 20:9 2.222
13\56 441.525 1460.430 9:4 2.250
36\155 441.744 1460.211 25:11 2.273
23\99 441.868 1460.087 16:7 2.286
33\142 442.004 1459.951 23:10 2.300
10\43 442.315 1459.640 7:3 2.333
37\159 442.593 1459.362 26:11 2.364
27\116 442.696 1459.259 19:8 2.375
44\189 442.783 1459.172 31:13 2.385
17\73 442.921 1459.034 12:5 2.400
41\176 443.069 1458.886 29:12 2.417
24\103 443.174 1458.781 17:7 2.429
31\133 443.313 1458.642 22:9 2.444
7\30 443.790 1458.166 5:2 2.500 Semihard 4L 5s⟨3/1⟩
32\137 444.252 1457.703 23:9 2.556
25\107 444.382 1457.573 18:7 2.571
43\184 444.479 1457.476 31:12 2.583
18\77 444.613 1457.342 13:5 2.600
47\201 444.736 1457.219 34:13 2.615
29\124 444.812 1457.143 21:8 2.625
40\171 444.902 1457.053 29:11 2.636
11\47 445.138 1456.817 8:3 2.667
37\158 445.395 1456.560 27:10 2.700
26\111 445.503 1456.452 19:7 2.714
41\175 445.601 1456.354 30:11 2.727
15\64 445.771 1456.184 11:4 2.750
34\145 445.976 1455.979 25:9 2.778
19\81 446.138 1455.817 14:5 2.800
23\98 446.377 1455.578 17:6 2.833
4\17 447.519 1454.436 3:1 3.000 Hard 4L 5s⟨3/1⟩
21\89 448.776 1453.179 16:5 3.200
17\72 449.073 1452.882 13:4 3.250
30\127 449.281 1452.674 23:7 3.286
13\55 449.553 1452.402 10:3 3.333
35\148 449.787 1452.168 27:8 3.375
22\93 449.925 1452.030 17:5 3.400
31\131 450.081 1451.874 24:7 3.429
9\38 450.463 1451.492 7:2 3.500
32\135 450.834 1451.121 25:7 3.571
23\97 450.979 1450.976 18:5 3.600
37\156 451.105 1450.850 29:8 3.625
14\59 451.311 1450.644 11:3 3.667
33\139 451.543 1450.412 26:7 3.714
19\80 451.714 1450.241 15:4 3.750
24\101 451.950 1450.005 19:5 3.800
5\21 452.846 1449.109 4:1 4.000 Superhard 4L 5s⟨3/1⟩
21\88 453.876 1448.079 17:4 4.250
16\67 454.198 1447.757 13:3 4.333
27\113 454.449 1447.506 22:5 4.400
11\46 454.815 1447.140 9:2 4.500
28\117 455.169 1446.786 23:5 4.600
17\71 455.398 1446.557 14:3 4.667
23\96 455.677 1446.278 19:4 4.750
6\25 456.469 1445.486 5:1 5.000
19\79 457.432 1444.523 16:3 5.333
13\54 457.878 1444.077 11:2 5.500
20\83 458.302 1443.653 17:3 5.667
7\29 459.093 1442.862 6:1 6.000
15\62 460.150 1441.805 13:2 6.500
8\33 461.080 1440.875 7:1 7.000
9\37 462.638 1439.317 8:1 8.000
1\4 475.489 1426.466 1:0 → ∞ Collapsed 4L 5s⟨3/1⟩

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¢
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