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4L 4s (3/1-equivalent)

Revision as of 11:38, 22 August 2026 by BudjarnLambeth (talk | contribs) (→3.5.7 subgroup)

4L 4s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 4 small steps, with a period of 1 large step and 1 small step that repeats every 475.5 ¢, or 4 times every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 237.7 ¢ to 475.5 ¢, or from 0 ¢ to 237.7 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

↖ 3L 3s⟨3/1⟩ ↑ 4L 3s⟨3/1⟩ 5L 3s⟨3/1⟩ ↗
← 3L 4s⟨3/1⟩ 4L 4s (3/1-equivalent) 5L 4s⟨3/1⟩ →
↙ 3L 5s⟨3/1⟩ ↓ 4L 5s⟨3/1⟩ 5L 5s⟨3/1⟩ ↘
Scale structure
Step pattern LsLsLsLs
sLsLsLsL
Equave 3/1 (1902.0 ¢)
Period 1\4 (475.5 ¢)
Generator size(edt)
Bright 1\8 to 1\4 (237.7 ¢ to 475.5 ¢)
Dark 0\4 to 1\8 (0.0 ¢ to 237.7 ¢)
Related MOS scales
Parent none
Sister 4L 4s⟨3/1⟩ (self)
Daughters 8L 4s⟨3/1⟩, 4L 8s⟨3/1⟩
Neutralized 8edt
2-Flought 12L 4s⟨3/1⟩, 4L 12s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 1\8 (237.7 ¢)
Supersoft (L:s = 4:3) 4\28 (271.7 ¢)
Soft (L:s = 3:2) 3\20 (285.3 ¢)
Semisoft (L:s = 5:3) 5\32 (297.2 ¢)
Basic (L:s = 2:1) 2\12 (317.0 ¢)
Semihard (L:s = 5:2) 5\28 (339.6 ¢)
Hard (L:s = 3:1) 3\16 (356.6 ¢)
Superhard (L:s = 4:1) 4\20 (380.4 ¢)
Collapsed (L:s = 1:0) 1\4 (475.5 ¢)
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This is a relatively unexplored MOS. All its neighbouring MOS sizes have seen some degree of exploration, but 4L 4s<3/1> hasn't yet.

Its scale shape is highly reminiscent of the familiar diminished scale from octave-bssed music theory (4L 4s).

In addition to the true mos form, LsLsLsLs, there are four near-mos forms – LLsLsLss, LLsLssLs, LLssLLss, and LLssLsLs – in which the period and its multiples are the only intervals with more than two varieties. The near-mos forms are only proper if the dark generator is larger than 1\12edt (158.5 ¢).

Theory

Modes

Modes of 4L 4s⟨3/1⟩
UDP Cyclic
order		 Step
pattern
4|0(4) 1 LsLsLsLs
0|4(4) 2 sLsLsLsL

Temperament interpretations

Potentially useful temperaments include:

3.5.7 subgroup

Intervals

Intervals of 4L 4s⟨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 237.7 ¢
Major 1-mosstep M1ms L 237.7 ¢ to 475.5 ¢
2-mosstep Perfect 2-mosstep P2ms L + s 475.5 ¢
3-mosstep Minor 3-mosstep m3ms L + 2s 475.5 ¢ to 713.2 ¢
Major 3-mosstep M3ms 2L + s 713.2 ¢ to 951.0 ¢
4-mosstep Perfect 4-mosstep P4ms 2L + 2s 951.0 ¢
5-mosstep Minor 5-mosstep m5ms 2L + 3s 951.0 ¢ to 1188.7 ¢
Major 5-mosstep M5ms 3L + 2s 1188.7 ¢ to 1426.5 ¢
6-mosstep Perfect 6-mosstep P6ms 3L + 3s 1426.5 ¢
7-mosstep Minor 7-mosstep m7ms 3L + 4s 1426.5 ¢ to 1664.2 ¢
Major 7-mosstep M7ms 4L + 3s 1664.2 ¢ to 1902.0 ¢
8-mosstep Perfect 8-mosstep P8ms 4L + 4s 1902.0 ¢

Scale degrees

Scale degrees of the modes of 4L 4s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8
4|0(4) 1 LsLsLsLs Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf.
0|4(4) 2 sLsLsLsL Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 4L 4s⟨3/1⟩
Generator(edt) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\8 237.744 237.744 1:1 1.000 Equalized 4L 4s⟨3/1⟩
6\44 259.358 216.131 6:5 1.200
5\36 264.160 211.328 5:4 1.250
9\64 267.462 208.026 9:7 1.286
4\28 271.708 203.781 4:3 1.333 Supersoft 4L 4s⟨3/1⟩
11\76 275.283 200.206 11:8 1.375
7\48 277.368 198.120 7:5 1.400
10\68 279.699 195.789 10:7 1.429
3\20 285.293 190.196 3:2 1.500 Soft 4L 4s⟨3/1⟩
11\72 290.576 184.912 11:7 1.571
8\52 292.608 182.880 8:5 1.600
13\84 294.350 181.139 13:8 1.625
5\32 297.180 178.308 5:3 1.667 Semisoft 4L 4s⟨3/1⟩
12\76 300.309 175.180 12:7 1.714
7\44 302.584 172.905 7:4 1.750
9\56 305.671 169.817 9:5 1.800
2\12 316.993 158.496 2:1 2.000 Basic 4L 4s⟨3/1⟩
9\52 329.185 146.304 9:4 2.250
7\40 332.842 142.647 7:3 2.333
12\68 335.639 139.850 12:5 2.400
5\28 339.635 135.854 5:2 2.500 Semihard 4L 4s⟨3/1⟩
13\72 343.409 132.080 13:5 2.600
8\44 345.810 129.679 8:3 2.667
11\60 348.692 126.797 11:4 2.750
3\16 356.617 118.872 3:1 3.000 Hard 4L 4s⟨3/1⟩
10\52 365.761 109.728 10:3 3.333
7\36 369.825 105.664 7:2 3.500
11\56 373.598 101.890 11:3 3.667
4\20 380.391 95.098 4:1 4.000 Superhard 4L 4s⟨3/1⟩
9\44 389.036 86.453 9:2 4.500
5\24 396.241 79.248 5:1 5.000
6\28 407.562 67.927 6:1 6.000
1\4 475.489 0.000 1:0 → ∞ Collapsed 4L 4s⟨3/1⟩


  Todo: expand
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