3L 2s (3/1-equivalent)

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↖ 2L 1s⟨3/1⟩ ↑ 3L 1s⟨3/1⟩ 4L 1s⟨3/1⟩ ↗
← 2L 2s⟨3/1⟩ 3L 2s (3/1-equivalent) 4L 2s⟨3/1⟩ →
↙ 2L 3s⟨3/1⟩ ↓ 3L 3s⟨3/1⟩ 4L 3s⟨3/1⟩ ↘
Scale structure
Step pattern LLsLs
sLsLL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 3\5 to 2\3 (1141.2 ¢ to 1268.0 ¢)
Dark 1\3 to 2\5 (634.0 ¢ to 760.8 ¢)
Related MOS scales
Parent 2L 1s⟨3/1⟩
Sister 2L 3s⟨3/1⟩
Daughters 5L 3s⟨3/1⟩, 3L 5s⟨3/1⟩
Neutralized 1L 4s⟨3/1⟩
2-Flought 8L 2s⟨3/1⟩, 3L 7s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 3\5 (1141.2 ¢)
Supersoft (L:s = 4:3) 11\18 (1162.3 ¢)
Soft (L:s = 3:2) 8\13 (1170.4 ¢)
Semisoft (L:s = 5:3) 13\21 (1177.4 ¢)
Basic (L:s = 2:1) 5\8 (1188.7 ¢)
Semihard (L:s = 5:2) 12\19 (1201.2 ¢)
Hard (L:s = 3:1) 7\11 (1210.3 ¢)
Superhard (L:s = 4:1) 9\14 (1222.7 ¢)
Collapsed (L:s = 1:0) 2\3 (1268.0 ¢)
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3L 2s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 3 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1141.2 ¢ to 1268 ¢, or from 634 ¢ to 760.8 ¢.

This MOS was first explored by hilarious_hilarion on the Xenharmonic Alliance Discord server in September 2026. Specifically they discovered a just intonation version of this MOS in the 3-limit:

Hilarion's just scale has a hardness somewhere around 2.45. It has generators 3/1 and 2/1 (bright) or 3/1 and 3/2 (dark). From an octave-equivalent point of view, its first tritave resembles an open voicing of the familiar 7sus4 chord from 12edo jazz harmony. Hilarion's just scale is especially well approximated in 19edt and 46edt. EDTs 27, 30, 35, 41 and 49 also approximate it decently.

A non-just 3L 2s⟨3/1⟩ scale occurs in 13edt (equal tempered Bohlen-Pierce), where it is generated by 8\13 with a hardness of exactly 1.5. Though that iteration of the scale is very different to Hilarion's version, as it completely avoids both the perfect fifth and the octave, and is much more dissonant but also much more tritave-equivalent in character.

Theory

Modes

Modes of 3L 2s⟨3/1⟩
UDP Cyclic
order		 Step
pattern
4|0 1 LLsLs
3|1 4 LsLLs
2|2 2 LsLsL
1|3 5 sLLsL
0|4 3 sLsLL

Temperament interpretations

Todo: expand

Intervals

Intervals of 3L 2s⟨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 380.4 ¢
Major 1-mosstep M1ms L 380.4 ¢ to 634.0 ¢
2-mosstep Perfect 2-mosstep P2ms L + s 634.0 ¢ to 760.8 ¢
Augmented 2-mosstep A2ms 2L 760.8 ¢ to 1268.0 ¢
3-mosstep Diminished 3-mosstep d3ms L + 2s 634.0 ¢ to 1141.2 ¢
Perfect 3-mosstep P3ms 2L + s 1141.2 ¢ to 1268.0 ¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 1268.0 ¢ to 1521.6 ¢
Major 4-mosstep M4ms 3L + s 1521.6 ¢ to 1902.0 ¢
5-mosstep Perfect 5-mosstep P5ms 3L + 2s 1902.0 ¢

Scale degrees

Scale degrees of the modes of 3L 2s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5
4|0 1 LLsLs Perf. Maj. Aug. Perf. Maj. Perf.
3|1 4 LsLLs Perf. Maj. Perf. Perf. Maj. Perf.
2|2 2 LsLsL Perf. Maj. Perf. Perf. Min. Perf.
1|3 5 sLLsL Perf. Min. Perf. Perf. Min. Perf.
0|4 3 sLsLL Perf. Min. Perf. Dim. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 3L 2s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
3\5 1141.173 760.782 1:1 1.000 Equalized 3L 2s⟨3/1⟩
17\28 1154.758 747.197 6:5 1.200
14\23 1157.712 744.243 5:4 1.250
25\41 1159.729 742.226 9:7 1.286
11\18 1162.306 739.649 4:3 1.333 Supersoft 3L 2s⟨3/1⟩
30\49 1164.462 737.493 11:8 1.375
19\31 1165.714 736.241 7:5 1.400
27\44 1167.109 734.846 10:7 1.429
8\13 1170.434 731.521 3:2 1.500 Soft 3L 2s⟨3/1⟩
29\47 1173.547 728.408 11:7 1.571
21\34 1174.737 727.218 8:5 1.600
34\55 1175.754 726.201 13:8 1.625
13\21 1177.401 724.554 5:3 1.667 Semisoft 3L 2s⟨3/1⟩
31\50 1179.212 722.743 12:7 1.714
18\29 1180.524 721.431 7:4 1.750
23\37 1182.296 719.659 9:5 1.800
5\8 1188.722 713.233 2:1 2.000 Basic 3L 2s⟨3/1⟩
Scales with tunings softer than this are proper
22\35 1195.515 706.440 9:4 2.250
17\27 1197.527 704.428 7:3 2.333
29\46 1199.059 702.896 12:5 2.400
12\19 1201.235 700.720 5:2 2.500 Semihard 3L 2s⟨3/1⟩
31\49 1203.278 698.677 13:5 2.600
19\30 1204.572 697.384 8:3 2.667
26\41 1206.118 695.837 11:4 2.750
7\11 1210.335 691.620 3:1 3.000 Hard 3L 2s⟨3/1⟩
23\36 1215.138 686.817 10:3 3.333
16\25 1217.251 684.704 7:2 3.500
25\39 1219.202 682.753 11:3 3.667
9\14 1222.685 679.270 4:1 4.000 Superhard 3L 2s⟨3/1⟩
20\31 1227.068 674.887 9:2 4.500
11\17 1230.677 671.278 5:1 5.000
13\20 1236.271 665.684 6:1 6.000
2\3 1267.970 633.985 1:0 → ∞ Collapsed 3L 2s⟨3/1⟩
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