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6L 2s (3/1-equivalent)

Revision as of 11:43, 30 September 2026 by BudjarnLambeth (talk | contribs) (I think this is a useful scale)
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6L 2s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 6 large steps and 2 small steps, with a period of 3 large steps and 1 small step that repeats every 951.0 ¢, or twice every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 237.7 ¢ to 317 ¢, or from 634 ¢ to 713.2 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

↖ 5L 1s⟨3/1⟩ ↑ 6L 1s⟨3/1⟩ 7L 1s⟨3/1⟩ ↗
← 5L 2s⟨3/1⟩ 6L 2s (3/1-equivalent) 7L 2s⟨3/1⟩ →
↙ 5L 3s⟨3/1⟩ ↓ 6L 3s⟨3/1⟩ 7L 3s⟨3/1⟩ ↘
Scale structure
Step pattern LLLsLLLs
sLLLsLLL
Equave 3/1 (1902.0 ¢)
Period 1\2 (951.0 ¢)
Generator size(edt)
Bright 1\8 to 1\6 (237.7 ¢ to 317.0 ¢)
Dark 2\6 to 3\8 (634.0 ¢ to 713.2 ¢)
Related MOS scales
Parent 2L 4s⟨3/1⟩
Sister 2L 6s⟨3/1⟩
Daughters 8L 6s⟨3/1⟩, 6L 8s⟨3/1⟩
Neutralized 4L 4s⟨3/1⟩
2-Flought 14L 2s⟨3/1⟩, 6L 10s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 1\8 (237.7 ¢)
Supersoft (L:s = 4:3) 4\30 (253.6 ¢)
Soft (L:s = 3:2) 3\22 (259.4 ¢)
Semisoft (L:s = 5:3) 5\36 (264.2 ¢)
Basic (L:s = 2:1) 2\14 (271.7 ¢)
Semihard (L:s = 5:2) 5\34 (279.7 ¢)
Hard (L:s = 3:1) 3\20 (285.3 ¢)
Superhard (L:s = 4:1) 4\26 (292.6 ¢)
Collapsed (L:s = 1:0) 1\6 (317.0 ¢)
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Theory

Modes

Modes of 6L 2s⟨3/1⟩
UDP Cyclic
order		 Step
pattern
6|0(2) 1 LLLsLLLs
4|2(2) 2 LLsLLLsL
2|4(2) 3 LsLLLsLL
0|6(2) 4 sLLLsLLL

Temperament interpretations

The temperament that joins 26edt and 32edt produces this MOS. Both 26edt and 32edt are no-twos zeta peaks.

In 26edt it is generated by 4\26 (~292.6¢) and has a hardness of exactly 4.0.

In 32edt it is generated by 5\32 (~297.2¢) and has a hardness of exactly 5.9.

In 26edt this MOS has the intervals (in cents):

  • 292.608
  • 365.761
  • 658.369
  • 950.978
  • 1243.586
  • 1316.738
  • 1609.347
  • 1901.955

(And rotations thereof.)

That roughly approximates this no-3s 19-limit JI scale:

  • 13/11
  • 11/9
  • 19/13
  • 19/11
  • 35/17 or 39/19
  • 15/7
  • 33/13
  • 3/1

Or this full 11-limit JI scale:

  • 6/5
  • 5/4
  • 16/11
  • 7/4
  • 33/16
  • 15/7
  • 5/2
  • 3/1

Intervals

Intervals of 6L 2s⟨3/1⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Diminished 1-mosstep d1ms s 0.0 ¢ to 237.7 ¢
Perfect 1-mosstep P1ms L 237.7 ¢ to 317.0 ¢
2-mosstep Minor 2-mosstep m2ms L + s 317.0 ¢ to 475.5 ¢
Major 2-mosstep M2ms 2L 475.5 ¢ to 634.0 ¢
3-mosstep Perfect 3-mosstep P3ms 2L + s 634.0 ¢ to 713.2 ¢
Augmented 3-mosstep A3ms 3L 713.2 ¢ to 951.0 ¢
4-mosstep Perfect 4-mosstep P4ms 3L + s 951.0 ¢
5-mosstep Diminished 5-mosstep d5ms 3L + 2s 951.0 ¢ to 1188.7 ¢
Perfect 5-mosstep P5ms 4L + s 1188.7 ¢ to 1268.0 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 1268.0 ¢ to 1426.5 ¢
Major 6-mosstep M6ms 5L + s 1426.5 ¢ to 1585.0 ¢
7-mosstep Perfect 7-mosstep P7ms 5L + 2s 1585.0 ¢ to 1664.2 ¢
Augmented 7-mosstep A7ms 6L + s 1664.2 ¢ to 1902.0 ¢
8-mosstep Perfect 8-mosstep P8ms 6L + 2s 1902.0 ¢

Scale degrees

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

Scale tree

Scale tree and tuning spectrum of 6L 2s⟨3/1⟩
Generator(edt) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\8 237.744 713.233 1:1 1.000 Equalized 6L 2s⟨3/1⟩
6\46 248.081 702.896 6:5 1.200
5\38 250.257 700.720 5:4 1.250
9\68 251.729 699.248 9:7 1.286
4\30 253.594 697.384 4:3 1.333 Supersoft 6L 2s⟨3/1⟩
11\82 255.140 695.837 11:8 1.375
7\52 256.032 694.945 7:5 1.400
10\74 257.021 693.957 10:7 1.429
3\22 259.358 691.620 3:2 1.500 Soft 6L 2s⟨3/1⟩
11\80 261.519 689.459 11:7 1.571
8\58 262.339 688.639 8:5 1.600
13\94 263.036 687.941 13:8 1.625
5\36 264.160 686.817 5:3 1.667 Semisoft 6L 2s⟨3/1⟩
12\86 265.389 685.588 12:7 1.714
7\50 266.274 684.704 7:4 1.750
9\64 267.462 683.515 9:5 1.800
2\14 271.708 679.270 2:1 2.000 Basic 6L 2s⟨3/1⟩
9\62 276.090 674.887 9:4 2.250
7\48 277.368 673.609 7:3 2.333
12\82 278.335 672.643 12:5 2.400
5\34 279.699 671.278 5:2 2.500 Semihard 6L 2s⟨3/1⟩
13\88 280.971 670.007 13:5 2.600
8\54 281.771 669.206 8:3 2.667
11\74 282.723 668.254 11:4 2.750
3\20 285.293 665.684 3:1 3.000 Hard 6L 2s⟨3/1⟩
10\66 288.175 662.803 10:3 3.333
7\46 289.428 661.550 7:2 3.500
11\72 290.576 660.401 11:3 3.667
4\26 292.608 658.369 4:1 4.000 Superhard 6L 2s⟨3/1⟩
9\58 295.131 655.847 9:2 4.500
5\32 297.180 653.797 5:1 5.000
6\38 300.309 650.669 6:1 6.000
1\6 316.993 633.985 1:0 → ∞ Collapsed 6L 2s⟨3/1⟩
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