4L 6s (5/1-equivalent)

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↖ 3L 5s⟨5/1⟩ ↑ 4L 5s⟨5/1⟩ 5L 5s⟨5/1⟩ ↗
← 3L 6s⟨5/1⟩ 4L 6s (5/1-equivalent) 5L 6s⟨5/1⟩ →
↙ 3L 7s⟨5/1⟩ ↓ 4L 7s⟨5/1⟩ 5L 7s⟨5/1⟩ ↘ 
┌╥┬╥┬┬╥┬╥┬┬┐
│║│║││║│║│││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLssLsLss
ssLsLssLsL
Equave 5/1 (2786.3¢)
Period 1\2ed5 (1393.2¢)
Generator size(ed5/1)
Bright 2\10 to 1\4 (557.3¢ to 696.6¢)
Dark 1\4 to 3\10 (696.6¢ to 835.9¢)
Related MOS scales
Parent 4L 2s⟨5/1⟩
Sister 6L 4s⟨5/1⟩
Daughters 10L 4s⟨5/1⟩, 4L 10s⟨5/1⟩
Neutralized 8L 2s⟨5/1⟩
2-Flought 14L 6s⟨5/1⟩, 4L 16s⟨5/1⟩
Equal tunings(ed5/1)
Equalized (L:s = 1:1) 2\10 (557.3¢)
Supersoft (L:s = 4:3) 7\34 (573.7¢)
Soft (L:s = 3:2) 5\24 (580.5¢)
Semisoft (L:s = 5:3) 8\38 (586.6¢)
Basic (L:s = 2:1) 3\14 (597.1¢)
Semihard (L:s = 5:2) 7\32 (609.5¢)
Hard (L:s = 3:1) 4\18 (619.2¢)
Superhard (L:s = 4:1) 5\22 (633.3¢)
Collapsed (L:s = 1:0) 1\4 (696.6¢)

4L 6s⟨5/1⟩ is a 5/1-equivalent (non-octave) moment of symmetry scale containing 4 large steps and 6 small steps, with a period of 2 large steps and 3 small steps that repeats every 1393.2¢, or twice every interval of 5/1 (2786.3¢). Generators that produce this scale range from 557.3¢ to 696.6¢, or from 696.6¢ to 835.9¢.

Modes

Modes of 4L 6s⟨5/1⟩
UDP Cyclic
order		 Step
pattern
8|0(2) 1 LsLssLsLss
6|2(2) 3 LssLsLssLs
4|4(2) 5 sLsLssLsLs
2|6(2) 2 sLssLsLssL
0|8(2) 4 ssLsLssLsL

Intervals

Intervals of 4L 6s⟨5/1⟩
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 278.6¢
Major 1-mosstep M1ms L 278.6¢ to 696.6¢
2-mosstep Diminished 2-mosstep d2ms 2s 0.0¢ to 557.3¢
Perfect 2-mosstep P2ms L + s 557.3¢ to 696.6¢
3-mosstep Perfect 3-mosstep P3ms L + 2s 696.6¢ to 835.9¢
Augmented 3-mosstep A3ms 2L + s 835.9¢ to 1393.2¢
4-mosstep Minor 4-mosstep m4ms L + 3s 696.6¢ to 1114.5¢
Major 4-mosstep M4ms 2L + 2s 1114.5¢ to 1393.2¢
5-mosstep Perfect 5-mosstep P5ms 2L + 3s 1393.2¢
6-mosstep Minor 6-mosstep m6ms 2L + 4s 1393.2¢ to 1671.8¢
Major 6-mosstep M6ms 3L + 3s 1671.8¢ to 2089.7¢
7-mosstep Diminished 7-mosstep d7ms 2L + 5s 1393.2¢ to 1950.4¢
Perfect 7-mosstep P7ms 3L + 4s 1950.4¢ to 2089.7¢
8-mosstep Perfect 8-mosstep P8ms 3L + 5s 2089.7¢ to 2229.1¢
Augmented 8-mosstep A8ms 4L + 4s 2229.1¢ to 2786.3¢
9-mosstep Minor 9-mosstep m9ms 3L + 6s 2089.7¢ to 2507.7¢
Major 9-mosstep M9ms 4L + 5s 2507.7¢ to 2786.3¢
10-mosstep Perfect 10-mosstep P10ms 4L + 6s 2786.3¢

Theory

The primary no-twos-or-threes temperament interpretation of this scale is as the decatonic scale of the 5.7.11 temperament juggernaut, in which the generator is 7/5 and the half-pentave period is 11/5. This is a very good and low-badness temperament on the 5.7.11 subgroup, although it suffers from a somewhat high error because of the smeary 11/5 at 1393 cents. This MOS can be viewed as a pentave analog of diatonic or lambda, since its basic tuning of 14ed5 is the smallest ed5 that can seriously claim to represent no-twos-or-threes harmony.

Scale tree

Scale Tree and Tuning Spectrum of 4L 6s⟨5/1⟩
Generator(ed5/1) Cents Step ratio Comments
Bright Dark L:s Hardness
2\10 557.263 835.894 1:1 1.000 Equalized 4L 6s⟨5/1⟩
11\54 567.582 825.574 6:5 1.200
9\44 569.928 823.229 5:4 1.250
16\78 571.552 821.605 9:7 1.286 Optimal tridecimal juggernaut
7\34 573.653 819.504 4:3 1.333 Supersoft 4L 6s⟨5/1⟩
19\92 575.434 817.723 11:8 1.375
12\58 576.479 816.678 7:5 1.400
17\82 577.650 815.506 10:7 1.429
5\24 580.482 812.675 3:2 1.500 Soft 4L 6s⟨5/1⟩
18\86 583.182 809.975 11:7 1.571 Optimal juggernaut, bright generator is near-7/5
13\62 584.227 808.930 8:5 1.600
21\100 585.126 808.031 13:8 1.625
8\38 586.592 806.564 5:3 1.667 Semisoft 4L 6s⟨5/1⟩
19\90 588.222 804.935 12:7 1.714
11\52 589.413 803.744 7:4 1.750
14\66 591.036 802.121 9:5 1.800
3\14 597.067 796.090 2:1 2.000 Basic 4L 6s⟨5/1⟩
Scales with tunings softer than this are proper
13\60 603.701 789.456 9:4 2.250
10\46 605.720 787.436 7:3 2.333
17\78 607.274 785.883 12:5 2.400
7\32 609.506 783.651 5:2 2.500 Semihard 4L 6s⟨5/1⟩
Dark generator is near-11/7
18\82 611.630 781.527 13:5 2.600
11\50 612.989 780.168 8:3 2.667
15\68 614.628 778.529 11:4 2.750
4\18 619.181 773.976 3:1 3.000 Hard 4L 6s⟨5/1⟩
13\58 624.519 768.638 10:3 3.333
9\40 626.921 766.236 7:2 3.500
14\62 629.168 763.989 11:3 3.667
5\22 633.253 759.904 4:1 4.000 Superhard 4L 6s⟨5/1⟩
11\48 638.530 754.627 9:2 4.500
6\26 642.995 750.161 5:1 5.000
7\30 650.140 743.017 6:1 6.000
1\4 696.578 696.578 1:0 → ∞ Collapsed 4L 6s⟨5/1⟩
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