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⟩
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 Rotational 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

Scale degree of 4L 6s⟨5/1⟩
Scale degree 14ed5 (Basic, L:s = 2:1) 18ed5 (Hard, L:s = 3:1) 24ed5 (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-mosdegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-mosdegree 1 199 1 154.8 2 232.2
Major 1-mosdegree 2 398 3 464.4 3 348.3
Diminished 2-mosdegree 2 398 2 309.6 4 464.4
Perfect 2-mosdegree 3 597.1 4 619.2 5 580.5
Perfect 3-mosdegree 4 796.1 5 774 7 812.7
Augmented 3-mosdegree 5 995.1 7 1083.6 8 928.8
Minor 4-mosdegree 5 995.1 6 928.8 9 1044.9
Major 4-mosdegree 6 1194.1 8 1238.4 10 1161
Perfect 5-mosdegree 7 1393.2 9 1393.2 12 1393.2
Minor 6-mosdegree 8 1592.2 10 1548 14 1625.3
Major 6-mosdegree 9 1791.2 12 1857.5 15 1741.4
Diminished 7-mosdegree 9 1791.2 11 1702.7 16 1857.5
Perfect 7-mosdegree 10 1990.2 13 2012.3 17 1973.6
Perfect 8-mosdegree 11 2189.2 14 2167.1 19 2205.8
Augmented 8-mosdegree 12 2388.3 16 2476.7 20 2321.9
Minor 9-mosdegree 12 2388.3 15 2321.9 21 2438
Major 9-mosdegree 13 2587.3 17 2631.5 22 2554.1
Perfect 10-mosdegree (equave) 14 2786.3 18 2786.3 24 2786.3 5/1 (exact)

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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