4L 2s

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↖ 3L 1s ↑4L 1s 5L 1s ↗
← 3L 2s4L 2s 5L 2s →
↙ 3L 3s ↓4L 3s 5L 3s ↘
┌╥╥┬╥╥┬┐
│║║│║║││
││││││││
└┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLs
sLLsLL
Equave 2/1 (1200.0¢)
Period 1\2 (600.0¢)
Generator size
Bright 1\6 to 1\4 (200.0¢ to 300.0¢)
Dark 1\4 to 2\6 (300.0¢ to 400.0¢)
TAMNAMS information
Name citric
Prefix citro-
Abbrev. cit
Related MOS scales
Parent 2L 2s
Sister 2L 4s
Daughters 6L 4s, 4L 6s
Neutralized 2L 4s
2-Flought 10L 2s, 4L 8s
Equal tunings
Equalized (L:s = 1:1) 1\6 (200.0¢)
Supersoft (L:s = 4:3) 4\22 (218.2¢)
Soft (L:s = 3:2) 3\16 (225.0¢)
Semisoft (L:s = 5:3) 5\26 (230.8¢)
Basic (L:s = 2:1) 2\10 (240.0¢)
Semihard (L:s = 5:2) 5\24 (250.0¢)
Hard (L:s = 3:1) 3\14 (257.1¢)
Superhard (L:s = 4:1) 4\18 (266.7¢)
Collapsed (L:s = 1:0) 1\4 (300.0¢)

4L 2s, named citric in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 2 small steps, with a period of 2 large steps and 1 small step that repeats every 600.0¢, or twice every octave. Generators that produce this scale range from 200¢ to 300¢, or from 300¢ to 400¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period. 4L 2s can be seen as a warped diatonic scale, where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale (6edo), but with two "whole tones" that are smaller than the others.

Scales with the true MOS pattern are always proper, because there is only one small step per period. In addition, there are near-MOS patterns, such as LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).

Name

TAMNAMS suggests the temperament-agnostic name citric for this scale.

Theory

Low harmonic entropy scales

There are three scales with this MOS pattern that are significant minima of harmonic entropy. The first is antikythera, or no-3's srutal/pajara, which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.

The second is decimal, in which two generators make a 4/3, and the third is Doublewide, in which the generator is 7/6 so the period minus the generator is 6/5.

Modes

Modes of 4L 2s
UDP Rotational order Step pattern
4 | 0(2) 1 LLsLLs
2 | 2(2) 2 LsLLsL
0 | 4(2) 3 sLLsLL

Scale tree

Scale Tree and Tuning Spectrum of 4L 2s
Generator(edo) Cents Step Ratio Comments(always proper)
Bright Dark L:s Hardness
1\6 200.000 400.000 1:1 1.000 Equalized 4L 2s
6\34 211.765 388.235 6:5 1.200
5\28 214.286 385.714 5:4 1.250 Antikythera
9\50 216.000 384.000 9:7 1.286
4\22 218.182 381.818 4:3 1.333 Supersoft 4L 2s
11\60 220.000 380.000 11:8 1.375
7\38 221.053 378.947 7:5 1.400
10\54 222.222 377.778 10:7 1.429
3\16 225.000 375.000 3:2 1.500 Soft 4L 2s
11\58 227.586 372.414 11:7 1.571
8\42 228.571 371.429 8:5 1.600
13\68 229.412 370.588 13:8 1.625 Golden lemba
5\26 230.769 369.231 5:3 1.667 Semisoft 4L 2s
12\62 232.258 367.742 12:7 1.714
7\36 233.333 366.667 7:4 1.750 Lemba is around here
9\46 234.783 365.217 9:5 1.800
2\10 240.000 360.000 2:1 2.000 Basic 4L 2s
Optimum rank range
9\44 245.455 354.545 9:4 2.250
7\34 247.059 352.941 7:3 2.333
12\58 248.276 351.724 12:5 2.400
5\24 250.000 350.000 5:2 2.500 Semihard 4L 2s
13\62 251.613 348.387 13:5 2.600
8\38 252.632 347.368 8:3 2.667
11\52 253.846 346.154 11:4 2.750
3\14 257.143 342.857 3:1 3.000 Hard 4L 2s
10\46 260.870 339.130 10:3 3.333
7\32 262.500 337.500 7:2 3.500
11\50 264.000 336.000 11:3 3.667
4\18 266.667 333.333 4:1 4.000 Superhard 4L 2s
9\40 270.000 330.000 9:2 4.500
5\22 272.727 327.273 5:1 5.000
6\26 276.923 323.077 6:1 6.000 Doublewide is around here
1\4 300.000 300.000 1:0 → ∞ Collapsed 4L 2s