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
Brightest mode LLsLLs
Equave (cents) 2/1 (1200.0¢)
Period (cents) 1\2edo (600.0¢)
Generator ranges
Bright 1\6 (200.0¢) to 1\4 (300.0¢)
Dark 1\4 (300.0¢) to 2\6 (400.0¢)
TAMNAMS information
Name citric
Prefix citro-
Abbrev. cit
Related scales
Parent 2L 2s
Sister 2L 4s
Daughters 6L 4s, 4L 6s
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

Generator Cents Comments
1\6 200
6\34 211.76
5\28 214.29 Antikythera is around here
4\22 218.18
3\16 225
227.56
8\42 228.57
600/(1+phi) Golden lemba
13\68 229.41
5\26 230.77
232.8
7\36 233.33 Lemba is around here
2\10 240 Boundary of propriety for near-MOS

Optimum rank range (L/s=2/1) for MOS

5\24 250 Decimal is around here
251.89
8\38 252.63
253.39 L/s = e
3\14 257.14 L/s = 3
258.81 L/s = pi
4\18 266.67 L/s = 4
5\22 272.73
6\26 276.92 Doublewide is around here
7\30 280
8\34 282.35
9\38 284.21
10\42 285.71
11\46 286.96
1\4 300