4L 2s

↖ 3L 1s	↑ 4L 1s	5L 1s ↗
← 3L 2s	4L 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 ¢).

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 ¢ 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.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 4L 2s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-citrostep	Perfect 0-citrostep	P0cits	0	0.0 ¢
1-citrostep	Diminished 1-citrostep	d1cits	s	0.0 ¢ to 200.0 ¢
	Perfect 1-citrostep	P1cits	L	200.0 ¢ to 300.0 ¢
2-citrostep	Perfect 2-citrostep	P2cits	L + s	300.0 ¢ to 400.0 ¢
	Augmented 2-citrostep	A2cits	2L	400.0 ¢ to 600.0 ¢
3-citrostep	Perfect 3-citrostep	P3cits	2L + s	600.0 ¢
4-citrostep	Diminished 4-citrostep	d4cits	2L + 2s	600.0 ¢ to 800.0 ¢
	Perfect 4-citrostep	P4cits	3L + s	800.0 ¢ to 900.0 ¢
5-citrostep	Perfect 5-citrostep	P5cits	3L + 2s	900.0 ¢ to 1000.0 ¢
	Augmented 5-citrostep	A5cits	4L + s	1000.0 ¢ to 1200.0 ¢
6-citrostep	Perfect 6-citrostep	P6cits	4L + 2s	1200.0 ¢

Generator chain

Generator chain of 4L 2s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.
4	Augmented 1-citrodegree	A1citd	Augmented 4-citrodegree	A4citd
3	Augmented 0-citrodegree	A0citd	Augmented 3-citrodegree	A3citd
2	Augmented 2-citrodegree	A2citd	Augmented 5-citrodegree	A5citd
1	Perfect 1-citrodegree	P1citd	Perfect 4-citrodegree	P4citd
0	Perfect 0-citrodegree Perfect 3-citrodegree	P0citd P3citd	Perfect 3-citrodegree Perfect 6-citrodegree	P3citd P6citd
−1	Perfect 2-citrodegree	P2citd	Perfect 5-citrodegree	P5citd
−2	Diminished 1-citrodegree	d1citd	Diminished 4-citrodegree	d4citd
−3	Diminished 3-citrodegree	d3citd	Diminished 6-citrodegree	d6citd
−4	Diminished 2-citrodegree	d2citd	Diminished 5-citrodegree	d5citd

Modes

Scale degrees of the modes of 4L 2s

UDP	Cyclic order	Step pattern	Scale degree (citrodegree)
			0	1	2	3	4	5	6
4|0(2)	1	LLsLLs	Perf.	Perf.	Aug.	Perf.	Perf.	Aug.	Perf.
2|2(2)	2	LsLLsL	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.
0|4(2)	3	sLLsLL	Perf.	Dim.	Perf.	Perf.	Dim.	Perf.	Perf.

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