2L 5s (5/3-equivalent)

↖ 1L 4s⟨5/3⟩	↑ 2L 4s⟨5/3⟩	3L 4s⟨5/3⟩ ↗
← 1L 5s⟨5/3⟩	2L 5s (5/3-equivalent)	3L 5s⟨5/3⟩ →
↙ 1L 6s⟨5/3⟩	↓ 2L 6s⟨5/3⟩	3L 6s⟨5/3⟩ ↘

Scale structure
Step pattern	LssLsss sssLssL
Equave	5/3 (884.4 ¢)
Period	5/3 (884.4 ¢)
Generator size(ed5/3)
Bright	3\7 to 1\2 (379.0 ¢ to 442.2 ¢)
Dark	1\2 to 4\7 (442.2 ¢ to 505.3 ¢)
Related MOS scales
Parent	2L 3s⟨5/3⟩
Sister	5L 2s⟨5/3⟩
Daughters	7L 2s⟨5/3⟩, 2L 7s⟨5/3⟩
Neutralized	4L 3s⟨5/3⟩
2-Flought	9L 5s⟨5/3⟩, 2L 12s⟨5/3⟩
Equal tunings(ed5/3)
Equalized (L:s = 1:1)	3\7 (379.0 ¢)
Supersoft (L:s = 4:3)	10\23 (384.5 ¢)
Soft (L:s = 3:2)	7\16 (386.9 ¢)
Semisoft (L:s = 5:3)	11\25 (389.1 ¢)
Basic (L:s = 2:1)	4\9 (393.0 ¢)
Semihard (L:s = 5:2)	9\20 (398.0 ¢)
Hard (L:s = 3:1)	5\11 (402.0 ¢)
Superhard (L:s = 4:1)	6\13 (408.2 ¢)
Collapsed (L:s = 1:0)	1\2 (442.2 ¢)
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2L 5s⟨5/3⟩ is a 5/3-equivalent (non-octave) moment of symmetry scale containing 2 large steps and 5 small steps, repeating every interval of 5/3 (884.4 ¢). Generators that produce this scale range from 379 ¢ to 442.2 ¢, or from 442.2 ¢ to 505.3 ¢.

2L 5s⟨5/3⟩, in its softer tunings, can be thought of as a MOS generated by a ~4/3 (or equivalently, 5/4), where the small step thereby represents 16/15. This is just at very close to the 3:2 tuning, at 16ed5/3. Tunings softer than that, however, support a 5/3.4/3.7/6-subgroup rank-2 temperament that tempers out 225/224, and therefore equates three 4/3 generators to 10/7 (whose equave complement is 7/6) and four generators to 8/7. This creates a structure extremely analogous to meantone, where tertian triads form 6:7:8 and its retroversion, which can be considered the primary consonance on equave 5/3.

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 2L 5s⟨5/3⟩

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 126.3 ¢
	Major 1-mosstep	M1ms	L	126.3 ¢ to 442.2 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 252.7 ¢
	Major 2-mosstep	M2ms	L + s	252.7 ¢ to 442.2 ¢
3-mosstep	Diminished 3-mosstep	d3ms	3s	0.0 ¢ to 379.0 ¢
	Perfect 3-mosstep	P3ms	L + 2s	379.0 ¢ to 442.2 ¢
4-mosstep	Perfect 4-mosstep	P4ms	L + 3s	442.2 ¢ to 505.3 ¢
	Augmented 4-mosstep	A4ms	2L + 2s	505.3 ¢ to 884.4 ¢
5-mosstep	Minor 5-mosstep	m5ms	L + 4s	442.2 ¢ to 631.7 ¢
	Major 5-mosstep	M5ms	2L + 3s	631.7 ¢ to 884.4 ¢
6-mosstep	Minor 6-mosstep	m6ms	L + 5s	442.2 ¢ to 758.0 ¢
	Major 6-mosstep	M6ms	2L + 4s	758.0 ¢ to 884.4 ¢
7-mosstep	Perfect 7-mosstep	P7ms	2L + 5s	884.4 ¢

Generator chain

Generator chain of 2L 5s⟨5/3⟩

Bright gens	Scale degree	Abbrev.
8	Augmented 3-mosdegree	A3md
7	Augmented 0-mosdegree	A0md
6	Augmented 4-mosdegree	A4md
5	Major 1-mosdegree	M1md
4	Major 5-mosdegree	M5md
3	Major 2-mosdegree	M2md
2	Major 6-mosdegree	M6md
1	Perfect 3-mosdegree	P3md
0	Perfect 0-mosdegree Perfect 7-mosdegree	P0md P7md
−1	Perfect 4-mosdegree	P4md
−2	Minor 1-mosdegree	m1md
−3	Minor 5-mosdegree	m5md
−4	Minor 2-mosdegree	m2md
−5	Minor 6-mosdegree	m6md
−6	Diminished 3-mosdegree	d3md
−7	Diminished 7-mosdegree	d7md
−8	Diminished 4-mosdegree	d4md

Modes

Scale degrees of the modes of 2L 5s⟨5/3⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7
6|0	1	LssLsss	Perf.	Maj.	Maj.	Perf.	Aug.	Maj.	Maj.	Perf.
5|1	4	LsssLss	Perf.	Maj.	Maj.	Perf.	Perf.	Maj.	Maj.	Perf.
4|2	7	sLssLss	Perf.	Min.	Maj.	Perf.	Perf.	Maj.	Maj.	Perf.
3|3	3	sLsssLs	Perf.	Min.	Maj.	Perf.	Perf.	Min.	Maj.	Perf.
2|4	6	ssLssLs	Perf.	Min.	Min.	Perf.	Perf.	Min.	Maj.	Perf.
1|5	2	ssLsssL	Perf.	Min.	Min.	Perf.	Perf.	Min.	Min.	Perf.
0|6	5	sssLssL	Perf.	Min.	Min.	Dim.	Perf.	Min.	Min.	Perf.

List of ed5/3s supporting this scale

Scale tree and tuning spectrum of 2L 5s⟨5/3⟩

Generator(ed5/3)							Cents		Step ratio		Comments
							Bright	Dark	L:s	Hardness
3\7							379.011	505.348	1:1	1.000	Equalized 2L 5s⟨5/3⟩
						19\44	381.882	502.477	7:6	1.167
					16\37		382.425	501.933	6:5	1.200
						29\67	382.782	501.577	11:9	1.222
				13\30			383.222	501.137	5:4	1.250
						36\83	383.577	500.781	14:11	1.273
					23\53		383.778	500.580	9:7	1.286
						33\76	383.998	500.361	13:10	1.300
			10\23				384.504	499.855	4:3	1.333	Supersoft 2L 5s⟨5/3⟩
						37\85	384.956	499.403	15:11	1.364
					27\62		385.124	499.235	11:8	1.375
						44\101	385.265	499.094	18:13	1.385
				17\39			385.490	498.869	7:5	1.400
						41\94	385.731	498.628	17:12	1.417
					24\55		385.902	498.457	10:7	1.429
						31\71	386.128	498.230	13:9	1.444
		7\16					386.907	497.452	3:2	1.500	Soft 2L 5s⟨5/3⟩
						32\73	387.664	496.695	14:9	1.556
					25\57		387.877	496.482	11:7	1.571
						43\98	388.035	496.324	19:12	1.583
				18\41			388.255	496.104	8:5	1.600
						47\107	388.457	495.902	21:13	1.615
					29\66		388.582	495.777	13:8	1.625
						40\91	388.729	495.630	18:11	1.636
			11\25				389.118	495.241	5:3	1.667	Semisoft 2L 5s⟨5/3⟩
						37\84	389.539	494.820	17:10	1.700
					26\59		389.717	494.641	12:7	1.714
						41\93	389.879	494.480	19:11	1.727
				15\34			390.158	494.200	7:4	1.750
						34\77	390.496	493.863	16:9	1.778
					19\43		390.763	493.596	9:5	1.800
						23\52	391.159	493.200	11:6	1.833
	4\9						393.048	491.310	2:1	2.000	Basic 2L 5s⟨5/3⟩ Scales with tunings softer than this are proper
						21\47	395.139	489.220	11:5	2.200
					17\38		395.634	488.725	9:4	2.250
						30\67	395.982	488.377	16:7	2.286
				13\29			396.437	487.922	7:3	2.333
						35\78	396.828	487.531	19:8	2.375
					22\49		397.059	487.300	12:5	2.400
						31\69	397.321	487.038	17:7	2.429
			9\20				397.961	486.397	5:2	2.500	Semihard 2L 5s⟨5/3⟩
						32\71	398.584	485.775	18:7	2.571
					23\51		398.828	485.530	13:5	2.600
						37\82	399.040	485.319	21:8	2.625
				14\31			399.388	484.971	8:3	2.667
						33\73	399.779	484.580	19:7	2.714
					19\42		400.067	484.292	11:4	2.750
						24\53	400.464	483.894	14:5	2.800
		5\11					401.981	482.377	3:1	3.000	Hard 2L 5s⟨5/3⟩
						21\46	403.729	480.630	13:4	3.250
					16\35		404.278	480.080	10:3	3.333
						27\59	404.707	479.652	17:5	3.400
				11\24			405.331	479.028	7:2	3.500
						28\61	405.935	478.424	18:5	3.600
					17\37		406.327	478.032	11:3	3.667
						23\50	406.805	477.554	15:4	3.750
			6\13				408.166	476.193	4:1	4.000	Superhard 2L 5s⟨5/3⟩
						19\41	409.825	474.534	13:3	4.333
					13\28		410.595	473.764	9:2	4.500
						20\43	411.330	473.029	14:3	4.667
				7\15			412.701	471.658	5:1	5.000
						15\32	414.543	469.816	11:2	5.500
					8\17		416.169	468.190	6:1	6.000
						9\19	418.907	465.452	7:1	7.000
1\2							442.179	442.179	1:0	→ ∞	Collapsed 2L 5s⟨5/3⟩

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