5L 2s (3/1-equivalent)

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

Scale structure
Step pattern	LLLsLLs sLLsLLL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	4\7 to 3\5 (1086.8 ¢ to 1141.2 ¢)
Dark	2\5 to 3\7 (760.8 ¢ to 815.1 ¢)
Related MOS scales
Parent	2L 3s⟨3/1⟩
Sister	2L 5s⟨3/1⟩
Daughters	7L 5s⟨3/1⟩, 5L 7s⟨3/1⟩
Neutralized	3L 4s⟨3/1⟩
2-Flought	12L 2s⟨3/1⟩, 5L 9s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	4\7 (1086.8 ¢)
Supersoft (L:s = 4:3)	15\26 (1097.3 ¢)
Soft (L:s = 3:2)	11\19 (1101.1 ¢)
Semisoft (L:s = 5:3)	18\31 (1104.4 ¢)
Basic (L:s = 2:1)	7\12 (1109.5 ¢)
Semihard (L:s = 5:2)	17\29 (1114.9 ¢)
Hard (L:s = 3:1)	10\17 (1118.8 ¢)
Superhard (L:s = 4:1)	13\22 (1123.9 ¢)
Collapsed (L:s = 1:0)	3\5 (1141.2 ¢)
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5L 2s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1086.8 ¢ to 1141.2 ¢, or from 760.8 ¢ to 815.1 ¢.

Theory

As a macrodiatonic scale

It is the macrodiatonic scale with the period of a tritave. This means it is a diatonic scale, but has octaves stretched out to the size of a tritave. Other intervals are also stretched in a way that makes the unrecognizable–the diatonic fifth is now the size of a major seventh. Interestingly, 19edt, an approximation of 12edo, has a tuning of this scale, meaning it contains both a diatonic scale (which approximates 12edo's diatonic scale) and a macrodiatonic scale.

Temperament interpretations

Although they have not been studied in detail, it is possible to construct no-twos rank-2 temperament interpretations of this scale, such as the as-of-yet unnamed b12 & b5 temperament in the 3.13.17 subgroup, in which the generator (the stretched counterpart of the fifth) is ~17/9 and a stack of 4 generators tritave-reduced (the stretched counterpart of the major third) is ~13/9. See also the page for 12edt.

Modes

The modes have step patterns which are the same as the modes of the diatonic scale.

Modes of 5L 2s⟨3/1⟩

UDP	Cyclic order	Step pattern
6|0	1	LLLsLLs
5|1	5	LLsLLLs
4|2	2	LLsLLsL
3|3	6	LsLLLsL
2|4	3	LsLLsLL
1|5	7	sLLLsLL
0|6	4	sLLsLLL

Scale degrees

Scale degrees of the modes of 5L 2s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7
6|0	1	LLLsLLs	Perf.	Maj.	Maj.	Aug.	Perf.	Maj.	Maj.	Perf.
5|1	5	LLsLLLs	Perf.	Maj.	Maj.	Perf.	Perf.	Maj.	Maj.	Perf.
4|2	2	LLsLLsL	Perf.	Maj.	Maj.	Perf.	Perf.	Maj.	Min.	Perf.
3|3	6	LsLLLsL	Perf.	Maj.	Min.	Perf.	Perf.	Maj.	Min.	Perf.
2|4	3	LsLLsLL	Perf.	Maj.	Min.	Perf.	Perf.	Min.	Min.	Perf.
1|5	7	sLLLsLL	Perf.	Min.	Min.	Perf.	Perf.	Min.	Min.	Perf.
0|6	4	sLLsLLL	Perf.	Min.	Min.	Perf.	Dim.	Min.	Min.	Perf.

Notation

Being a macrodiatonic scale, it can notated using the traditional diatonic notation, if all intervals are reinterpreted as their stretched versions (like octaves as tritaves). However, this approach involves 1-based indexing for a non-diatonic MOS which is generally discouraged. Alternatively, a generic MOS notation may be used like diamond MOS notation.

Scale tree

Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead. Details: Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: {{MOS tuning spectrum | 3/2 = Example comment | 4/3 = Another example comment }} The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.

Scale tree and tuning spectrum of 5L 2s⟨3/1⟩

Generator(edt)							Cents		Step ratio		Comments
							Bright	Dark	L:s	Hardness
4\7							1086.831	815.124	1:1	1.000	Equalized 5L 2s⟨3/1⟩
						27\47	1092.612	809.343	7:6	1.167
					23\40		1093.624	808.331	6:5	1.200
						42\73	1094.275	807.680	11:9	1.222
				19\33			1095.065	806.890	5:4	1.250
						53\92	1095.691	806.264	14:11	1.273
					34\59		1096.042	805.913	9:7	1.286
						49\85	1096.421	805.534	13:10	1.300
			15\26				1097.282	804.673	4:3	1.333	Supersoft 5L 2s⟨3/1⟩
						56\97	1098.036	803.919	15:11	1.364
					41\71		1098.312	803.643	11:8	1.375
						67\116	1098.543	803.412	18:13	1.385
				26\45			1098.907	803.048	7:5	1.400
						63\109	1099.295	802.660	17:12	1.417
					37\64		1099.568	802.387	10:7	1.429
						48\83	1099.926	802.029	13:9	1.444
		11\19					1101.132	800.823	3:2	1.500	Soft 5L 2s⟨3/1⟩
						51\88	1102.269	799.686	14:9	1.556
					40\69		1102.583	799.372	11:7	1.571
						69\119	1102.814	799.141	19:12	1.583
				29\50			1103.134	798.821	8:5	1.600
						76\131	1103.424	798.531	21:13	1.615
					47\81		1103.604	798.351	13:8	1.625
						65\112	1103.813	798.142	18:11	1.636
			18\31				1104.361	797.594	5:3	1.667	Semisoft 5L 2s⟨3/1⟩
						61\105	1104.945	797.010	17:10	1.700
					43\74		1105.190	796.765	12:7	1.714
						68\117	1105.410	796.545	19:11	1.727
				25\43			1105.788	796.167	7:4	1.750
						57\98	1106.239	795.716	16:9	1.778
					32\55		1106.592	795.363	9:5	1.800
						39\67	1107.108	794.847	11:6	1.833
	7\12						1109.474	792.481	2:1	2.000	Basic 5L 2s⟨3/1⟩ Scales with tunings softer than this are proper CTE tuning for the b12 & b5 temperament (1109.689)
						38\65	1111.912	790.043	11:5	2.200
					31\53		1112.464	789.491	9:4	2.250
						55\94	1112.846	789.109	16:7	2.286
				24\41			1113.340	788.615	7:3	2.333
						65\111	1113.757	788.198	19:8	2.375
					41\70		1114.002	787.953	12:5	2.400
						58\99	1114.277	787.678	17:7	2.429
			17\29				1114.939	787.016	5:2	2.500	Semihard 5L 2s⟨3/1⟩
						61\104	1115.570	786.385	18:7	2.571
					44\75		1115.814	786.141	13:5	2.600
						71\121	1116.023	785.932	21:8	2.625
				27\46			1116.365	785.590	8:3	2.667
						64\109	1116.744	785.211	19:7	2.714
					37\63		1117.021	784.934	11:4	2.750
						47\80	1117.399	784.556	14:5	2.800
		10\17					1118.797	783.158	3:1	3.000	Hard 5L 2s⟨3/1⟩
						43\73	1120.330	781.625	13:4	3.250
					33\56		1120.795	781.160	10:3	3.333
						56\95	1121.152	780.803	17:5	3.400
				23\39			1121.666	780.289	7:2	3.500
						59\100	1122.153	779.802	18:5	3.600
					36\61		1122.465	779.490	11:3	3.667
						49\83	1122.841	779.114	15:4	3.750
			13\22				1123.883	778.073	4:1	4.000	Superhard 5L 2s⟨3/1⟩
						42\71	1125.100	776.855	13:3	4.333
					29\49		1125.647	776.308	9:2	4.500
						45\76	1126.158	775.797	14:3	4.667
				16\27			1127.084	774.871	5:1	5.000
						35\59	1128.278	773.677	11:2	5.500
					19\32		1129.286	772.669	6:1	6.000
						22\37	1130.892	771.063	7:1	7.000
3\5							1141.173	760.782	1:0	→ ∞	Collapsed 5L 2s⟨3/1⟩