6L 3s (3/1-equivalent)

6L 3s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 6 large steps and 3 small steps, with a period of 2 large steps and 1 small step that repeats every 634.0 ¢, or 3 times every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 211.3 ¢ to 317 ¢, or from 317 ¢ to 422.7 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

↖ 5L 2s⟨3/1⟩	↑ 6L 2s⟨3/1⟩	7L 2s⟨3/1⟩ ↗
← 5L 3s⟨3/1⟩	6L 3s (3/1-equivalent)	7L 3s⟨3/1⟩ →
↙ 5L 4s⟨3/1⟩	↓ 6L 4s⟨3/1⟩	7L 4s⟨3/1⟩ ↘

Scale structure
Step pattern	LLsLLsLLs sLLsLLsLL
Equave	3/1 (1902.0 ¢)
Period	1\3 (634.0 ¢)
Generator size(edt)
Bright	1\9 to 1\6 (211.3 ¢ to 317.0 ¢)
Dark	1\6 to 2\9 (317.0 ¢ to 422.7 ¢)
Related MOS scales
Parent	3L 3s⟨3/1⟩
Sister	3L 6s⟨3/1⟩
Daughters	9L 6s⟨3/1⟩, 6L 9s⟨3/1⟩
Neutralized	3L 6s⟨3/1⟩
2-Flought	15L 3s⟨3/1⟩, 6L 12s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	1\9 (211.3 ¢)
Supersoft (L:s = 4:3)	4\33 (230.5 ¢)
Soft (L:s = 3:2)	3\24 (237.7 ¢)
Semisoft (L:s = 5:3)	5\39 (243.8 ¢)
Basic (L:s = 2:1)	2\15 (253.6 ¢)
Semihard (L:s = 5:2)	5\36 (264.2 ¢)
Hard (L:s = 3:1)	3\21 (271.7 ¢)
Superhard (L:s = 4:1)	4\27 (281.8 ¢)
Collapsed (L:s = 1:0)	1\6 (317.0 ¢)
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Theory

Modes

Modes of 6L 3s⟨3/1⟩

UDP	Cyclic order	Step pattern
6|0(3)	1	LLsLLsLLs
3|3(3)	2	LsLLsLLsL
0|6(3)	3	sLLsLLsLL

Temperament interpretations

The temperament that joins 15edt and 39edt produces this MOS. Both 15edt and 39edt are no-twos zeta peaks. 69edt, another no-twos zeta peak, also supports this temperament.

In 15edt it is generated by 2\15 (~253.6¢) and has a hardness of exactly 2.0.

In 69edt it is generated by 9\69 (~248.1¢) and has a hardness of exactly 1.8.

In 39edt (’Triple Bohlen-Pierce’) it is generated by 5\39 (~243.8¢) and has a hardness of about 1.67.

In 39edt this MOS has the intervals (in cents):

• 243.84
• 390.145
• 633.985
• 877.825
• 1024.13
• 1267.97
• 1511.81
• 1658.115
• 1901.955

(And rotations thereof.)

That roughly approximates this 13-limit JI scale:

• 7/6
• 5/4
• 13/9
• 5/3
• 9/5
• 25/12
• 12/5
• 13/5
• 3/1

(The intervals with multiple-of-3 denominators gain consonance in higher tritaves.)

Intervals

Intervals of 6L 3s⟨3/1⟩

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Diminished 1-mosstep	d1ms	s	0.0 ¢ to 211.3 ¢
	Perfect 1-mosstep	P1ms	L	211.3 ¢ to 317.0 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	317.0 ¢ to 422.7 ¢
	Augmented 2-mosstep	A2ms	2L	422.7 ¢ to 634.0 ¢
3-mosstep	Perfect 3-mosstep	P3ms	2L + s	634.0 ¢
4-mosstep	Diminished 4-mosstep	d4ms	2L + 2s	634.0 ¢ to 845.3 ¢
	Perfect 4-mosstep	P4ms	3L + s	845.3 ¢ to 951.0 ¢
5-mosstep	Perfect 5-mosstep	P5ms	3L + 2s	951.0 ¢ to 1056.6 ¢
	Augmented 5-mosstep	A5ms	4L + s	1056.6 ¢ to 1268.0 ¢
6-mosstep	Perfect 6-mosstep	P6ms	4L + 2s	1268.0 ¢
7-mosstep	Diminished 7-mosstep	d7ms	4L + 3s	1268.0 ¢ to 1479.3 ¢
	Perfect 7-mosstep	P7ms	5L + 2s	1479.3 ¢ to 1585.0 ¢
8-mosstep	Perfect 8-mosstep	P8ms	5L + 3s	1585.0 ¢ to 1690.6 ¢
	Augmented 8-mosstep	A8ms	6L + 2s	1690.6 ¢ to 1902.0 ¢
9-mosstep	Perfect 9-mosstep	P9ms	6L + 3s	1902.0 ¢

Scale degrees

Scale degrees of the modes of 6L 3s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9
6|0(3)	1	LLsLLsLLs	Perf.	Perf.	Aug.	Perf.	Perf.	Aug.	Perf.	Perf.	Aug.	Perf.
3|3(3)	2	LsLLsLLsL	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.	Perf.
0|6(3)	3	sLLsLLsLL	Perf.	Dim.	Perf.	Perf.	Dim.	Perf.	Perf.	Dim.	Perf.	Perf.

Scale tree

Scale tree and tuning spectrum of 6L 3s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\9						211.328	422.657	1:1	1.000	Equalized 6L 3s⟨3/1⟩
					6\51	223.759	410.226	6:5	1.200
				5\42		226.423	407.562	5:4	1.250
					9\75	228.235	405.750	9:7	1.286
			4\33			230.540	403.445	4:3	1.333	Supersoft 6L 3s⟨3/1⟩
					11\90	232.461	401.524	11:8	1.375
				7\57		233.573	400.412	7:5	1.400
					10\81	234.809	399.176	10:7	1.429
		3\24				237.744	396.241	3:2	1.500	Soft 6L 3s⟨3/1⟩
					11\87	240.477	393.508	11:7	1.571
				8\63		241.518	392.467	8:5	1.600
					13\102	242.406	391.579	13:8	1.625
			5\39			243.840	390.145	5:3	1.667	Semisoft 6L 3s⟨3/1⟩
					12\93	245.414	388.571	12:7	1.714
				7\54		246.550	387.435	7:4	1.750
					9\69	248.081	385.904	9:5	1.800
	2\15					253.594	380.391	2:1	2.000	Basic 6L 3s⟨3/1⟩
					9\66	259.358	374.628	9:4	2.250
				7\51		261.053	372.932	7:3	2.333
					12\87	262.339	371.646	12:5	2.400
			5\36			264.160	369.825	5:2	2.500	Semihard 6L 3s⟨3/1⟩
					13\93	265.865	368.120	13:5	2.600
				8\57		266.941	367.044	8:3	2.667
					11\78	268.224	365.761	11:4	2.750
		3\21				271.708	362.277	3:1	3.000	Hard 6L 3s⟨3/1⟩
					10\69	275.646	358.339	10:3	3.333
				7\48		277.368	356.617	7:2	3.500
					11\75	278.953	355.032	11:3	3.667
			4\27			281.771	352.214	4:1	4.000	Superhard 6L 3s⟨3/1⟩
					9\60	285.293	348.692	9:2	4.500
				5\33		288.175	345.810	5:1	5.000
					6\39	292.608	341.377	6:1	6.000
1\6						316.993	316.993	1:0	→ ∞	Collapsed 6L 3s⟨3/1⟩