3L 6s (3/1-equivalent)

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

Scale structure
Step pattern	LssLssLss ssLssLssL
Equave	3/1 (1902.0 ¢)
Period	1\3 (634.0 ¢)
Generator size(edt)
Bright	2\9 to 1\3 (422.7 ¢ to 634.0 ¢)
Dark	0\3 to 1\9 (0.0 ¢ to 211.3 ¢)
Related MOS scales
Parent	3L 3s⟨3/1⟩
Sister	6L 3s⟨3/1⟩
Daughters	9L 3s⟨3/1⟩, 3L 9s⟨3/1⟩
Neutralized	6L 3s⟨3/1⟩
2-Flought	12L 6s⟨3/1⟩, 3L 15s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	2\9 (422.7 ¢)
Supersoft (L:s = 4:3)	7\30 (443.8 ¢)
Soft (L:s = 3:2)	5\21 (452.8 ¢)
Semisoft (L:s = 5:3)	8\33 (461.1 ¢)
Basic (L:s = 2:1)	3\12 (475.5 ¢)
Semihard (L:s = 5:2)	7\27 (493.1 ¢)
Hard (L:s = 3:1)	4\15 (507.2 ¢)
Superhard (L:s = 4:1)	5\18 (528.3 ¢)
Collapsed (L:s = 1:0)	1\3 (634.0 ¢)
View • Talk • Edit

3L 6s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 3 large steps and 6 small steps, with a period of 1 large step and 2 small steps that repeats every 634.0 ¢, or 3 times every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 422.7 ¢ to 634 ¢, or from 0 ¢ to 211.3 ¢.

Theory

Modes

Modes of 3L 6s⟨3/1⟩

UDP	Cyclic order	Step pattern
6|0(3)	1	LssLssLss
3|3(3)	3	sLssLssLs
0|6(3)	2	ssLssLssL

Temperament interpretations

There are two interesting temperaments with this MOS: the one that joins 12edt&57edt, and the one that joins 12edt&39edt. Both are generated by neutral second intervals of different sizes.

Both are equated in 12edt, where they are generated by 3\12 with a hardness of 2.0.

12&57

12&57 temperament is supported by 12edt and 57edt.

In 57edt, this temperament is generated by 14\57 (~166.8¢) with a hardness of 1.8, and has a porcupine-like sound.

In 57edt this MOS has the intervals (in cents):

• 166.838
• 467.147
• 633.985
• 800.823
• 1101.132
• 1267.970
• 1434.808
• 1735.117
• 1901.955

(And rotations thereof.)

One possible JI interpretation in the 19-limit is as:

• 10/9
• 13/10
• 13/9
• 19/12
• 17/9
• 25/12
• 16/7
• 11/4
• 3/1

Where intervals with multiple-of-3 denominators become more consonant in higher tritaves.

12&39

12&39 temperament is supported by 12edt, 51edt and 39edt (’triple Bohlen-Pierce’).

In 51edt it is generated by 13\51 (~149.2¢) with a hardness of 2.25.

In 39edt it is generated by 10\39 (~146.3¢) with a hardness of about 2.33.

In 51edt this MOS has the intervals (in cents):

• 149.173
• 484.812
• 633.985
• 783.158
• 1118.797
• 1267.970
• 1417.143
• 1752.782
• 1901.955

(And rotations thereof.)

One possible JI interpretation in the 17-limit is as:

• 12/11
• 4/3
• 13/9
• 11/7
• 17/9
• 25/12
• 9/4
• 11/4
• 3/1

Where intervals with multiple-of-3 denominators become more consonant in higher tritaves.

Intervals

Intervals of 3L 6s⟨3/1⟩

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Perfect 1-mosstep	P1ms	s	0.0 ¢ to 211.3 ¢
	Augmented 1-mosstep	A1ms	L	211.3 ¢ to 634.0 ¢
2-mosstep	Diminished 2-mosstep	d2ms	2s	0.0 ¢ to 422.7 ¢
	Perfect 2-mosstep	P2ms	L + s	422.7 ¢ to 634.0 ¢
3-mosstep	Perfect 3-mosstep	P3ms	L + 2s	634.0 ¢
4-mosstep	Perfect 4-mosstep	P4ms	L + 3s	634.0 ¢ to 845.3 ¢
	Augmented 4-mosstep	A4ms	2L + 2s	845.3 ¢ to 1268.0 ¢
5-mosstep	Diminished 5-mosstep	d5ms	L + 4s	634.0 ¢ to 1056.6 ¢
	Perfect 5-mosstep	P5ms	2L + 3s	1056.6 ¢ to 1268.0 ¢
6-mosstep	Perfect 6-mosstep	P6ms	2L + 4s	1268.0 ¢
7-mosstep	Perfect 7-mosstep	P7ms	2L + 5s	1268.0 ¢ to 1479.3 ¢
	Augmented 7-mosstep	A7ms	3L + 4s	1479.3 ¢ to 1902.0 ¢
8-mosstep	Diminished 8-mosstep	d8ms	2L + 6s	1268.0 ¢ to 1690.6 ¢
	Perfect 8-mosstep	P8ms	3L + 5s	1690.6 ¢ to 1902.0 ¢
9-mosstep	Perfect 9-mosstep	P9ms	3L + 6s	1902.0 ¢

Scale degrees

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

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

Scale tree

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

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
2\9						422.657	211.328	1:1	1.000	Equalized 3L 6s⟨3/1⟩
					11\48	435.865	198.120	6:5	1.200
				9\39		438.913	195.072	5:4	1.250
					16\69	441.033	192.952	9:7	1.286
			7\30			443.790	190.196	4:3	1.333	Supersoft 3L 6s⟨3/1⟩
					19\81	446.138	187.847	11:8	1.375
				12\51		447.519	186.466	7:5	1.400
					17\72	449.073	184.912	10:7	1.429
		5\21				452.846	181.139	3:2	1.500	Soft 3L 6s⟨3/1⟩
					18\75	456.469	177.516	11:7	1.571
				13\54		457.878	176.107	8:5	1.600
					21\87	459.093	174.892	13:8	1.625
			8\33			461.080	172.905	5:3	1.667	Semisoft 3L 6s⟨3/1⟩
					19\78	463.297	170.688	12:7	1.714
				11\45		464.922	169.063	7:4	1.750
					14\57	467.147	166.838	9:5	1.800
	3\12					475.489	158.496	2:1	2.000	Basic 3L 6s⟨3/1⟩ Scales with tunings softer than this are proper
					13\51	484.812	149.173	9:4	2.250
				10\39		487.681	146.304	7:3	2.333
					17\66	489.898	144.088	12:5	2.400
			7\27			493.099	140.886	5:2	2.500	Semihard 3L 6s⟨3/1⟩
					18\69	496.162	137.823	13:5	2.600
				11\42		498.131	135.854	8:3	2.667
					15\57	500.514	133.471	11:4	2.750
		4\15				507.188	126.797	3:1	3.000	Hard 3L 6s⟨3/1⟩
					13\48	515.113	118.872	10:3	3.333
				9\33		518.715	115.270	7:2	3.500
					14\51	522.105	111.880	11:3	3.667
			5\18			528.321	105.664	4:1	4.000	Superhard 3L 6s⟨3/1⟩
					11\39	536.449	97.536	9:2	4.500
				6\21		543.416	90.569	5:1	5.000
					7\24	554.737	79.248	6:1	6.000
1\3						633.985	0.000	1:0	→ ∞	Collapsed 3L 6s⟨3/1⟩