4L 7s (3/1-equivalent)

4L 7s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 7 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1383.2 ¢ to 1426.5 ¢, or from 475.5 ¢ to 518.7 ¢.

↖ 3L 6s⟨3/1⟩	↑ 4L 6s⟨3/1⟩	5L 6s⟨3/1⟩ ↗
← 3L 7s⟨3/1⟩	4L 7s (3/1-equivalent)	5L 7s⟨3/1⟩ →
↙ 3L 8s⟨3/1⟩	↓ 4L 8s⟨3/1⟩	5L 8s⟨3/1⟩ ↘

Scale structure
Step pattern	LsLssLssLss ssLssLssLsL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	8\11 to 3\4 (1383.2 ¢ to 1426.5 ¢)
Dark	1\4 to 3\11 (475.5 ¢ to 518.7 ¢)
Related MOS scales
Parent	4L 3s⟨3/1⟩
Sister	7L 4s⟨3/1⟩
Daughters	11L 4s⟨3/1⟩, 4L 11s⟨3/1⟩
Neutralized	8L 3s⟨3/1⟩
2-Flought	15L 7s⟨3/1⟩, 4L 18s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	8\11 (1383.2 ¢)
Supersoft (L:s = 4:3)	27\37 (1387.9 ¢)
Soft (L:s = 3:2)	19\26 (1389.9 ¢)
Semisoft (L:s = 5:3)	30\41 (1391.7 ¢)
Basic (L:s = 2:1)	11\15 (1394.8 ¢)
Semihard (L:s = 5:2)	25\34 (1398.5 ¢)
Hard (L:s = 3:1)	14\19 (1401.4 ¢)
Superhard (L:s = 4:1)	17\23 (1405.8 ¢)
Collapsed (L:s = 1:0)	3\4 (1426.5 ¢)
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Theory

Modes

Modes of 4L 7s⟨3/1⟩

UDP	Cyclic order	Step pattern
10|0	1	LsLssLssLss
9|1	9	LssLsLssLss
8|2	6	LssLssLsLss
7|3	3	LssLssLssLs
6|4	11	sLsLssLssLs
5|5	8	sLssLsLssLs
4|6	5	sLssLssLsLs
3|7	2	sLssLssLssL
2|8	10	ssLsLssLssL
1|9	7	ssLssLsLssL
0|10	4	ssLssLssLsL

Temperament interpretations

The temperament that joins 15edt and 26edt produces this MOS. Both 15edt and 26edt are no-twos zeta peaks. 56edt, another no-twos zeta peak, also supports this temperament.

Other EDTs supporting it include: 41, 67, 71…

In 15edt it is generated by 4\15 (~507.2¢) and has a hardness of exactly 2.0.

In 56edt it is generated by 15\56 (~509.5¢) and has a hardness of exactly 1.75.

In 26edt (’double Bohlen-Pierce’) it is generated by 7\26 (~512.1) and has a hardness of exactly 1.5.

In 56edt this MOS has the intervals (in cents):

• 136.045
• 373.455
• 509.500
• 645.545
• 882.955
• 1019.000
• 1256.410
• 1392.455
• 1528.500
• 1765.910
• 1901.955

(And rotations thereof.)

That roughly approximates this 13-limit JI scale:

• 13/12
• 5/4
• 4/3
• 13/9
• 5/3
• 9/5
• 25/12
• 20/9
• 12/5
• 25/9
• 3/1

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

Intervals

Intervals of 4L 7s⟨3/1⟩

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 172.9 ¢
	Major 1-mosstep	M1ms	L	172.9 ¢ to 475.5 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 345.8 ¢
	Major 2-mosstep	M2ms	L + s	345.8 ¢ to 475.5 ¢
3-mosstep	Perfect 3-mosstep	P3ms	L + 2s	475.5 ¢ to 518.7 ¢
	Augmented 3-mosstep	A3ms	2L + s	518.7 ¢ to 951.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	L + 3s	475.5 ¢ to 691.6 ¢
	Major 4-mosstep	M4ms	2L + 2s	691.6 ¢ to 951.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	L + 4s	475.5 ¢ to 864.5 ¢
	Major 5-mosstep	M5ms	2L + 3s	864.5 ¢ to 951.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	2L + 4s	951.0 ¢ to 1037.4 ¢
	Major 6-mosstep	M6ms	3L + 3s	1037.4 ¢ to 1426.5 ¢
7-mosstep	Minor 7-mosstep	m7ms	2L + 5s	951.0 ¢ to 1210.3 ¢
	Major 7-mosstep	M7ms	3L + 4s	1210.3 ¢ to 1426.5 ¢
8-mosstep	Diminished 8-mosstep	d8ms	2L + 6s	951.0 ¢ to 1383.2 ¢
	Perfect 8-mosstep	P8ms	3L + 5s	1383.2 ¢ to 1426.5 ¢
9-mosstep	Minor 9-mosstep	m9ms	3L + 6s	1426.5 ¢ to 1556.1 ¢
	Major 9-mosstep	M9ms	4L + 5s	1556.1 ¢ to 1902.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	3L + 7s	1426.5 ¢ to 1729.1 ¢
	Major 10-mosstep	M10ms	4L + 6s	1729.1 ¢ to 1902.0 ¢
11-mosstep	Perfect 11-mosstep	P11ms	4L + 7s	1902.0 ¢

Scale degrees

Scale degrees of the modes of 4L 7s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9	10	11
10|0	1	LsLssLssLss	Perf.	Maj.	Maj.	Aug.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Perf.
9|1	9	LssLsLssLss	Perf.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Perf.
8|2	6	LssLssLsLss	Perf.	Maj.	Maj.	Perf.	Maj.	Maj.	Min.	Maj.	Perf.	Maj.	Maj.	Perf.
7|3	3	LssLssLssLs	Perf.	Maj.	Maj.	Perf.	Maj.	Maj.	Min.	Maj.	Perf.	Min.	Maj.	Perf.
6|4	11	sLsLssLssLs	Perf.	Min.	Maj.	Perf.	Maj.	Maj.	Min.	Maj.	Perf.	Min.	Maj.	Perf.
5|5	8	sLssLsLssLs	Perf.	Min.	Maj.	Perf.	Min.	Maj.	Min.	Maj.	Perf.	Min.	Maj.	Perf.
4|6	5	sLssLssLsLs	Perf.	Min.	Maj.	Perf.	Min.	Maj.	Min.	Min.	Perf.	Min.	Maj.	Perf.
3|7	2	sLssLssLssL	Perf.	Min.	Maj.	Perf.	Min.	Maj.	Min.	Min.	Perf.	Min.	Min.	Perf.
2|8	10	ssLsLssLssL	Perf.	Min.	Min.	Perf.	Min.	Maj.	Min.	Min.	Perf.	Min.	Min.	Perf.
1|9	7	ssLssLsLssL	Perf.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Perf.
0|10	4	ssLssLssLsL	Perf.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Dim.	Min.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 4L 7s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
8\11						1383.240	518.715	1:1	1.000	Equalized 4L 7s⟨3/1⟩
					43\59	1386.171	515.784	6:5	1.200
				35\48		1386.842	515.113	5:4	1.250
					62\85	1387.308	514.647	9:7	1.286
			27\37			1387.913	514.042	4:3	1.333	Supersoft 4L 7s⟨3/1⟩
					73\100	1388.427	513.528	11:8	1.375
				46\63		1388.729	513.226	7:5	1.400
					65\89	1389.068	512.887	10:7	1.429
		19\26				1389.890	512.065	3:2	1.500	Soft 4L 7s⟨3/1⟩
					68\93	1390.677	511.278	11:7	1.571
				49\67		1390.982	510.973	8:5	1.600
					79\108	1391.245	510.710	13:8	1.625
			30\41			1391.674	510.281	5:3	1.667	Semisoft 4L 7s⟨3/1⟩
					71\97	1392.153	509.802	12:7	1.714
				41\56		1392.503	509.452	7:4	1.750
					52\71	1392.981	508.974	9:5	1.800
	11\15					1394.767	507.188	2:1	2.000	Basic 4L 7s⟨3/1⟩ Scales with tunings softer than this are proper
					47\64	1396.748	505.207	9:4	2.250
				36\49		1397.355	504.600	7:3	2.333
					61\83	1397.822	504.133	12:5	2.400
			25\34			1398.496	503.459	5:2	2.500	Semihard 4L 7s⟨3/1⟩
					64\87	1399.139	502.816	13:5	2.600
				39\53		1399.552	502.403	8:3	2.667
					53\72	1400.050	501.905	11:4	2.750
		14\19				1401.441	500.514	3:1	3.000	Hard 4L 7s⟨3/1⟩
					45\61	1403.082	498.873	10:3	3.333
				31\42		1403.824	498.131	7:2	3.500
					48\65	1404.521	497.434	11:3	3.667
			17\23			1405.793	496.162	4:1	4.000	Superhard 4L 7s⟨3/1⟩
					37\50	1407.447	494.508	9:2	4.500
				20\27		1408.856	493.099	5:1	5.000
					23\31	1411.128	490.827	6:1	6.000
3\4						1426.466	475.489	1:0	→ ∞	Collapsed 4L 7s⟨3/1⟩