6L 3s

6L 3s, named hyrulic in TAMNAMS, is a 2/1-equivalent (octave-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 400.0 ¢, or 3 times every octave. Generators that produce this scale range from 133.3 ¢ to 200 ¢, or from 200 ¢ to 266.7 ¢. 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	↑ 6L 2s	7L 2s ↗
← 5L 3s	6L 3s	7L 3s →
↙ 5L 4s	↓ 6L 4s	7L 4s ↘

Scale structure
Step pattern	LLsLLsLLs sLLsLLsLL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	1\9 to 1\6 (133.3 ¢ to 200.0 ¢)
Dark	1\6 to 2\9 (200.0 ¢ to 266.7 ¢)
TAMNAMS information
Name	hyrulic
Prefix	hyru-
Abbrev.	hy
Related MOS scales
Parent	3L 3s
Sister	3L 6s
Daughters	9L 6s, 6L 9s
Neutralized	3L 6s
2-Flought	15L 3s, 6L 12s
Equal tunings
Equalized (L:s = 1:1)	1\9 (133.3 ¢)
Supersoft (L:s = 4:3)	4\33 (145.5 ¢)
Soft (L:s = 3:2)	3\24 (150.0 ¢)
Semisoft (L:s = 5:3)	5\39 (153.8 ¢)
Basic (L:s = 2:1)	2\15 (160.0 ¢)
Semihard (L:s = 5:2)	5\36 (166.7 ¢)
Hard (L:s = 3:1)	3\21 (171.4 ¢)
Superhard (L:s = 4:1)	4\27 (177.8 ¢)
Collapsed (L:s = 1:0)	1\6 (200.0 ¢)
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The tiny region between 26\171 and 281\1848 is the terrain region, giving the albitonic scale for the remarkable terrain temperament chromatic pair. If you feel that is too small a region to allocate or prefer EDOs of a more usual size, you can call it the region between 5\33 and 6\39, but that misses the extreme accuracy of this temperament.

For a more approachable temperament, try triforce.

Intervals

Intervals of 6L 3s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-hyrustep	Perfect 0-hyrustep	P0hys	0	0.0 ¢
1-hyrustep	Diminished 1-hyrustep	d1hys	s	0.0 ¢ to 133.3 ¢
	Perfect 1-hyrustep	P1hys	L	133.3 ¢ to 200.0 ¢
2-hyrustep	Perfect 2-hyrustep	P2hys	L + s	200.0 ¢ to 266.7 ¢
	Augmented 2-hyrustep	A2hys	2L	266.7 ¢ to 400.0 ¢
3-hyrustep	Perfect 3-hyrustep	P3hys	2L + s	400.0 ¢
4-hyrustep	Diminished 4-hyrustep	d4hys	2L + 2s	400.0 ¢ to 533.3 ¢
	Perfect 4-hyrustep	P4hys	3L + s	533.3 ¢ to 600.0 ¢
5-hyrustep	Perfect 5-hyrustep	P5hys	3L + 2s	600.0 ¢ to 666.7 ¢
	Augmented 5-hyrustep	A5hys	4L + s	666.7 ¢ to 800.0 ¢
6-hyrustep	Perfect 6-hyrustep	P6hys	4L + 2s	800.0 ¢
7-hyrustep	Diminished 7-hyrustep	d7hys	4L + 3s	800.0 ¢ to 933.3 ¢
	Perfect 7-hyrustep	P7hys	5L + 2s	933.3 ¢ to 1000.0 ¢
8-hyrustep	Perfect 8-hyrustep	P8hys	5L + 3s	1000.0 ¢ to 1066.7 ¢
	Augmented 8-hyrustep	A8hys	6L + 2s	1066.7 ¢ to 1200.0 ¢
9-hyrustep	Perfect 9-hyrustep	P9hys	6L + 3s	1200.0 ¢

Modes

Scale degrees of the modes of 6L 3s

UDP	Cyclic order	Step pattern	Scale degree (hyrudegree)
			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.

Scales

• Triphi – L:s = φ tuning
• Terrain9 – 3558edo tuning

Scale tree

Scale tree and tuning spectrum of 6L 3s

Generator(edo)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\9						133.333	266.667	1:1	1.000	Equalized 6L 3s
					6\51	141.176	258.824	6:5	1.200
				5\42		142.857	257.143	5:4	1.250
					9\75	144.000	256.000	9:7	1.286
			4\33			145.455	254.545	4:3	1.333	Supersoft 6L 3s
					11\90	146.667	253.333	11:8	1.375
				7\57		147.368	252.632	7:5	1.400
					10\81	148.148	251.852	10:7	1.429
		3\24				150.000	250.000	3:2	1.500	Soft 6L 3s
					11\87	151.724	248.276	11:7	1.571	Hemimist
				8\63		152.381	247.619	8:5	1.600
					13\102	152.941	247.059	13:8	1.625	Golden triforce (152.7864 ¢)
			5\39			153.846	246.154	5:3	1.667	Semisoft 6L 3s Triforce
					12\93	154.839	245.161	12:7	1.714
				7\54		155.556	244.444	7:4	1.750
					9\69	156.522	243.478	9:5	1.800
	2\15					160.000	240.000	2:1	2.000	Basic 6L 3s
					9\66	163.636	236.364	9:4	2.250
				7\51		164.706	235.294	7:3	2.333
					12\87	165.517	234.483	12:5	2.400
			5\36			166.667	233.333	5:2	2.500	Semihard 6L 3s
					13\93	167.742	232.258	13:5	2.600	Unnamed golden tuning (167.9285¢)
				8\57		168.421	231.579	8:3	2.667
					11\78	169.231	230.769	11:4	2.750
		3\21				171.429	228.571	3:1	3.000	Hard 6L 3s
					10\69	173.913	226.087	10:3	3.333
				7\48		175.000	225.000	7:2	3.500
					11\75	176.000	224.000	11:3	3.667
			4\27			177.778	222.222	4:1	4.000	Superhard 6L 3s Oodako
					9\60	180.000	220.000	9:2	4.500
				5\33		181.818	218.182	5:1	5.000
					6\39	184.615	215.385	6:1	6.000	Mirkat
1\6						200.000	200.000	1:0	→ ∞	Collapsed 6L 3s