User:MisterShafXen/6L 15s

6L 15s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 6 large steps and 15 small steps, with a period of 2 large steps and 5 small steps that repeats every 400.0 ¢, or 3 times every octave. 6L 15s is a grandchild scale of 6L 3s, expanding it by 12 tones. Generators that produce this scale range from 171.4 ¢ to 200 ¢, or from 200 ¢ to 228.6 ¢.

↖ 5L 14s	↑ 6L 14s	7L 14s ↗
← 5L 15s	6L 15s	7L 15s →
↙ 5L 16s	↓ 6L 16s	7L 16s ↘

Scale structure
Step pattern	LssLsssLssLsssLssLsss sssLssLsssLssLsssLssL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	3\21 to 1\6 (171.4 ¢ to 200.0 ¢)
Dark	1\6 to 4\21 (200.0 ¢ to 228.6 ¢)
TAMNAMS information
Related to	6L 3s (hyrulic)
With tunings	3:1 to 1:0 (hard)
Related MOS scales
Parent	6L 9s
Sister	15L 6s
Daughters	21L 6s, 6L 21s
Neutralized	12L 9s
2-Flought	27L 15s, 6L 36s
Equal tunings
Equalized (L:s = 1:1)	3\21 (171.4 ¢)
Supersoft (L:s = 4:3)	10\69 (173.9 ¢)
Soft (L:s = 3:2)	7\48 (175.0 ¢)
Semisoft (L:s = 5:3)	11\75 (176.0 ¢)
Basic (L:s = 2:1)	4\27 (177.8 ¢)
Semihard (L:s = 5:2)	9\60 (180.0 ¢)
Hard (L:s = 3:1)	5\33 (181.8 ¢)
Superhard (L:s = 4:1)	6\39 (184.6 ¢)
Collapsed (L:s = 1:0)	1\6 (200.0 ¢)
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Scale properties

Intervals of 6L 15s

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 57.1 ¢
	Major 1-mosstep	M1ms	L	57.1 ¢ to 200.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 114.3 ¢
	Major 2-mosstep	M2ms	L + s	114.3 ¢ to 200.0 ¢
3-mosstep	Diminished 3-mosstep	d3ms	3s	0.0 ¢ to 171.4 ¢
	Perfect 3-mosstep	P3ms	L + 2s	171.4 ¢ to 200.0 ¢
4-mosstep	Perfect 4-mosstep	P4ms	L + 3s	200.0 ¢ to 228.6 ¢
	Augmented 4-mosstep	A4ms	2L + 2s	228.6 ¢ to 400.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	L + 4s	200.0 ¢ to 285.7 ¢
	Major 5-mosstep	M5ms	2L + 3s	285.7 ¢ to 400.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	L + 5s	200.0 ¢ to 342.9 ¢
	Major 6-mosstep	M6ms	2L + 4s	342.9 ¢ to 400.0 ¢
7-mosstep	Perfect 7-mosstep	P7ms	2L + 5s	400.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	2L + 6s	400.0 ¢ to 457.1 ¢
	Major 8-mosstep	M8ms	3L + 5s	457.1 ¢ to 600.0 ¢
9-mosstep	Minor 9-mosstep	m9ms	2L + 7s	400.0 ¢ to 514.3 ¢
	Major 9-mosstep	M9ms	3L + 6s	514.3 ¢ to 600.0 ¢
10-mosstep	Diminished 10-mosstep	d10ms	2L + 8s	400.0 ¢ to 571.4 ¢
	Perfect 10-mosstep	P10ms	3L + 7s	571.4 ¢ to 600.0 ¢
11-mosstep	Perfect 11-mosstep	P11ms	3L + 8s	600.0 ¢ to 628.6 ¢
	Augmented 11-mosstep	A11ms	4L + 7s	628.6 ¢ to 800.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	3L + 9s	600.0 ¢ to 685.7 ¢
	Major 12-mosstep	M12ms	4L + 8s	685.7 ¢ to 800.0 ¢
13-mosstep	Minor 13-mosstep	m13ms	3L + 10s	600.0 ¢ to 742.9 ¢
	Major 13-mosstep	M13ms	4L + 9s	742.9 ¢ to 800.0 ¢
14-mosstep	Perfect 14-mosstep	P14ms	4L + 10s	800.0 ¢
15-mosstep	Minor 15-mosstep	m15ms	4L + 11s	800.0 ¢ to 857.1 ¢
	Major 15-mosstep	M15ms	5L + 10s	857.1 ¢ to 1000.0 ¢
16-mosstep	Minor 16-mosstep	m16ms	4L + 12s	800.0 ¢ to 914.3 ¢
	Major 16-mosstep	M16ms	5L + 11s	914.3 ¢ to 1000.0 ¢
17-mosstep	Diminished 17-mosstep	d17ms	4L + 13s	800.0 ¢ to 971.4 ¢
	Perfect 17-mosstep	P17ms	5L + 12s	971.4 ¢ to 1000.0 ¢
18-mosstep	Perfect 18-mosstep	P18ms	5L + 13s	1000.0 ¢ to 1028.6 ¢
	Augmented 18-mosstep	A18ms	6L + 12s	1028.6 ¢ to 1200.0 ¢
19-mosstep	Minor 19-mosstep	m19ms	5L + 14s	1000.0 ¢ to 1085.7 ¢
	Major 19-mosstep	M19ms	6L + 13s	1085.7 ¢ to 1200.0 ¢
20-mosstep	Minor 20-mosstep	m20ms	5L + 15s	1000.0 ¢ to 1142.9 ¢
	Major 20-mosstep	M20ms	6L + 14s	1142.9 ¢ to 1200.0 ¢
21-mosstep	Perfect 21-mosstep	P21ms	6L + 15s	1200.0 ¢

Harmonic entropy info for 6L 15s

Interval	Range in cents	Average of HE (from HE Calc)	Min of HE
Perfect 0-mosstep	0.0 ¢	~2.4654 nats	~2.4654 nats
Minor 1-mosstep	0.0 ¢ to 57.1 ¢	~4.7090 nats	~4.4127 nats
Major 1-mosstep	57.1 ¢ to 200.0 ¢	~4.6998 nats	~4.6486 nats
Minor 2-mosstep	0.0 ¢ to 114.3 ¢	~4.7106 nats	~4.6700 nats
Major 2-mosstep	114.3 ¢ to 200.0 ¢	~4.6367 nats	~4.6197 nats
Diminished 3-mosstep	0.0 ¢ to 171.4 ¢	~4.6448 nats	~4.6172 nats
Perfect 3-mosstep	171.4 ¢ to 200.0 ¢	~4.5984 nats	~4.5938 nats
Perfect 4-mosstep	200.0 ¢ to 228.6 ¢	~4.5841 nats	~4.5840 nats
Augmented 4-mosstep	228.6 ¢ to 400.0 ¢	~4.5725 nats	~4.5463 nats
Minor 5-mosstep	200.0 ¢ to 285.7 ¢	~4.5712 nats	~4.5636 nats
Major 5-mosstep	285.7 ¢ to 400.0 ¢	~4.5628 nats	~4.5404 nats
Minor 6-mosstep	200.0 ¢ to 342.9 ¢	~4.5607 nats	~4.5404 nats
Major 6-mosstep	342.9 ¢ to 400.0 ¢	~4.6049 nats	~4.5624 nats
Perfect 7-mosstep	400.0 ¢	~4.5262 nats	~4.5262 nats
Minor 8-mosstep	400.0 ¢ to 457.1 ¢	~4.6061 nats	~4.6002 nats
Major 8-mosstep	457.1 ¢ to 600.0 ¢	~4.4917 nats	~4.3665 nats
Minor 9-mosstep	400.0 ¢ to 514.3 ¢	~4.4638 nats	~4.3665 nats
Major 9-mosstep	514.3 ¢ to 600.0 ¢	~4.6059 nats	~4.5553 nats
Diminished 10-mosstep	400.0 ¢ to 571.4 ¢	~4.5486 nats	~4.3905 nats
Perfect 10-mosstep	571.4 ¢ to 600.0 ¢	~4.5656 nats	~4.5596 nats
Perfect 11-mosstep	600.0 ¢ to 628.6 ¢	~4.6009 nats	~4.5975 nats
Augmented 11-mosstep	628.6 ¢ to 800.0 ¢	~4.4950 nats	~4.1464 nats
Minor 12-mosstep	600.0 ¢ to 685.7 ¢	~4.6106 nats	~4.5237 nats
Major 12-mosstep	685.7 ¢ to 800.0 ¢	~4.3065 nats	~4.1230 nats
Minor 13-mosstep	600.0 ¢ to 742.9 ¢	~4.3630 nats	~4.1319 nats
Major 13-mosstep	742.9 ¢ to 800.0 ¢	~4.6355 nats	~4.6155 nats
Perfect 14-mosstep	800.0 ¢	~4.5856 nats	~4.5856 nats
Minor 15-mosstep	800.0 ¢ to 857.1 ¢	~4.6083 nats	~4.5928 nats
Major 15-mosstep	857.1 ¢ to 1000.0 ¢	~4.5021 nats	~4.4255 nats
Minor 16-mosstep	800.0 ¢ to 914.3 ¢	~4.4871 nats	~4.4255 nats
Major 16-mosstep	914.3 ¢ to 1000.0 ¢	~4.6059 nats	~4.5713 nats
Diminished 17-mosstep	800.0 ¢ to 971.4 ¢	~4.5691 nats	~4.4407 nats
Perfect 17-mosstep	971.4 ¢ to 1000.0 ¢	~4.5421 nats	~4.5323 nats
Perfect 18-mosstep	1000.0 ¢ to 1028.6 ¢	~4.5804 nats	~4.5781 nats
Augmented 18-mosstep	1028.6 ¢ to 1200.0 ¢	~4.6107 nats	~4.6048 nats
Minor 19-mosstep	1000.0 ¢ to 1085.7 ¢	~4.6078 nats	~4.6063 nats
Major 19-mosstep	1085.7 ¢ to 1200.0 ¢	~4.6402 nats	~4.6148 nats
Minor 20-mosstep	1000.0 ¢ to 1142.9 ¢	~4.6337 nats	~4.6084 nats
Major 20-mosstep	1142.9 ¢ to 1200.0 ¢	~4.6967 nats	~4.5785 nats
Perfect 21-mosstep	1200.0 ¢	~3.3273 nats	~3.3273 nats

Simple Tunings of 6L 15s

Scale degree	Abbrev.	Basic (2:1) 27edo		Hard (3:1) 33edo		Soft (3:2) 48edo
		Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\27	0.0	0\33	0.0	0\48	0.0
Minor 1-mosdegree	m1md	1\27	44.4	1\33	36.4	2\48	50.0
Major 1-mosdegree	M1md	2\27	88.9	3\33	109.1	3\48	75.0
Minor 2-mosdegree	m2md	2\27	88.9	2\33	72.7	4\48	100.0
Major 2-mosdegree	M2md	3\27	133.3	4\33	145.5	5\48	125.0
Diminished 3-mosdegree	d3md	3\27	133.3	3\33	109.1	6\48	150.0
Perfect 3-mosdegree	P3md	4\27	177.8	5\33	181.8	7\48	175.0
Perfect 4-mosdegree	P4md	5\27	222.2	6\33	218.2	9\48	225.0
Augmented 4-mosdegree	A4md	6\27	266.7	8\33	290.9	10\48	250.0
Minor 5-mosdegree	m5md	6\27	266.7	7\33	254.5	11\48	275.0
Major 5-mosdegree	M5md	7\27	311.1	9\33	327.3	12\48	300.0
Minor 6-mosdegree	m6md	7\27	311.1	8\33	290.9	13\48	325.0
Major 6-mosdegree	M6md	8\27	355.6	10\33	363.6	14\48	350.0
Perfect 7-mosdegree	P7md	9\27	400.0	11\33	400.0	16\48	400.0
Minor 8-mosdegree	m8md	10\27	444.4	12\33	436.4	18\48	450.0
Major 8-mosdegree	M8md	11\27	488.9	14\33	509.1	19\48	475.0
Minor 9-mosdegree	m9md	11\27	488.9	13\33	472.7	20\48	500.0
Major 9-mosdegree	M9md	12\27	533.3	15\33	545.5	21\48	525.0
Diminished 10-mosdegree	d10md	12\27	533.3	14\33	509.1	22\48	550.0
Perfect 10-mosdegree	P10md	13\27	577.8	16\33	581.8	23\48	575.0
Perfect 11-mosdegree	P11md	14\27	622.2	17\33	618.2	25\48	625.0
Augmented 11-mosdegree	A11md	15\27	666.7	19\33	690.9	26\48	650.0
Minor 12-mosdegree	m12md	15\27	666.7	18\33	654.5	27\48	675.0
Major 12-mosdegree	M12md	16\27	711.1	20\33	727.3	28\48	700.0
Minor 13-mosdegree	m13md	16\27	711.1	19\33	690.9	29\48	725.0
Major 13-mosdegree	M13md	17\27	755.6	21\33	763.6	30\48	750.0
Perfect 14-mosdegree	P14md	18\27	800.0	22\33	800.0	32\48	800.0
Minor 15-mosdegree	m15md	19\27	844.4	23\33	836.4	34\48	850.0
Major 15-mosdegree	M15md	20\27	888.9	25\33	909.1	35\48	875.0
Minor 16-mosdegree	m16md	20\27	888.9	24\33	872.7	36\48	900.0
Major 16-mosdegree	M16md	21\27	933.3	26\33	945.5	37\48	925.0
Diminished 17-mosdegree	d17md	21\27	933.3	25\33	909.1	38\48	950.0
Perfect 17-mosdegree	P17md	22\27	977.8	27\33	981.8	39\48	975.0
Perfect 18-mosdegree	P18md	23\27	1022.2	28\33	1018.2	41\48	1025.0
Augmented 18-mosdegree	A18md	24\27	1066.7	30\33	1090.9	42\48	1050.0
Minor 19-mosdegree	m19md	24\27	1066.7	29\33	1054.5	43\48	1075.0
Major 19-mosdegree	M19md	25\27	1111.1	31\33	1127.3	44\48	1100.0
Minor 20-mosdegree	m20md	25\27	1111.1	30\33	1090.9	45\48	1125.0
Major 20-mosdegree	M20md	26\27	1155.6	32\33	1163.6	46\48	1150.0
Perfect 21-mosdegree	P21md	27\27	1200.0	33\33	1200.0	48\48	1200.0

Generator chain of 6L 15s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
8	Augmented 3-mosdegree	A3md	Augmented 10-mosdegree	A10md	Augmented 17-mosdegree	A17md
7	Augmented 0-mosdegree	A0md	Augmented 7-mosdegree	A7md	Augmented 14-mosdegree	A14md
6	Augmented 4-mosdegree	A4md	Augmented 11-mosdegree	A11md	Augmented 18-mosdegree	A18md
5	Major 1-mosdegree	M1md	Major 8-mosdegree	M8md	Major 15-mosdegree	M15md
4	Major 5-mosdegree	M5md	Major 12-mosdegree	M12md	Major 19-mosdegree	M19md
3	Major 2-mosdegree	M2md	Major 9-mosdegree	M9md	Major 16-mosdegree	M16md
2	Major 6-mosdegree	M6md	Major 13-mosdegree	M13md	Major 20-mosdegree	M20md
1	Perfect 3-mosdegree	P3md	Perfect 10-mosdegree	P10md	Perfect 17-mosdegree	P17md
0	Perfect 0-mosdegree Perfect 7-mosdegree	P0md P7md	Perfect 7-mosdegree Perfect 14-mosdegree	P7md P14md	Perfect 14-mosdegree Perfect 21-mosdegree	P14md P21md
−1	Perfect 4-mosdegree	P4md	Perfect 11-mosdegree	P11md	Perfect 18-mosdegree	P18md
−2	Minor 1-mosdegree	m1md	Minor 8-mosdegree	m8md	Minor 15-mosdegree	m15md
−3	Minor 5-mosdegree	m5md	Minor 12-mosdegree	m12md	Minor 19-mosdegree	m19md
−4	Minor 2-mosdegree	m2md	Minor 9-mosdegree	m9md	Minor 16-mosdegree	m16md
−5	Minor 6-mosdegree	m6md	Minor 13-mosdegree	m13md	Minor 20-mosdegree	m20md
−6	Diminished 3-mosdegree	d3md	Diminished 10-mosdegree	d10md	Diminished 17-mosdegree	d17md
−7	Diminished 7-mosdegree	d7md	Diminished 14-mosdegree	d14md	Diminished 21-mosdegree	d21md
−8	Diminished 4-mosdegree	d4md	Diminished 11-mosdegree	d11md	Diminished 18-mosdegree	d18md

Modes of 6L 15s

UDP	Cyclic order	Step pattern	Mode names
18|0(3)	1	LssLsssLssLsssLssLsss	Triple Antilocrian
15|3(3)	4	LsssLssLsssLssLsssLss	Triple Antilydian
12|6(3)	7	sLssLsssLssLsssLssLss	Triple Antionian
9|9(3)	3	sLsssLssLsssLssLsssLs	Triple Antimixolydian
6|12(3)	6	ssLssLsssLssLsssLssLs	Triple Antidorian
3|15(3)	2	ssLsssLssLsssLssLsssL	Triple Antiaeolian
0|18(3)	5	sssLssLsssLssLsssLssL	Triple Antiphrygian

Scale tree

Scale tree and tuning spectrum of 6L 15s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
3\21						171.429	228.571	1:1	1.000	Equalized 6L 15s
					16\111	172.973	227.027	6:5	1.200
				13\90		173.333	226.667	5:4	1.250
					23\159	173.585	226.415	9:7	1.286
			10\69			173.913	226.087	4:3	1.333	Supersoft 6L 15s
					27\186	174.194	225.806	11:8	1.375
				17\117		174.359	225.641	7:5	1.400
					24\165	174.545	225.455	10:7	1.429
		7\48				175.000	225.000	3:2	1.500	Soft 6L 15s
					25\171	175.439	224.561	11:7	1.571
				18\123		175.610	224.390	8:5	1.600
					29\198	175.758	224.242	13:8	1.625
			11\75			176.000	224.000	5:3	1.667	Semisoft 6L 15s
					26\177	176.271	223.729	12:7	1.714
				15\102		176.471	223.529	7:4	1.750
					19\129	176.744	223.256	9:5	1.800
	4\27					177.778	222.222	2:1	2.000	Basic 6L 15s Scales with tunings softer than this are proper
					17\114	178.947	221.053	9:4	2.250
				13\87		179.310	220.690	7:3	2.333
					22\147	179.592	220.408	12:5	2.400
			9\60			180.000	220.000	5:2	2.500	Semihard 6L 15s
					23\153	180.392	219.608	13:5	2.600
				14\93		180.645	219.355	8:3	2.667
					19\126	180.952	219.048	11:4	2.750
		5\33				181.818	218.182	3:1	3.000	Hard 6L 15s
					16\105	182.857	217.143	10:3	3.333
				11\72		183.333	216.667	7:2	3.500
					17\111	183.784	216.216	11:3	3.667
			6\39			184.615	215.385	4:1	4.000	Superhard 6L 15s
					13\84	185.714	214.286	9:2	4.500
				7\45		186.667	213.333	5:1	5.000
					8\51	188.235	211.765	6:1	6.000
1\6						200.000	200.000	1:0	→ ∞	Collapsed 6L 15s