3L 6s

↖ 2L 5s	↑ 3L 5s	4L 5s ↗
← 2L 6s	3L 6s	4L 6s →
↙ 2L 7s	↓ 3L 7s	4L 7s ↘

┌╥┬┬╥┬┬╥┬┬┐ │║││║││║│││ │││││││││││ └┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LssLssLss ssLssLssL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	2\9 to 1\3 (266.7 ¢ to 400.0 ¢)
Dark	0\3 to 1\9 (0.0 ¢ to 133.3 ¢)
TAMNAMS information
Name	tcherepnin
Prefix	cher-
Abbrev.	ch
Related MOS scales
Parent	3L 3s
Sister	6L 3s
Daughters	9L 3s, 3L 9s
Neutralized	6L 3s
2-Flought	12L 6s, 3L 15s
Equal tunings
Equalized (L:s = 1:1)	2\9 (266.7 ¢)
Supersoft (L:s = 4:3)	7\30 (280.0 ¢)
Soft (L:s = 3:2)	5\21 (285.7 ¢)
Semisoft (L:s = 5:3)	8\33 (290.9 ¢)
Basic (L:s = 2:1)	3\12 (300.0 ¢)
Semihard (L:s = 5:2)	7\27 (311.1 ¢)
Hard (L:s = 3:1)	4\15 (320.0 ¢)
Superhard (L:s = 4:1)	5\18 (333.3 ¢)
Collapsed (L:s = 1:0)	1\3 (400.0 ¢)

3L 6s, named tcherepnin in TAMNAMS, is a 2/1-equivalent (octave-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 400.0 ¢, or 3 times every octave. Generators that produce this scale range from 266.7 ¢ to 400 ¢, or from 0 ¢ to 133.3 ¢.

Name

TAMNAMS suggests the temperaement-agnostic name tcherepnin for this scale. This is named after Alexander Tcherepnin's 9-note scale which, in 12edo, is notated as C Db Eb E F G Ab A B C, forming the step pattern of HWHHWHHWH in standard notation (assuming a root of C).

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 6s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-cherstep	Perfect 0-cherstep	P0chs	0	0.0 ¢
1-cherstep	Perfect 1-cherstep	P1chs	s	0.0 ¢ to 133.3 ¢
	Augmented 1-cherstep	A1chs	L	133.3 ¢ to 400.0 ¢
2-cherstep	Diminished 2-cherstep	d2chs	2s	0.0 ¢ to 266.7 ¢
	Perfect 2-cherstep	P2chs	L + s	266.7 ¢ to 400.0 ¢
3-cherstep	Perfect 3-cherstep	P3chs	L + 2s	400.0 ¢
4-cherstep	Perfect 4-cherstep	P4chs	L + 3s	400.0 ¢ to 533.3 ¢
	Augmented 4-cherstep	A4chs	2L + 2s	533.3 ¢ to 800.0 ¢
5-cherstep	Diminished 5-cherstep	d5chs	L + 4s	400.0 ¢ to 666.7 ¢
	Perfect 5-cherstep	P5chs	2L + 3s	666.7 ¢ to 800.0 ¢
6-cherstep	Perfect 6-cherstep	P6chs	2L + 4s	800.0 ¢
7-cherstep	Perfect 7-cherstep	P7chs	2L + 5s	800.0 ¢ to 933.3 ¢
	Augmented 7-cherstep	A7chs	3L + 4s	933.3 ¢ to 1200.0 ¢
8-cherstep	Diminished 8-cherstep	d8chs	2L + 6s	800.0 ¢ to 1066.7 ¢
	Perfect 8-cherstep	P8chs	3L + 5s	1066.7 ¢ to 1200.0 ¢
9-cherstep	Perfect 9-cherstep	P9chs	3L + 6s	1200.0 ¢

Generator chain

Generator chain of 3L 6s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
3	Augmented 0-cherdegree	A0chd	Augmented 3-cherdegree	A3chd	Augmented 6-cherdegree	A6chd
2	Augmented 1-cherdegree	A1chd	Augmented 4-cherdegree	A4chd	Augmented 7-cherdegree	A7chd
1	Perfect 2-cherdegree	P2chd	Perfect 5-cherdegree	P5chd	Perfect 8-cherdegree	P8chd
0	Perfect 0-cherdegree Perfect 3-cherdegree	P0chd P3chd	Perfect 3-cherdegree Perfect 6-cherdegree	P3chd P6chd	Perfect 6-cherdegree Perfect 9-cherdegree	P6chd P9chd
−1	Perfect 1-cherdegree	P1chd	Perfect 4-cherdegree	P4chd	Perfect 7-cherdegree	P7chd
−2	Diminished 2-cherdegree	d2chd	Diminished 5-cherdegree	d5chd	Diminished 8-cherdegree	d8chd
−3	Diminished 3-cherdegree	d3chd	Diminished 6-cherdegree	d6chd	Diminished 9-cherdegree	d9chd

Modes

Scale degrees of the modes of 3L 6s

UDP	Cyclic order	Step pattern	Scale degree (cherdegree)
			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

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
2\9						266.667	133.333	1:1	1.000	Equalized 3L 6s
					11\48	275.000	125.000	6:5	1.200
				9\39		276.923	123.077	5:4	1.250
					16\69	278.261	121.739	9:7	1.286
			7\30			280.000	120.000	4:3	1.333	Supersoft 3L 6s
					19\81	281.481	118.519	11:8	1.375
				12\51		282.353	117.647	7:5	1.400
					17\72	283.333	116.667	10:7	1.429
		5\21				285.714	114.286	3:2	1.500	Soft 3L 6s
					18\75	288.000	112.000	11:7	1.571
				13\54		288.889	111.111	8:5	1.600
					21\87	289.655	110.345	13:8	1.625
			8\33			290.909	109.091	5:3	1.667	Semisoft 3L 6s
					19\78	292.308	107.692	12:7	1.714
				11\45		293.333	106.667	7:4	1.750
					14\57	294.737	105.263	9:5	1.800	August
	3\12					300.000	100.000	2:1	2.000	Basic 3L 6s Scales with tunings softer than this are proper
					13\51	305.882	94.118	9:4	2.250
				10\39		307.692	92.308	7:3	2.333
					17\66	309.091	90.909	12:5	2.400	Augene
			7\27			311.111	88.889	5:2	2.500	Semihard 3L 6s
					18\69	313.043	86.957	13:5	2.600
				11\42		314.286	85.714	8:3	2.667
					15\57	315.789	84.211	11:4	2.750	August
		4\15				320.000	80.000	3:1	3.000	Hard 3L 6s
					13\48	325.000	75.000	10:3	3.333	Inflated
				9\33		327.273	72.727	7:2	3.500
					14\51	329.412	70.588	11:3	3.667
			5\18			333.333	66.667	4:1	4.000	Superhard 3L 6s
					11\39	338.462	61.538	9:2	4.500
				6\21		342.857	57.143	5:1	5.000
					7\24	350.000	50.000	6:1	6.000	Hemiug, hemiaug ↓
1\3						400.000	0.000	1:0	→ ∞	Collapsed 3L 6s

External links

• Piano Encyclopedia, on Tcherepnin's 9-note scale: https://pianoencyclopedia.com/scales/tcherepnin-nine-note-mode-i/