4L 7s

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Scale structure
Step pattern	LsLssLssLss ssLssLssLsL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	8\11 to 3\4 (872.7 ¢ to 900.0 ¢)
Dark	1\4 to 3\11 (300.0 ¢ to 327.3 ¢)
TAMNAMS information
Related to	4L 3s (smitonic)
With tunings	2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent	4L 3s
Sister	7L 4s
Daughters	11L 4s, 4L 11s
Neutralized	8L 3s
2-Flought	15L 7s, 4L 18s
Equal tunings
Equalized (L:s = 1:1)	8\11 (872.7 ¢)
Supersoft (L:s = 4:3)	27\37 (875.7 ¢)
Soft (L:s = 3:2)	19\26 (876.9 ¢)
Semisoft (L:s = 5:3)	30\41 (878.0 ¢)
Basic (L:s = 2:1)	11\15 (880.0 ¢)
Semihard (L:s = 5:2)	25\34 (882.4 ¢)
Hard (L:s = 3:1)	14\19 (884.2 ¢)
Superhard (L:s = 4:1)	17\23 (887.0 ¢)
Collapsed (L:s = 1:0)	3\4 (900.0 ¢)

4L 7s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 7 small steps, repeating every octave. 4L 7s is a child scale of 4L 3s, expanding it by 4 tones. Generators that produce this scale range from 872.7 ¢ to 900 ¢, or from 300 ¢ to 327.3 ¢. One of the harmonic entropy minimums in this range is Kleismic/Hanson.

Name

TAMNAMS formerly used the name kleistonic for the name of this scale (prefix klei-). Other names include p-chro smitonic or smipechromic.

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 4L 7s

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 109.1 ¢
	Major 1-mosstep	M1ms	L	109.1 ¢ to 300.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 218.2 ¢
	Major 2-mosstep	M2ms	L + s	218.2 ¢ to 300.0 ¢
3-mosstep	Perfect 3-mosstep	P3ms	L + 2s	300.0 ¢ to 327.3 ¢
	Augmented 3-mosstep	A3ms	2L + s	327.3 ¢ to 600.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	L + 3s	300.0 ¢ to 436.4 ¢
	Major 4-mosstep	M4ms	2L + 2s	436.4 ¢ to 600.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	L + 4s	300.0 ¢ to 545.5 ¢
	Major 5-mosstep	M5ms	2L + 3s	545.5 ¢ to 600.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	2L + 4s	600.0 ¢ to 654.5 ¢
	Major 6-mosstep	M6ms	3L + 3s	654.5 ¢ to 900.0 ¢
7-mosstep	Minor 7-mosstep	m7ms	2L + 5s	600.0 ¢ to 763.6 ¢
	Major 7-mosstep	M7ms	3L + 4s	763.6 ¢ to 900.0 ¢
8-mosstep	Diminished 8-mosstep	d8ms	2L + 6s	600.0 ¢ to 872.7 ¢
	Perfect 8-mosstep	P8ms	3L + 5s	872.7 ¢ to 900.0 ¢
9-mosstep	Minor 9-mosstep	m9ms	3L + 6s	900.0 ¢ to 981.8 ¢
	Major 9-mosstep	M9ms	4L + 5s	981.8 ¢ to 1200.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	3L + 7s	900.0 ¢ to 1090.9 ¢
	Major 10-mosstep	M10ms	4L + 6s	1090.9 ¢ to 1200.0 ¢
11-mosstep	Perfect 11-mosstep	P11ms	4L + 7s	1200.0 ¢

Generator chain

Generator chain of 4L 7s

Bright gens	Scale degree	Abbrev.
14	Augmented 2-mosdegree	A2md
13	Augmented 5-mosdegree	A5md
12	Augmented 8-mosdegree	A8md
11	Augmented 0-mosdegree	A0md
10	Augmented 3-mosdegree	A3md
9	Major 6-mosdegree	M6md
8	Major 9-mosdegree	M9md
7	Major 1-mosdegree	M1md
6	Major 4-mosdegree	M4md
5	Major 7-mosdegree	M7md
4	Major 10-mosdegree	M10md
3	Major 2-mosdegree	M2md
2	Major 5-mosdegree	M5md
1	Perfect 8-mosdegree	P8md
0	Perfect 0-mosdegree Perfect 11-mosdegree	P0md P11md
−1	Perfect 3-mosdegree	P3md
−2	Minor 6-mosdegree	m6md
−3	Minor 9-mosdegree	m9md
−4	Minor 1-mosdegree	m1md
−5	Minor 4-mosdegree	m4md
−6	Minor 7-mosdegree	m7md
−7	Minor 10-mosdegree	m10md
−8	Minor 2-mosdegree	m2md
−9	Minor 5-mosdegree	m5md
−10	Diminished 8-mosdegree	d8md
−11	Diminished 11-mosdegree	d11md
−12	Diminished 3-mosdegree	d3md
−13	Diminished 6-mosdegree	d6md
−14	Diminished 9-mosdegree	d9md

Modes

Scale degrees of the modes of 4L 7s

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.

Tuning ranges

Soft range

The soft range for tunings of 4L 7s encompasses parasoft and hyposoft tunings. This implies step ratios smaller than 2/1, meaning a generator sharper than 4\15 = 320 ¢.

This is the range associated with extensions of Orgone[7]. The small step is recognizable as a near diatonic semitone, while the large step is in the ambiguous area of neutral seconds.

Soft edos include 15edo and 26edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various soft tunings:

	15edo (basic)	26edo (soft)	Some JI approximations
generator (g)	4\15, 320.00	7\26, 323.08	77/64, 6/5
L (octave - 3g)	2\15, 160.00	3\26, 138.46	12/11, 13/12
s (4g - octave)	1\15, 80.00	2\19, 92.31	21/20, 22/21, 20/19

Hypohard

Hypohard tunings of 4L 7s have step ratios between 2/1 and 3/1, implying a generator sharper than 5\19 = 315.79 ¢ and flatter than 4\15 = 320 ¢.

This range represents one of the harmonic entropy minimums, where 6 generators make a just diatonic fifth (3/2), an octave above. This is the range associated with the eponymous Kleismic (aka Hanson) temperament and its extensions.

Hypohard edos include 15edo, 19edo, and 34edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various hypohard tunings:

	15edo (basic)	19edo (hard)	34edo (semihard)	Some JI approximations
generator (g)	4\15, 320.00	5\19, 315.79	9\34, 317.65	6/5
L (octave − 3g)	2\15, 160.00	3\19, 189.47	5\34, 176.47	10/9, 11/10 (in 15edo)
s (4g − octave)	1\15, 80.00	1\19, 63.16	2\34, 70.59	25/24, 26/25 (in better kleismic tunings)

Parahard

Parahard tunings of 4L 7s have step ratios between 3/1 and 4/1, implying a generator sharper than 6\23 = 313.04¢ and flatter than 5\19 = 315.79¢.

The minor third is at its purest here, but the resulting scales tend to approximate intervals that employ a much higher limit harmony, especially in the case of the superhard 23edo. However, the large step is recognizable as a regular diatonic whole step, approximating both 10/9 and 9/8, while the small step is a slightly sharp of a quarter tone.

Parahard edos include 19edo, 23edo, and 42edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various parahard tunings:

	19edo (hard)	23edo (superhard)	42edo (parahard)	Some JI approximations
generator (g)	5\19, 315.79	6\23, 313.04	11\42, 314.29	6/5
L (octave − 3g)	3\19, 189.47	4\23, 208.70	7\42, 200.00	10/9, 9/8
s (4g − octave)	1\19, 63.16	1\23, 52.17	2\42, 57.14	28/27, 33/32

Hyperhard

Hyperhard tunings of 4L 7s have step ratios between 4/1 and 6/1, implying a generator sharper than 8\31 = 309.68¢ and flatter than 6\23 = 313.04¢.

The temperament known as Myna (a pun on "minor third") resides here, as this is the range where 10 generators make a just diatonic fifth (3/2), two octaves above. These scales are stacked with simple intervals, but are melodically difficult due to the extreme step size disparity, where the small step is generally flat of a quarter tone.

Hyperhard edos include 23edo, 31edo, and 27edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various hyperhard tunings:

	23edo (superhard)	31edo (extrahard)	27edo (pentahard)	Some JI approximations
generator (g)	6\23, 313.04	8\31, 309.68	7\27, 311.11	6/5
L (octave − 3g)	4\23, 208.70	6\31, 232.26	5\27, 222.22	8/7, 9/8
s (4g − octave)	1\23, 52.17	1\31, 38.71	1\27, 44.44	36/35, 45/44

Scales

• Oregon11
• Orgone11
• Magicaltet11
• Cata11
• Starlingtet11
• Myna11

Scale tree

Scale tree and tuning spectrum of 4L 7s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
8\11						872.727	327.273	1:1	1.000	Equalized 4L 7s
					43\59	874.576	325.424	6:5	1.200	Oregon
				35\48		875.000	325.000	5:4	1.250
					62\85	875.294	324.706	9:7	1.286
			27\37			875.676	324.324	4:3	1.333	Supersoft 4L 7s
					73\100	876.000	324.000	11:8	1.375
				46\63		876.190	323.810	7:5	1.400
					65\89	876.404	323.596	10:7	1.429	Orgone
		19\26				876.923	323.077	3:2	1.500	Soft 4L 7s
					68\93	877.419	322.581	11:7	1.571	Magicaltet
				49\67		877.612	322.388	8:5	1.600
					79\108	877.778	322.222	13:8	1.625	Golden superklesimic
			30\41			878.049	321.951	5:3	1.667	Semisoft 4L 7s Superkleismic
					71\97	878.351	321.649	12:7	1.714
				41\56		878.571	321.429	7:4	1.750
					52\71	878.873	321.127	9:5	1.800
	11\15					880.000	320.000	2:1	2.000	Basic 4L 7s Scales with tunings softer than this are proper
					47\64	881.250	318.750	9:4	2.250
				36\49		881.633	318.367	7:3	2.333	Catalan
					61\83	881.928	318.072	12:5	2.400
			25\34			882.353	317.647	5:2	2.500	Semihard 4L 7s
					64\87	882.759	317.241	13:5	2.600	Countercata
				39\53		883.019	316.981	8:3	2.667	Hanson/cata
					53\72	883.333	316.667	11:4	2.750	Catakleismic
		14\19				884.211	315.789	3:1	3.000	Hard 4L 7s
					45\61	885.246	314.754	10:3	3.333	Parakleismic
				31\42		885.714	314.286	7:2	3.500
					48\65	886.154	313.846	11:3	3.667
			17\23			886.957	313.043	4:1	4.000	Superhard 4L 7s
					37\50	888.000	312.000	9:2	4.500	Oolong
				20\27		888.889	311.111	5:1	5.000	Starlingtet
					23\31	890.323	309.677	6:1	6.000	Myna
3\4						900.000	300.000	1:0	→ ∞	Collapsed 4L 7s

Gallery

Cheat sheet for 19EDO, a hard tuning for 4L 7s (or kleistonic).