9L 3s

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← 8L 3s 9L 3s 10L 3s →
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Scale structure
Step pattern LLLsLLLsLLLs
sLLLsLLLsLLL
Equave 2/1 (1200.0 ¢)
Period 1\3 (400.0 ¢)
Generator size
Bright 1\12 to 1\9 (100.0 ¢ to 133.3 ¢)
Dark 2\9 to 3\12 (266.7 ¢ to 300.0 ¢)
TAMNAMS information
Related to 3L 6s (tcherepnin)
With tunings 1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent 3L 6s
Sister 3L 9s
Daughters 12L 9s, 9L 12s
Neutralized 6L 6s
2-Flought 21L 3s, 9L 15s
Equal tunings
Equalized (L:s = 1:1) 1\12 (100.0 ¢)
Supersoft (L:s = 4:3) 4\45 (106.7 ¢)
Soft (L:s = 3:2) 3\33 (109.1 ¢)
Semisoft (L:s = 5:3) 5\54 (111.1 ¢)
Basic (L:s = 2:1) 2\21 (114.3 ¢)
Semihard (L:s = 5:2) 5\51 (117.6 ¢)
Hard (L:s = 3:1) 3\30 (120.0 ¢)
Superhard (L:s = 4:1) 4\39 (123.1 ¢)
Collapsed (L:s = 1:0) 1\9 (133.3 ¢)

9L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 9 large steps and 3 small steps, with a period of 3 large steps and 1 small step that repeats every 400.0 ¢, or 3 times every octave. 9L 3s is a child scale of 3L 6s, expanding it by 3 tones. Generators that produce this scale range from 100 ¢ to 133.3 ¢, or from 266.7 ¢ to 300 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

9L 3s represents the chromatic scale of August temperament and its higher-limit extensions.

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 9L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Diminished 1-mosstep d1ms s 0.0 ¢ to 100.0 ¢
Perfect 1-mosstep P1ms L 100.0 ¢ to 133.3 ¢
2-mosstep Minor 2-mosstep m2ms L + s 133.3 ¢ to 200.0 ¢
Major 2-mosstep M2ms 2L 200.0 ¢ to 266.7 ¢
3-mosstep Perfect 3-mosstep P3ms 2L + s 266.7 ¢ to 300.0 ¢
Augmented 3-mosstep A3ms 3L 300.0 ¢ to 400.0 ¢
4-mosstep Perfect 4-mosstep P4ms 3L + s 400.0 ¢
5-mosstep Diminished 5-mosstep d5ms 3L + 2s 400.0 ¢ to 500.0 ¢
Perfect 5-mosstep P5ms 4L + s 500.0 ¢ to 533.3 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 533.3 ¢ to 600.0 ¢
Major 6-mosstep M6ms 5L + s 600.0 ¢ to 666.7 ¢
7-mosstep Perfect 7-mosstep P7ms 5L + 2s 666.7 ¢ to 700.0 ¢
Augmented 7-mosstep A7ms 6L + s 700.0 ¢ to 800.0 ¢
8-mosstep Perfect 8-mosstep P8ms 6L + 2s 800.0 ¢
9-mosstep Diminished 9-mosstep d9ms 6L + 3s 800.0 ¢ to 900.0 ¢
Perfect 9-mosstep P9ms 7L + 2s 900.0 ¢ to 933.3 ¢
10-mosstep Minor 10-mosstep m10ms 7L + 3s 933.3 ¢ to 1000.0 ¢
Major 10-mosstep M10ms 8L + 2s 1000.0 ¢ to 1066.7 ¢
11-mosstep Perfect 11-mosstep P11ms 8L + 3s 1066.7 ¢ to 1100.0 ¢
Augmented 11-mosstep A11ms 9L + 2s 1100.0 ¢ to 1200.0 ¢
12-mosstep Perfect 12-mosstep P12ms 9L + 3s 1200.0 ¢

Generator chain

Generator chain of 9L 3s
Bright gens Scale degree Abbrev. Scale degree Abbrev. Scale degree Abbrev.
6 Augmented 2-mosdegree A2md Augmented 6-mosdegree A6md Augmented 10-mosdegree A10md
5 Augmented 1-mosdegree A1md Augmented 5-mosdegree A5md Augmented 9-mosdegree A9md
4 Augmented 0-mosdegree A0md Augmented 4-mosdegree A4md Augmented 8-mosdegree A8md
3 Augmented 3-mosdegree A3md Augmented 7-mosdegree A7md Augmented 11-mosdegree A11md
2 Major 2-mosdegree M2md Major 6-mosdegree M6md Major 10-mosdegree M10md
1 Perfect 1-mosdegree P1md Perfect 5-mosdegree P5md Perfect 9-mosdegree P9md
0 Perfect 0-mosdegree
Perfect 4-mosdegree		 P0md
P4md		 Perfect 4-mosdegree
Perfect 8-mosdegree		 P4md
P8md		 Perfect 8-mosdegree
Perfect 12-mosdegree		 P8md
P12md
−1 Perfect 3-mosdegree P3md Perfect 7-mosdegree P7md Perfect 11-mosdegree P11md
−2 Minor 2-mosdegree m2md Minor 6-mosdegree m6md Minor 10-mosdegree m10md
−3 Diminished 1-mosdegree d1md Diminished 5-mosdegree d5md Diminished 9-mosdegree d9md
−4 Diminished 4-mosdegree d4md Diminished 8-mosdegree d8md Diminished 12-mosdegree d12md
−5 Diminished 3-mosdegree d3md Diminished 7-mosdegree d7md Diminished 11-mosdegree d11md
−6 Diminished 2-mosdegree d2md Diminished 6-mosdegree d6md Diminished 10-mosdegree d10md

Modes

Scale degrees of the modes of 9L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12
9|0(3) 1 LLLsLLLsLLLs Perf. Perf. Maj. Aug. Perf. Perf. Maj. Aug. Perf. Perf. Maj. Aug. Perf.
6|3(3) 2 LLsLLLsLLLsL Perf. Perf. Maj. Perf. Perf. Perf. Maj. Perf. Perf. Perf. Maj. Perf. Perf.
3|6(3) 3 LsLLLsLLLsLL Perf. Perf. Min. Perf. Perf. Perf. Min. Perf. Perf. Perf. Min. Perf. Perf.
0|9(3) 4 sLLLsLLLsLLL Perf. Dim. Min. Perf. Perf. Dim. Min. Perf. Perf. Dim. Min. Perf. Perf.

Scale tree

Scale tree and tuning spectrum of 9L 3s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\12 100.000 300.000 1:1 1.000 Equalized 9L 3s
6\69 104.348 295.652 6:5 1.200
5\57 105.263 294.737 5:4 1.250
9\102 105.882 294.118 9:7 1.286
4\45 106.667 293.333 4:3 1.333 Supersoft 9L 3s
11\123 107.317 292.683 11:8 1.375
7\78 107.692 292.308 7:5 1.400 August
10\111 108.108 291.892 10:7 1.429
3\33 109.091 290.909 3:2 1.500 Soft 9L 3s
11\120 110.000 290.000 11:7 1.571
8\87 110.345 289.655 8:5 1.600
13\141 110.638 289.362 13:8 1.625
5\54 111.111 288.889 5:3 1.667 Semisoft 9L 3s
12\129 111.628 288.372 12:7 1.714
7\75 112.000 288.000 7:4 1.750
9\96 112.500 287.500 9:5 1.800 11-limit August
2\21 114.286 285.714 2:1 2.000 Basic 9L 3s
13-limit August
9\93 116.129 283.871 9:4 2.250
7\72 116.667 283.333 7:3 2.333
12\123 117.073 282.927 12:5 2.400
5\51 117.647 282.353 5:2 2.500 Semihard 9L 3s
13\132 118.182 281.818 13:5 2.600
8\81 118.519 281.481 8:3 2.667
11\111 118.919 281.081 11:4 2.750
3\30 120.000 280.000 3:1 3.000 Hard 9L 3s
10\99 121.212 278.788 10:3 3.333
7\69 121.739 278.261 7:2 3.500
11\108 122.222 277.778 11:3 3.667
4\39 123.077 276.923 4:1 4.000 Superhard 9L 3s
9\87 124.138 275.862 9:2 4.500
5\48 125.000 275.000 5:1 5.000
6\57 126.316 273.684 6:1 6.000
1\9 133.333 266.667 1:0 → ∞ Collapsed 9L 3s
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