9L 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
Descends from
3L 6s (tcherepnin)
Ancestor's step ratio range
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¢)
| ↖ 8L 2s | ↑ 9L 2s | 10L 2s ↗ |
| ← 8L 3s | 9L 3s | 10L 3s → |
| ↙ 8L 4s | ↓ 9L 4s | 10L 4s ↘ |
┌╥╥╥┬╥╥╥┬╥╥╥┬┐ │║║║│║║║│║║║││ ││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┘
sLLLsLLLsLLL
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.
Modes
| UDP | Cyclic order |
Step pattern |
|---|---|---|
| 9|0(3) | 1 | LLLsLLLsLLLs |
| 6|3(3) | 2 | LLsLLLsLLLsL |
| 3|6(3) | 3 | LsLLLsLLLsLL |
| 0|9(3) | 4 | sLLLsLLLsLLL |
Intervals
| 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¢ |
Scale tree
| 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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