15L 4s
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Step pattern
LLLLsLLLLsLLLLsLLLs
sLLLsLLLLsLLLLsLLLL
Equave
2/1 (1200.0 ¢)
Period
2/1 (1200.0 ¢)
Bright
5\19 to 4\15 (315.8 ¢ to 320.0 ¢)
Dark
11\15 to 14\19 (880.0 ¢ to 884.2 ¢)
Descends from
4L 3s (smitonic)
Ancestor's step ratio range
3:1 to 4:1 (parahard)
Parent
4L 11s
Sister
4L 15s
Daughters
19L 15s, 15L 19s
Neutralized
11L 8s
2-Flought
34L 4s, 15L 23s
Equalized (L:s = 1:1)
5\19 (315.8 ¢)
Supersoft (L:s = 4:3)
19\72 (316.7 ¢)
Soft (L:s = 3:2)
14\53 (317.0 ¢)
Semisoft (L:s = 5:3)
23\87 (317.2 ¢)
Basic (L:s = 2:1)
9\34 (317.6 ¢)
Semihard (L:s = 5:2)
22\83 (318.1 ¢)
Hard (L:s = 3:1)
13\49 (318.4 ¢)
Superhard (L:s = 4:1)
17\64 (318.8 ¢)
Collapsed (L:s = 1:0)
4\15 (320.0 ¢)
| ↖ 14L 3s | ↑ 15L 3s | 16L 3s ↗ |
| ← 14L 4s | 15L 4s | 16L 4s → |
| ↙ 14L 5s | ↓ 15L 5s | 16L 5s ↘ |
┌╥╥╥╥┬╥╥╥╥┬╥╥╥╥┬╥╥╥┬┐ │║║║║│║║║║│║║║║│║║║││ │││││││││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
sLLLsLLLLsLLLLsLLLL
Generator size
TAMNAMS information
Related MOS scales
Equal tunings
15L 4s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 15 large steps and 4 small steps, repeating every octave. 15L 4s is a great-grandchild scale of 4L 3s, expanding it by 12 tones. Generators that produce this scale range from 315.8 ¢ to 320 ¢, or from 880 ¢ to 884.2 ¢.
It is notable as the MOS scale of Kleismic[19].
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 diatonic interval categories.
Intervals
The intervals of 15L 4s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-mosstep and perfect 19-mosstep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.
| 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 63.2 ¢ |
| Major 1-mosstep | M1ms | L | 63.2 ¢ to 80.0 ¢ | |
| 2-mosstep | Minor 2-mosstep | m2ms | L + s | 80.0 ¢ to 126.3 ¢ |
| Major 2-mosstep | M2ms | 2L | 126.3 ¢ to 160.0 ¢ | |
| 3-mosstep | Minor 3-mosstep | m3ms | 2L + s | 160.0 ¢ to 189.5 ¢ |
| Major 3-mosstep | M3ms | 3L | 189.5 ¢ to 240.0 ¢ | |
| 4-mosstep | Minor 4-mosstep | m4ms | 3L + s | 240.0 ¢ to 252.6 ¢ |
| Major 4-mosstep | M4ms | 4L | 252.6 ¢ to 320.0 ¢ | |
| 5-mosstep | Diminished 5-mosstep | d5ms | 3L + 2s | 240.0 ¢ to 315.8 ¢ |
| Perfect 5-mosstep | P5ms | 4L + s | 315.8 ¢ to 320.0 ¢ | |
| 6-mosstep | Minor 6-mosstep | m6ms | 4L + 2s | 320.0 ¢ to 378.9 ¢ |
| Major 6-mosstep | M6ms | 5L + s | 378.9 ¢ to 400.0 ¢ | |
| 7-mosstep | Minor 7-mosstep | m7ms | 5L + 2s | 400.0 ¢ to 442.1 ¢ |
| Major 7-mosstep | M7ms | 6L + s | 442.1 ¢ to 480.0 ¢ | |
| 8-mosstep | Minor 8-mosstep | m8ms | 6L + 2s | 480.0 ¢ to 505.3 ¢ |
| Major 8-mosstep | M8ms | 7L + s | 505.3 ¢ to 560.0 ¢ | |
| 9-mosstep | Minor 9-mosstep | m9ms | 7L + 2s | 560.0 ¢ to 568.4 ¢ |
| Major 9-mosstep | M9ms | 8L + s | 568.4 ¢ to 640.0 ¢ | |
| 10-mosstep | Minor 10-mosstep | m10ms | 7L + 3s | 560.0 ¢ to 631.6 ¢ |
| Major 10-mosstep | M10ms | 8L + 2s | 631.6 ¢ to 640.0 ¢ | |
| 11-mosstep | Minor 11-mosstep | m11ms | 8L + 3s | 640.0 ¢ to 694.7 ¢ |
| Major 11-mosstep | M11ms | 9L + 2s | 694.7 ¢ to 720.0 ¢ | |
| 12-mosstep | Minor 12-mosstep | m12ms | 9L + 3s | 720.0 ¢ to 757.9 ¢ |
| Major 12-mosstep | M12ms | 10L + 2s | 757.9 ¢ to 800.0 ¢ | |
| 13-mosstep | Minor 13-mosstep | m13ms | 10L + 3s | 800.0 ¢ to 821.1 ¢ |
| Major 13-mosstep | M13ms | 11L + 2s | 821.1 ¢ to 880.0 ¢ | |
| 14-mosstep | Perfect 14-mosstep | P14ms | 11L + 3s | 880.0 ¢ to 884.2 ¢ |
| Augmented 14-mosstep | A14ms | 12L + 2s | 884.2 ¢ to 960.0 ¢ | |
| 15-mosstep | Minor 15-mosstep | m15ms | 11L + 4s | 880.0 ¢ to 947.4 ¢ |
| Major 15-mosstep | M15ms | 12L + 3s | 947.4 ¢ to 960.0 ¢ | |
| 16-mosstep | Minor 16-mosstep | m16ms | 12L + 4s | 960.0 ¢ to 1010.5 ¢ |
| Major 16-mosstep | M16ms | 13L + 3s | 1010.5 ¢ to 1040.0 ¢ | |
| 17-mosstep | Minor 17-mosstep | m17ms | 13L + 4s | 1040.0 ¢ to 1073.7 ¢ |
| Major 17-mosstep | M17ms | 14L + 3s | 1073.7 ¢ to 1120.0 ¢ | |
| 18-mosstep | Minor 18-mosstep | m18ms | 14L + 4s | 1120.0 ¢ to 1136.8 ¢ |
| Major 18-mosstep | M18ms | 15L + 3s | 1136.8 ¢ to 1200.0 ¢ | |
| 19-mosstep | Perfect 19-mosstep | P19ms | 15L + 4s | 1200.0 ¢ |
Generator chain
A chain of bright generators, each a perfect 5-mosstep, produces the following scale degrees. A chain of 19 bright generators contains the scale degrees of one of the modes of 15L 4s. Expanding the chain to 34 scale degrees produces the modes of either 19L 15s (for soft-of-basic tunings) or 15L 19s (for hard-of-basic tunings).
| Bright gens | Scale degree | Abbrev. |
|---|---|---|
| 33 | Augmented 13-mosdegree | A13md |
| 32 | Augmented 8-mosdegree | A8md |
| 31 | Augmented 3-mosdegree | A3md |
| 30 | Augmented 17-mosdegree | A17md |
| 29 | Augmented 12-mosdegree | A12md |
| 28 | Augmented 7-mosdegree | A7md |
| 27 | Augmented 2-mosdegree | A2md |
| 26 | Augmented 16-mosdegree | A16md |
| 25 | Augmented 11-mosdegree | A11md |
| 24 | Augmented 6-mosdegree | A6md |
| 23 | Augmented 1-mosdegree | A1md |
| 22 | Augmented 15-mosdegree | A15md |
| 21 | Augmented 10-mosdegree | A10md |
| 20 | Augmented 5-mosdegree | A5md |
| 19 | Augmented 0-mosdegree | A0md |
| 18 | Augmented 14-mosdegree | A14md |
| 17 | Major 9-mosdegree | M9md |
| 16 | Major 4-mosdegree | M4md |
| 15 | Major 18-mosdegree | M18md |
| 14 | Major 13-mosdegree | M13md |
| 13 | Major 8-mosdegree | M8md |
| 12 | Major 3-mosdegree | M3md |
| 11 | Major 17-mosdegree | M17md |
| 10 | Major 12-mosdegree | M12md |
| 9 | Major 7-mosdegree | M7md |
| 8 | Major 2-mosdegree | M2md |
| 7 | Major 16-mosdegree | M16md |
| 6 | Major 11-mosdegree | M11md |
| 5 | Major 6-mosdegree | M6md |
| 4 | Major 1-mosdegree | M1md |
| 3 | Major 15-mosdegree | M15md |
| 2 | Major 10-mosdegree | M10md |
| 1 | Perfect 5-mosdegree | P5md |
| 0 | Perfect 0-mosdegree Perfect 19-mosdegree |
P0md P19md |
| −1 | Perfect 14-mosdegree | P14md |
| −2 | Minor 9-mosdegree | m9md |
| −3 | Minor 4-mosdegree | m4md |
| −4 | Minor 18-mosdegree | m18md |
| −5 | Minor 13-mosdegree | m13md |
| −6 | Minor 8-mosdegree | m8md |
| −7 | Minor 3-mosdegree | m3md |
| −8 | Minor 17-mosdegree | m17md |
| −9 | Minor 12-mosdegree | m12md |
| −10 | Minor 7-mosdegree | m7md |
| −11 | Minor 2-mosdegree | m2md |
| −12 | Minor 16-mosdegree | m16md |
| −13 | Minor 11-mosdegree | m11md |
| −14 | Minor 6-mosdegree | m6md |
| −15 | Minor 1-mosdegree | m1md |
| −16 | Minor 15-mosdegree | m15md |
| −17 | Minor 10-mosdegree | m10md |
| −18 | Diminished 5-mosdegree | d5md |
| −19 | Diminished 19-mosdegree | d19md |
| −20 | Diminished 14-mosdegree | d14md |
| −21 | Diminished 9-mosdegree | d9md |
| −22 | Diminished 4-mosdegree | d4md |
| −23 | Diminished 18-mosdegree | d18md |
| −24 | Diminished 13-mosdegree | d13md |
| −25 | Diminished 8-mosdegree | d8md |
| −26 | Diminished 3-mosdegree | d3md |
| −27 | Diminished 17-mosdegree | d17md |
| −28 | Diminished 12-mosdegree | d12md |
| −29 | Diminished 7-mosdegree | d7md |
| −30 | Diminished 2-mosdegree | d2md |
| −31 | Diminished 16-mosdegree | d16md |
| −32 | Diminished 11-mosdegree | d11md |
| −33 | Diminished 6-mosdegree | d6md |
Modes
| UDP | Cyclic order |
Step pattern |
Scale degree (mosdegree) | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | |||
| 18|0 | 1 | LLLLsLLLLsLLLLsLLLs | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Maj. | Maj. | Maj. | Maj. | Aug. | Maj. | Maj. | Maj. | Maj. | Perf. |
| 17|1 | 6 | LLLLsLLLLsLLLsLLLLs | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. |
| 16|2 | 11 | LLLLsLLLsLLLLsLLLLs | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Min. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. |
| 15|3 | 16 | LLLsLLLLsLLLLsLLLLs | Perf. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Maj. | Min. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. |
| 14|4 | 2 | LLLsLLLLsLLLLsLLLsL | Perf. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Maj. | Min. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Min. | Perf. |
| 13|5 | 7 | LLLsLLLLsLLLsLLLLsL | Perf. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Maj. | Min. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Maj. | Min. | Perf. |
| 12|6 | 12 | LLLsLLLsLLLLsLLLLsL | Perf. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Maj. | Min. | Perf. |
| 11|7 | 17 | LLsLLLLsLLLLsLLLLsL | Perf. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Maj. | Min. | Perf. |
| 10|8 | 3 | LLsLLLLsLLLLsLLLsLL | Perf. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Maj. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Perf. |
| 9|9 | 8 | LLsLLLLsLLLsLLLLsLL | Perf. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Perf. |
| 8|10 | 13 | LLsLLLsLLLLsLLLLsLL | Perf. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Perf. |
| 7|11 | 18 | LsLLLLsLLLLsLLLLsLL | Perf. | Maj. | Min. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Maj. | Min. | Min. | Perf. |
| 6|12 | 4 | LsLLLLsLLLLsLLLsLLL | Perf. | Maj. | Min. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Maj. | Maj. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Perf. |
| 5|13 | 9 | LsLLLLsLLLsLLLLsLLL | Perf. | Maj. | Min. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Maj. | Min. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Perf. |
| 4|14 | 14 | LsLLLsLLLLsLLLLsLLL | Perf. | Maj. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Maj. | Min. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Perf. |
| 3|15 | 19 | sLLLLsLLLLsLLLLsLLL | Perf. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Maj. | Min. | Min. | Min. | Perf. | Maj. | Min. | Min. | Min. | Perf. |
| 2|16 | 5 | sLLLLsLLLLsLLLsLLLL | Perf. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Maj. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Perf. |
| 1|17 | 10 | sLLLLsLLLsLLLLsLLLL | Perf. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Perf. |
| 0|18 | 15 | sLLLsLLLLsLLLLsLLLL | Perf. | Min. | Min. | Min. | Min. | Dim. | Min. | Min. | Min. | Min. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Perf. |
Scale tree
| Generator(edo) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 5\19 | 315.789 | 884.211 | 1:1 | 1.000 | Equalized 15L 4s | |||||
| 29\110 | 316.364 | 883.636 | 6:5 | 1.200 | ||||||
| 24\91 | 316.484 | 883.516 | 5:4 | 1.250 | ||||||
| 43\163 | 316.564 | 883.436 | 9:7 | 1.286 | ||||||
| 19\72 | 316.667 | 883.333 | 4:3 | 1.333 | Supersoft 15L 4s | |||||
| 52\197 | 316.751 | 883.249 | 11:8 | 1.375 | Catakleismic | |||||
| 33\125 | 316.800 | 883.200 | 7:5 | 1.400 | ||||||
| 47\178 | 316.854 | 883.146 | 10:7 | 1.429 | ||||||
| 14\53 | 316.981 | 883.019 | 3:2 | 1.500 | Soft 15L 4s | |||||
| 51\193 | 317.098 | 882.902 | 11:7 | 1.571 | ||||||
| 37\140 | 317.143 | 882.857 | 8:5 | 1.600 | ||||||
| 60\227 | 317.181 | 882.819 | 13:8 | 1.625 | ||||||
| 23\87 | 317.241 | 882.759 | 5:3 | 1.667 | Semisoft 15L 4s | |||||
| 55\208 | 317.308 | 882.692 | 12:7 | 1.714 | ||||||
| 32\121 | 317.355 | 882.645 | 7:4 | 1.750 | ||||||
| 41\155 | 317.419 | 882.581 | 9:5 | 1.800 | ||||||
| 9\34 | 317.647 | 882.353 | 2:1 | 2.000 | Basic 15L 4s Scales with tunings softer than this are proper | |||||
| 40\151 | 317.881 | 882.119 | 9:4 | 2.250 | ||||||
| 31\117 | 317.949 | 882.051 | 7:3 | 2.333 | ||||||
| 53\200 | 318.000 | 882.000 | 12:5 | 2.400 | ||||||
| 22\83 | 318.072 | 881.928 | 5:2 | 2.500 | Semihard 15L 4s | |||||
| 57\215 | 318.140 | 881.860 | 13:5 | 2.600 | ||||||
| 35\132 | 318.182 | 881.818 | 8:3 | 2.667 | ||||||
| 48\181 | 318.232 | 881.768 | 11:4 | 2.750 | ||||||
| 13\49 | 318.367 | 881.633 | 3:1 | 3.000 | Hard 15L 4s | |||||
| 43\162 | 318.519 | 881.481 | 10:3 | 3.333 | ||||||
| 30\113 | 318.584 | 881.416 | 7:2 | 3.500 | ||||||
| 47\177 | 318.644 | 881.356 | 11:3 | 3.667 | ||||||
| 17\64 | 318.750 | 881.250 | 4:1 | 4.000 | Superhard 15L 4s | |||||
| 38\143 | 318.881 | 881.119 | 9:2 | 4.500 | ||||||
| 21\79 | 318.987 | 881.013 | 5:1 | 5.000 | ||||||
| 25\94 | 319.149 | 880.851 | 6:1 | 6.000 | ||||||
| 4\15 | 320.000 | 880.000 | 1:0 | → ∞ | Collapsed 15L 4s | |||||
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