12L 12s
12L 12s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 12 large steps and 12 small steps, with a period of 1 large step and 1 small step that repeats every 100.0 ¢, or 12 times every octave. Generators that produce this scale range from 50 ¢ to 100 ¢, or from 0 ¢ to 50 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.
| ↖ 11L 11s | ↑ 12L 11s | 13L 11s ↗ |
| ← 11L 12s | 12L 12s | 13L 12s → |
| ↙ 11L 13s | ↓ 12L 13s | 13L 13s ↘ |
sLsLsLsLsLsLsLsLsLsLsLsL
It is the 24-note mos scale of the compton, catler, and duodecim temperaments of the compton family, which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of 12edo, and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map 3/2 to 7 steps of 12edo, thus tempering out the Pythagorean comma. Compton uses the 3-limit of 12edo, and adds an independent generator for 5/4 to improve the accuracy of 5-limit harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds 7/4 as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full 7-limit of 12edo, and uses 11/8 as an independent generator.
Using the TAMNAMS extension, it can be named dodecawood, since it has 12 periods per octave, each with one large step and one small step.
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
While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also make sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the 81/80 comma, and inflecting 12edo intervals by this step produces more accurate approximations of 5-limit intervals. In catler, the small step represents 64/63 and 36/35, and inflecting a 12edo minor seventh down by this step gives a more accurate ~7/4, with other ratios involving harmonic 7 also improved, while intervals within the 5-limit are represented as in 12edo.
| 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 50.0 ¢ |
| Major 1-mosstep | M1ms | L | 50.0 ¢ to 100.0 ¢ | |
| 2-mosstep | Perfect 2-mosstep | P2ms | L + s | 100.0 ¢ |
| 3-mosstep | Minor 3-mosstep | m3ms | L + 2s | 100.0 ¢ to 150.0 ¢ |
| Major 3-mosstep | M3ms | 2L + s | 150.0 ¢ to 200.0 ¢ | |
| 4-mosstep | Perfect 4-mosstep | P4ms | 2L + 2s | 200.0 ¢ |
| 5-mosstep | Minor 5-mosstep | m5ms | 2L + 3s | 200.0 ¢ to 250.0 ¢ |
| Major 5-mosstep | M5ms | 3L + 2s | 250.0 ¢ to 300.0 ¢ | |
| 6-mosstep | Perfect 6-mosstep | P6ms | 3L + 3s | 300.0 ¢ |
| 7-mosstep | Minor 7-mosstep | m7ms | 3L + 4s | 300.0 ¢ to 350.0 ¢ |
| Major 7-mosstep | M7ms | 4L + 3s | 350.0 ¢ to 400.0 ¢ | |
| 8-mosstep | Perfect 8-mosstep | P8ms | 4L + 4s | 400.0 ¢ |
| 9-mosstep | Minor 9-mosstep | m9ms | 4L + 5s | 400.0 ¢ to 450.0 ¢ |
| Major 9-mosstep | M9ms | 5L + 4s | 450.0 ¢ to 500.0 ¢ | |
| 10-mosstep | Perfect 10-mosstep | P10ms | 5L + 5s | 500.0 ¢ |
| 11-mosstep | Minor 11-mosstep | m11ms | 5L + 6s | 500.0 ¢ to 550.0 ¢ |
| Major 11-mosstep | M11ms | 6L + 5s | 550.0 ¢ to 600.0 ¢ | |
| 12-mosstep | Perfect 12-mosstep | P12ms | 6L + 6s | 600.0 ¢ |
| 13-mosstep | Minor 13-mosstep | m13ms | 6L + 7s | 600.0 ¢ to 650.0 ¢ |
| Major 13-mosstep | M13ms | 7L + 6s | 650.0 ¢ to 700.0 ¢ | |
| 14-mosstep | Perfect 14-mosstep | P14ms | 7L + 7s | 700.0 ¢ |
| 15-mosstep | Minor 15-mosstep | m15ms | 7L + 8s | 700.0 ¢ to 750.0 ¢ |
| Major 15-mosstep | M15ms | 8L + 7s | 750.0 ¢ to 800.0 ¢ | |
| 16-mosstep | Perfect 16-mosstep | P16ms | 8L + 8s | 800.0 ¢ |
| 17-mosstep | Minor 17-mosstep | m17ms | 8L + 9s | 800.0 ¢ to 850.0 ¢ |
| Major 17-mosstep | M17ms | 9L + 8s | 850.0 ¢ to 900.0 ¢ | |
| 18-mosstep | Perfect 18-mosstep | P18ms | 9L + 9s | 900.0 ¢ |
| 19-mosstep | Minor 19-mosstep | m19ms | 9L + 10s | 900.0 ¢ to 950.0 ¢ |
| Major 19-mosstep | M19ms | 10L + 9s | 950.0 ¢ to 1000.0 ¢ | |
| 20-mosstep | Perfect 20-mosstep | P20ms | 10L + 10s | 1000.0 ¢ |
| 21-mosstep | Minor 21-mosstep | m21ms | 10L + 11s | 1000.0 ¢ to 1050.0 ¢ |
| Major 21-mosstep | M21ms | 11L + 10s | 1050.0 ¢ to 1100.0 ¢ | |
| 22-mosstep | Perfect 22-mosstep | P22ms | 11L + 11s | 1100.0 ¢ |
| 23-mosstep | Minor 23-mosstep | m23ms | 11L + 12s | 1100.0 ¢ to 1150.0 ¢ |
| Major 23-mosstep | M23ms | 12L + 11s | 1150.0 ¢ to 1200.0 ¢ | |
| 24-mosstep | Perfect 24-mosstep | P24ms | 12L + 12s | 1200.0 ¢ |
Modes
Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because 5/4, 7/4, and 11/8 are all closer to the 12edo step above it than the step below it.
| 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 | 20 | 21 | 22 | 23 | 24 | |||
| 12|0(12) | 1 | LsLsLsLsLsLsLsLsLsLsLsLs | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. | Maj. | Perf. |
| 0|12(12) | 2 | sLsLsLsLsLsLsLsLsLsLsLsL | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. | Min. | Perf. |
Scale tree
Softer tunings of 12L 12s are closer to an unequal derivative of 24edo, while harder tunings are closer to two rings of 12edo a comma step apart.
| Generator(edo) | Cents | Step ratio | Comments(always proper) | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 1\24 | 50.000 | 50.000 | 1:1 | 1.000 | Equalized 12L 12s | |||||
| 6\132 | 54.545 | 45.455 | 6:5 | 1.200 | ||||||
| 5\108 | 55.556 | 44.444 | 5:4 | 1.250 | ||||||
| 9\192 | 56.250 | 43.750 | 9:7 | 1.286 | ||||||
| 4\84 | 57.143 | 42.857 | 4:3 | 1.333 | Supersoft 12L 12s | |||||
| 11\228 | 57.895 | 42.105 | 11:8 | 1.375 | ||||||
| 7\144 | 58.333 | 41.667 | 7:5 | 1.400 | ||||||
| 10\204 | 58.824 | 41.176 | 10:7 | 1.429 | ||||||
| 3\60 | 60.000 | 40.000 | 3:2 | 1.500 | Soft 12L 12s Duodecim | |||||
| 11\216 | 61.111 | 38.889 | 11:7 | 1.571 | ||||||
| 8\156 | 61.538 | 38.462 | 8:5 | 1.600 | ||||||
| 13\252 | 61.905 | 38.095 | 13:8 | 1.625 | ||||||
| 5\96 | 62.500 | 37.500 | 5:3 | 1.667 | Semisoft 12L 12s | |||||
| 12\228 | 63.158 | 36.842 | 12:7 | 1.714 | ||||||
| 7\132 | 63.636 | 36.364 | 7:4 | 1.750 | ||||||
| 9\168 | 64.286 | 35.714 | 9:5 | 1.800 | ||||||
| 2\36 | 66.667 | 33.333 | 2:1 | 2.000 | Basic 12L 12s | |||||
| 9\156 | 69.231 | 30.769 | 9:4 | 2.250 | ||||||
| 7\120 | 70.000 | 30.000 | 7:3 | 2.333 | ||||||
| 12\204 | 70.588 | 29.412 | 12:5 | 2.400 | ||||||
| 5\84 | 71.429 | 28.571 | 5:2 | 2.500 | Semihard 12L 12s Catler | |||||
| 13\216 | 72.222 | 27.778 | 13:5 | 2.600 | ||||||
| 8\132 | 72.727 | 27.273 | 8:3 | 2.667 | ||||||
| 11\180 | 73.333 | 26.667 | 11:4 | 2.750 | ||||||
| 3\48 | 75.000 | 25.000 | 3:1 | 3.000 | Hard 12L 12s | |||||
| 10\156 | 76.923 | 23.077 | 10:3 | 3.333 | ||||||
| 7\108 | 77.778 | 22.222 | 7:2 | 3.500 | ||||||
| 11\168 | 78.571 | 21.429 | 11:3 | 3.667 | ||||||
| 4\60 | 80.000 | 20.000 | 4:1 | 4.000 | Superhard 12L 12s | |||||
| 9\132 | 81.818 | 18.182 | 9:2 | 4.500 | ||||||
| 5\72 | 83.333 | 16.667 | 5:1 | 5.000 | ||||||
| 6\84 | 85.714 | 14.286 | 6:1 | 6.000 | Compton | |||||
| 1\12 | 100.000 | 0.000 | 1:0 | → ∞ | Collapsed 12L 12s | |||||