7L 3s
| ↖ 6L 2s | ↑7L 2s | 8L 2s ↗ |
| ← 6L 3s | 7L 3s | 8L 3s → |
| ↙ 6L 4s | ↓7L 4s | 8L 4s ↘ |
┌╥╥╥┬╥╥┬╥╥┬┐ │║║║│║║│║║││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sLLsLLsLLL
7L 3s, named dicoid in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 840¢ to 857.1¢, or from 342.9¢ to 360¢. 7L 3s represents temperaments such as mohajira/mohaha/mohoho, among others, whose generators are around a neutral 3rd. The seven and ten-note forms of mohaha/mohoho form a chromatic pair.
Name
TAMNAMS suggests the temperament-agnostic name dicoid (from dicot, an exotemperament) for the name of this scale.
Intervals and degrees
- This article assumes TAMNAMS for naming step ratios, intervals, and scale degrees.
Under TAMNAMS, names for this scale's degrees, the positions of the scale's tones, are called mosdegrees, or dicodegrees. Its intervals, the pitch difference between any two tones, are based on the number of large and small steps between them and are thus called mossteps, or dicosteps. Both mosdegrees and mossteps, are 0-indexed, as opposed to 1-indexed; such names, such as mos-1st instead of 0-mosstep, are discouraged for non-diatonic MOS scales.
| Intervals | Steps subtended | Range in cents | Average of HE (from HE Calc) |
Min of HE | ||
|---|---|---|---|---|---|---|
| Generic[1] | Specific[2] | Abbrev.[3] | ||||
| 0-dicostep | Perfect 0-dicostep | P0ms | 0 | 0.0¢ | ~2.4654 nats | ~2.4654 nats |
| 1-dicostep | Minor 1-dicostep | m1ms | s | 0.0¢ to 120.0¢ | ~4.7342 nats | ~4.6803 nats |
| Major 1-dicostep | M1ms | L | 120.0¢ to 171.4¢ | ~4.6306 nats | ~4.6188 nats | |
| 2-dicostep | Minor 2-dicostep | m2ms | L + s | 171.4¢ to 240.0¢ | ~4.5850 nats | ~4.5840 nats |
| Major 2-dicostep | M2ms | 2L | 240.0¢ to 342.9¢ | ~4.5661 nats | ~4.5425 nats | |
| 3-dicostep | Perfect 3-dicostep | P3ms | 2L + s | 342.9¢ to 360.0¢ | ~4.6180 nats | ~4.6140 nats |
| Augmented 3-dicostep | A3ms | 3L | 360.0¢ to 514.3¢ | ~4.5740 nats | ~4.4854 nats | |
| 4-dicostep | Minor 4-dicostep | m4ms | 2L + 2s | 342.9¢ to 480.0¢ | ~4.5736 nats | ~4.4846 nats |
| Major 4-dicostep | M4ms | 3L + s | 480.0¢ to 514.3¢ | ~4.3933 nats | ~4.3665 nats | |
| 5-dicostep | Minor 5-dicostep | m5ms | 3L + 2s | 514.3¢ to 600.0¢ | ~4.5951 nats | ~4.5596 nats |
| Major 5-dicostep | M5ms | 4L + s | 600.0¢ to 685.7¢ | ~4.6198 nats | ~4.5980 nats | |
| 6-dicostep | Minor 6-dicostep | m6ms | 4L + 2s | 685.7¢ to 720.0¢ | ~4.1626 nats | ~4.1225 nats |
| Major 6-dicostep | M6ms | 5L + s | 720.0¢ to 857.1¢ | ~4.6108 nats | ~4.5711 nats | |
| 7-dicostep | Diminished 7-dicostep | d7ms | 4L + 3s | 685.7¢ to 840.0¢ | ~4.6078 nats | ~4.5720 nats |
| Perfect 7-dicostep | P7ms | 5L + 2s | 840.0¢ to 857.1¢ | ~4.6136 nats | ~4.6107 nats | |
| 8-dicostep | Minor 8-dicostep | m8ms | 5L + 3s | 857.1¢ to 960.0¢ | ~4.5618 nats | ~4.4308 nats |
| Major 8-dicostep | M8ms | 6L + 2s | 960.0¢ to 1028.6¢ | ~4.5638 nats | ~4.5313 nats | |
| 9-dicostep | Minor 9-dicostep | m9ms | 6L + 3s | 1028.6¢ to 1080.0¢ | ~4.6075 nats | ~4.6058 nats |
| Major 9-dicostep | M9ms | 7L + 2s | 1080.0¢ to 1200.0¢ | ~4.6698 nats | ~4.6202 nats | |
| 10-dicostep | Perfect 10-dicostep | P10ms | 7L + 3s | 1200.0¢ | ~3.3273 nats | ~3.3273 nats |
- Generic intervals are denoted solely by the number of steps they subtend.
- Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
- Abbreviations can be further shortened to 'ms' if context allows.
Modes
| UDP | Rotational Order | Step pattern | Scale degree (dicodegree) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
| 9|0 | 1 | LLLsLLsLLs | Perf. | Maj. | Maj. | Aug. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
| 8|1 | 8 | LLsLLLsLLs | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
| 7|2 | 5 | LLsLLsLLLs | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Min. | Perf. | Maj. | Maj. | Perf. |
| 6|3 | 2 | LLsLLsLLsL | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Min. | Perf. | Maj. | Min. | Perf. |
| 5|4 | 9 | LsLLLsLLsL | Perf. | Maj. | Min. | Perf. | Maj. | Maj. | Min. | Perf. | Maj. | Min. | Perf. |
| 4|5 | 6 | LsLLsLLLsL | Perf. | Maj. | Min. | Perf. | Maj. | Min. | Min. | Perf. | Maj. | Min. | Perf. |
| 3|6 | 3 | LsLLsLLsLL | Perf. | Maj. | Min. | Perf. | Maj. | Min. | Min. | Perf. | Min. | Min. | Perf. |
| 2|7 | 10 | sLLLsLLsLL | Perf. | Min. | Min. | Perf. | Maj. | Min. | Min. | Perf. | Min. | Min. | Perf. |
| 1|8 | 7 | sLLsLLLsLL | Perf. | Min. | Min. | Perf. | Min. | Min. | Min. | Perf. | Min. | Min. | Perf. |
| 0|9 | 4 | sLLsLLsLLL | Perf. | Min. | Min. | Perf. | Min. | Min. | Min. | Dim. | Min. | Min. | Perf. |
Theory
Neutral intervals
7L 3s combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals, thus making it compatible with Arabic and Turkish scales, but not with traditional Western scales. Notable intervals include:
- The perfect 3-mosstep, the scale's dark generator, whose range is around that of a neutral third.
- The perfect 7-mosstep, the scale's bright generator, the inversion of the perfect 3-mosstep, whose range is around that of a neutral sixth.
- The minor mosstep, or small step, which ranges form a quartertone to a minor second.
- The major mosstep, or large step, which ranges from a submajor second to a sinaic, or trienthird (around 128¢).
- The major 4-mosstep, whose range coincides with that of a perfect fourth.
- The minor 6-mosstep, the inversion of the major 4-mosstep, whose range coincides with that of a perfect 5th.
Quartertone and tetrachordal analysis
Due to the presence of quartertone-like intervals, Graham Breed has proposed the terms tone (abbreviated as t) and quartertone (abbreviated as q) as alternatives for large and small steps. This interpretation only makes sense for step ratios in which the small step approximates a quartertone. Additionally, Breed has also proposed a larger tone size, abbreviated using a capital T, to refer to the combination of t and q. Through this addition of a larger step, 7-note subsets of 7L 3s can be constructed. Some of these subsets are identical to that of 3L 4s, such as T-t-T-t-T-t-t, but Breed states that non-MOS patterns are possible, such as T-t-t-T-t-t-T.
Additionally, due to the presence of fourth and fifth-like intervals, 7L 3s can be analyzed as a tetrachordal scale. Since the major 4-dicostep, the fourth-like interval, is reached using 4 steps rather than 3 (3 tones and 1 quartertone), Andrew Heathwaite offers an additional step A, for augmented second, to refer to the combination of two tones (t). Thus, the possible tetrachords can be built as a combination of a (large) tone and two (regular) tones (T-t-t), or an augmented step, small tone, and quartertone (A-t-q).
Scale tree
| Generator(edo) | Cents | Step Ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 7\10 | 840.000 | 360.000 | 1:1 | 1.000 | Equalized 7L 3s | |||||
| 40\57 | 842.105 | 357.895 | 6:5 | 1.200 | Restles ↑ | |||||
| 33\47 | 842.553 | 357.447 | 5:4 | 1.250 | ||||||
| 59\84 | 842.857 | 357.143 | 9:7 | 1.286 | ||||||
| 26\37 | 843.243 | 356.757 | 4:3 | 1.333 | Supersoft 7L 3s | |||||
| 71\101 | 843.564 | 356.436 | 11:8 | 1.375 | ||||||
| 45\64 | 843.750 | 356.250 | 7:5 | 1.400 | Beatles | |||||
| 64\91 | 843.956 | 356.044 | 10:7 | 1.429 | ||||||
| 19\27 | 844.444 | 355.556 | 3:2 | 1.500 | Soft 7L 3s Suhajira / ringo | |||||
| 69\98 | 844.898 | 355.102 | 11:7 | 1.571 | ||||||
| 50\71 | 845.070 | 354.930 | 8:5 | 1.600 | ||||||
| 81\115 | 845.217 | 354.783 | 13:8 | 1.625 | Golden suhajira | |||||
| 31\44 | 845.455 | 354.545 | 5:3 | 1.667 | Semisoft 7L 3s | |||||
| 74\105 | 845.714 | 354.286 | 12:7 | 1.714 | ||||||
| 43\61 | 845.902 | 354.098 | 7:4 | 1.750 | ||||||
| 55\78 | 846.154 | 353.846 | 9:5 | 1.800 | ||||||
| 12\17 | 847.059 | 352.941 | 2:1 | 2.000 | Basic 7L 3s Scales with tunings softer than this are proper | |||||
| 53\75 | 848.000 | 352.000 | 9:4 | 2.250 | ||||||
| 41\58 | 848.276 | 351.724 | 7:3 | 2.333 | ||||||
| 70\99 | 848.485 | 351.515 | 12:5 | 2.400 | Hemif / hemififths | |||||
| 29\41 | 848.780 | 351.220 | 5:2 | 2.500 | Semihard 7L 3s Mohaha / neutrominant | |||||
| 75\106 | 849.057 | 350.943 | 13:5 | 2.600 | Hemif / salsa / karadeniz | |||||
| 46\65 | 849.231 | 350.769 | 8:3 | 2.667 | Mohaha / mohamaq | |||||
| 63\89 | 849.438 | 350.562 | 11:4 | 2.750 | ||||||
| 17\24 | 850.000 | 350.000 | 3:1 | 3.000 | Hard 7L 3s | |||||
| 56\79 | 850.633 | 349.367 | 10:3 | 3.333 | ||||||
| 39\55 | 850.909 | 349.091 | 7:2 | 3.500 | ||||||
| 61\86 | 851.163 | 348.837 | 11:3 | 3.667 | ||||||
| 22\31 | 851.613 | 348.387 | 4:1 | 4.000 | Superhard 7L 3s Mohaha / migration / mohajira | |||||
| 49\69 | 852.174 | 347.826 | 9:2 | 4.500 | ||||||
| 27\38 | 852.632 | 347.368 | 5:1 | 5.000 | ||||||
| 32\45 | 853.333 | 346.667 | 6:1 | 6.000 | Mohaha / ptolemy | |||||
| 5\7 | 857.143 | 342.857 | 1:0 | → ∞ | Collapsed 7L 3s | |||||
External links
- Graham Breed's page on 7L 3s (which covers 3L 7s to an extent)