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 as the name of 7L 3s. The name derives from dichotic and dicot temperament. Although this name is directly based off of a temperament, tunings of dichotic and dicot cover the entire tuning range of 7L 3s; see TAMNAMS/Appendix #Dicoid (7L 3s) for more information.
Scale properties
Intervals
The intervals of 7L 3s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-dicostep and perfect 10-dicostep), 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-dicostep | Perfect 0-dicostep | P0dis | 0 | 0.0¢ |
| 1-dicostep | Minor 1-dicostep | m1dis | s | 0.0¢ to 120.0¢ |
| Major 1-dicostep | M1dis | L | 120.0¢ to 171.4¢ | |
| 2-dicostep | Minor 2-dicostep | m2dis | L + s | 171.4¢ to 240.0¢ |
| Major 2-dicostep | M2dis | 2L | 240.0¢ to 342.9¢ | |
| 3-dicostep | Perfect 3-dicostep | P3dis | 2L + s | 342.9¢ to 360.0¢ |
| Augmented 3-dicostep | A3dis | 3L | 360.0¢ to 514.3¢ | |
| 4-dicostep | Minor 4-dicostep | m4dis | 2L + 2s | 342.9¢ to 480.0¢ |
| Major 4-dicostep | M4dis | 3L + s | 480.0¢ to 514.3¢ | |
| 5-dicostep | Minor 5-dicostep | m5dis | 3L + 2s | 514.3¢ to 600.0¢ |
| Major 5-dicostep | M5dis | 4L + s | 600.0¢ to 685.7¢ | |
| 6-dicostep | Minor 6-dicostep | m6dis | 4L + 2s | 685.7¢ to 720.0¢ |
| Major 6-dicostep | M6dis | 5L + s | 720.0¢ to 857.1¢ | |
| 7-dicostep | Diminished 7-dicostep | d7dis | 4L + 3s | 685.7¢ to 840.0¢ |
| Perfect 7-dicostep | P7dis | 5L + 2s | 840.0¢ to 857.1¢ | |
| 8-dicostep | Minor 8-dicostep | m8dis | 5L + 3s | 857.1¢ to 960.0¢ |
| Major 8-dicostep | M8dis | 6L + 2s | 960.0¢ to 1028.6¢ | |
| 9-dicostep | Minor 9-dicostep | m9dis | 6L + 3s | 1028.6¢ to 1080.0¢ |
| Major 9-dicostep | M9dis | 7L + 2s | 1080.0¢ to 1200.0¢ | |
| 10-dicostep | Perfect 10-dicostep | P10dis | 7L + 3s | 1200.0¢ |
Generator chain
A chain of bright generators, each a perfect 7-dicostep, produces the following scale degrees. A chain of 10 bright generators contains the scale degrees of one of the modes of 7L 3s. Expanding the chain to 17 scale degrees produces the modes of either 10L 7s (for soft-of-basic tunings) or 7L 10s (for hard-of-basic tunings).
| Bright gens | Scale Degree | Abbrev. |
|---|---|---|
| 16 | Augmented 2-dicodegree | A2did |
| 15 | Augmented 5-dicodegree | A5did |
| 14 | Augmented 8-dicodegree | A8did |
| 13 | Augmented 1-dicodegree | A1did |
| 12 | Augmented 4-dicodegree | A4did |
| 11 | Augmented 7-dicodegree | A7did |
| 10 | Augmented 0-dicodegree | A0did |
| 9 | Augmented 3-dicodegree | A3did |
| 8 | Major 6-dicodegree | M6did |
| 7 | Major 9-dicodegree | M9did |
| 6 | Major 2-dicodegree | M2did |
| 5 | Major 5-dicodegree | M5did |
| 4 | Major 8-dicodegree | M8did |
| 3 | Major 1-dicodegree | M1did |
| 2 | Major 4-dicodegree | M4did |
| 1 | Perfect 7-dicodegree | P7did |
| 0 | Perfect 0-dicodegree Perfect 10-dicodegree |
P0did P10did |
| -1 | Perfect 3-dicodegree | P3did |
| -2 | Minor 6-dicodegree | m6did |
| -3 | Minor 9-dicodegree | m9did |
| -4 | Minor 2-dicodegree | m2did |
| -5 | Minor 5-dicodegree | m5did |
| -6 | Minor 8-dicodegree | m8did |
| -7 | Minor 1-dicodegree | m1did |
| -8 | Minor 4-dicodegree | m4did |
| -9 | Diminished 7-dicodegree | d7did |
| -10 | Diminished 10-dicodegree | d10did |
| -11 | Diminished 3-dicodegree | d3did |
| -12 | Diminished 6-dicodegree | d6did |
| -13 | Diminished 9-dicodegree | d9did |
| -14 | Diminished 2-dicodegree | d2did |
| -15 | Diminished 5-dicodegree | d5did |
| -16 | Diminished 8-dicodegree | d8did |
Modes
| UDP | Cyclic 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)