7L 3s

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↖ 6L 2s ↑7L 2s 8L 2s ↗
← 6L 3s7L 3s 8L 3s →
↙ 6L 4s ↓7L 4s 8L 4s ↘
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 7\10 to 5\7 (840.0¢ to 857.1¢)
Dark 2\7 to 3\10 (342.9¢ to 360.0¢)
TAMNAMS information
Name dicoid
Prefix dico-
Abbrev. dico
Related MOS scales
Parent 3L 4s
Sister 3L 7s
Daughters 10L 7s, 7L 10s
Neutralized 4L 6s
2-Flought 17L 3s, 7L 13s
Equal tunings
Equalized (L:s = 1:1) 7\10 (840.0¢)
Supersoft (L:s = 4:3) 26\37 (843.2¢)
Soft (L:s = 3:2) 19\27 (844.4¢)
Semisoft (L:s = 5:3) 31\44 (845.5¢)
Basic (L:s = 2:1) 12\17 (847.1¢)
Semihard (L:s = 5:2) 29\41 (848.8¢)
Hard (L:s = 3:1) 17\24 (850.0¢)
Superhard (L:s = 4:1) 22\31 (851.6¢)
Collapsed (L:s = 1:0) 5\7 (857.1¢)

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 of 7L 3s
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

  1. Generic intervals are denoted solely by the number of steps they subtend.
  2. Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
  3. Abbreviations can be further shortened to 'ms' if context allows.

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

Scale Tree and Tuning Spectrum of 7L 3s
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