7L 3s

↖ 6L 2s	↑ 7L 2s	8L 2s ↗
← 6L 3s	7L 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.	di
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 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

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

Intervals of 7L 3s

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 ¢

Modes

Scale degrees of the modes of 7L 3s

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).

External links

• Graham Breed's page on 7L 3s (which covers 3L 7s to an extent)