2L 5s

Todo: expand Add hard-of-basic tunings, include JI ratio approximations

↖ 1L 4s	↑ 2L 4s	3L 4s ↗
← 1L 5s	2L 5s	3L 5s →
↙ 1L 6s	↓ 2L 6s	3L 6s ↘

┌╥┬┬╥┬┬┬┐ │║││║││││ │││││││││ └┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LssLsss sssLssL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	3\7 to 1\2 (514.3 ¢ to 600.0 ¢)
Dark	1\2 to 4\7 (600.0 ¢ to 685.7 ¢)
TAMNAMS information
Name	antidiatonic
Prefix	pel-
Abbrev.	pel
Related MOS scales
Parent	2L 3s
Sister	5L 2s
Daughters	7L 2s, 2L 7s
Neutralized	4L 3s
2-Flought	9L 5s, 2L 12s
Equal tunings
Equalized (L:s = 1:1)	3\7 (514.3 ¢)
Supersoft (L:s = 4:3)	10\23 (521.7 ¢)
Soft (L:s = 3:2)	7\16 (525.0 ¢)
Semisoft (L:s = 5:3)	11\25 (528.0 ¢)
Basic (L:s = 2:1)	4\9 (533.3 ¢)
Semihard (L:s = 5:2)	9\20 (540.0 ¢)
Hard (L:s = 3:1)	5\11 (545.5 ¢)
Superhard (L:s = 4:1)	6\13 (553.8 ¢)
Collapsed (L:s = 1:0)	1\2 (600.0 ¢)

2L 5s, named antidiatonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 514.3 ¢ to 600 ¢, or from 600 ¢ to 685.7 ¢.

Name

TAMNAMS suggests the temperament-agnostic name antidiatonic for this scale, adopted from the common use of the term to refer to diatonic (5L 2s) but with the large and small steps switched.

Notation

This article assumes TAMNAMS for naming step ratios, intervals, and scale degrees, and diamond-MOS notation for note names.

Intervals and degrees

Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.

Being a moment-of-symmetry scale, every interval class of 2L 5s, except for the unison and octave, has two varieties – large and small – whose relative qualities are denoted as major or minor, or augmented, perfect, and diminished for the generators.

Intervals of 2L 5s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-pelstep	Perfect 0-pelstep	P0pels	0	0.0 ¢
1-pelstep	Minor 1-pelstep	m1pels	s	0.0 ¢ to 171.4 ¢
	Major 1-pelstep	M1pels	L	171.4 ¢ to 600.0 ¢
2-pelstep	Minor 2-pelstep	m2pels	2s	0.0 ¢ to 342.9 ¢
	Major 2-pelstep	M2pels	L + s	342.9 ¢ to 600.0 ¢
3-pelstep	Diminished 3-pelstep	d3pels	3s	0.0 ¢ to 514.3 ¢
	Perfect 3-pelstep	P3pels	L + 2s	514.3 ¢ to 600.0 ¢
4-pelstep	Perfect 4-pelstep	P4pels	L + 3s	600.0 ¢ to 685.7 ¢
	Augmented 4-pelstep	A4pels	2L + 2s	685.7 ¢ to 1200.0 ¢
5-pelstep	Minor 5-pelstep	m5pels	L + 4s	600.0 ¢ to 857.1 ¢
	Major 5-pelstep	M5pels	2L + 3s	857.1 ¢ to 1200.0 ¢
6-pelstep	Minor 6-pelstep	m6pels	L + 5s	600.0 ¢ to 1028.6 ¢
	Major 6-pelstep	M6pels	2L + 4s	1028.6 ¢ to 1200.0 ¢
7-pelstep	Perfect 7-pelstep	P7pels	2L + 5s	1200.0 ¢

Note names

For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern sLsssLs and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 2L 5s is the following:

J, K, K&/L@, L, M, N, O, O&/P@, P, J

Theory

Antidiatonic is similar to diatonic (5L 2s) except interval classes are flipped. For example, there are natural, harmonic, and melodic major scales instead of minor scales, and its locrian scale, called "antilocrian", has an augmented "fifth" instead of a diminished fifth. The flatter the "fifth" (or 4-mosstep) gets, the less the scale resembles diatonic. Additionally, there are temperaments associated with this MOS, such as score, that do not have intervals that resemble a diatonic 5th, flattened or otherwise.

Low harmonic entropy scales

There is one notable harmonic entropy minimum: Liese/triton, in which the generator is 7/5 (582.5¢) and three of them make a 3/1 (1902¢).

Temperament interpretations

2L 5s has several rank-2 temperament interpretations, such as:

• Mavila, with generators around 679.8¢.
• Casablanca, with generators around 657.8¢.
• Liese, with generators around 632.4¢.

Tuning ranges

Simple tunings

The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 9edo, 11edo, and 16edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.

Scale degree of 2L 5s

Scale degree	9edo (Basic, L:s = 2:1)		11edo (Hard, L:s = 3:1)		16edo (Soft, L:s = 3:2)		Approx. JI Ratios
	Steps	Cents	Steps	Cents	Steps	Cents
Perfect 0-peldegree (unison)	0	0	0	0	0	0	1/1 (exact)
Minor 1-peldegree	1	133.3	1	109.1	2	150
Major 1-peldegree	2	266.7	3	327.3	3	225
Minor 2-peldegree	2	266.7	2	218.2	4	300
Major 2-peldegree	3	400	4	436.4	5	375
Diminished 3-peldegree	3	400	3	327.3	6	450
Perfect 3-peldegree	4	533.3	5	545.5	7	525
Perfect 4-peldegree	5	666.7	6	654.5	9	675
Augmented 4-peldegree	6	800	8	872.7	10	750
Minor 5-peldegree	6	800	7	763.6	11	825
Major 5-peldegree	7	933.3	9	981.8	12	900
Minor 6-peldegree	7	933.3	8	872.7	13	975
Major 6-peldegree	8	1066.7	10	1090.9	14	1050
Perfect 7-peldegree (octave)	9	1200	11	1200	16	1200	2/1 (exact)

Soft-of-basic tunings

Main article: Mavila

Much of the range for soft-of-basic antidiatonic tunings (1:1 to 2:1) corresponds to mavila temperament. Edos include 9edo (not shown), 16edo, and 23edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.

Scale degree of 2L 5s

Scale degree	16edo (Soft, L:s = 3:2)		23edo (Supersoft, L:s = 4:3)		Approx. JI Ratios
	Steps	Cents	Steps	Cents
Perfect 0-peldegree (unison)	0	0	0	0	1/1 (exact)
Minor 1-peldegree	2	150	3	156.5
Major 1-peldegree	3	225	4	208.7
Minor 2-peldegree	4	300	6	313
Major 2-peldegree	5	375	7	365.2
Diminished 3-peldegree	6	450	9	469.6
Perfect 3-peldegree	7	525	10	521.7
Perfect 4-peldegree	9	675	13	678.3
Augmented 4-peldegree	10	750	14	730.4
Minor 5-peldegree	11	825	16	834.8
Major 5-peldegree	12	900	17	887
Minor 6-peldegree	13	975	19	991.3
Major 6-peldegree	14	1050	20	1043.5
Perfect 7-peldegree (octave)	16	1200	23	1200	2/1 (exact)

Modes

Modes of the antidiatonic scale are usually named as "anti-" combined with the opposite mode of the diatonic scale, e.g. 4|2 being called "antiaeolian". CompactStar also gave original names based on regions of France to mirror how modes of the diatonic scale are named on regions of Greece and Turkey.

Modes of 2L 5s

UDP	Cyclic order	Step pattern
6|0	1	LssLsss
5|1	4	LsssLss
4|2	7	sLssLss
3|3	3	sLsssLs
2|4	6	ssLssLs
1|5	2	ssLsssL
0|6	5	sssLssL