3L 4s

↖ 2L 3s ↑ 3L 3s 4L 3s ↗

← 2L 4s 3L 4s 4L 4s →

↙ 2L 5s ↓ 3L 5s 4L 5s ↘
	Scale structure
Step pattern	LsLsLss; ssLsLsL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
	Generator size
Bright	2\7 to 1\3 (342.9 ¢ to 400.0 ¢)
Dark	2\3 to 5\7 (800.0 ¢ to 857.1 ¢)
	TAMNAMS information
Name	mosh
Prefix	mosh-
Abbrev.	mosh
	Related MOS scales
Parent	3L 1s
Sister	4L 3s
Daughters	7L 3s, 3L 7s
Neutralized	6L 1s
2-Flought	10L 4s, 3L 11s
	Equal tunings
Equalized (L:s = 1:1)	2\7 (342.9 ¢)
Supersoft (L:s = 4:3)	7\24 (350.0 ¢)
Soft (L:s = 3:2)	5\17 (352.9 ¢)
Semisoft (L:s = 5:3)	8\27 (355.6 ¢)
Basic (L:s = 2:1)	3\10 (360.0 ¢)
Semihard (L:s = 5:2)	7\23 (365.2 ¢)
Hard (L:s = 3:1)	4\13 (369.2 ¢)
Superhard (L:s = 4:1)	5\16 (375.0 ¢)
Collapsed (L:s = 1:0)	1\3 (400.0 ¢)
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3L 4s, named mosh in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 4 small steps, repeating every octave. Generators that produce this scale range from 342.9 ¢ to 400 ¢, or from 800 ¢ to 857.1 ¢.

Name

TAMNAMS suggests the temperament-agnostic name mosh for this scale, adopted from an older mos naming scheme by Graham Breed. The name is a contraction of "mohajira-ish".

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 3L 4s
Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.	Steps subtended	Range in cents
0-moshstep	Perfect 0-moshstep	P0moshs	0	0.0 ¢
1-moshstep	Minor 1-moshstep	m1moshs	s	0.0 ¢ to 171.4 ¢
1-moshstep	Major 1-moshstep	M1moshs	L	171.4 ¢ to 400.0 ¢
2-moshstep	Diminished 2-moshstep	d2moshs	2s	0.0 ¢ to 342.9 ¢
2-moshstep	Perfect 2-moshstep	P2moshs	L + s	342.9 ¢ to 400.0 ¢
3-moshstep	Minor 3-moshstep	m3moshs	L + 2s	400.0 ¢ to 514.3 ¢
3-moshstep	Major 3-moshstep	M3moshs	2L + s	514.3 ¢ to 800.0 ¢
4-moshstep	Minor 4-moshstep	m4moshs	L + 3s	400.0 ¢ to 685.7 ¢
4-moshstep	Major 4-moshstep	M4moshs	2L + 2s	685.7 ¢ to 800.0 ¢
5-moshstep	Perfect 5-moshstep	P5moshs	2L + 3s	800.0 ¢ to 857.1 ¢
5-moshstep	Augmented 5-moshstep	A5moshs	3L + 2s	857.1 ¢ to 1200.0 ¢
6-moshstep	Minor 6-moshstep	m6moshs	2L + 4s	800.0 ¢ to 1028.6 ¢
6-moshstep	Major 6-moshstep	M6moshs	3L + 3s	1028.6 ¢ to 1200.0 ¢
7-moshstep	Perfect 7-moshstep	P7moshs	3L + 4s	1200.0 ¢

Generator chain

Generator chain of 3L 4s
Bright gens	Scale degree	Abbrev.
9	Augmented 4-moshdegree	A4moshd
8	Augmented 2-moshdegree	A2moshd
7	Augmented 0-moshdegree	A0moshd
6	Augmented 5-moshdegree	A5moshd
5	Major 3-moshdegree	M3moshd
4	Major 1-moshdegree	M1moshd
3	Major 6-moshdegree	M6moshd
2	Major 4-moshdegree	M4moshd
1	Perfect 2-moshdegree	P2moshd
0	Perfect 0-moshdegree Perfect 7-moshdegree	P0moshd P7moshd
−1	Perfect 5-moshdegree	P5moshd
−2	Minor 3-moshdegree	m3moshd
−3	Minor 1-moshdegree	m1moshd
−4	Minor 6-moshdegree	m6moshd
−5	Minor 4-moshdegree	m4moshd
−6	Diminished 2-moshdegree	d2moshd
−7	Diminished 7-moshdegree	d7moshd
−8	Diminished 5-moshdegree	d5moshd
−9	Diminished 3-moshdegree	d3moshd

Modes

Scale degrees of the modes of 3L 4s
UDP	Cyclic order	Step pattern	Scale degree (moshdegree)
UDP	Cyclic order	Step pattern	0	1	2	3	4	5	6	7
6\|0	1	LsLsLss	Perf.	Maj.	Perf.	Maj.	Maj.	Aug.	Maj.	Perf.
5\|1	3	LsLssLs	Perf.	Maj.	Perf.	Maj.	Maj.	Perf.	Maj.	Perf.
4\|2	5	LssLsLs	Perf.	Maj.	Perf.	Min.	Maj.	Perf.	Maj.	Perf.
3\|3	7	sLsLsLs	Perf.	Min.	Perf.	Min.	Maj.	Perf.	Maj.	Perf.
2\|4	2	sLsLssL	Perf.	Min.	Perf.	Min.	Maj.	Perf.	Min.	Perf.
1\|5	4	sLssLsL	Perf.	Min.	Perf.	Min.	Min.	Perf.	Min.	Perf.
0\|6	6	ssLsLsL	Perf.	Min.	Dim.	Min.	Min.	Perf.	Min.	Perf.

Proposed names

The first set of mode nicknames was coined by Andrew Heathwaite. The other set was coined by CellularAutomaton and follows the diatonic modes' naming convention by using ancient Greek toponyms that sound similar to Heathwaite's names. The third shows which modes are a mixture of which diatonic modes, as discussed in #Theory.

Modes of 3L 4s
UDP	Cyclic order	Step pattern	Mode names (Heathwaite)	Mode names (CA)	Mixed diatonic modes
6\|0	1	LsLsLss	Dril	Dalmatian	Dorian + Lydian
5\|1	3	LsLssLs	Gil	Galatian	Aeolian + Lydian
4\|2	5	LssLsLs	Kleeth	Cilician	Aeolian + Ionian
3\|3	7	sLsLsLs	Bish	Bithynian	Phrygian + Ionian
2\|4	2	sLsLssL	Fish	Pisidian	Phrygian + Mixolydian
1\|5	4	sLssLsL	Jwl	Illyrian	Locrian + Mixolydian
0\|6	6	ssLsLsL	Led	Lycian	Locrian + Dorian

Theory

Mosh can be thought of as a midpoint between two diatonic scales which are two cyclic orders away from each other. For example, LsLsLss is the midpoint between LsLLLsL and LLLsLLs. You can prove this by simple addition.

  2 1 2 2 2 1 2 (LsLLLsL)
+ 2 2 2 1 2 2 1 (LLLsLLs)
= 4 3 4 3 4 3 3 (LsLsLss)

The rest of the equivalencies are listed in #Proposed names. 

Low harmonic entropy scales
There are two notable harmonic entropy minima:

* Neutral third scales, such as dicot, hemififth, and mohajira, in which the generator is a neutral 3rd (around 350 ¢) and two of them make a 3/2 (702 ¢).
* Magic, in which the generator is 5/4 (386 ¢) and 5 of them make a 3/1 (1902 ¢).

Tuning ranges
3\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make mos 7L 3s (dicoid); the other scales make mos 3L 7s (sephiroid).

In dicoid, the generators are all "neutral thirds," and two of them make an approximation of the "perfect fifth." Additionally, the L of the scale is somewhere around a "whole tone" and the s of the scale is somewhere around a "neutral tone".

In sephiroid, the generators are major thirds (including some very flat ones), and two generators are definitely sharp of a perfect fifth. L ranges from a "supermajor second" to a "major third" and s is a "semitone" or smaller.

Ultrasoft
Ultrasoft mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than 7\24 = 350 ¢.

Ultrasoft mosh can be considered "meantone mosh". This is because the large step is a "meantone" in these tunings, somewhere between near-10/9 (as in 38edo) and near-9/8 (as in 24edo).

Ultrasoft mosh edos include 24edo, 31edo, 38edo, and 55edo.
* 24edo can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.
* 38edo can be used to tune the diminished and perfect mosthirds near 6/5 and 11/9, respectively.

These identifications are associated with mohajira temperament.

The sizes of the generator, large step and small step of mosh are as follows in various ultrasoft mosh tunings. 





24edo (supersoft)

31edo

38edo

55edo

JI intervals represented


generator (g)

7\24, 350.00

9\31, 348.39

11\38, 347.37

16\55, 349.09

11/9


L (4g − octave)

4\24, 200.00

5\31, 193.55

6\38, 189.47

9\55, 196.36

9/8, 10/9


s (octave − 3g)

3\24, 150.00

4\31, 154.84

5\38, 157.89

7\55, 152.72

11/10, 12/11


Quasisoft
Quasisoft tunings of mosh have a step ratio between 3/2 and 5/3, implying a generator sharper than 5\17 = 352.94 ¢ and flatter than 8\27 = 355.56 ¢.

The large step is a sharper major second in these tunings than in ultrasoft tunings. These tunings could be considered "parapyth mosh" or "archy mosh", in analogy to ultrasoft mosh being meantone mosh.

These identifications are associated with beatles and suhajira temperaments.





17edo (soft)

27edo (semisoft)

44edo

JI intervals represented


generator (g)

5\17, 352.94

8\27, 355.56

13\44, 354.55

16/13, 11/9


L (4g − octave)

3\17, 211.76

5\27, 222.22

8\44, 218.18

9/8, 8/7


s (octave − 3g)

2\17, 141.18

3\27, 133.33

5\44, 137.37

12/11, 13/12, 14/13


Hypohard
Hypohard tunings of mosh have a step ratio between 2 and 3, implying a generator sharper than 3\10 = 360 ¢ and flatter than 4\13 = 369.23 ¢.

The large step ranges from a semifourth to a subminor third in these tunings. The small step is now clearly a semitone, ranging from 1\10 (120 ¢) to 1\13 (92.31 ¢).

The symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.

This range is associated with sephiroth temperament.





10edo (basic)

13edo (hard)

23edo (semihard)


generator (g)

3\10, 360.00

4\13, 369.23

7\23, 365.22


L (4g − octave)

2\10, 240.00

3\13, 276.92

5\23, 260.87


s (octave − 3g)

1\10, 120.00

1\13, 92.31

2\23, 104.35


Ultrahard
Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than 5\16 = 375 ¢. The generator is thus near a 5/4 major third, five of which add up to an approximate 3/1. The 7-note mos only has two perfect fifths, so extending the chain to bigger mosses, such as the 3L 7s 10-note mos, is suggested for getting 5-limit harmony.

This range is associated with magic temperament.





16edo (superhard)

19edo

22edo

41edo

JI intervals represented


generator (g)

5\16, 375.00

6\19, 378.95

7\22, 381.82

13\41, 380.49

5/4


L (4g − octave)

4\16, 300.00

5\19, 315.79

6\22, 327.27

11\41, 321.95

6/5


s (octave − 3g)

1\16, 75.00

1\19, 63.16

1\22, 54.54

2\41, 58.54

25/24


Scales
* Mohaha7 – 38\131 tuning
* Neutral7 – 111\380 tuning
* Namo7 – 128\437 tuning
* Rastgross1 – POTE tuning of namo
* Hemif7 – 17\58 tuning
* Suhajira7 – POTE tuning of suhajira
* Sephiroth7 – 9\29 tuning
* Magic7 – 46\145 tuning

Scale tree
Generator ranges: 
* Chroma-positive generator: 342.8571 ¢ (2\7) to 400.0000 ¢ (1\3)
* Chroma-negative generator: 800.0000 ¢ (2\3) to 857.1429 ¢ (5\7)


Scale tree and tuning spectrum of 3L 4s


Generator^(edo)

Cents

Step ratio

Comments


Bright

Dark

L:s

Hardness


2\7











342.857

857.143

1:1

1.000

Equalized 3L 4s













11\38

347.368

852.632

6:5

1.200

Mohaha / ptolemy ↑










9\31



348.387

851.613

5:4

1.250

Mohaha / migration / mohajira












16\55

349.091

850.909

9:7

1.286










7\24





350.000

850.000

4:3

1.333

Supersoft 3L 4s













19\65

350.769

849.231

11:8

1.375

Mohaha / mohamaq










12\41



351.220

848.780

7:5

1.400

Mohaha / neutrominant












17\58

351.724

848.276

10:7

1.429

Hemif / hemififths






5\17







352.941

847.059

3:2

1.500

Soft 3L 4s













18\61

354.098

845.902

11:7

1.571

Suhajira










13\44



354.545

845.455

8:5

1.600














21\71

354.930

845.070

13:8

1.625

Golden suhajira (354.8232 ¢)








8\27





355.556

844.444

5:3

1.667

Semisoft 3L 4s
Suhajira / ringo












19\64

356.250

843.750

12:7

1.714

Beatles










11\37



356.757

843.243

7:4

1.750














14\47

357.447

842.553

9:5

1.800






3\10









360.000

840.000

2:1

2.000

Basic 3L 4s
Scales with tunings softer than this are proper













13\43

362.791

837.209

9:4

2.250












10\33



363.636

836.364

7:3

2.333














17\56

364.286

835.714

12:5

2.400










7\23





365.217

834.783

5:2

2.500

Semihard 3L 4s













18\59

366.102

833.898

13:5

2.600

Unnamed golden tuning (366.2564 ¢)










11\36



366.667

833.333

8:3

2.667














15\49

367.347

832.653

11:4

2.750








4\13







369.231

830.769

3:1

3.000

Hard 3L 4s













13\42

371.429

828.571

10:3

3.333












9\29



372.414

827.586

7:2

3.500

Sephiroth












14\45

373.333

826.667

11:3

3.667










5\16





375.000

825.000

4:1

4.000

Superhard 3L 4s













11\35

377.143

822.857

9:2

4.500

Muggles










6\19



378.947

821.053

5:1

5.000

Magic












7\22

381.818

818.182

6:1

6.000

Würschmidt ↓


1\3











400.000

800.000

1:0

→ ∞

Collapsed 3L 4s

	24edo (supersoft)	31edo	38edo	55edo	JI intervals represented
generator (g)	7\24, 350.00	9\31, 348.39	11\38, 347.37	16\55, 349.09	11/9
L (4g − octave)	4\24, 200.00	5\31, 193.55	6\38, 189.47	9\55, 196.36	9/8, 10/9
s (octave − 3g)	3\24, 150.00	4\31, 154.84	5\38, 157.89	7\55, 152.72	11/10, 12/11

	17edo (soft)	27edo (semisoft)	44edo	JI intervals represented
generator (g)	5\17, 352.94	8\27, 355.56	13\44, 354.55	16/13, 11/9
L (4g − octave)	3\17, 211.76	5\27, 222.22	8\44, 218.18	9/8, 8/7
s (octave − 3g)	2\17, 141.18	3\27, 133.33	5\44, 137.37	12/11, 13/12, 14/13

	10edo (basic)	13edo (hard)	23edo (semihard)
generator (g)	3\10, 360.00	4\13, 369.23	7\23, 365.22
L (4g − octave)	2\10, 240.00	3\13, 276.92	5\23, 260.87
s (octave − 3g)	1\10, 120.00	1\13, 92.31	2\23, 104.35

	16edo (superhard)	19edo	22edo	41edo	JI intervals represented
generator (g)	5\16, 375.00	6\19, 378.95	7\22, 381.82	13\41, 380.49	5/4
L (4g − octave)	4\16, 300.00	5\19, 315.79	6\22, 327.27	11\41, 321.95	6/5
s (octave − 3g)	1\16, 75.00	1\19, 63.16	1\22, 54.54	2\41, 58.54	25/24

Generator^(edo)						Cents		Step ratio		Comments
Generator^(edo)						Bright	Dark	L:s	Hardness	Comments
2\7						342.857	857.143	1:1	1.000	Equalized 3L 4s
					11\38	347.368	852.632	6:5	1.200	Mohaha / ptolemy ↑
				9\31		348.387	851.613	5:4	1.250	Mohaha / migration / mohajira
					16\55	349.091	850.909	9:7	1.286
			7\24			350.000	850.000	4:3	1.333	Supersoft 3L 4s
					19\65	350.769	849.231	11:8	1.375	Mohaha / mohamaq
				12\41		351.220	848.780	7:5	1.400	Mohaha / neutrominant
					17\58	351.724	848.276	10:7	1.429	Hemif / hemififths
		5\17				352.941	847.059	3:2	1.500	Soft 3L 4s
					18\61	354.098	845.902	11:7	1.571	Suhajira
				13\44		354.545	845.455	8:5	1.600
					21\71	354.930	845.070	13:8	1.625	Golden suhajira (354.8232 ¢)
			8\27			355.556	844.444	5:3	1.667	Semisoft 3L 4s Suhajira / ringo
					19\64	356.250	843.750	12:7	1.714	Beatles
				11\37		356.757	843.243	7:4	1.750
					14\47	357.447	842.553	9:5	1.800
	3\10					360.000	840.000	2:1	2.000	Basic 3L 4s Scales with tunings softer than this are proper
					13\43	362.791	837.209	9:4	2.250
				10\33		363.636	836.364	7:3	2.333
					17\56	364.286	835.714	12:5	2.400
			7\23			365.217	834.783	5:2	2.500	Semihard 3L 4s
					18\59	366.102	833.898	13:5	2.600	Unnamed golden tuning (366.2564 ¢)
				11\36		366.667	833.333	8:3	2.667
					15\49	367.347	832.653	11:4	2.750
		4\13				369.231	830.769	3:1	3.000	Hard 3L 4s
					13\42	371.429	828.571	10:3	3.333
				9\29		372.414	827.586	7:2	3.500	Sephiroth
					14\45	373.333	826.667	11:3	3.667
			5\16			375.000	825.000	4:1	4.000	Superhard 3L 4s
					11\35	377.143	822.857	9:2	4.500	Muggles
				6\19		378.947	821.053	5:1	5.000	Magic
					7\22	381.818	818.182	6:1	6.000	Würschmidt ↓
1\3						400.000	800.000	1:0	→ ∞	Collapsed 3L 4s