3L 4s

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↖ 2L 3s ↑3L 3s 4L 3s ↗
← 2L 4s3L 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¢)

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".

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 3L 4s, 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 3L 4s
Intervals Steps subtended Range in cents Average of HE
(from HE Calc)
Min of HE
Generic[1] Specific[2] Abbrev.[3]
0-moshstep Perfect 0-moshstep P0ms 0 0.0¢ ~2.4654 nats ~2.4654 nats
1-moshstep Minor 1-moshstep m1ms s 0.0¢ to 171.4¢ ~4.6629 nats ~4.6233 nats
Major 1-moshstep M1ms L 171.4¢ to 400.0¢ ~4.5777 nats ~4.5654 nats
2-moshstep Diminished 2-moshstep d2ms 2s 0.0¢ to 342.9¢ ~4.5848 nats ~4.5636 nats
Perfect 2-moshstep P2ms L + s 342.9¢ to 400.0¢ ~4.5908 nats ~4.5278 nats
3-moshstep Minor 3-moshstep m3ms L + 2s 400.0¢ to 514.3¢ ~4.5004 nats ~4.3665 nats
Major 3-moshstep M3ms 2L + s 514.3¢ to 800.0¢ ~4.5961 nats ~4.5653 nats
4-moshstep Minor 4-moshstep m4ms L + 3s 400.0¢ to 685.7¢ ~4.6012 nats ~4.5728 nats
Major 4-moshstep M4ms 2L + 2s 685.7¢ to 800.0¢ ~4.3812 nats ~4.1313 nats
5-moshstep Perfect 5-moshstep P5ms 2L + 3s 800.0¢ to 857.1¢ ~4.6039 nats ~4.5814 nats
Augmented 5-moshstep A5ms 3L + 2s 857.1¢ to 1200.0¢ ~4.5762 nats ~4.4980 nats
6-moshstep Minor 6-moshstep m6ms 2L + 4s 800.0¢ to 1028.6¢ ~4.5670 nats ~4.4980 nats
Major 6-moshstep M6ms 3L + 3s 1028.6¢ to 1200.0¢ ~4.6181 nats ~4.6074 nats
7-moshstep Perfect 7-moshstep P7ms 3L + 4s 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.

Note names

For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern sLsLsLs and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 3L 4s is the following:

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

Theory

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 MOSes, 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

Modes

The various modes of 3L 4s (with Modal UDP Notation and nicknames coined by Andrew Heathwaite) are:


Modes of 3L 4s
UDP Rotational order Step pattern Mode names
6|0 1 LsLsLss dril
5|1 3 LsLssLs gil
4|2 5 LssLsLs kleeth
3|3 7 sLsLsLs bish
2|4 2 sLsLssL fish
1|5 4 sLssLsL jwl
0|6 6 ssLsLsL led

Scale tree

Generator ranges:

  • Chroma-positive generator: 342.8571 cents (2\7) to 400 cents (1\3)
  • Chroma-negative generator: 800 cents (2\3) to 857.1429 cents (5\7)
Generator Cents L s L/s Comments
2\7 342.857 1 1 1.000
11\38 347.368 6 5 1.200 Mohaha / ptolemy↑
9\31 348.387 5 4 1.250 Mohaha / migration / mohajira
16\55 349.091 9 7 1.286
7\24 350.000 4 3 1.333
19\65 350.769 11 8 1.375 Mohaha / mohamaq
12\41 351.220 7 5 1.400 Mohaha / neutrominant
17\58 351.724 10 7 1.429 Hemif / Hemififths
5\17 352.941 3 2 1.500
18\61 354.098 11 7 1.571 Suhajira
13\44 354.545 8 5 1.600
21\71 354.930 13 8 1.625 Golden suhajira (354.8232¢)
8\27 355.556 5 3 1.667 Suhajira / ringo
19\64 356.250 12 7 1.714 Beatles
11\37 356.757 7 4 1.750
14\47 357.447 9 5 1.800
3\10 360.000 2 1 2.000 Basic mosh
(Generators smaller than this are proper)
13\43 362.791 9 4 2.250
10\33 363.636 7 3 2.333
17\56 364.286 12 5 2.400
7\23 365.217 5 2 2.500
18\59 366.102 13 5 2.600 Unnamed golden tuning (366.2564¢)
11\36 366.667 8 3 2.667
15\49 367.347 11 4 2.750
4\13 369.231 3 1 3.000
13\42 371.429 10 3 3.333
9\29 372.414 7 2 3.500 Sephiroth
14\45 373.333 11 3 3.667
5\16 375.000 4 1 4.000
11\35 377.143 9 2 4.500 Muggles
6\19 378.947 5 1 5.000 Magic
7\22 381.818 6 1 6.000 Wuerschmidt
1\3 400.000 1 0 → inf