3L 4s
| ↖ 2L 3s | ↑3L 3s | 4L 3s ↗ |
| ← 2L 4s | 3L 4s | 4L 4s → |
| ↙ 2L 5s | ↓3L 5s | 4L 5s ↘ |
┌╥┬╥┬╥┬┬┐ │║│║│║│││ │││││││││ └┴┴┴┴┴┴┴┘
ssLsLsL
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 | 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 |
- Generic intervals are denoted solely by the number of steps they subtend.
- Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
- 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
| UDP | Rotational Order | Step pattern | Scale degree (moshdegree) | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 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
These nicknames were coined by Andrew Heathwaite:
| UDP | Rotational order | Step pattern | Mode names |
|---|---|---|---|
| 5|0 | 1 | LsLsLss | dril |
| 4|1 | 3 | LsLssLs | gil |
| 3|2 | 5 | LssLsLs | kleeth |
| 2|3 | 7 | sLsLsLs | bish |
| 1|4 | 2 | sLsLssL | fish |
| 0|5 | 4 | sLssLsL | jwl |
| -1|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 | ||||||