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".
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 diatonic interval categories.
Intervals
The intervals of 3L 4s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-moshstep and perfect 7-moshstep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.
| Intervals | Steps subtended |
Range in cents | ||
|---|---|---|---|---|
| Generic | Specific | Abbrev. | ||
| 0-moshstep | Perfect 0-moshstep | P0moshs | 0 | 0.0 ¢ |
| 1-moshstep | Minor 1-moshstep | m1moshs | s | 0.0 ¢ to 171.4 ¢ |
| Major 1-moshstep | M1moshs | L | 171.4 ¢ to 400.0 ¢ | |
| 2-moshstep | Diminished 2-moshstep | d2moshs | 2s | 0.0 ¢ to 342.9 ¢ |
| Perfect 2-moshstep | P2moshs | L + s | 342.9 ¢ to 400.0 ¢ | |
| 3-moshstep | Minor 3-moshstep | m3moshs | L + 2s | 400.0 ¢ to 514.3 ¢ |
| Major 3-moshstep | M3moshs | 2L + s | 514.3 ¢ to 800.0 ¢ | |
| 4-moshstep | Minor 4-moshstep | m4moshs | L + 3s | 400.0 ¢ to 685.7 ¢ |
| Major 4-moshstep | M4moshs | 2L + 2s | 685.7 ¢ to 800.0 ¢ | |
| 5-moshstep | Perfect 5-moshstep | P5moshs | 2L + 3s | 800.0 ¢ to 857.1 ¢ |
| Augmented 5-moshstep | A5moshs | 3L + 2s | 857.1 ¢ to 1200.0 ¢ | |
| 6-moshstep | Minor 6-moshstep | m6moshs | 2L + 4s | 800.0 ¢ to 1028.6 ¢ |
| Major 6-moshstep | M6moshs | 3L + 3s | 1028.6 ¢ to 1200.0 ¢ | |
| 7-moshstep | Perfect 7-moshstep | P7moshs | 3L + 4s | 1200.0 ¢ |
Generator chain
A chain of bright generators, each a perfect 2-moshstep, produces the following scale degrees. A chain of 7 bright generators contains the scale degrees of one of the modes of 3L 4s. Expanding the chain to 10 scale degrees produces the modes of either 7L 3s (for soft-of-basic tunings) or 3L 7s (for hard-of-basic tunings).
| 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
| UDP | Cyclic 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
One 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 the Heathwaite names.
| UDP | Cyclic order |
Step pattern |
Mode names (Heathwaite) |
Mode names (CA) |
|---|---|---|---|---|
| 6|0 | 1 | LsLsLss | Dril | Dalmatian |
| 5|1 | 3 | LsLssLs | Gil | Galatian |
| 4|2 | 5 | LssLsLs | Kleeth | Cilician |
| 3|3 | 7 | sLsLsLs | Bish | Bithynian |
| 2|4 | 2 | sLsLssL | Fish | Pisidian |
| 1|5 | 4 | sLssLsL | Jwl | Illyrian |
| 0|6 | 6 | ssLsLsL | Led | Lycian |
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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 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 |
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)
| 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 | |||||