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
| 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¢ |
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
The following mode nicknames were coined by Andrew Heathwaite:
| UDP | Cyclic 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 |
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 |
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 | ||||||