# 3L 4s

↖2L 3s | ↑3L 3s | 4L 3s↗ |

←2L 4s | 3L 4s | 4L 4s→ |

↙2L 5s | ↓3L 5s | 4L 5s↘ |

**Equal tunings****3L 4s** is the MOS scale built from a generator that falls between 1\3 (one degree of 3edo – 400 cents) and 2\7 (two degrees of 7edo – 343 cents).

## Standing assumptions

The TAMNAMS system is used in this article to name 3L 4s intervals and step size ratios and step ratio ranges.

The notation used in this article is sLsLsLs = JKLMNOPJ unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)

Thus the 10edo gamut is as follows:

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

## Names

TAMNAMS (used by this article) uses the name **mosh** for this pattern. This is taken from an older MOS naming scheme by Graham Breed.

## Intervals

Note: In TAMNAMS, a k-step interval class in mosh may be called a "k-step", "k-mosstep", or "k-moshstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

## Low-harmonic-entropy scales

The two notable harmonic entropy minima with this pattern are neutral third scales ("dicot" / "hemififth" / "mohajira") where two generators make a 3/2, and magic, where the generator is a 5/4 but five of them make a 3/1.

## 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:

UDP | Step pattern | Mode names |
---|---|---|

6|0 | LsLsLss | dril |

5|1 | LsLssLs | gil |

4|2 | LssLsLs | kleeth |

3|3 | sLsLsLs | bish |

2|4 | sLsLssL | fish |

1|5 | sLssLsL | jwl |

0|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 |