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3L 4s

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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 or mosh 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).

Notation

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

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 minor and major mosthirds near 6/5 and 11/9, respectively

The sizes of the generator, large step and small step of mosh are as follows in various ultrasoft mosh tunings.

24edo 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.

17edo 27edo 44edo
generator (g) 5\17, 352.94 8\27, 355.56 13\44, 354.55
L (4g - octave) 3\17, 211.76 5\27, 222.22 8\44, 218.18
s (octave - 3g) 2\17, 141.18 3\27, 133.33 5\44, 137.37

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.

10edo 13edo 23edo
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.

16edo 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:

Mode UDP Nickname
s L s L s L s 3|3 bish
L s L s L s s 6|0 dril
s L s L s s L 2|4 fish
L s L s s L s 5|1 gil
s L s s L s L 1|5 jwl
L s s L s L s 4|2 kleeth
s s L s L s L 0|6 led

Scale tree

The spectrum looks like this:

g 2g 3g 4g (-1200) comments
1\3 400.000 800.000 1200.000 400.000
15\46 391.304 782.609 1173.913 365.217
14\43 390.698 781.395 1172.093 362.791
13\40 390.000 780.000 1170.000 360.000
12\37 389.189 778.378 1167.568 356.757
11\34 388.235 776.471 1164.706 352.941
10\31 387.097 774.194 1161.290 348.387
19\59 386.441 772.881 1159.322 345.763
9\28 385.714 771.429 1157.143 342.857
8\25 384.000 768.000 1152.000 336.000
23\72 383.333 766.667 1150.000 333.333
15\47 382.988 765.957 1148.936 331.915
7\22 381.818 763.636 1145.455 327.273
13\41 380.488 760.976 1141.463 321.951
19\60 380.000 760.000 1140.000 320.000
25\79 379.747 759.494 1139.2405 318.987
6\19 378.947 757.895 1136.842 315.789
11\35 377.143 754.286 1131.429 308.571
16\51 376.471 752.941 1129.412 305.882
5\16 375.000 750.000 1125.000 300.000
24\77 374.026 748.052 1122.078 296.104
19\61 373.7705 747.541 1121.3115 295.082
14\45 373.333 746.667 1120.000 293.333
9\29 372.414 744.828 1117.241 289.655
13\42 371.429 742.857 1114.286 285.714
17\55 370.909 741.818 1112.727 283.636
4\13 369.231 738.462 1107.692 276.923 L/s = 3
23\75 368.000 736.000 1104.000 272.000

19\62 367.742 735.484 1103.226 270.968
15\49 367.347 734.694 1102.041 269.388
11\36 366.667 733.333 1100.000 266.667
366.256 732.513 1198.77 265.026
7\23 365.217 730.435 1095.652 260.870
17\56 364.286 728.571 1092.857 257.143
10\33 363.636 727.272 1090.909 254.545
13\43 362.791 725.581 1088.372 251.163
16\53 362.264 724.528 1086.7925 249.057
19\63 361.905 723.8095 1085.714 247.619
3\10 360.000 720.000 1080.000 240.000 Boundary of propriety(generators smaller than this are proper)
38\127 359.055 718.110 1077.165 236.2205
35\117 358.974 717.949 1076.923 235.898
32\107 358.8785 717.757 1076.6355 235.514
29\97 358.763 717.526 1076.289 235.0515
26\87 358.621 717.241 1075.862 234.483
23\77 358.442 716.883 1075.325 233.767
20\67 358.209 716.418 1074.627 232.836
17\57 357.895 715.7895 1073.684 231.579
14\47 357.447 714.894 1072.340 229.787
11\37 356.757 713.514 1070.270 227.027
356.5035 713.007 1069.511 226.014
8\27 355.556 711.111 1066.667 222.222
354.930 709.859 1064.789 219.718 Golden mosh
21\71 354.783 709.565 1064.348 219.13
13\44 354.5455 709.091 1063.636 218.182
354.088 708.177 1062.266 216.354
5\17 352.941 705.882 1058.824 211.765 Optimum rank range (L/s=3/2)
12\41 351.220 702.439 1053.659 204.878
7\24 350.000 700.000 1050.000 200.000
16\55 349.091 698.182 1047.273 196.364
9\31 348.387 696.774 1045.161 193.548
11\38 347.368 694.737 1042.105 189.474
2\7 342.857 685.714 1028.571 171.429
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