2L 8s

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2L 8s
Pattern ssLssssLss
Period 1\2
Generator range 0\2 (0.0¢) to 1\10 (120.0¢)
Parent MOS 2L 6s
Daughter MOSes 10L 2s, 2L 10s
Sister MOS 8L 2s
TAMNAMS name antidimanic
Equal tunings
Supersoft (L:s = 4:3) 3\32 (112.5¢)
Soft (L:s = 3:2) 2\22 (109.1¢)
Semisoft (L:s = 5:3) 3\34 (105.9¢)
Basic (L:s = 2:1) 1\12 (100.0¢)
Semihard (L:s = 5:2) 2\26 (92.3¢)
Hard (L:s = 3:1) 1\14 (85.7¢)
Superhard (L:s = 4:1) 1\16 (75.0¢)

2L 8s is the MOS pattern of the decatonic scale of Paul Erlich and others.

The only significant harmonic entropy minimum that is proper is the decatonic scale itself (pajara[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes 4/3, and another generator down makes 5/4. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:

http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf

http://www.lumma.org/tuning/erlich/

Improper harmonic entropy minima include injera (which is similar to pajara except that 5/4 is now four generators up and no periods) and shrutar (which is basically pajara with the generator divided in two).

In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the "symmetric" scale and LsssssLsss is called the "pentachordal" scale (because it has two identical "pentachords" in the same way that the diatonic scale has two identical tetrachords).

Notation

The notation used in this article is ssLssssLss = JKLMNOPQRSJ 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 12edo gamut is as follows:

J K L L&/M@ M N O P Q Q&/R@ R S J

Scale tree

Generator Cents L s L/s Comments
Chroma-positive Chroma-negative
4\10 480.000 120.000 1 1 1.000
21\52 484.615 115.385 6 5 1.200 Semimiracle↑
17\42 485.714 114.286 5 4 1.250
30\74 486.486 113.514 9 7 1.286
13\32 487.500 112.500 4 3 1.333
35\86 488.372 111.628 11 8 1.375
22\54 488.889 111.111 7 5 1.400
31\76 489.474 110.526 10 7 1.428
9\22 490.909 107.692 3 2 1.500 L/s = 3/2, pajara
32\78 492.308 131.250 11 7 1.571
23\56 492.857 107.143 8 5 1.600 Keen
37\90 493.333 106.667 13 8 1.625 Golden pajaroid
14\34 494.118 105.882 5 3 1.667
33\80 495.000 105.000 12 7 1.714 Srutal
19\46 495.652 104.348 7 4 1.750
24\58 496.552 103.448 9 5 1.800 Diaschismic
5\12 500.000 100.000 2 1 2.000 Basic 2L 8s
(Generators smaller than this are proper)
21\50 504.000 96.000 9 4 2.250 Bimeantone
16\38 505.263 94.737 7 3 2.333
27\64 506.250 93.750 12 5 2.400
11\26 507.692 92.308 5 2 2.500 Injera
28\66 509.091 90.909 13 5 2.600 Golden injeroid
17\40 510.000 90.000 8 3 2.667
23\54 511.111 88.889 11 4 2.750
6\14 514.286 85.714 3 1 3.000 L/s = 3/1
19\44 518.182 81.818 10 3 3.333
13\30 520.000 80.000 7 2 3.500
20\46 521.739 78.261 11 3 3.667
7\16 525.000 75.000 4 1 4.000
15\34 529.412 70.588 9 2 4.500 Vishnuzmic (incomplete)
8\18 533.333 66.667 5 1 5.000
9\20 540.000 60.000 6 1 6.000 Shrutar, teff/pombe (incomplete)↓
1\2 600.000 0.000 1 0 → inf