2L 8s
| ↖1L 7s | ↑2L 7s | 3L 7s↗ |
| ←1L 8s | 2L 8s | 3L 8s→ |
| ↙1L 9s | ↓2L 9s | 3L 9s↘ |
┌╥┬┬┬┬╥┬┬┬┬┐ │║││││║│││││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
ssssLssssL
2L 10s
2L 8s, named jaric in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 8 small steps, with a period of 1 large step and 4 small steps that repeats every 600.0¢, or twice every octave. Generators that produce this scale range from 480¢ to 600¢, or from 0¢ to 120¢.
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 N O P Q Q& R S J
Modes
- 8|0(2) LssssLssss, J-K&-L&-M-N-O-P&-Q&-R-S-J
- 6|2(2) sLssssLsss, J-K-L&-M-N-O-P-Q&-R-S-J
- 4|4(2) ssLssssLss, J-K-L-M-N-O-P-Q-R-S-J
- 2|6(2) sssLssssLs, J-K-L-M@-N-O-P-Q-R@-S-J
- 0|8(2) ssssLssssL, J-K-L-M@-N@-O-P-Q-R@-S@-J
Scale tree
| Generator(edo) | Cents | Step Ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 4\10 | 480.000 | 120.000 | 1:1 | 1.000 | Equalized 2L 8s | |||||
| 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 | Supersoft 2L 8s | |||||
| 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 | 1.429 | ||||||
| 9\22 | 490.909 | 109.091 | 3:2 | 1.500 | Soft 2L 8s Pajara | |||||
| 32\78 | 492.308 | 107.692 | 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 | ||||||
| 14\34 | 494.118 | 105.882 | 5:3 | 1.667 | Semisoft 2L 8s | |||||
| 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 Scales with tunings softer 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 | Semihard 2L 8s Injera | |||||
| 28\66 | 509.091 | 90.909 | 13:5 | 2.600 | ||||||
| 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 | Hard 2L 8s | |||||
| 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 | Superhard 2L 8s | |||||
| 15\34 | 529.412 | 70.588 | 9:2 | 4.500 | Vishnu (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 | → ∞ | Collapsed 2L 8s | |||||