2L 5s
2L 5s, pelogic, or antidiatonic refers to the structure of octave-equivalent MOS scales with generators ranging from 3\7 (3 degrees of 7edo = 514.29¢) to 1\2 (one degree of 2edo = 600¢). In the case of 7edo, L and s are the same size; in the case of 2edo, s becomes so small it disappears (and all that remains are the two equal L's).
While antidiatonic is closely associated with mavila temperament, not every 2L 5s scale an instance of "mavila", since some of them extend to 2L 7s scales (like the 2L 5s generated by 11edo's 6\11 = 656.5657¢), not 7L 2s mavila superdiatonic scales.
In terms of harmonic entropy, the most significant minimum is at Liese/Triton, in which the generator is about 7/5 and three of them make a 3/1.
| generator in degrees of an edo | generator in cents | tetrachord | L in cents | s in cents | L to s ratio | comments | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3\7 | 514.3 | 1 1 1 | 171.4 | 171.4 | 1.00 | |||||||
| 19\44 | 518.2 | 6 6 7 | 190.9 | 163.6 | 1.17 | |||||||
| 16\37 | 518.9 | 5 5 6 | 194.6 | 162.2 | 1.20 | |||||||
| 13\30 | 520.0 | 4 4 5 | 200.0 | 160.0 | 1.25 | Mavila extends from here... | ||||||
| 10\23 | 521.7 | 3 3 4 | 208.7 | 156.5 | 1.33 | |||||||
| 17\39 | 523.1 | 5 5 7 | 215.4 | 153.8 | 1.40 | |||||||
| 7\16 | 525.0 | 2 2 3 | 225.0 | 150.0 | 1.50 | Mavila in Armodue
Optimum rank range (L/s=3/2) | ||||||
| 526.3 | 2 2 pi | 231.5 | 147.4 | pi/2 | ||||||||
| 18\41 | 526.8 | 5 5 8 | 234.1 | 146.3 | 1.60 | |||||||
| 1200*5/(13-phi) | 1 1 phi | 235.7 | 145.7 | phi | Golden mavila | |||||||
| 29\66 | 527.3 | 8 8 13 | 236.4 | 145.5 | 1.625 | |||||||
| 11\25 | 528.0 | 3 3 5 | 240.0 | 144.0 | 1.67 | |||||||
| 529.1 | 1 1 √3 | 245.6 | 141.8 | √3 | ||||||||
| 15\34 | 529.4 | 4 4 7 | 247.1 | 141.2 | 1.75 | ...to somewhere around here | ||||||
| 4\9 | 533.3 | 1 1 2 | 266.7 | 133.3 | 2.00 | Boundary of propriety (generators
smaller than this are proper) | ||||||
| 13\29 | 537.9 | 3 3 7 | 289.7 | 124.1 | 2.33 | |||||||
| 9\20 | 540.0 | 2 2 5 | 300.0 | 120.0 | 2.50 | |||||||
| 541.4 | 1 1 phi+1 | 306.9 | 117.2 | 1 1 phi+1 | ||||||||
| 14\31 | 541.9 | 3 3 8 | 309.7 | 116.1 | 2.66 | |||||||
| 542.5 | 1 1 e | 321.55 | 115.0 | e | L/s = e | |||||||
| 5\11 | 545.5 | 1 1 3 | 327.3 | 109.1 | 3.00 | L/s = 3 | ||||||
| 546.8 | 1 1 pi | 334.1 | 106.35 | pi | L/s = pi | |||||||
| 11\24 | 550.0 | 2 2 7 | 350.0 | 100.0 | 3.50 | |||||||
| 6\13 | 553.8 | 1 1 4 | 369.2 | 92.3 | 4.00 | Thuja is optimal around here
L/s = 4 | ||||||
| 7\15 | 560.0 | 1 1 5 | 400.0 | 80.0 | 5.00 | ie. (11/8)^5 = 5/1 | ||||||
| 8\17 | 564.7 | 1 1 6 | 423.5 | 70.6 | 6.00 | |||||||
| 9\19 | 568.4 | 1 1 7 | 442.1 | 63.2 | 7.00 | Liese/Triton is around here | ||||||
| 1\2 | 600.0 | 0 0 1 | 600.0 | 0 | — | |||||||