2L 5s
2L 5s refers to a Moment of Symmetry scale with two large steps and five small steps. Common names for such a tuning are mavila and anti-diatonic. The generator is a sharp fourth (or flat fifth), falling between 3\7 (3 degrees of 7edo) and 1\2 (1 degree of 2edo — the half-octave tritone that appears in every even-numbered edo).
The word "mavila" is used in different ways by different folks. Not every user of the word would consider every 2L 5s scale an instance of "mavila." In particular, between 13\29 and 14\31, and centered on 9\20, is the albitonic scale for the 2.7.11.13 subgroup temperament score, which is not intended to be treated as having any kind of fifth, flat or otherwise.
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 | |||||||
| 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 | — | |||||||