2L 5s
| ↖ 1L 4s | ↑ 2L 4s | 3L 4s ↗ |
| ← 1L 5s | 2L 5s | 3L 5s → |
| ↙ 1L 6s | ↓ 2L 6s | 3L 6s ↘ |
┌╥┬┬╥┬┬┬┐ │║││║││││ │││││││││ └┴┴┴┴┴┴┴┘
sssLssL
2L 5s 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 and 7L 2s, not every 2L 5s scale is 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 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.
The antidiatonic scale is similar to the normal diatonic scale except interval classes are flipped. For example, there are natural, harmonic, and melodic major scales instead of minor scales, and the antilocrian scale has an augmented fifth instead of a diminished fifth. The flatter the "fifth" gets, the less the scale resembles the diatonic scale, with mavila scales generated by flat fifths appearing like a flipped diatonic, but casablanca and triton scales generated by subfifths or even tritones much less so.
Names
It is often called antidiatonic, because it has the step pattern of diatonic (5L 2s), but with the large and small steps switched. TAMNAMS uses this name as well.
Notation
We'll use the convention DEFGABCD (D Antidorian, sLsssLs); D = 293.665 Hz, &/@ = raise and lower by one chroma. The chain of mavila fifths becomes … E& B& F C G D A E B F@ C@ … Note that 7 fifths up flattens a note by a chroma, rather than sharpening it as in diatonic (5L 2s).
Intervals
Note: In TAMNAMS, a k-step interval class in antidiatonic may be called a "k-step", "k-mosstep", or "k-pelstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
Modes
Modes of the antidiatonic scale are usually named as "anti-" combined with the opposite mode of the diatonic scale, e.g. 4|2 being called "antiaeolian". CompactStar also gave original names based on regions of France to mirror how modes of the diatonic scale are named on regions of Greece and Turkey.
- 6|0 LssLsss (anti-locrian, corsican)
- 5|1 LsssLss (anti-phrygian, breton)
- 4|2 sLssLss (anti-aeolian, burgundian)
- 3|3 sLsssLs (anti-dorian, picardian)
- 2|4 ssLssLs (anti-mixolydian, norman)
- 1|5 ssLsssL (anti-ionian, provencal)
- 0|6 sssLssL (anti-lydian, alsatian)
Scale tree
Generator ranges:
- Chroma-positive generator: 514.2857 cents (3\7) to 600 cents (1\2)
- Chroma-negative generator: 600 cents (1\2) to 685.7143 cents (4\7)
| Generator | Cents | L | s | L/s | Comments | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| 3\7 | 514.286 | 1 | 1 | 1.000 | ||||||
| 16\37 | 518.919 | 6 | 5 | 1.200 | Gravity↑ | |||||
| 13\30 | 520.000 | 5 | 4 | 1.250 | ||||||
| 23\53 | 520.755 | 9 | 7 | 1.286 | ||||||
| 10\23 | 521.739 | 4 | 3 | 1.333 | ||||||
| 27\62 | 522.581 | 11 | 8 | 1.375 | ||||||
| 17\39 | 523.077 | 7 | 5 | 1.400 | ||||||
| 24\55 | 523.636 | 10 | 7 | 1.428 | ||||||
| 7\16 | 525.000 | 3 | 2 | 1.500 | Mavila is in this region | |||||
| 25\57 | 526.316 | 11 | 7 | 1.571 | ||||||
| 18\41 | 526.829 | 8 | 5 | 1.600 | ||||||
| 29\66 | 527.273 | 13 | 8 | 1.625 | Golden mavila (527.1497¢) | |||||
| 11\25 | 528.000 | 5 | 3 | 1.667 | ||||||
| 26\59 | 528.814 | 12 | 7 | 1.714 | ||||||
| 15\34 | 529.412 | 7 | 4 | 1.750 | ||||||
| 19\43 | 530.233 | 9 | 5 | 1.800 | Mabila / Amavil | |||||
| 4\9 | 533.333 | 2 | 1 | 2.000 | Basic antidiatonic (Generators smaller than this are proper) | |||||
| 17\38 | 536.842 | 9 | 4 | 2.250 | ||||||
| 13\29 | 537.931 | 7 | 3 | 2.333 | ||||||
| 22\49 | 538.776 | 12 | 5 | 2.400 | ||||||
| 9\20 | 540.000 | 5 | 2 | 2.500 | Score | |||||
| 23\51 | 541.176 | 13 | 5 | 2.600 | Unnamed golden tuning (541.3837¢) | |||||
| 14\31 | 541.935 | 8 | 3 | 2.667 | Casablanca is around here | |||||
| 19\42 | 542.857 | 11 | 4 | 2.750 | ||||||
| 5\11 | 545.455 | 3 | 1 | 3.000 | ||||||
| 16\35 | 548.571 | 10 | 3 | 3.333 | ||||||
| 11\24 | 550.000 | 7 | 2 | 3.500 | ||||||
| 17\37 | 551.351 | 11 | 3 | 3.667 | Freivald / emka | |||||
| 6\13 | 553.846 | 4 | 1 | 4.000 | ||||||
| 13\28 | 557.143 | 9 | 2 | 4.500 | ||||||
| 7\15 | 560.000 | 5 | 1 | 5.000 | Thuja is around here | |||||
| 8\17 | 564.706 | 6 | 1 | 6.000 | Liese↓, triton↓ | |||||
| 1\2 | 600.000 | 1 | 0 | → inf | ||||||