2L 5s
| ↖ 1L 4s | ↑ 2L 4s | 3L 4s ↗ |
| ← 1L 5s | 2L 5s | 3L 5s → |
| ↙ 1L 6s | ↓ 2L 6s | 3L 6s ↘ |
┌╥┬┬╥┬┬┬┐ │║││║││││ │││││││││ └┴┴┴┴┴┴┴┘
sssLssL
2L 5s, named antidiatonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 514.3 ¢ to 600 ¢, or from 600 ¢ to 685.7 ¢.
Antidiatonic is similar to diatonic except interval classes are flipped. For example, there are natural, harmonic, and melodic major scales instead of minor scales, and its locrian scale, called "antilocrian", has an augmented fifth instead of a diminished fifth. The flatter the fifth, the less this scale resembles diatonic.
The most well-known forms of this scale are produced by mavila, with fifths sharp enough to resemble diatonic. Other temperaments that produce this scale include score, casablanca, and triton, whose fifths are so flat that they cannot be interpreted as a diatonic 5th, flattened or otherwise.
Name
TAMNAMS suggests the temperament-agnostic name antidiatonic for this scale, adopted from the common use of the term to refer to diatonic (5L 2s) but with the large and small steps switched.
The scale is also often called peletonic or peltonic, based on its prefix.
Intervals
- This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.
| Intervals | Steps subtended |
Range in cents | ||
|---|---|---|---|---|
| Generic | Specific | Abbrev. | ||
| 0-pelstep | Perfect 0-pelstep | P0pels | 0 | 0.0 ¢ |
| 1-pelstep | Minor 1-pelstep | m1pels | s | 0.0 ¢ to 171.4 ¢ |
| Major 1-pelstep | M1pels | L | 171.4 ¢ to 600.0 ¢ | |
| 2-pelstep | Minor 2-pelstep | m2pels | 2s | 0.0 ¢ to 342.9 ¢ |
| Major 2-pelstep | M2pels | L + s | 342.9 ¢ to 600.0 ¢ | |
| 3-pelstep | Diminished 3-pelstep | d3pels | 3s | 0.0 ¢ to 514.3 ¢ |
| Perfect 3-pelstep | P3pels | L + 2s | 514.3 ¢ to 600.0 ¢ | |
| 4-pelstep | Perfect 4-pelstep | P4pels | L + 3s | 600.0 ¢ to 685.7 ¢ |
| Augmented 4-pelstep | A4pels | 2L + 2s | 685.7 ¢ to 1200.0 ¢ | |
| 5-pelstep | Minor 5-pelstep | m5pels | L + 4s | 600.0 ¢ to 857.1 ¢ |
| Major 5-pelstep | M5pels | 2L + 3s | 857.1 ¢ to 1200.0 ¢ | |
| 6-pelstep | Minor 6-pelstep | m6pels | L + 5s | 600.0 ¢ to 1028.6 ¢ |
| Major 6-pelstep | M6pels | 2L + 4s | 1028.6 ¢ to 1200.0 ¢ | |
| 7-pelstep | Perfect 7-pelstep | P7pels | 2L + 5s | 1200.0 ¢ |
Notation
The most common way of notating this scale, particularly when working with mavila, is to use the same note names and accidentals as that of diatonic (CDEFGAB, #, and b), but read as antidiatonic instead.
2L 5s can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.
The first, melodic notation, defines sharp/flat, major/minor, and aug/dim in terms of the antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. M2 + M2 is not M3, and D + M2 is not E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Note that the notes that form chords are different from in diatonic: for example, a major chord, P1–M3–P5, is approximately 4:5:6 as would be expected, but is notated C–E♯–G on C.
Alternatively, one can essentially pretend the antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim, known as harmonic notation. The primary purpose of doing this is to allow music notated in 12edo or another diatonic system to be directly translated on the fly, or to allow support for 2L 5s in tools that only allow chain-of-fifths notation, and it carries over the way interval arithmetic works from diatonic notation, at the cost of notating the sizes of intervals and the shapes of chords incorrectly: that is, a major chord, P1–M3–P5, is notated C–E–G on C, but is no longer ~4:5:6 (since the third is closer to a minor third).
For the sake of clarity, the first notation is commonly called melodic notation, and the second is called harmonic notation, but this is a bit of a misnomer as both preserve different features of the notation of harmony.
| Notation | P1–M3–P5 ~ 4:5:6 | P1–M3–P5 = C–E–G on C |
|---|---|---|
| Diatonic | No | Yes |
| Antidiatonic | Yes | No |
Under harmonic antidiatonic notation, the basic gamut (for D anti-dorian) is the following: D, E, Eb/F#, F, G, A, B, Bb/C#, C, D.
Under melodic antidiatonic notation, the basic gamut is the following: D, E, E#/Fb, F, G, A, B, B#/Cb, C, D.
Theory
Low harmonic entropy scales
There is one notable harmonic entropy minimum: Liese/triton, in which the generator is 10/7 (632.5 ¢) and three of them make a 3/1 (1897.6 ¢).
Temperament interpretations
2L 5s has several rank-2 temperament interpretations, such as:
- Mavila, with generators around 679.8 ¢.
- Casablanca, with generators around 657.8 ¢.
- Liese, with generators around 632.4 ¢.
Tuning ranges
Simple tunings
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 9edo, 11edo, and 16edo.
| Scale degree | Abbrev. | Basic (2:1) 9edo |
Hard (3:1) 11edo |
Soft (3:2) 16edo | |||
|---|---|---|---|---|---|---|---|
| Steps | ¢ | Steps | ¢ | Steps | ¢ | ||
| Perfect 0-peldegree | P0peld | 0\9 | 0.0 | 0\11 | 0.0 | 0\16 | 0.0 |
| Minor 1-peldegree | m1peld | 1\9 | 133.3 | 1\11 | 109.1 | 2\16 | 150.0 |
| Major 1-peldegree | M1peld | 2\9 | 266.7 | 3\11 | 327.3 | 3\16 | 225.0 |
| Minor 2-peldegree | m2peld | 2\9 | 266.7 | 2\11 | 218.2 | 4\16 | 300.0 |
| Major 2-peldegree | M2peld | 3\9 | 400.0 | 4\11 | 436.4 | 5\16 | 375.0 |
| Diminished 3-peldegree | d3peld | 3\9 | 400.0 | 3\11 | 327.3 | 6\16 | 450.0 |
| Perfect 3-peldegree | P3peld | 4\9 | 533.3 | 5\11 | 545.5 | 7\16 | 525.0 |
| Perfect 4-peldegree | P4peld | 5\9 | 666.7 | 6\11 | 654.5 | 9\16 | 675.0 |
| Augmented 4-peldegree | A4peld | 6\9 | 800.0 | 8\11 | 872.7 | 10\16 | 750.0 |
| Minor 5-peldegree | m5peld | 6\9 | 800.0 | 7\11 | 763.6 | 11\16 | 825.0 |
| Major 5-peldegree | M5peld | 7\9 | 933.3 | 9\11 | 981.8 | 12\16 | 900.0 |
| Minor 6-peldegree | m6peld | 7\9 | 933.3 | 8\11 | 872.7 | 13\16 | 975.0 |
| Major 6-peldegree | M6peld | 8\9 | 1066.7 | 10\11 | 1090.9 | 14\16 | 1050.0 |
| Perfect 7-peldegree | P7peld | 9\9 | 1200.0 | 11\11 | 1200.0 | 16\16 | 1200.0 |
Soft-of-basic tunings
Much of the range for soft-of-basic antidiatonic tunings (1:1 to 2:1) corresponds to mavila temperament. Edos include 9edo (not shown), 16edo, and 23edo.
| Scale degree | Abbrev. | Supersoft (4:3) 23edo |
Soft (3:2) 16edo |
Basic (2:1) 9edo | |||
|---|---|---|---|---|---|---|---|
| Steps | ¢ | Steps | ¢ | Steps | ¢ | ||
| Perfect 0-peldegree | P0peld | 0\23 | 0.0 | 0\16 | 0.0 | 0\9 | 0.0 |
| Minor 1-peldegree | m1peld | 3\23 | 156.5 | 2\16 | 150.0 | 1\9 | 133.3 |
| Major 1-peldegree | M1peld | 4\23 | 208.7 | 3\16 | 225.0 | 2\9 | 266.7 |
| Minor 2-peldegree | m2peld | 6\23 | 313.0 | 4\16 | 300.0 | 2\9 | 266.7 |
| Major 2-peldegree | M2peld | 7\23 | 365.2 | 5\16 | 375.0 | 3\9 | 400.0 |
| Diminished 3-peldegree | d3peld | 9\23 | 469.6 | 6\16 | 450.0 | 3\9 | 400.0 |
| Perfect 3-peldegree | P3peld | 10\23 | 521.7 | 7\16 | 525.0 | 4\9 | 533.3 |
| Perfect 4-peldegree | P4peld | 13\23 | 678.3 | 9\16 | 675.0 | 5\9 | 666.7 |
| Augmented 4-peldegree | A4peld | 14\23 | 730.4 | 10\16 | 750.0 | 6\9 | 800.0 |
| Minor 5-peldegree | m5peld | 16\23 | 834.8 | 11\16 | 825.0 | 6\9 | 800.0 |
| Major 5-peldegree | M5peld | 17\23 | 887.0 | 12\16 | 900.0 | 7\9 | 933.3 |
| Minor 6-peldegree | m6peld | 19\23 | 991.3 | 13\16 | 975.0 | 7\9 | 933.3 |
| Major 6-peldegree | M6peld | 20\23 | 1043.5 | 14\16 | 1050.0 | 8\9 | 1066.7 |
| Perfect 7-peldegree | P7peld | 23\23 | 1200.0 | 16\16 | 1200.0 | 9\9 | 1200.0 |
Hypohard tunings
The range of hard-of-basic tunings correspond to temperaments that have significantly flattened antidiatonic 5ths, such as score and casablanca. 20edo and 31edo represent these two temperaments quite well.
| Scale degree | Abbrev. | Basic (2:1) 9edo |
Semihard (5:2) 20edo |
Hard (3:1) 11edo | |||
|---|---|---|---|---|---|---|---|
| Steps | ¢ | Steps | ¢ | Steps | ¢ | ||
| Perfect 0-peldegree | P0peld | 0\9 | 0.0 | 0\20 | 0.0 | 0\11 | 0.0 |
| Minor 1-peldegree | m1peld | 1\9 | 133.3 | 2\20 | 120.0 | 1\11 | 109.1 |
| Major 1-peldegree | M1peld | 2\9 | 266.7 | 5\20 | 300.0 | 3\11 | 327.3 |
| Minor 2-peldegree | m2peld | 2\9 | 266.7 | 4\20 | 240.0 | 2\11 | 218.2 |
| Major 2-peldegree | M2peld | 3\9 | 400.0 | 7\20 | 420.0 | 4\11 | 436.4 |
| Diminished 3-peldegree | d3peld | 3\9 | 400.0 | 6\20 | 360.0 | 3\11 | 327.3 |
| Perfect 3-peldegree | P3peld | 4\9 | 533.3 | 9\20 | 540.0 | 5\11 | 545.5 |
| Perfect 4-peldegree | P4peld | 5\9 | 666.7 | 11\20 | 660.0 | 6\11 | 654.5 |
| Augmented 4-peldegree | A4peld | 6\9 | 800.0 | 14\20 | 840.0 | 8\11 | 872.7 |
| Minor 5-peldegree | m5peld | 6\9 | 800.0 | 13\20 | 780.0 | 7\11 | 763.6 |
| Major 5-peldegree | M5peld | 7\9 | 933.3 | 16\20 | 960.0 | 9\11 | 981.8 |
| Minor 6-peldegree | m6peld | 7\9 | 933.3 | 15\20 | 900.0 | 8\11 | 872.7 |
| Major 6-peldegree | M6peld | 8\9 | 1066.7 | 18\20 | 1080.0 | 10\11 | 1090.9 |
| Perfect 7-peldegree | P7peld | 9\9 | 1200.0 | 20\20 | 1200.0 | 11\11 | 1200.0 |
Ultrahard tunings
Ultrahard tunings, particularly with the harder end of the spectrum, correspond to liese temperament, represent by edos such as 17edo, 19edo, and larger edos such as 55edo.
| Scale degree | Abbrev. | Superhard (4:1) 13edo |
5:1 15edo |
6:1 17edo |
7:1 19edo | ||||
|---|---|---|---|---|---|---|---|---|---|
| Steps | ¢ | Steps | ¢ | Steps | ¢ | Steps | ¢ | ||
| Perfect 0-peldegree | P0peld | 0\13 | 0.0 | 0\15 | 0.0 | 0\17 | 0.0 | 0\19 | 0.0 |
| Minor 1-peldegree | m1peld | 1\13 | 92.3 | 1\15 | 80.0 | 1\17 | 70.6 | 1\19 | 63.2 |
| Major 1-peldegree | M1peld | 4\13 | 369.2 | 5\15 | 400.0 | 6\17 | 423.5 | 7\19 | 442.1 |
| Minor 2-peldegree | m2peld | 2\13 | 184.6 | 2\15 | 160.0 | 2\17 | 141.2 | 2\19 | 126.3 |
| Major 2-peldegree | M2peld | 5\13 | 461.5 | 6\15 | 480.0 | 7\17 | 494.1 | 8\19 | 505.3 |
| Diminished 3-peldegree | d3peld | 3\13 | 276.9 | 3\15 | 240.0 | 3\17 | 211.8 | 3\19 | 189.5 |
| Perfect 3-peldegree | P3peld | 6\13 | 553.8 | 7\15 | 560.0 | 8\17 | 564.7 | 9\19 | 568.4 |
| Perfect 4-peldegree | P4peld | 7\13 | 646.2 | 8\15 | 640.0 | 9\17 | 635.3 | 10\19 | 631.6 |
| Augmented 4-peldegree | A4peld | 10\13 | 923.1 | 12\15 | 960.0 | 14\17 | 988.2 | 16\19 | 1010.5 |
| Minor 5-peldegree | m5peld | 8\13 | 738.5 | 9\15 | 720.0 | 10\17 | 705.9 | 11\19 | 694.7 |
| Major 5-peldegree | M5peld | 11\13 | 1015.4 | 13\15 | 1040.0 | 15\17 | 1058.8 | 17\19 | 1073.7 |
| Minor 6-peldegree | m6peld | 9\13 | 830.8 | 10\15 | 800.0 | 11\17 | 776.5 | 12\19 | 757.9 |
| Major 6-peldegree | M6peld | 12\13 | 1107.7 | 14\15 | 1120.0 | 16\17 | 1129.4 | 18\19 | 1136.8 |
| Perfect 7-peldegree | P7peld | 13\13 | 1200.0 | 15\15 | 1200.0 | 17\17 | 1200.0 | 19\19 | 1200.0 |
Modes
| UDP | Cyclic order |
Step pattern |
Scale degree (peldegree) | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||
| 6|0 | 1 | LssLsss | Perf. | Maj. | Maj. | Perf. | Aug. | Maj. | Maj. | Perf. |
| 5|1 | 4 | LsssLss | Perf. | Maj. | Maj. | Perf. | Perf. | Maj. | Maj. | Perf. |
| 4|2 | 7 | sLssLss | Perf. | Min. | Maj. | Perf. | Perf. | Maj. | Maj. | Perf. |
| 3|3 | 3 | sLsssLs | Perf. | Min. | Maj. | Perf. | Perf. | Min. | Maj. | Perf. |
| 2|4 | 6 | ssLssLs | Perf. | Min. | Min. | Perf. | Perf. | Min. | Maj. | Perf. |
| 1|5 | 2 | ssLsssL | Perf. | Min. | Min. | Perf. | Perf. | Min. | Min. | Perf. |
| 0|6 | 5 | sssLssL | Perf. | Min. | Min. | Dim. | Perf. | Min. | Min. | Perf. |
Proposed names
Modes of antidiatonic are usually named as "anti-" combined with the corresponding mode of the diatonic scale, where anti-locrian is the brightest mode and anti-lydian is the darkest mode. 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.
| UDP | Cyclic order |
Step pattern |
Mode names | CompactStar's names |
|---|---|---|---|---|
| 6|0 | 1 | LssLsss | Anti-locrian | Corsican |
| 5|1 | 4 | LsssLss | Anti-phrygian | Breton |
| 4|2 | 7 | sLssLss | Anti-aeolian | Burgundian |
| 3|3 | 3 | sLsssLs | Anti-dorian | Picardian |
| 2|4 | 6 | ssLssLs | Anti-mixolydian | Norman |
| 1|5 | 2 | ssLsssL | Anti-ionian | Provencal |
| 0|6 | 5 | sssLssL | Anti-lydian | Alsatian |
Scale tree
| Generator(edo) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 3\7 | 514.286 | 685.714 | 1:1 | 1.000 | Equalized 2L 5s | |||||
| 16\37 | 518.919 | 681.081 | 6:5 | 1.200 | Gravity ↑ | |||||
| 13\30 | 520.000 | 680.000 | 5:4 | 1.250 | ||||||
| 23\53 | 520.755 | 679.245 | 9:7 | 1.286 | ||||||
| 10\23 | 521.739 | 678.261 | 4:3 | 1.333 | Supersoft 2L 5s | |||||
| 27\62 | 522.581 | 677.419 | 11:8 | 1.375 | ||||||
| 17\39 | 523.077 | 676.923 | 7:5 | 1.400 | ||||||
| 24\55 | 523.636 | 676.364 | 10:7 | 1.429 | ||||||
| 7\16 | 525.000 | 675.000 | 3:2 | 1.500 | Soft 2L 5s Mavila | |||||
| 25\57 | 526.316 | 673.684 | 11:7 | 1.571 | ||||||
| 18\41 | 526.829 | 673.171 | 8:5 | 1.600 | ||||||
| 29\66 | 527.273 | 672.727 | 13:8 | 1.625 | Golden mavila (527.1497 ¢) | |||||
| 11\25 | 528.000 | 672.000 | 5:3 | 1.667 | Semisoft 2L 5s | |||||
| 26\59 | 528.814 | 671.186 | 12:7 | 1.714 | ||||||
| 15\34 | 529.412 | 670.588 | 7:4 | 1.750 | ||||||
| 19\43 | 530.233 | 669.767 | 9:5 | 1.800 | Mabila/amavil | |||||
| 4\9 | 533.333 | 666.667 | 2:1 | 2.000 | Basic 2L 5s Scales with tunings softer than this are proper Pelog | |||||
| 17\38 | 536.842 | 663.158 | 9:4 | 2.250 | ||||||
| 13\29 | 537.931 | 662.069 | 7:3 | 2.333 | ||||||
| 22\49 | 538.776 | 661.224 | 12:5 | 2.400 | ||||||
| 9\20 | 540.000 | 660.000 | 5:2 | 2.500 | Semihard 2L 5s Score | |||||
| 23\51 | 541.176 | 658.824 | 13:5 | 2.600 | Unnamed golden tuning (541.3837 ¢) | |||||
| 14\31 | 541.935 | 658.065 | 8:3 | 2.667 | Casablanca | |||||
| 19\42 | 542.857 | 657.143 | 11:4 | 2.750 | ||||||
| 5\11 | 545.455 | 654.545 | 3:1 | 3.000 | Hard 2L 5s | |||||
| 16\35 | 548.571 | 651.429 | 10:3 | 3.333 | ||||||
| 11\24 | 550.000 | 650.000 | 7:2 | 3.500 | ||||||
| 17\37 | 551.351 | 648.649 | 11:3 | 3.667 | Freivald/emka | |||||
| 6\13 | 553.846 | 646.154 | 4:1 | 4.000 | Superhard 2L 5s | |||||
| 13\28 | 557.143 | 642.857 | 9:2 | 4.500 | ||||||
| 7\15 | 560.000 | 640.000 | 5:1 | 5.000 | ||||||
| 8\17 | 564.706 | 635.294 | 6:1 | 6.000 | Liese ↓, triton ↓ | |||||
| 1\2 | 600.000 | 600.000 | 1:0 | → ∞ | Collapsed 2L 5s | |||||