2L 5s

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↖ 1L 4s ↑ 2L 4s 3L 4s ↗
← 1L 5s 2L 5s 3L 5s →
↙ 1L 6s ↓ 2L 6s 3L 6s ↘ 
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Scale structure
Step pattern LssLsss
sssLssL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 3\7 to 1\2 (514.3¢ to 600.0¢)
Dark 1\2 to 4\7 (600.0¢ to 685.7¢)
TAMNAMS information
Name antidiatonic
Prefix pel-
Abbrev. pel
Related MOS scales
Parent 2L 3s
Sister 5L 2s
Daughters 7L 2s, 2L 7s
Neutralized 4L 3s
2-Flought 9L 5s, 2L 12s
Equal tunings
Equalized (L:s = 1:1) 3\7 (514.3¢)
Supersoft (L:s = 4:3) 10\23 (521.7¢)
Soft (L:s = 3:2) 7\16 (525.0¢)
Semisoft (L:s = 5:3) 11\25 (528.0¢)
Basic (L:s = 2:1) 4\9 (533.3¢)
Semihard (L:s = 5:2) 9\20 (540.0¢)
Hard (L:s = 3:1) 5\11 (545.5¢)
Superhard (L:s = 4:1) 6\13 (553.8¢)
Collapsed (L:s = 1:0) 1\2 (600.0¢)

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 temperament, 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.

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 diatonic interval categories.
Intervals of 2L 5s
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 temperament, is to use the same note names and accidentals as that of diatonic (CDEFGAB, #, and b), but read as antidiatonic instead. There are, however, two ways of notating accidentals:

  • Harmonic antidiatonic notation, where the sharps and flats of diatonic switch roles: sharps flatten and flats sharpen.
  • Melodic antidiatonic notation, where the meaning of sharps and flats is preserved: sharps sharpen and flats flatten.

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.

Lua error in Module:MOS_tunings at line 299: attempt to call field 'search_by_args_within_equave' (a nil value).

Soft-of-basic tunings

Main article: Mavila

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.

Lua error in Module:MOS_tunings at line 299: attempt to call field 'search_by_args_within_equave' (a nil value).

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.

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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.

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Modes

Scale degrees of the modes of 2L 5s 
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.

Modes of 2L 5s
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

Scale Tree and Tuning Spectrum of 2L 5s
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
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
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