2L 5s

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↖ 1L 4s ↑2L 4s 3L 4s ↗
← 1L 5s2L 5s 3L 5s →
↙ 1L 6s ↓2L 6s 3L 6s ↘
┌╥┬┬╥┬┬┬┐
│║││║││││
│││││││││
└┴┴┴┴┴┴┴┘
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 assumes TAMNAMS for naming step ratios, mossteps, and mosdegrees.

Names for this scale's intervals are the same as that as diatonic intervals (prefixed with pel- to distinguish them from diatonic intervals), but with the harmonic qualities of diatonic switched: major and minor are switched, as are augmented and diminished.

Intervals of 2L 5s
Intervals Steps subtended Range in cents Average of HE
(from HE Calc)
Min of HE
Generic[1] Specific[2] Abbrev.[3]
0-pelstep Perfect 0-pelstep P0ms 0 0.0¢ ~2.4654 nats ~2.4654 nats
1-pelstep Minor 1-pelstep m1ms s 0.0¢ to 171.4¢ ~4.6448 nats ~4.6172 nats
Major 1-pelstep M1ms L 171.4¢ to 600.0¢ ~4.5726 nats ~4.5616 nats
2-pelstep Minor 2-pelstep m2ms 2s 0.0¢ to 342.9¢ ~4.5733 nats ~4.5385 nats
Major 2-pelstep M2ms L + s 342.9¢ to 600.0¢ ~4.5641 nats ~4.4885 nats
3-pelstep Diminished 3-pelstep d3ms 3s 0.0¢ to 514.3¢ ~4.5816 nats ~4.5262 nats
Perfect 3-pelstep P3ms L + 2s 514.3¢ to 600.0¢ ~4.6059 nats ~4.5553 nats
4-pelstep Perfect 4-pelstep P4ms L + 3s 600.0¢ to 685.7¢ ~4.6106 nats ~4.5237 nats
Augmented 4-pelstep A4ms 2L + 2s 685.7¢ to 1200.0¢ ~4.5886 nats ~4.4823 nats
5-pelstep Minor 5-pelstep m5ms L + 4s 600.0¢ to 857.1¢ ~4.6003 nats ~4.5712 nats
Major 5-pelstep M5ms 2L + 3s 857.1¢ to 1200.0¢ ~4.5427 nats ~4.4212 nats
6-pelstep Minor 6-pelstep m6ms L + 5s 600.0¢ to 1028.6¢ ~4.5499 nats ~4.4823 nats
Major 6-pelstep M6ms 2L + 4s 1028.6¢ to 1200.0¢ ~4.6107 nats ~4.6048 nats
7-pelstep Perfect 7-pelstep P7ms 2L + 5s 1200.0¢ ~3.3273 nats ~3.3273 nats

  1. Generic intervals are denoted solely by the number of steps they subtend.
  2. Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
  3. Abbreviations can be further shortened to 'ms' if context allows.

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 7/5 (582.5¢) and three of them make a 3/1 (1902¢).

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 of 2L 5s
Scale degree 9edo (Basic, L:s = 2:1) 11edo (Hard, L:s = 3:1) 16edo (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-peldegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-peldegree 1 133.3 1 109.1 2 150
Major 1-peldegree 2 266.7 3 327.3 3 225
Minor 2-peldegree 2 266.7 2 218.2 4 300
Major 2-peldegree 3 400 4 436.4 5 375
Diminished 3-peldegree 3 400 3 327.3 6 450
Perfect 3-peldegree 4 533.3 5 545.5 7 525
Perfect 4-peldegree 5 666.7 6 654.5 9 675
Augmented 4-peldegree 6 800 8 872.7 10 750
Minor 5-peldegree 6 800 7 763.6 11 825
Major 5-peldegree 7 933.3 9 981.8 12 900
Minor 6-peldegree 7 933.3 8 872.7 13 975
Major 6-peldegree 8 1066.7 10 1090.9 14 1050
Perfect 7-peldegree (octave) 9 1200 11 1200 16 1200 2/1 (exact)

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 of 2L 5s
Scale degree 16edo (Soft, L:s = 3:2) 23edo (Supersoft, L:s = 4:3) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-peldegree (unison) 0 0 0 0 1/1 (exact)
Minor 1-peldegree 2 150 3 156.5
Major 1-peldegree 3 225 4 208.7
Minor 2-peldegree 4 300 6 313
Major 2-peldegree 5 375 7 365.2
Diminished 3-peldegree 6 450 9 469.6
Perfect 3-peldegree 7 525 10 521.7
Perfect 4-peldegree 9 675 13 678.3
Augmented 4-peldegree 10 750 14 730.4
Minor 5-peldegree 11 825 16 834.8
Major 5-peldegree 12 900 17 887
Minor 6-peldegree 13 975 19 991.3
Major 6-peldegree 14 1050 20 1043.5
Perfect 7-peldegree (octave) 16 1200 23 1200 2/1 (exact)

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 of 2L 5s
Scale degree 20edo (Semihard, L:s = 5:2) 31edo (L:s = 8:3) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-peldegree (unison) 0 0 0 0 1/1 (exact)
Minor 1-peldegree 2 120 3 116.1
Major 1-peldegree 5 300 8 309.7
Minor 2-peldegree 4 240 6 232.3
Major 2-peldegree 7 420 11 425.8
Diminished 3-peldegree 6 360 9 348.4
Perfect 3-peldegree 9 540 14 541.9
Perfect 4-peldegree 11 660 17 658.1
Augmented 4-peldegree 14 840 22 851.6
Minor 5-peldegree 13 780 20 774.2
Major 5-peldegree 16 960 25 967.7
Minor 6-peldegree 15 900 23 890.3
Major 6-peldegree 18 1080 28 1083.9
Perfect 7-peldegree (octave) 20 1200 31 1200 2/1 (exact)

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 of 2L 5s
Scale degree 17edo (L:s = 6:1) 19edo (L:s = 7:1) 55edo (L:s = 20:3) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-peldegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-peldegree 1 70.6 1 63.2 3 65.5
Major 1-peldegree 6 423.5 7 442.1 20 436.4
Minor 2-peldegree 2 141.2 2 126.3 6 130.9
Major 2-peldegree 7 494.1 8 505.3 23 501.8
Diminished 3-peldegree 3 211.8 3 189.5 9 196.4
Perfect 3-peldegree 8 564.7 9 568.4 26 567.3
Perfect 4-peldegree 9 635.3 10 631.6 29 632.7
Augmented 4-peldegree 14 988.2 16 1010.5 46 1003.6
Minor 5-peldegree 10 705.9 11 694.7 32 698.2
Major 5-peldegree 15 1058.8 17 1073.7 49 1069.1
Minor 6-peldegree 11 776.5 12 757.9 35 763.6
Major 6-peldegree 16 1129.4 18 1136.8 52 1134.5
Perfect 7-peldegree (octave) 17 1200 19 1200 55 1200 2/1 (exact)

Modes

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 Rotational 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