2L 5s

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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, 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 interval regions.
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, 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.


Simple Tunings of 2L 5s
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

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.


Soft-of-basic Tunings of 2L 5s
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.


Hypohard Tunings of 2L 5s
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.


Ultrahard Tunings of 2L 5s
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

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
6|0 1 LssLsss
5|1 4 LsssLss
4|2 7 sLssLss
3|3 3 sLsssLs
2|4 6 ssLssLs
1|5 2 ssLsssL
0|6 5 sssLssL

Scale tree

Template:Scale tree

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