2L 5s
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Todo: expand
Add hard-of-basic tunings, include JI ratio approximations |
| ↖ 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 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.
Notation
- This article assumes TAMNAMS for naming step ratios, with TAMNAMS-prefixed ordinal names for intervals and scale degrees.
Intervals and degrees
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 | 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 ¢ |
Note names
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. This article uses this interpretation of sharps and flats.
- Melodic antidiatonic notation, where the meaning of sharps and flats is preserved: sharps sharpen and flats flatten.
For this article, the naturals DEFGABC are applied to the step pattern sLsssLs, the antidorian mode on D. Thus, the basic gamut for 2L 5s is the following:
D, E, Eb/F#, F, G, A, B, Bb/C#, 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.
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MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| Scale degree | On D | 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) | D | 0 | 0 | 0 | 0 | 0 | 0 | 1/1 (exact) |
| Minor 1-peldegree | E | 1 | 133.3 | 1 | 109.1 | 2 | 150 | |
| Major 1-peldegree | Eb | 2 | 266.7 | 3 | 327.3 | 3 | 225 | |
| Minor 2-peldegree | F# | 2 | 266.7 | 2 | 218.2 | 4 | 300 | |
| Major 2-peldegree | F | 3 | 400 | 4 | 436.4 | 5 | 375 | |
| Diminished 3-peldegree | G# | 3 | 400 | 3 | 327.3 | 6 | 450 | |
| Perfect 3-peldegree | G | 4 | 533.3 | 5 | 545.5 | 7 | 525 | |
| Perfect 4-peldegree | A | 5 | 666.7 | 6 | 654.5 | 9 | 675 | |
| Augmented 4-peldegree | Ab | 6 | 800 | 8 | 872.7 | 10 | 750 | |
| Minor 5-peldegree | B | 6 | 800 | 7 | 763.6 | 11 | 825 | |
| Major 5-peldegree | Bb | 7 | 933.3 | 9 | 981.8 | 12 | 900 | |
| Minor 6-peldegree | C# | 7 | 933.3 | 8 | 872.7 | 13 | 975 | |
| Major 6-peldegree | C | 8 | 1066.7 | 10 | 1090.9 | 14 | 1050 | |
| Perfect 7-peldegree (octave) | D | 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.
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MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| Scale degree | On D | 16edo (Soft, L:s = 3:2) | 23edo (Supersoft, L:s = 4:3) | Approx. JI Ratios | ||
|---|---|---|---|---|---|---|
| Steps | Cents | Steps | Cents | |||
| Perfect 0-peldegree (unison) | D | 0 | 0 | 0 | 0 | 1/1 (exact) |
| Minor 1-peldegree | E | 2 | 150 | 3 | 156.5 | |
| Major 1-peldegree | Eb | 3 | 225 | 4 | 208.7 | |
| Minor 2-peldegree | F# | 4 | 300 | 6 | 313 | |
| Major 2-peldegree | F | 5 | 375 | 7 | 365.2 | |
| Diminished 3-peldegree | G# | 6 | 450 | 9 | 469.6 | |
| Perfect 3-peldegree | G | 7 | 525 | 10 | 521.7 | |
| Perfect 4-peldegree | A | 9 | 675 | 13 | 678.3 | |
| Augmented 4-peldegree | Ab | 10 | 750 | 14 | 730.4 | |
| Minor 5-peldegree | B | 11 | 825 | 16 | 834.8 | |
| Major 5-peldegree | Bb | 12 | 900 | 17 | 887 | |
| Minor 6-peldegree | C# | 13 | 975 | 19 | 991.3 | |
| Major 6-peldegree | C | 14 | 1050 | 20 | 1043.5 | |
| Perfect 7-peldegree (octave) | D | 16 | 1200 | 23 | 1200 | 2/1 (exact) |
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.
| 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
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Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead.
Details: Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: {{MOS tuning spectrum
| 3/2 = Example comment
| 4/3 = Another example comment
}}
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| 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 | ||||||
| 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 | |||||
| 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 | ||||||
| 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 | ||||||
| 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 | |||||
| 23\51 | 541.176 | 658.824 | 13:5 | 2.600 | ||||||
| 14\31 | 541.935 | 658.065 | 8:3 | 2.667 | ||||||
| 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 | ||||||
| 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 | ||||||
| 1\2 | 600.000 | 600.000 | 1:0 | → ∞ | Collapsed 2L 5s | |||||