Chirality
Chirality is a property of asymmetry which can be applied to periodic scales.
A scale is called chiral if reversing the order of the steps results in a different scale (which is not a mode of the original scale). The two scales form a chiral pair and are right/left-handed. Handedness is determined by lexicographically comparing all modes of each chirality (i.e. treat scale step size sequences as words to be arranged in "alphabetical order", where this alphabetical order is from bigger step to smaller step) and then lexicographically comparing the respective lexicographically first modes. We choose the convention that the chirality that comes lexicographically first at the end is called the right-handed chirality.
The smallest example of a chiral pair in an edo is 321/312, with the former being right-handed and the latter being left-handed. Similarly, the simplest chiral pair for abstract patterns is Lms/Lsm.
Scales for which this property does not hold are called achiral. For example, the diatonic scale of 12edo is achiral because 2221221 reverses to 1221222, which is identical to the original scale up to cyclical permutation.
Properties
- Chiral scales have at least 3 notes;
- Chiral scales are at least max-variety 3 (they cannot be MOS or DE)
- Chiral scales have a density of 1 (see table below)
- Chiral scales with rational step ratios can only exist in edos larger than 5edo
- A CPS made by choosing n/2 out of n elements (for n even) is achiral. Otherwise it may be chiral (for example, the 1 3 5 7 9 11 pentadekany is chiral).
Chiral scales in edos up to 20edo
| EDO | Number of Chiral Scales |
Percentage of Chiral Scales |
Corresponding Ratio |
|---|---|---|---|
| 1 | 0 | 0.0% | 0/1 |
| 2 | 0 | 0.0% | 0/1 |
| 3 | 0 | 0.0% | 0/1 |
| 4 | 0 | 0.0% | 0/1 |
| 5 | 0 | 0.0% | 0/1 |
| 6 | 2 | 22.2% | 2/9 |
| 7 | 4 | 22.2% | 2/9 |
| 8 | 12 | 40.0% | 2/5 |
| 9 | 28 | 50.0% | 1/2 |
| 10 | 60 | 60.6% | 20/33 |
| 11 | 124 | 66.7% | 2/3 |
| 12 | 254 | 75.8% | 254/335 |
| 13 | 504 | 80.0% | 4/5 |
| 14 | 986 | 84.9% | 986/1161 |
| 15 | 1936 | 88.7% | 968/1091 |
| 16 | 3720 | 91.2% | 31/34 |
| 17 | 7200 | 93.4% | 240/257 |
| 18 | 13804 | 95.0% | 493/519 |
| 19 | 26572 | 96.3% | 26/27 |
| 20 | 50892 | 97.2% | 16964/17459 |
Chirality in MV3 scales
Assume a scale is MV3 and is of the form ax by bz. Additionally assume that the mos ax 2bY that results from equating y and z is not a multimos. Then the scale must be chiral because there are no rotations that will make the two equivalent (each mode of the mos ax 2bY corresponding to two chiral variants). This is how you determine the handedness of the scale:
- Identify y and z to Y to get a mos. Take the mode that has the most x's at the beginning. When you undo the identification, the mv3 mode will start with either y or z first.
- If y is bigger than z then the chiral variant beginning with y is right-handed. Otherwise, it is left-handed.
Such chiral mv3's could be named in the format [handedness] [mv3 name] [the corresponding mode of ax 2bY]. For example, the chiral diasem mode 331323132 can be named "Left-Handed Diasem 8|0", where 8|0 is the UDP notation for the 5L 4s mode LLSLSLSLS.