# 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 as follows:

- Lexicographically compare 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). For each chirality, record the mode that comes first (among all the modes of the chirality) according to this alphabetical order. You should end up with two modes M and M'.
- Lexicographically compare M and M'. We choose the convention that if M lexicographically comes before M', then M and all its modes are
*lexicographically right-handed*(lex-RH), and M' and all its modes are*lexicographically left-handed*(lex-LH).

The smallest example of a chiral pair in an edo is 321/312, with the former being lexicographically right-handed and the latter being lexicographically 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.

## Examples

- Odd-sized GO scales, such as the Zarlino scale and diasem, are chiral; even-sized GO scales, such as blackdye, are not.
- 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).

## 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

## 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).