Chirality: Difference between revisions
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{{Wikipedia}} | |||
'''Chirality''' is a property of asymmetry which can be applied to [[periodic scale]]s. | '''Chirality''' is a property of asymmetry which can be applied to [[periodic scale]]s. | ||
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: | 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 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'' | # 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. | 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 [[5L 2s|diatonic scale]] of 12edo is achiral because 2221221 reverses to 1221222, which is identical to the original scale up to cyclical permutation. | Scales for which this property does not hold are called ''achiral''. For example, the [[5L 2s|diatonic scale]] of 12edo is achiral because 2221221 reverses to 1221222, which is identical to the original scale up to cyclical permutation. Odd-sized achiral scales have a '''symmetric mode''', wherein the [[inverse interval|inverse]] of each interval (about the period) also exists in the mode. | ||
== Examples == | == Examples == | ||
* All [[mos scale]]s are achiral. | |||
* Odd-sized [[GO]] scales, such as the [[Zarlino]] scale and [[diasem]], are chiral; even-sized GO scales, such as [[blackdye]], are not. | * 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). | * 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 == | == Properties == | ||
* Chiral scales have at least 3 notes; | * Chiral scales have at least 3 notes; | ||
* Chiral scales are at least max-variety 3 | * Chiral scales are at least max-variety 3; | ||
* Chiral scales with positive rational step ratios can only exist in edos larger than [[5edo]]. | |||
* Chiral scales with rational step ratios can only exist in edos larger than [[5edo]] | |||
== | == Conjectures == | ||
{| class="wikitable | * With respect to edos, chiral scales with positive rational step ratios have a {{w|natural density|density}} of 1 (see table below). | ||
! | |||
! Number of <br> | {| class="wikitable center-all right-2 right-3" | ||
! | |+Chiral scales in edos up to 20edo | ||
! | ! Edo | ||
! Number of<br>chiral scales | |||
! Number of<br>scales | |||
! Percentage of<br>chiral scales | |||
|- | |- | ||
| 1 | | 1 | ||
| 0 | | 0 | ||
| 1 | |||
| 0.0% | | 0.0% | ||
|- | |- | ||
| 2 | | 2 | ||
| 0 | | 0 | ||
| 2 | |||
| 0.0% | | 0.0% | ||
|- | |- | ||
| 3 | | 3 | ||
| 0 | | 0 | ||
| 3 | |||
| 0.0% | | 0.0% | ||
|- | |- | ||
| 4 | | 4 | ||
| 0 | | 0 | ||
| 5 | |||
| 0.0% | | 0.0% | ||
|- | |- | ||
| 5 | | 5 | ||
| 0 | | 0 | ||
| 7 | |||
| 0.0% | | 0.0% | ||
|- | |- | ||
| 6 | | 6 | ||
| 2 | | 2 | ||
| 9 | |||
| 22.2% | | 22.2% | ||
|- | |- | ||
| 7 | | 7 | ||
| 4 | | 4 | ||
| 18 | |||
| 22.2% | | 22.2% | ||
|- | |- | ||
| 8 | | 8 | ||
| 12 | | 12 | ||
| 30 | |||
| 40.0% | | 40.0% | ||
|- | |- | ||
| 9 | | 9 | ||
| 28 | | 28 | ||
| 56 | |||
| 50.0% | | 50.0% | ||
|- | |- | ||
| 10 | | 10 | ||
| 60 | | 60 | ||
| 99 | |||
| 60.6% | | 60.6% | ||
|- | |- | ||
| 11 | | 11 | ||
| 124 | | 124 | ||
| 186 | |||
| 66.7% | | 66.7% | ||
|- | |- | ||
| 12 | | 12 | ||
| 254 | | 254 | ||
| 335 | |||
| 75.8% | | 75.8% | ||
|- | |- | ||
| 13 | | 13 | ||
| 504 | | 504 | ||
| 630 | |||
| 80.0% | | 80.0% | ||
|- | |- | ||
| 14 | | 14 | ||
| 986 | | 986 | ||
| 1161 | |||
| 84.9% | | 84.9% | ||
|- | |- | ||
| 15 | | 15 | ||
| 1936 | | 1936 | ||
| 2182 | |||
| 88.7% | | 88.7% | ||
|- | |- | ||
| 16 | | 16 | ||
| 3720 | | 3720 | ||
| 4080 | |||
| 91.2% | | 91.2% | ||
|- | |- | ||
| 17 | | 17 | ||
| 7200 | | 7200 | ||
| 7710 | |||
| 93.4% | | 93.4% | ||
|- | |- | ||
| 18 | | 18 | ||
| 13804 | | 13804 | ||
| 14532 | |||
| 95.0% | | 95.0% | ||
|- | |- | ||
| 19 | | 19 | ||
| 26572 | | 26572 | ||
| 27594 | |||
| 96.3% | | 96.3% | ||
|- | |- | ||
| 20 | | 20 | ||
| 50892 | | 50892 | ||
| 52377 | |||
| 97.2% | | 97.2% | ||
|} | |} | ||
== Chirality in MV3 scales == | == 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). | 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). | ||
[[Category:Scale | [[Category:Scale]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Pages with open problems]] | |||