Chirality: Difference between revisions
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== Examples == | == Examples == | ||
* 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 (they cannot be [[MOS]] or [[DE]]) | * Chiral scales are at least max-variety 3 (they cannot be [[MOS]] or [[DE]]); | ||
* Chiral scales have a | * Chiral scales have a {{w|natural density|density}} of 1 (see table below); | ||
* Chiral scales with 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]]. | ||
== Chiral scales in edos up to 20edo == | == Chiral scales in edos up to 20edo == | ||
{| class="wikitable | {| class="wikitable right-all center-4" | ||
! | ! Edo | ||
! Number of <br> | ! Number of <br> chiral scales | ||
! Percentage of <br> | ! Percentage of <br> chiral scales | ||
! Corresponding <br> | ! Corresponding <br> ratio | ||
|- | |- | ||
| 1 | | 1 | ||
| 0 | | 0 | ||
| 0.0% | | 0.0% | ||
| 0/1 | |||
|- | |- | ||
| 2 | | 2 | ||
| 0 | | 0 | ||
| 0.0% | | 0.0% | ||
| 0/1 | |||
|- | |- | ||
| 3 | | 3 | ||
| 0 | | 0 | ||
| 0.0% | | 0.0% | ||
| 0/1 | |||
|- | |- | ||
| 4 | | 4 | ||
| 0 | | 0 | ||
| 0.0% | | 0.0% | ||
| 0/1 | |||
|- | |- | ||
| 5 | | 5 | ||
| 0 | | 0 | ||
| 0.0% | | 0.0% | ||
| 0/1 | |||
|- | |- | ||
| 6 | | 6 | ||
| 2 | | 2 | ||
| 22.2% | | 22.2% | ||
| 2/9 | |||
|- | |- | ||
| 7 | | 7 | ||
| 4 | | 4 | ||
| 22.2% | | 22.2% | ||
| 2/9 | |||
|- | |- | ||
| 8 | | 8 | ||
| 12 | | 12 | ||
| 40.0% | | 40.0% | ||
| 2/5 | |||
|- | |- | ||
| 9 | | 9 | ||
| 28 | | 28 | ||
| 50.0% | | 50.0% | ||
| 1/2 | |||
|- | |- | ||
| 10 | | 10 | ||
| 60 | | 60 | ||
| 60.6% | | 60.6% | ||
| 20/33 | |||
|- | |- | ||
| 11 | | 11 | ||
| 124 | | 124 | ||
| 66.7% | | 66.7% | ||
| 2/3 | |||
|- | |- | ||
| 12 | | 12 | ||
| 254 | | 254 | ||
| 75.8% | | 75.8% | ||
| 254/335 | |||
|- | |- | ||
| 13 | | 13 | ||
| 504 | | 504 | ||
| 80.0% | | 80.0% | ||
| 4/5 | |||
|- | |- | ||
| 14 | | 14 | ||
| 986 | | 986 | ||
| 84.9% | | 84.9% | ||
| 986/1161 | |||
|- | |- | ||
| 15 | | 15 | ||
| 1936 | | 1936 | ||
| 88.7% | | 88.7% | ||
| 968/1091 | |||
|- | |- | ||
| 16 | | 16 | ||
| 3720 | | 3720 | ||
| 91.2% | | 91.2% | ||
| 31/34 | |||
|- | |- | ||
| 17 | | 17 | ||
| 7200 | | 7200 | ||
| 93.4% | | 93.4% | ||
| 240/257 | |||
|- | |- | ||
| 18 | | 18 | ||
| 13804 | | 13804 | ||
| 95.0% | | 95.0% | ||
| 493/519 | |||
|- | |- | ||
| 19 | | 19 | ||
| 26572 | | 26572 | ||
| 96.3% | | 96.3% | ||
| 26/27 | |||
|- | |- | ||
| 20 | | 20 | ||
| 50892 | | 50892 | ||
| 97.2% | | 97.2% | ||
| 16964/17459 | |||
|} | |} | ||
== 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]] | ||