Chirality - Revision history

Fredg999: Mention symmetric mode, misc. edits

2025-07-21T05:56:06Z

Mention symmetric mode, misc. edits

FloraC: Note the condition of positive scale steps

2025-07-07T12:45:50Z

Note the condition of positive scale steps

← Older revision		Revision as of 12:45, 7 July 2025
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	* 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 rational step ratios can only exist in edos larger than [[5edo]].		* Chiral scales with positive rational step ratios can only exist in edos larger than [[5edo]].

	== Conjectures ==		== Conjectures ==
	* With respect to edos, chiral scales have a {{w\|natural density\|density}} of 1 (see table below).		* With respect to edos, chiral scales with positive rational step ratios have a {{w\|natural density\|density}} of 1 (see table below).

	{\| class="wikitable center-all right-2 right-3"		{\| class="wikitable center-all right-2 right-3"

FloraC: /* Conjectures */ the fraction is useless noise. Present the actual number of scales instead

2025-07-07T12:42:46Z

Conjectures: the fraction is useless noise. Present the actual number of scales instead

← Older revision		Revision as of 12:42, 7 July 2025
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	== Conjectures ==		== Conjectures ==
	* With respect to edos, chiral scales have a {{w\|natural density\|density}} of 1 (see table below);		* With respect to edos, chiral scales have a {{w\|natural density\|density}} of 1 (see table below).

	{\| class="wikitable right-~~all center~~-4"		{\| class="wikitable center-all right-2 right-3"
	\|+Chiral scales in edos up to 20edo		\|+Chiral scales in edos up to 20edo
	! Edo		! Edo
	! Number of <br> chiral scales		! Number of<br>chiral scales
	! ~~Percentage~~ of <br> ~~chiral~~ scales		! Number of<br>scales
	! ~~Corresponding~~ <br> ~~ratio~~		! Percentage of<br>chiral scales
	\|-		\|-
	\| 1		\| 1
	\| 0		\| 0
			\| 1
	\| 0.0%		\| 0.0%
	~~\| 0/1~~
	\|-		\|-
	\| 2		\| 2
	\| 0		\| 0
			\| 2
	\| 0.0%		\| 0.0%
	~~\| 0/1~~
	\|-		\|-
	\| 3		\| 3
	\| 0		\| 0
			\| 3
	\| 0.0%		\| 0.0%
	~~\| 0/1~~
	\|-		\|-
	\| 4		\| 4
	\| 0		\| 0
			\| 5
	\| 0.0%		\| 0.0%
	~~\| 0/1~~
	\|-		\|-
	\| 5		\| 5
	\| 0		\| 0
			\| 7
	\| 0.0%		\| 0.0%
	~~\| 0/1~~
	\|-		\|-
	\| 6		\| 6
	\| 2		\| 2
			\| 9
	\| 22.2%		\| 22.2%
	~~\| 2/9~~
	\|-		\|-
	\| 7		\| 7
	\| 4		\| 4
			\| 18
	\| 22.2%		\| 22.2%
	~~\| 2/9~~
	\|-		\|-
	\| 8		\| 8
	\| 12		\| 12
			\| 30
	\| 40.0%		\| 40.0%
	~~\| 2/5~~
	\|-		\|-
	\| 9		\| 9
	\| 28		\| 28
			\| 56
	\| 50.0%		\| 50.0%
	~~\| 1/2~~
	\|-		\|-
	\| 10		\| 10
	\| 60		\| 60
			\| 99
	\| 60.6%		\| 60.6%
	~~\| 20/33~~
	\|-		\|-
	\| 11		\| 11
	\| 124		\| 124
			\| 186
	\| 66.7%		\| 66.7%
	~~\| 2/3~~
	\|-		\|-
	\| 12		\| 12
	\| 254		\| 254
			\| 335
	\| 75.8%		\| 75.8%
	~~\| 254/335~~
	\|-		\|-
	\| 13		\| 13
	\| 504		\| 504
			\| 630
	\| 80.0%		\| 80.0%
	~~\| 4/5~~
	\|-		\|-
	\| 14		\| 14
	\| 986		\| 986
			\| 1161
	\| 84.9%		\| 84.9%
	~~\| 986/1161~~
	\|-		\|-
	\| 15		\| 15
	\| 1936		\| 1936
			\| 2182
	\| 88.7%		\| 88.7%
	~~\| 968/1091~~
	\|-		\|-
	\| 16		\| 16
	\| 3720		\| 3720
			\| 4080
	\| 91.2%		\| 91.2%
	~~\| 31/34~~
	\|-		\|-
	\| 17		\| 17
	\| 7200		\| 7200
			\| 7710
	\| 93.4%		\| 93.4%
	~~\| 240/257~~
	\|-		\|-
	\| 18		\| 18
	\| 13804		\| 13804
			\| 14532
	\| 95.0%		\| 95.0%
	~~\| 493/519~~
	\|-		\|-
	\| 19		\| 19
	\| 26572		\| 26572
			\| 27594
	\| 96.3%		\| 96.3%
	~~\| 26/27~~
	\|-		\|-
	\| 20		\| 20
	\| 50892		\| 50892
			\| 52377
	\| 97.2%		\| 97.2%
	~~\| 16964/17459~~
	\|}		\|}

FloraC: Move unproved properties to a distinct section

2025-07-07T12:36:02Z

Move unproved properties to a distinct section

← Older revision		Revision as of 12:36, 7 July 2025
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	* 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 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~~ ==		== Conjectures ==
			* With respect to edos, chiral scales have a {{w\|natural density\|density}} of 1 (see table below);

	{\| class="wikitable right-all center-4"		{\| class="wikitable right-all center-4"
			\|+Chiral scales in edos up to 20edo
	! Edo		! Edo
	! Number of <br> chiral scales		! Number of <br> chiral scales
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	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]
			[[Category:Pages with open problems]]

FloraC: Try to make it clearer that mos scales are achiral

2025-01-08T12:20:06Z

Try to make it clearer that mos scales are achiral

← Older revision		Revision as of 12:20, 8 January 2025
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	== 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).
Line 16:		Line 17:
	== 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;
	* Chiral scales have a {{w\|natural density\|density}} of 1 (see table below);		* 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]].

FloraC: Style and misc. cleanup

2025-01-08T12:18:43Z

Style and misc. cleanup

← Older revision		Revision as of 12:18, 8 January 2025
Line 12:		Line 12:
	== 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 ~~[http://en.wikipedia.org/wiki/Natural_density~~ density] of 1 (see table below)		* 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~~" style="text~~-~~align:right~~"		{\| class="wikitable right-all center-4"
	! ~~EDO~~		! Edo
	! Number of <br> ~~Chiral Scales~~		! Number of <br> chiral scales
	! Percentage of <br> ~~Chiral Scales~~		! Percentage of <br> chiral scales
	! Corresponding <br> ~~Ratio~~		! Corresponding <br> ratio
	\|-		\|-
	\| 1		\| 1
	\| 0		\| 0
	\| 0.0%		\| 0.0%
	~~\| style="text-align:center"~~ \| 0/1		\| 0/1
	\|-		\|-
	\| 2		\| 2
	\| 0		\| 0
	\| 0.0%		\| 0.0%
	~~\| style="text-align:center"~~ \| 0/1		\| 0/1
	\|-		\|-
	\| 3		\| 3
	\| 0		\| 0
	\| 0.0%		\| 0.0%
	~~\| style="text-align:center"~~ \| 0/1		\| 0/1
	\|-		\|-
	\| 4		\| 4
	\| 0		\| 0
	\| 0.0%		\| 0.0%
	~~\| style="text-align:center"~~ \| 0/1		\| 0/1
	\|-		\|-
	\| 5		\| 5
	\| 0		\| 0
	\| 0.0%		\| 0.0%
	~~\| style="text-align:center"~~ \| 0/1		\| 0/1
	\|-		\|-
	\| 6		\| 6
	\| 2		\| 2
	\| 22.2%		\| 22.2%
	~~\| style="text-align:center"~~ \| 2/9		\| 2/9
	\|-		\|-
	\| 7		\| 7
	\| 4		\| 4
	\| 22.2%		\| 22.2%
	~~\| style="text-align:center"~~ \| 2/9		\| 2/9
	\|-		\|-
	\| 8		\| 8
	\| 12		\| 12
	\| 40.0%		\| 40.0%
	~~\| style="text-align:center"~~ \| 2/5		\| 2/5
	\|-		\|-
	\| 9		\| 9
	\| 28		\| 28
	\| 50.0%		\| 50.0%
	~~\| style="text-align:center"~~ \| 1/2		\| 1/2
	\|-		\|-
	\| 10		\| 10
	\| 60		\| 60
	\| 60.6%		\| 60.6%
	~~\| style="text-align:center"~~ \| 20/33		\| 20/33
	\|-		\|-
	\| 11		\| 11
	\| 124		\| 124
	\| 66.7%		\| 66.7%
	~~\| style="text-align:center"~~ \| 2/3		\| 2/3
	\|-		\|-
	\| 12		\| 12
	\| 254		\| 254
	\| 75.8%		\| 75.8%
	~~\| style="text-align:center"~~ \| 254/335		\| 254/335
	\|-		\|-
	\| 13		\| 13
	\| 504		\| 504
	\| 80.0%		\| 80.0%
	~~\| style="text-align:center"~~ \| 4/5		\| 4/5
	\|-		\|-
	\| 14		\| 14
	\| 986		\| 986
	\| 84.9%		\| 84.9%
	~~\| style="text-align:center"~~ \| 986/1161		\| 986/1161
	\|-		\|-
	\| 15		\| 15
	\| 1936		\| 1936
	\| 88.7%		\| 88.7%
	~~\| style="text-align:center"~~ \| 968/1091		\| 968/1091
	\|-		\|-
	\| 16		\| 16
	\| 3720		\| 3720
	\| 91.2%		\| 91.2%
	~~\| style="text-align:center"~~ \| 31/34		\| 31/34
	\|-		\|-
	\| 17		\| 17
	\| 7200		\| 7200
	\| 93.4%		\| 93.4%
	~~\| style="text-align:center"~~ \| 240/257		\| 240/257
	\|-		\|-
	\| 18		\| 18
	\| 13804		\| 13804
	\| 95.0%		\| 95.0%
	~~\| style="text-align:center"~~ \| 493/519		\| 493/519
	\|-		\|-
	\| 19		\| 19
	\| 26572		\| 26572
	\| 96.3%		\| 96.3%
	~~\| style="text-align:center"~~ \| 26/27		\| 26/27
	\|-		\|-
	\| 20		\| 20
	\| 50892		\| 50892
	\| 97.2%		\| 97.2%
	~~\| style="text-align:center"~~ \| 16964/17459		\| 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).

	~~== External links ==~~
	* [[Wikipedia: Chirality]]

	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]

Xenwolf: display WP link on top

2022-11-07T08:28:51Z

display WP link on top

← Older revision		Revision as of 08:28, 7 November 2022
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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.

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:50Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
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	* [[Wikipedia: Chirality]]		* [[Wikipedia: Chirality]]

	[[Category:Scale ~~theory~~]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]

Inthar at 07:19, 24 September 2021

2021-09-24T07:19:43Z

← Older revision		Revision as of 07:19, 24 September 2021
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	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'' (lex-RH)~~. Otherwise~~ M and all its modes are ''lexicographically left-handed'' (lex-LH).		# 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.

Inthar: /* Chirality in MV3 scales */

2021-09-24T04:52:51Z

Chirality in MV3 scales

← Older revision		Revision as of 04:52, 24 September 2021
Line 128:		Line 128:

	== 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)~~. This defines a definition of handedness specifically for MV3 scales of this type:~~		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).
	~~# 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 ''MV3-right-handed''. Otherwise, it is ''MV3-left-handed''.~~

	~~Note that this definition of LH/RH won't agree with the lexicographical definition if x is not the biggest step. For example, LMMMSMM is lexicographically RH but MV3-LH.~~

	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.

	== External links ==		== External links ==

← Older revision		Revision as of 05:56, 21 July 2025
Line 2:		Line 2:
	'''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'' (lex-RH), and M' and all its modes are ''lexicographically left-handed'' (lex-LH).		# 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).
Line 8:		Line 8:
	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 ==
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	* 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 positive rational step ratios can only exist in edos larger than [[5edo]].

	== Conjectures ==		== Conjectures ==