Dual list - Revision history

Sintel: -legacy, todo intro, inaccessible

2025-03-29T17:59:21Z

-legacy, todo intro, inaccessible

← Older revision		Revision as of 17:59, 29 March 2025
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	{{~~Legacy~~}}		{{inacc}}
	This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].		{{todo\|intro\|inline=1}}
			: ''This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].''

	== Definition ==		== Definition ==

ArrowHead294: Further de-escaping

2025-02-26T13:55:39Z

Further de-escaping

← Older revision		Revision as of 13:55, 26 February 2025
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	== Examples ==		== Examples ==
	dl[{{map\| 31 49 72 87 107 }}, {{map\| 41 65 95 115 142 }}, {{map\| 72 114 167 202 249 }}]) {{nowrap\|{{=}} ~~{{!(}}~~{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}~~{{)!}}~~}}		dl[{{map\| 31 49 72 87 107 }}, {{map\| 41 65 95 115 142 }}, {{map\| 72 114 167 202 249 }}]) {{nowrap\|{{=}} [{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]}}

	In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap\|31 + 41 {{=}} 72\|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).		In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap\|31 + 41 {{=}} 72\|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).

ArrowHead294: Remove escapes as they are not needed

2025-02-26T13:54:54Z

Remove escapes as they are not needed

← Older revision		Revision as of 13:54, 26 February 2025
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	In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap\|31 + 41 {{=}} 72\|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).		In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap\|31 + 41 {{=}} 72\|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).

	dl[{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]) {{nowrap\|{{=}} ~~{{!(}}~~{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}~~{{)!}}~~}}		dl[{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]) {{nowrap\|{{=}} [{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]}}

	dl[{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]) {{nowrap\|{{=}} ~~{{!(}}~~{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}~~{{)!}}~~}}		dl[{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]) {{nowrap\|{{=}} [{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]}}

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]

ArrowHead294 at 15:40, 19 February 2025

2025-02-19T15:40:44Z

← Older revision		Revision as of 15:40, 19 February 2025
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	== Definition ==		== Definition ==
			If '''A''' is an integral matrix, denote by Saturate('''A''') the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of '''A''', defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by [[LLL reduction]], so let {{nowrap\|Sat('''A''') {{=}} LLL(Saturate('''A'''))}}. The square matrix with rational coefficients '''A'''{{+}}'''A''', where {{+}} denotes the {{w\|Moore–Penrose pseudoinverse}}, is an ''n''×''n'' projection matrix, where ''n'' is the number of columns of '''A'''. Let us also define clr('''P''') to be the rational number matrix '''P''' cleared of denominators by multiplying through the rows by the LCM of the denominators, and hrm('''A''') to be the Hermite reduction of '''A''', with any rows consisting of zeros removed. We may combine these into the dual list function: {{nowrap\|dl('''A''') {{=}} Sat(hrm(clr('''I''' − '''A'''{{+}}'''A''')))}}, where '''I''' is the ''n''×''n'' identity matrix.

	If A is an integral matrix, denote by Saturate (A) the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by [[LLL reduction]], so let Sat (A) = LLL (Saturate (A)). The square matrix with rational coefficients A<sup>+</sup>A, where A<sup>+</sup> is the pseudoinverse of A, is an ''n''×''n'' projection matrix, where ''n'' is the number of columns of A. Let us also define Clear (P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm (A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist (A) = Sat (Hrm (Clear (I - A<sup>+</sup>A))), where I is the ''n''×''n'' identity matrix.		The interest of the dual list stems from the fact that if '''A''' is a list of vals defining a temperament, then dl('''A''') is a list of monzos defining the same temperament, and conversely, Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.

	The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then ~~Dulist~~ (A) is a list of monzos defining the same temperament, and conversely – Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.

	== Examples ==		== Examples ==
			dl[{{map\| 31 49 72 87 107 }}, {{map\| 41 65 95 115 142 }}, {{map\| 72 114 167 202 249 }}]) {{nowrap\|{{=}} {{!(}}{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}{{)!}}}}

	~~Dulist ([{{map\| 31 49 72 87 107 }}, {{map\| 41 65 95 115 142 }}, {{map\| 72 114 167 202 249 }}]) = [{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]~~		In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap\|31 + 41 {{=}} 72\|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).

	In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two (31 + 41 = 72, 49 + 65 = 114, etc.). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).

	~~Dulist (~~[{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]) = [{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]		dl[{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]) {{nowrap\|{{=}} {{!(}}{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}{{)!}}}}

	~~Dulist (~~[{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]) = [{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]		dl[{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]) {{nowrap\|{{=}} {{!(}}{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}{{)!}}}}

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]

Sintel: link LLL

2025-02-17T17:21:16Z

link LLL

← Older revision		Revision as of 17:21, 17 February 2025
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	== Definition ==		== Definition ==

	If A is an integral matrix, denote by Saturate (A) the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat (A) = LLL (Saturate (A)). The square matrix with rational coefficients A<sup>+</sup>A, where A<sup>+</sup> is the pseudoinverse of A, is an ''n''×''n'' projection matrix, where ''n'' is the number of columns of A. Let us also define Clear (P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm (A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist (A) = Sat (Hrm (Clear (I - A<sup>+</sup>A))), where I is the ''n''×''n'' identity matrix.		If A is an integral matrix, denote by Saturate (A) the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by [[LLL reduction]], so let Sat (A) = LLL (Saturate (A)). The square matrix with rational coefficients A<sup>+</sup>A, where A<sup>+</sup> is the pseudoinverse of A, is an ''n''×''n'' projection matrix, where ''n'' is the number of columns of A. Let us also define Clear (P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm (A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist (A) = Sat (Hrm (Clear (I - A<sup>+</sup>A))), where I is the ''n''×''n'' identity matrix.

	The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist (A) is a list of monzos defining the same temperament, and conversely – Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.		The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist (A) is a list of monzos defining the same temperament, and conversely – Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.

Lériendil at 16:54, 17 February 2025

2025-02-17T16:54:02Z

← Older revision		Revision as of 16:54, 17 February 2025
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	{{Legacy~~\|Dual_list~~}}		{{Legacy}}
	This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].		This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].

Lériendil: deploying new cat

2025-02-17T16:07:57Z

deploying new cat

← Older revision		Revision as of 16:07, 17 February 2025
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			{{Legacy\|Dual_list}}
	This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].		This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].

Lériendil: Changed protection level for "Dual list" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

2025-02-17T16:00:34Z

Changed protection level for "Dual list" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:00, 17 February 2025
(No difference)

FloraC: Improve readability and +categories

2021-11-16T10:05:04Z

Improve readability and +categories

← Older revision		Revision as of 10:05, 16 November 2021
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	This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].		This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].

	=Definition=		== Definition ==

	If A is an integral matrix, denote by Saturate(A) the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat(A) = LLL(Saturate(A)). The square matrix with rational coefficients A`A, where A` is the pseudoinverse of A, is an ~~nxn~~ projection matrix, where n is the number of columns of A. Let us also define Clear(P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm(A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist(A) = Sat(Hrm(Clear(I - A`A))), where I is the ~~nxn~~ identity matrix.		If A is an integral matrix, denote by Saturate (A) the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat (A) = LLL (Saturate (A)). The square matrix with rational coefficients A<sup>+</sup>A, where A<sup>+</sup> is the pseudoinverse of A, is an ''n''×''n'' projection matrix, where ''n'' is the number of columns of A. Let us also define Clear (P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm (A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist (A) = Sat (Hrm (Clear (I - A<sup>+</sup>A))), where I is the ''n''×''n'' identity matrix.

	The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist(A) is a list of monzos defining the same temperament, and conversely--Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.		The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist (A) is a list of monzos defining the same temperament, and conversely – Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.

	=Examples=		== Examples ==

	Dulist([{{map\|31 49 72 87 107}}, {{map\|41 65 95 115 142}}, {{map\|72 114 167 202 249}}]) = [{{vector\|-3 2 -1 2 -1}}, {{vector\|-2 -3 -1 2 1}}, {{vector\|-2 0 3 -3 1}}]		Dulist ([{{map\| 31 49 72 87 107 }}, {{map\| 41 65 95 115 142 }}, {{map\| 72 114 167 202 249 }}]) = [{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]

	In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two (31+41=72, 49+65=114, etc.). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).		In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two (31 + 41 = 72, 49 + 65 = 114, etc.). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).

	Dulist([{{vector\|-3 2 -1 2 -1}}, {{vector\|-2 -3 -1 2 1}}, {{vector\|-2 0 3 -3 1}}]) = [{{map\|1 1 3 3 2}}, {{map\|0 6 -7 -2 15}}]		Dulist ([{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]) = [{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]

	Dulist([{{map\|1 1 3 3 2}}, {{map\|0 6 -7 -2 15}}]) = [{{vector\|-3 2 -1 2 -1}}, {{vector\|-2 -3 -1 2 1}}, {{vector\|-2 0 3 -3 1}}]		Dulist ([{{map\| 1 1 3 3 2 }}, {{map\| 0 6 -7 -2 15 }}]) = [{{vector\| -3 2 -1 2 -1 }}, {{vector\| -2 -3 -1 2 1 }}, {{vector\| -2 0 3 -3 1 }}]

			[[Category:Regular temperament theory]]
			[[Category:Math]]

Mike Battaglia: /* Definition */

2021-11-16T04:13:46Z

Definition

← Older revision		Revision as of 04:13, 16 November 2021
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	=Definition=		=Definition=

	If A is an integral matrix, denote by Saturate(A) the integral matrix which is the [[~~Saturation~~\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat(A) = LLL(Saturate(A)). The square matrix with rational coefficients A`A, where A` is the pseudoinverse of A, is an nxn projection matrix, where n is the number of columns of A. Let us also define Clear(P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm(A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist(A) = Sat(Hrm(Clear(I - A`A))), where I is the nxn identity matrix.		If A is an integral matrix, denote by Saturate(A) the integral matrix which is the [[Mathematical theory of saturation\|saturation]] of A, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat(A) = LLL(Saturate(A)). The square matrix with rational coefficients A`A, where A` is the pseudoinverse of A, is an nxn projection matrix, where n is the number of columns of A. Let us also define Clear(P) to be the rational number matrix P cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm(A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist(A) = Sat(Hrm(Clear(I - A`A))), where I is the nxn identity matrix.

	The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist(A) is a list of monzos defining the same temperament, and conversely--Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.		The interest of the dual list stems from the fact that if A is a list of vals defining a temperament, then Dulist(A) is a list of monzos defining the same temperament, and conversely--Dulist applied to a list of monzos defining a temperament gives a list of vals defining the same temperament.