Comma basis - Revision history

Dummy index: add interwiki

2025-10-22T15:03:56Z

add interwiki

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			{{interwiki
			\| de =
			\| en = Comma basis
			\| es =
			\| ja = コンマ基底
			}}
	{{Beginner\|Dual list}}		{{Beginner\|Dual list}}
	A '''comma basis''' is a list of {{w\|linearly independent}} [[comma]]s that characterizes a [[regular temperament\|temperament]].		A '''comma basis''' is a list of {{w\|linearly independent}} [[comma]]s that characterizes a [[regular temperament\|temperament]].

FloraC: Cleanup

2025-10-07T15:32:40Z

Cleanup

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	{{Beginner\|Dual list}}		{{Beginner\|Dual list}}
	A '''comma basis''' is a list of linearly independent ~~commas~~ that characterizes a temperament.		A '''comma basis''' is a list of {{w\|linearly independent}} [[comma]]s that characterizes a [[regular temperament\|temperament]].

	For example, septimal meantone tempers out [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ((225/224)*(126/125)=(81/80)~~, for example~~). This means that if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Thus, we only need to pick two of the three commas; the third is implied. So we may write meantone's comma basis as (81/80, 225/224). This can be written in matrix form using the monzos of the commas as columns: [{{~~vector~~\|-4 4 -1 0~~}}, {{vector~~\|-5 2 2 -1}}], or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[~~Normal lists~~ #Normal ~~interval lists~~\|normal forms]] have been developed as identifiers of temperaments.		For example, septimal meantone [[tempering out\|tempers out]] [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ({{nowrap\| (225/224)⋅(126/125) {{=}} (81/80) }}). This means that if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Thus, we only need to pick two of the three commas; the third is implied. So we may write meantone's comma basis as {81/80, 225/224}. This can be written in matrix form using the monzos of the commas as columns: {{monzo list\| -4 4 -1 0 \| -5 2 2 -1 }}, or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[normal forms #Normal forms for commas\|normal forms]] have been developed as identifiers of temperaments.

	Mathematically, it is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' ~~{{w\|~~linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out. The nullspace forms a subgroup of the domain of the mapping, and every linear combination of basis vectors ~~(for example, (81/80)^n * (225/224)^m)~~ is also tempered out.		Mathematically, a comma basis is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' linearly independent vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out. The nullspace forms a subgroup of the domain of the mapping, and as a result every {{w\|linear combination}} of basis vectors is also tempered out.

	== With respect to the mapping ==		== With respect to the mapping ==

Dummy index: verbosely

2025-10-07T14:56:47Z

verbosely

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	For example, septimal meantone tempers out [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ((225/224)*(126/125)=(81/80), for example). This means that if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Thus, we only need to pick two of the three commas; the third is implied. So we may write meantone's comma basis as (81/80, 225/224). This can be written in matrix form using the monzos of the commas as columns: [{{vector\|-4 4 -1 0}}, {{vector\|-5 2 2 -1}}], or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists\|normal forms]] have been developed as identifiers of temperaments.		For example, septimal meantone tempers out [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ((225/224)*(126/125)=(81/80), for example). This means that if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Thus, we only need to pick two of the three commas; the third is implied. So we may write meantone's comma basis as (81/80, 225/224). This can be written in matrix form using the monzos of the commas as columns: [{{vector\|-4 4 -1 0}}, {{vector\|-5 2 2 -1}}], or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists\|normal forms]] have been developed as identifiers of temperaments.

	Mathematically, it is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{w\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out.		Mathematically, it is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{w\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out. The nullspace forms a subgroup of the domain of the mapping, and every linear combination of basis vectors (for example, (81/80)^n * (225/224)^m) is also tempered out.

	== With respect to the mapping ==		== With respect to the mapping ==

VectorGraphics at 21:22, 22 August 2025

2025-08-22T21:22:23Z

← Older revision		Revision as of 21:22, 22 August 2025
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	{{Beginner\|Dual list}}		{{Beginner\|Dual list}}
	A '''comma basis''' is a ~~[[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel")~~ of ~~a [[regular temperament\|temperament]]. It consists of ''n'' {{w\|~~linearly independent~~}} vectors, where ''n'' is the [[nullity]], each one representing~~ a ~~[[comma]] that is made to [[vanish]]~~.		A '''comma basis''' is a list of linearly independent commas that characterizes a temperament.

	~~Linear independence means that no comma can be found as the sum of any multiples of the other commas.~~ For example, ~~consider the set of three commas 81~~/80, 126/125, and ~~225~~/~~224. As vectors those are {{vector\|-4 4 -1 0}}~~, ~~{{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that~~ the third ~~comma is actually the difference between the other two; {{nowrap\|{{vector\|-4 4 -1 0}} − {{vector\|1 2 -3 1}} {{~~=~~}} {{vector\|-5 2 2 -1}}}}~~, ~~or as cents, {{nowrap\|21~~.~~51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So,~~ if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. ~~Therefore~~, we only need to pick two of ~~these~~ commas ~~to put in our comma basis~~; the third ~~one would~~ be ~~implied~~.		For example, septimal meantone tempers out [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ((225/224)*(126/125)=(81/80), for example). This means that if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Thus, we only need to pick two of the three commas; the third is implied. So we may write meantone's comma basis as (81/80, 225/224). This can be written in matrix form using the monzos of the commas as columns: [{{vector\|-4 4 -1 0}}, {{vector\|-5 2 2 -1}}], or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists\|normal forms]] have been developed as identifiers of temperaments.

	~~The comma~~ basis ~~can be thought~~ of ~~either as~~ a ~~list~~ of vectors ~~or as a matrix formed by putting these vectors (as columns) together. Besides~~, it is ~~often presented in terms of ratios for convenience. Various~~ [[~~Normal lists #Normal interval lists\|normal forms~~]] ~~have been developed as identifiers~~ of ~~temperaments~~.		Mathematically, it is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{w\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out.

	== With respect to the mapping ==		== With respect to the mapping ==
	The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.		The comma basis is considered the dual of the temperament's [[mapping]] matrix, similar to how a val is considered dual to a monzo. Temperaments may be identified by either their mapping or comma basis.

	Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn about finding the nullspace by hand, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments #Nullspace]].		Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn about finding the nullspace by hand, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments #Nullspace]].

	To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.		To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.

			When applied to convert between [[wedgies]] and "multimonzos" (the wedgie version of a comma basis), the equivalent operation is the [[Hodge dual\|Hodge star]].

	Some math libraries, such as [https://www.sagemath.org/ Sage], provide functions for both directions; in Sage, to go from a mapping to a comma basis, use <code>left_kernel()</code>, and to go from a comma basis to a mapping, use <code>right_kernel()</code>. In other math libraries, such as [https://www.wolfram.com/language/ Wolfram Language], the nullspace operation <code>NullSpace[]</code> is primarily designed to work for mappings, and so if you want correct results, you must transform the basis for the nullspace into a mapping-like form, perform the nullspace operation, and then undo the initial transformation.		Some math libraries, such as [https://www.sagemath.org/ Sage], provide functions for both directions; in Sage, to go from a mapping to a comma basis, use <code>left_kernel()</code>, and to go from a comma basis to a mapping, use <code>right_kernel()</code>. In other math libraries, such as [https://www.wolfram.com/language/ Wolfram Language], the nullspace operation <code>NullSpace[]</code> is primarily designed to work for mappings, and so if you want correct results, you must transform the basis for the nullspace into a mapping-like form, perform the nullspace operation, and then undo the initial transformation.

FloraC: Clarify and - here-links

2025-06-25T17:35:24Z

Clarify and - here-links

← Older revision		Revision as of 17:35, 25 June 2025
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	The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.		The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.

	Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn ~~more~~ about finding the nullspace, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace~~\|here~~]].		Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn about finding the nullspace by hand, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments #Nullspace]].

	To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.		To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.

ArrowHead294 at 15:31, 19 February 2025

2025-02-19T15:31:19Z

← Older revision		Revision as of 15:31, 19 February 2025
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	{{Beginner\|Dual list}}		{{Beginner\|Dual list}}
	A '''comma basis''' is a [[basis]] for the ~~[[wikipedia:~~ Kernel (linear algebra)\|nullspace]] (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].		A '''comma basis''' is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{w\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].

	Linear independence means that no comma can be found as the sum of any multiples of the other commas. For example, consider the set of three commas 81/80, 126/125, and 225/224. As vectors those are {{vector\|-4 4 -1 0}}, {{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that the third comma is actually the difference between the other two; {{nowrap\|{{vector\|-4 4 -1 0}} − {{vector\|1 2 -3 1}} {{=}} {{vector\|-5 2 2 -1}}}}, or as cents, {{nowrap\|21.51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So, if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.		Linear independence means that no comma can be found as the sum of any multiples of the other commas. For example, consider the set of three commas 81/80, 126/125, and 225/224. As vectors those are {{vector\|-4 4 -1 0}}, {{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that the third comma is actually the difference between the other two; {{nowrap\|{{vector\|-4 4 -1 0}} − {{vector\|1 2 -3 1}} {{=}} {{vector\|-5 2 2 -1}}}}, or as cents, {{nowrap\|21.51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So, if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.

ArrowHead294 at 15:30, 19 February 2025

2025-02-19T15:30:46Z

← Older revision		Revision as of 15:30, 19 February 2025
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	{{Beginner\|Dual list}}		{{Beginner\|Dual list}}
	A '''comma basis''' is a [[basis]] for the [[wikipedia: Kernel (linear algebra)\|nullspace]] (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ~~<math>~~n~~</math> [[wikipedia: Linear independence~~\|linearly independent]] vectors, where ~~<math>~~n~~</math>~~ is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].		A '''comma basis''' is a [[basis]] for the [[wikipedia: Kernel (linear algebra)\|nullspace]] (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].

	Linear independence means that no comma can be found as the sum of any multiples of the other commas. For example, consider the set of three commas 81/80, 126/125, and 225/224. As vectors those are {{vector\|-4 4 -1 0}}, {{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that the third comma is actually the difference between the other two; {{vector\|-4 4 -1 0}} - {{vector\|1 2 -3 1}} = {{vector\|-5 2 2 -1}}, or as cents, 21.~~51¢ -~~ 13.~~80¢~~ = 7.~~71¢~~. So, if two of these three commas are ever made to vanish (mapped to 0¢), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.		Linear independence means that no comma can be found as the sum of any multiples of the other commas. For example, consider the set of three commas 81/80, 126/125, and 225/224. As vectors those are {{vector\|-4 4 -1 0}}, {{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that the third comma is actually the difference between the other two; {{nowrap\|{{vector\|-4 4 -1 0}} − {{vector\|1 2 -3 1}} {{=}} {{vector\|-5 2 2 -1}}}}, or as cents, {{nowrap\|21.51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So, if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.

	The comma basis can be thought of either as a list of vectors or as a matrix formed by putting these vectors (as columns) together. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists\|normal forms]] have been developed as identifiers of temperaments.		The comma basis can be thought of either as a list of vectors or as a matrix formed by putting these vectors (as columns) together. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists\|normal forms]] have been developed as identifiers of temperaments.
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	Some math libraries, such as [https://www.sagemath.org/ Sage], provide functions for both directions; in Sage, to go from a mapping to a comma basis, use <code>left_kernel()</code>, and to go from a comma basis to a mapping, use <code>right_kernel()</code>. In other math libraries, such as [https://www.wolfram.com/language/ Wolfram Language], the nullspace operation <code>NullSpace[]</code> is primarily designed to work for mappings, and so if you want correct results, you must transform the basis for the nullspace into a mapping-like form, perform the nullspace operation, and then undo the initial transformation.		Some math libraries, such as [https://www.sagemath.org/ Sage], provide functions for both directions; in Sage, to go from a mapping to a comma basis, use <code>left_kernel()</code>, and to go from a comma basis to a mapping, use <code>right_kernel()</code>. In other math libraries, such as [https://www.wolfram.com/language/ Wolfram Language], the nullspace operation <code>NullSpace[]</code> is primarily designed to work for mappings, and so if you want correct results, you must transform the basis for the nullspace into a mapping-like form, perform the nullspace operation, and then undo the initial transformation.

	This transformation that relates the two directions of nullspace operations is called the ~~[[wikipedia: Transpose~~\|transpose]]. It works by reflecting a matrix's values across its ''main'' diagonal, i.e. either the diagonal running from the top-left corner toward the bottom-right, or the diagonal running from the bottom-right corner toward the top-left.		This transformation that relates the two directions of nullspace operations is called the {{w\|transpose}}. It works by reflecting a matrix's values across its ''main'' diagonal, i.e. either the diagonal running from the top-left corner toward the bottom-right, or the diagonal running from the bottom-right corner toward the top-left.

	However, transposing a comma basis, using a mapping-style nullspace function, then transposing again, will return a mapping in a strange form, with all of its zeros in the top-right corner, rather than the bottom-left as is preferred. The solution for this problem is to use the anti-transpose instead of the transpose. This is the same you reflect the matrix's entries across its ''anti-''diagonal (starting from either the top-right or bottom-left corner).		However, transposing a comma basis, using a mapping-style nullspace function, then transposing again, will return a mapping in a strange form, with all of its zeros in the top-right corner, rather than the bottom-left as is preferred. The solution for this problem is to use the anti-transpose instead of the transpose. This is the same you reflect the matrix's entries across its ''anti-''diagonal (starting from either the top-right or bottom-left corner).

ArrowHead294 at 18:44, 2 November 2024

2024-11-02T18:44:33Z

← Older revision		Revision as of 18:44, 2 November 2024
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	The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.		The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.

	Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn more about finding the nullspace, see [[Dave Keenan ~~%26~~ Douglas Blumeyer~~%27s~~ guide to RTT/Exploring temperaments#Nullspace\|here]].		Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn more about finding the nullspace, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace\|here]].

	To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.		To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.

ArrowHead294: Update links

2024-11-02T18:41:04Z

Update links

← Older revision		Revision as of 18:41, 2 November 2024
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	== With respect to the mapping ==		== With respect to the mapping ==

	The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.		The comma basis is considered the dual of the temperament's [[mapping]] matrix. Temperaments may be identified by either their mapping or comma basis.

	Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn more about finding the nullspace, see [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT~~: exploring~~ temperaments #Nullspace]].		Functions for finding the nullspace of a matrix are readily available in many math libraries. All you need to do to get a comma basis for a mapping is to find the nullspace. To learn more about finding the nullspace, see [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Exploring temperaments#Nullspace\|here]].

	To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.		To reverse the nullspace operation, that is, to find a mapping from a comma basis, you can also use the nullspace operation; the relationship between a matrix and its nullspace essentially works both ways.

Fredg999: Beginner template

2023-07-08T18:59:23Z

Beginner template

← Older revision		Revision as of 18:59, 8 July 2023
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	~~This is a basic introduction to this concept. For a more mathematical take, see [[dual~~ list~~]].~~		{{Beginner\|Dual list}}

	A '''comma basis''' is a [[basis]] for the [[wikipedia: Kernel (linear algebra)\|nullspace]] (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of <math>n</math> [[wikipedia: Linear independence\|linearly independent]] vectors, where <math>n</math> is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].		A '''comma basis''' is a [[basis]] for the [[wikipedia: Kernel (linear algebra)\|nullspace]] (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of <math>n</math> [[wikipedia: Linear independence\|linearly independent]] vectors, where <math>n</math> is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].

← Older revision		Revision as of 15:31, 19 February 2025
Line 1:		Line 1:
	{{Beginner\|Dual list}}		{{Beginner\|Dual list}}
	A '''comma basis''' is a [[basis]] for the ~~[[wikipedia:~~ Kernel (linear algebra)\|nullspace]] (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].		A '''comma basis''' is a [[basis]] for the {{w\|Kernel (linear algebra)\|nullspace}} (sometimes also called the "kernel") of a [[regular temperament\|temperament]]. It consists of ''n'' {{w\|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].

	Linear independence means that no comma can be found as the sum of any multiples of the other commas. For example, consider the set of three commas 81/80, 126/125, and 225/224. As vectors those are {{vector\|-4 4 -1 0}}, {{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that the third comma is actually the difference between the other two; {{nowrap\|{{vector\|-4 4 -1 0}} − {{vector\|1 2 -3 1}} {{=}} {{vector\|-5 2 2 -1}}}}, or as cents, {{nowrap\|21.51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So, if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.		Linear independence means that no comma can be found as the sum of any multiples of the other commas. For example, consider the set of three commas 81/80, 126/125, and 225/224. As vectors those are {{vector\|-4 4 -1 0}}, {{vector\|1 2 -3 1}}, and {{vector\|-5 2 2 -1}}. Notice that the third comma is actually the difference between the other two; {{nowrap\|{{vector\|-4 4 -1 0}} − {{vector\|1 2 -3 1}} {{=}} {{vector\|-5 2 2 -1}}}}, or as cents, {{nowrap\|21.51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So, if two of these three commas are ever made to vanish (mapped to 0{{c}}), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.