Comma basis: Difference between revisions

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| ja = コンマ基底

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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.

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 ==

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+{{interwiki
+| de =
+| en = Comma basis
+| es =
+| ja = コンマ基底
+}}
 {{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.
+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 ==