Comma basis: Difference between revisions

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This is a basic introduction to this concept. For a more mathematical take, see [[dual list]].

A '''comma basis''' is a basis for the [[wikipedia: Kernel (linear algebra)|null-space]] (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 [[tempered out]].

A '''comma basis''' is a [[basis]] for the [[wikipedia: Kernel (linear algebra)|null-space]] (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 [[tempered out]].

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 tempered out (mapped to 0¢), then the third one necessarily is also tempered out. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.

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 This is a basic introduction to this concept. For a more mathematical take, see [[dual list]].
-A '''comma basis''' is a basis for the [[wikipedia: Kernel (linear algebra)|null-space]] (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 [[tempered out]].
+A '''comma basis''' is a [[basis]] for the [[wikipedia: Kernel (linear algebra)|null-space]] (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 [[tempered out]].
 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 tempered out (mapped to 0¢), then the third one necessarily is also tempered out. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.