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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To reverse the null-space operation, that is, to find a mapping from a comma basis, you can also use the null-space operation; the relationship between a matrix and its null-space essentially works both ways. However, math libraries' null-space operation is designed to work for mappings, and so if you want correct results, you must transform the basis for the null-space into a mapping-like form, perform the null-space operation, and then undo the initial transformation. This initial transformation you must do and undo is called the anti-transpose, which is just like the typical transpose of a matrix, except instead of reflecting the matrix's values across the main diagonal (starting from either the top-left or bottom-right corner), you reflect them across the anti-diagonal (starting from either the top-right or bottom-left corner). You can remember this because most mappings and comma bases have zeroes in the bottom-left corner, and you want to keep them there; some kind of transpose is necessary to convert the constituent comma vectors columns of the comma basis into rows as if they were constituent generator mapping rows of a mapping, but a normal transpose of the comma basis would flip its zeroes into the top-right corner instead.

[[Category:Regular temperament theory]]

[[Category:Monzo]]

@@ Line 1: / Line 1: @@
 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.
@@ Line 14: / Line 14: @@
 To reverse the null-space operation, that is, to find a mapping from a comma basis, you can also use the null-space operation; the relationship between a matrix and its null-space essentially works both ways. However, math libraries' null-space operation is designed to work for mappings, and so if you want correct results, you must transform the basis for the null-space into a mapping-like form, perform the null-space operation, and then undo the initial transformation. This initial transformation you must do and undo is called the anti-transpose, which is just like the typical transpose of a matrix, except instead of reflecting the matrix's values across the main diagonal (starting from either the top-left or bottom-right corner), you reflect them across the anti-diagonal (starting from either the top-right or bottom-left corner). You can remember this because most mappings and comma bases have zeroes in the bottom-left corner, and you want to keep them there; some kind of transpose is necessary to convert the constituent comma vectors columns of the comma basis into rows as if they were constituent generator mapping rows of a mapping, but a normal transpose of the comma basis would flip its zeroes into the top-right corner instead.
+[[Category:Regular temperament theory]]
+[[Category:Monzo]]