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)|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 [[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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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 ~~null-space~~ 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 ~~null-space~~. To learn more about finding the ~~null-space~~, see [[Douglas Blumeyer's RTT How-To #The other side of duality]].

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 [[Douglas Blumeyer's RTT How-To #The other side of duality]].

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.

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.

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 ~~null-space~~ 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 ~~null-space~~ into a mapping-like form, perform the ~~null-space~~ 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 ~~null-space~~ 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 [[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.

However, transposing a comma basis, using a mapping-style ~~null-space~~ 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).

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 maps (rows) of a mapping, but a normal transpose of the comma basis would flip its zeroes into the top-right corner instead.

@@ 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)|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 [[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 11: / Line 11: @@
 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 null-space 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 null-space. To learn more about finding the null-space, see [[Douglas Blumeyer's RTT How-To #The other side of duality]].
+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 [[Douglas Blumeyer's RTT How-To #The other side of duality]].
-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.
+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.
-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 null-space 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 null-space into a mapping-like form, perform the null-space 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 null-space 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 [[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.
-However, transposing a comma basis, using a mapping-style null-space 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).
 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 maps (rows) of a mapping, but a normal transpose of the comma basis would flip its zeroes into the top-right corner instead.