Comma basis: Difference between revisions
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| en = Comma basis | |||
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| ja = コンマ基底 | |||
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{{Beginner|Dual list}} | {{Beginner|Dual list}} | ||
A '''comma basis''' is a | A '''comma basis''' is a list of {{w|linearly independent}} [[comma]]s that characterizes a [[regular temperament|temperament]]. | ||
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, 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 == | ||
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 | 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. | ||
This transformation that relates the two directions of nullspace operations is called the | 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). | ||