Comma basis

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A comma basis is a basis for the nullspace (sometimes also called the "kernel") of a temperament. It consists of [math]n[/math] linearly independent vectors, where [math]n[/math] is the nullity, each one representing a comma that is made to vanish.

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 [-4 4 -1 0, [1 2 -3 1, and [-5 2 2 -1. Notice that the third comma is actually the difference between the other two; [-4 4 -1 0 - [1 2 -3 1 = [-5 2 2 -1, or as cents, 21.51¢ - 13.80¢ = 7.71¢. So, if two of these three commas are ever made to vanish (mapped to 0¢), then the third one necessarily is also made to vanish. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.

The comma basis can be thought of either as a list of vectors or as a matrix formed by putting these vectors (as columns) together. Besides, it is often presented in terms of ratios for convenience. Various normal forms have been developed as identifiers of temperaments.

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.

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 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.

Some math libraries, such as Sage, provide functions for both directions; in Sage, to go from a mapping to a comma basis, use left_kernel(), and to go from a comma basis to a mapping, use right_kernel(). In other math libraries, such as Wolfram Language, the nullspace operation NullSpace[] 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 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).

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.