Comma basis: Difference between revisions

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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'' {{w|~~linearly independent~~}} vectors, where ''n'' is the [[nullity]], each one representing~~ a ~~[[comma]] that is made to [[vanish]]~~.

A '''comma basis''' is a list of linearly independent commas that characterizes a temperament.

~~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; {{nowrap|{{vector|-4 4 -1 0}} − {{vector|1 2 -3 1}} {{~~=~~}} {{vector|-5 2 2 -1}}}}~~, ~~or as cents, {{nowrap|21~~.~~51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So,~~ 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. ~~Therefore~~, we only need to pick two of ~~these~~ commas ~~to put in our comma basis~~; the third ~~one would~~ be ~~implied~~.

For example, septimal meantone tempers out [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ((225/224)*(126/125)=(81/80), for example). 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: [{{vector|-4 4 -1 0}}, {{vector|-5 2 2 -1}}], or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists|normal forms]] have been developed as identifiers of temperaments.

~~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 lists #Normal interval lists|normal forms~~]] ~~have been developed as identifiers~~ of ~~temperaments~~.

Mathematically, it is a [[basis]] for the {{w|Kernel (linear algebra)|nullspace}} (sometimes also called the "kernel") of a [[regular temperament|temperament]]. It consists of ''n'' {{w|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out.

== 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 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.

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.

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 {{Beginner|Dual list}}
-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'' {{w|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].
+A '''comma basis''' is a list of linearly independent commas that characterizes a temperament.
-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; {{nowrap|{{vector|-4 4 -1 0}} − {{vector|1 2 -3 1}} {{=}} {{vector|-5 2 2 -1}}}}, or as cents, {{nowrap|21.51{{c}} − 13.80{{c}} {{=}} 7.71{{c}}}}. So, 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. Therefore, we only need to pick two of these commas to put in our comma basis; the third one would be implied.
+For example, septimal meantone tempers out [[225/224]], [[126/125]], and [[81/80]], but from any two of these commas can be derived the third ((225/224)*(126/125)=(81/80), for example). 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: [{{vector|-4 4 -1 0}}, {{vector|-5 2 2 -1}}], or equivalently as a list of monzos. Besides, it is often presented in terms of ratios for convenience. Various [[Normal lists #Normal interval lists|normal forms]] have been developed as identifiers of temperaments.
-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 lists #Normal interval lists|normal forms]] have been developed as identifiers of temperaments.
+Mathematically, it is a [[basis]] for the {{w|Kernel (linear algebra)|nullspace}} (sometimes also called the "kernel") of a [[regular temperament|temperament]]. It consists of ''n'' {{w|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing one of the commas that is tempered out.
 == 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 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.
+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.