Comma basis: Difference between revisions

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

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]].

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.

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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 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~~|here~~]].

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.

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 {{Beginner|Dual list}}
-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 ''n'' {{|linearly independent}} vectors, where ''n'' is the [[nullity]], each one representing a [[comma]] that is made to [[vanish]].
+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]].
 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.
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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 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|here]].
+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.