Temperament addition: Difference between revisions

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Meet and join does not ''necessarily'' involve linear dependence. Linear dependence only matters for meet and join when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].

==Negation==

[[File:Very simple illustration of temperament sum vs diff.png|500px|thumb|left|Equivalences of temperament arithmetic depending on negativity. ]]

The temperament difference can be understood as being the same operation as the temperament sum except with one of the two temperaments negated.

For single vectors (and multivectors), negation is as simple as changing the sign of every entry; for matrices, negation is accomplished by choosing a single row (in the case of mappings) or column (in the case of comma bases) and changing the sign of every entry in it.

Suppose you have a matrix representing temperament T₁ and another matrix representing T₂. If you want to find both their sum and difference, you can calculate both T₁ + T₂ and T₁ + -T₂. There's no need to also find -T₁ + T₂; this will merely give the negation of T₁ + -T₂. The same goes for -T₁ + -T₂, which is the negation of T₁ + T₂.

But a question remains: which result between T₁ + T₂ and T₁ + -T₂ is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that T₁ or T₂ wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.

The check is related to canonicalization of varianced multivectors as are used in exterior algebra for RTT. Essentially we take the minors of the matrix, and then look at their leading or trailing entry (leading in the case of a covariant matrix, like a mapping; trailing in the case of a contravariant matrix, like a comma basis): if this entry is positive, so is the temperament, and vice versa.

==Beyond min-grade-1==

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 Meet and join does not ''necessarily'' involve linear dependence. Linear dependence only matters for meet and join when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].
+==Negation==
+[[File:Very simple illustration of temperament sum vs diff.png|500px|thumb|left|Equivalences of temperament arithmetic depending on negativity. ]]
+The temperament difference can be understood as being the same operation as the temperament sum except with one of the two temperaments negated.
+For single vectors (and multivectors), negation is as simple as changing the sign of every entry; for matrices, negation is accomplished by choosing a single row (in the case of mappings) or column (in the case of comma bases) and changing the sign of every entry in it.
+Suppose you have a matrix representing temperament T₁ and another matrix representing T₂. If you want to find both their sum and difference, you can calculate both T₁ + T₂ and T₁ + -T₂. There's no need to also find -T₁ + T₂; this will merely give the negation of T₁ + -T₂. The same goes for -T₁ + -T₂, which is the negation of T₁ + T₂.
+But a question remains: which result between T₁ + T₂ and T₁ + -T₂ is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that T₁ or T₂ wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.
+The check is related to canonicalization of varianced multivectors as are used in exterior algebra for RTT. Essentially we take the minors of the matrix, and then look at their leading or trailing entry (leading in the case of a covariant matrix, like a mapping; trailing in the case of a contravariant matrix, like a comma basis): if this entry is positive, so is the temperament, and vice versa.
 ==Beyond min-grade-1==