Exterior algebra: Difference between revisions

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In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).

Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

=== Wedge product ===

The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a1 a2 a3 ... an] and [b1 b2 b3 ... bn], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry ci,j of the wedge product is aibj - biaj. cj,i is equal to -(ci,j), and ci,i where the two indices are the same is 0.

The wedge product is used in regular temperament theory to combine [[vals]] into [[Wedgie|multivals]], hence why multivals are called "wedgies". For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.

The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.

For a higher-dimensional multivector with elements ci, j, k..., the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.

== See also ==

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 In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).
 Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.
+=== Wedge product ===
+The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.
+The wedge product is used in regular temperament theory to combine [[vals]] into [[Wedgie|multivals]], hence why multivals are called "wedgies". For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.
+The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.
+For a higher-dimensional multivector with elements c<sub>i, j, k...</sub>, the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.
 == See also ==