Exterior algebra

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Exterior algebra

Exterior algebra is a type of algebra which has a product, called exterior product or wedge product and denoted with [math]\displaystyle{ \wedge }[/math], such that [math]\displaystyle{ v \wedge v = 0 }[/math] for every vector [math]\displaystyle{ v }[/math] in the vector space [math]\displaystyle{ V }[/math].

In regular temperament theory, exterior algebra is typically applied to the vector space of vals (or maps). The exterior product of two or more vals is called a multival, and its canonical form is called a wedgie (or Plücker coordinates), which can be used to uniquely identify a regular temperament.

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.

See also

• Plücker coordinates
• Hodge dual
• Dave Keenan & Douglas Blumeyer's guide to EA for RTT
• Interior product
• Recoverability
• User:Mike Battaglia/Exterior Norm Conjecture Table