# bimonzo, bivector

[Joe Monzo, with Paul Erlich]

A set of numbers notated in the form [[a b c >> , which is the list of coefficients found by calculating the wedge-product of two monzos. It is a specific cardinality referring to the 2-dimensional case of the more general term multimonzo. The bimonzo provides a unique means of identifying any set of intervals which form a 2-dimensional sub-lattice of the prime-space (or tonespace). To reflect this, the bimonzo always uses a double pair of brackets. Prime-factor 2 may not be ignored when calculating a bimonzo (or any other multimonzo). The most typical use of bimonzos is to identify the 2-dimensional set of vapros -- independent unison-vectors which are tempered out ("vanish") -- in a certain temperament.

The bimonzo is calculated as the cross-product of two monzos. As an example, we may examine a case where 12-et is intended to represent a 2,3,5-prime-space (or tonespace), in which the vapros defined by the syntonic comma and the enharmonic diesis are tempered out.

```12-et, 2,3,5-prime-space matrix of vapros:

[  7  0, -3 >
[ -4  4, -1 >

cross-products of the three minors:

[ 7   0]
[-4   4]  =  (7 *  4) - (-4 *  0)  =  28 -   0  =   28

[ 7  -3]
[-4  -1]  =  (7 * -1) - (-4 * -3)  =  -7 -  12  =  -19

[ 0  -3]
[ 4  -1]  =  (0 * -1) - ( 4 * -3)  =   0 - -12  =   12

therefore, bimonzo = [[ 28, -19, 12 >>
```

There is one element in the bimonzo for every possible unit bimonzo (or every possible pair of unit monzos). Geometrically, the bimonzo may be thought of as a directed area. If it vanishes, this directed area lies in the plane of (or parallel to) the plane describing the kernel of a temperament. (If additional vapros are present in the temperament, then the kernel is no longer described by a plane, but rather by something with more dimensions, such as a "space", "realm", etc.)

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