bibreed / bival

A bibreed (or bival) is the wedge product of two breeds (vals); in particular, a linear temperament wedgie.

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The bibreed (or bival) is 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 vals (or breeds). It is a specific cardinality referring to the 2-dimensional case of the more general term multibreed (wedgie). The enclosing brackets << || are for 3-dimensional space which is what a bibreed always describes. The number of brackets on both sides is the same, and is one less than the dimensionality of the assumed prime-space (tone space).

The bibreed shows the common vanishing unison-vectors shared by any set of ETs which approximate a 3-dimensional prime-space (or tone space), thus its use is valuable in grouping temperaments into families.

The most typical use of bibreeds is in describing the approximations of equal-temperaments to 5-limit just intonation, where the three dimensions of the prime-space (or tone space) represent the prime-factors 2, 3, and 5. In 4 dimensions, the bibreed is the complement of the bimonzo.

[Joe Monzo]
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