multimonzo

[Joe Monzo]

A set of numbers notated in the form [a b c> , which is the list of coefficients found by calculating the wedge-product of a certain number of monzos.

The multimonzo provides a unique means of identifying the entire infinite set of unison-vectors, within a given prime-space, which are tempered out ("vanish") in a certain temperament. Prime-factor 2 may not be ignored when calculating a multimonzo (as it often is in other tuning calculations).

Multimonzo is the generic term which covers all numbers of independent monzos being wedged; the trivial 1-dimensional case with enclosing brackets [ > is equivalent to the simple monzo.

The double enclosing brackets [[ >> describe a bimonzo, which represents a 2-dimensional set of monzos; [[[ >>> describes a trimonzo, which represents a 3-dimensional set of monzos, etc. The number of brackets on both sides is the same and designates the dimensionality of the set of monzos under consideration.

There is one element in the multimonzo for every possible unit multimonzo (or every possible pair of unit monzos).

The complement of the multimonzo is the multibreed (wedgie), which describes a temperament's entire mapping. Thus, by using the multimonzo the mapping can be calculated from any set of independent unison-vectors which vanish in that temperament.

For an equal-temperament, where d = the number of dimensions in the JI lattice which are approximated by that ET (including prime-factor 2), there must be d - 1 independent promos (that is, d - 1 independent unison-vectors which vanish in that temperament). Each independent promo introduced reduces the dimensionality of the temperament by 1. The dimension of the multimonzo is the codimension of the temperament. Some examples:

• For a 1-dimensional temperament of a 2-dimensional prime-space, the breed and the monzo are complements of each other, thus forming the complementary pair < breed | monzo > .

For example, if one considers a 3-limit (pythagorean) tuning, and tempers out the pythagorean-comma, then the breed is < 12 19 ] , and the vanishing monzo is [ -19 12, > . They may be written together in bra-ket form as < 12 19 | -19 12 > .

• In 3-dimensional prime-space, the complementary pairs are: [bibreed (bival), monzo] and [bimonzo, breed (val)].

As an example of a 2-dimensional temperament of a 3-dimensional prime-space: the vanishing multimonzo in all 5-limit temperaments belonging to the meantone family is the simple monzo [ -4 4, -1 > , because all meantones which map 5-limit JI, temper out the syntonic-comma, whose 2,3,5-monzo is [ -4 4, -1 > , and all of its multiples and submultiples. The bibreed is << 4 -4 1 ]] .

As an example of a 1-dimensional temperament of a 3-dimensional prime-space: the vanishing multimonzo in 5-limit 12-et is the bimonzo [[ 28 -19 12 >> (if the larger unison-vector comes first; if the order of the unison-vectors in the wedge calculation is reversed, the signs of the bimonzo elements are reversed), and the complementary multibreed is the simple breed (val) < 12 19 28 ] .

A special case of the multimonzo is when it is tempered-out (or "vanishes"), in which case it is, or is represented by, a vapro.

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