matrix

A mathematical construct, a generalization of the concept of number.

It can be used to do calculations on the prime-factors of ratios.

A more complex operation, calculating the determinant, indicates how many pitch-classes there are in a periodicity block.

Matrix was the term I formerly used for the diagrams I created to represent pitch-structures (see my first paper on my harmonic theories). As these diagrams changed and I became more familiar with similar work done by others, I changed the name to my lattice model of musical harmony.

In matrix calculations the inverse matrix, 1/matrix or matrix^-1, is often required. A simple example of a 2 x 2 matrix is illustrated below; finding the inverse of a higher-dimensional matrix is more complicated.

The inverse of matrix

   [a  b]
   [    ]
   [c  d]

is

    1    [ d -b]
   --- * [     ]
   det   [-c  a]

where

   det = (a * d) - (b * c).

The part represented as

    [ d -b]
    [     ]
    [-c  a]

is known as the adjoint.

see also transformation, determinant, interval matrix

[from Joe Monzo, JustMusic: A New Harmony]

with Paul Erlich]

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In general a matrix is a representation for a function acting on vectors. In other words it is a transformation from one vector space into another. The determinant is the amount of change for the measures of objects under this transformation. (measure is for instance surface area for 2 dimensions and volume for 3 dimensions)

A matrix is a block of n x m numbers for a transformation from a n-dimensional into an m-dimensional space. The determinant is only defined for 'square' matrices. The only use of matrices in your definitions is the measure of the periodicity block as far as I can think of. In this case the n unison vectors (bridges) of a n-dimensional 'lattice' space (n = 2 for 5-limit, n = 3 for 7-limit etc.) are taken to be the images of a transformation of n unit vectors and gathered in a matrix. The volume of the transformation of the unit cube is thus the volume of the periodicity block and can be calculated by taking the determinant of the resulting matrix. It doesn't matter if you gather the unison vectors as columns or rows in the matrix, because that doesn't influence the value of the determinant.

[from Kees van Prooijen, personal communication]

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See Graham Breed's excellent A Matrix Tutorial

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Updated: 2002-1-3

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