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Encyclopedia of Microtonal Music Theory

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[Joe Monzo, with Paul Erlich and John Chalmers]

A calculation made on a square matrix.

A 2 x 2 matrix presents a simple example; higher-dimensional matrices are more complex.

For matrix

   [a  b]
   [    ]
   [c  d]

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

Paul Erlich gives an example of how to calculate the determinant for a 3-dimensional matrix in his A Gentle Introduction to Fokker Periodicity Blocks, part 3:

		... imagine that the matrix wraps around and the left and right edges
		are joined together. Then add all products of three elements along
		diagonals that slant downward and to the right, and subtract all the
		products along diagonals sloping the other way. So:

		 [a b c]
		 [d e f] = a*e*i + b*f*g + c*d*h - c*e*g - a*f*h - b*d*i
		 [g h i]

When the rows of the matrix represent independent unison vectors, the absolute value of the determinant indicates the number of pitches contained within the corresponding periodicity-block. This was documented by Fokker.

Interestingly, the same numbers which delimit ET's with good rational implication also specify the number of pitch-classes in just-intonation systems which incorporate bridging to delineate the finity of the system.

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