source file: mills2.txt
Date: Wed, 9 Jul 1997 19:52:40 +0200

Subject: Re: Fokker

From: gbreed@cix.compulink.co.uk (Graham Breed)

John Chalmers wrote:

>BTW, Fokker used the idea of defining intervals in 2 and 3-D tones spaces
>in a somewhat different way. He chose sets of "unison vectors," which he
>set to 0, in 3x5 and 3x5x7 tone spaces to define temperaments as
>"periodicity blocks" in the space. For example, the diesis and the syntonic
>comma define a repeating block of 12 pitches as the area defined by the
>absolute value of the cross product of the vectors describing these
>intervals. (The box product is used for 3 intervals in 3-D space.) I
>believe the concept can be extended to higher dimensions  by computing the
>absolute values of the determinants of the square matrices whose row
>vectors are the defining intervals. 

12tet is the temperament where octaves are perfect, and both the
syntonic comma (-4 4 -1)H and the diesis (0 -1 2)H are zero.  In
my matrix representation, these conditions can be written:

                 ( 1 0  0)     (1 0 0) 
                 ( 7 0 -3)H' (0 0 0)H
                 (-4 4 -1)     (0 0 0)

Where H is the metric of pitch space, and H' is its approximation
in 12tet.  To get H', you multiply the right hand side of the
equation by the inverse of the matrix on the left.  This general
method works for all ETs that approximate some kind of JI.  For
the perfect octave case, you only need the left hand column of
the inverted matrix and the number of steps in an octave is +/-
the determinant of the original matrix.  This will also be one
dimension of the cross product of the two zeroed intervals. It is
also the determinant of the square matrix formed by the zeroed
intervals without the octave column, and this will work for any
dimension.  So, it looks like I was thinking along the same lines
as Fokker.  Did he come up with any other stuff like this?  Is
there a book I can read?

                      Graham

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