Fokker's Matrices

I have only been able to find two books of Fokker's writings -- _New
Music with 31 Notes_ and the big one with all his compositions in it,
edited by Rudolf Rasch, I think. The matrix/determinant stuff sounds
like something I need to seriously study. Can someone tell me more about
what Fokker wrote, or perhaps send me some photocopies, for educational
purposes only, and all other disclaimers relevant to copyright
violation? My first question would be which ETs can and cannot be
generated by this procedure and by analagous procedures with axes 5/3
and 3, or 5/3 and 5, and analagously in higher dimensions.

On Tue, 9 Jun 1998, Patrick Ozzard-Low wrote:
> Could anyone tell me - is there a unique theoretical limit at which a
> comma (of different kinds?) may be said to disappear? Are there
> strict and not so strict definitions of which systems (or musical
> sequences?) of which this can be said to pertain/occur? Is there a
> single accepted definitive reference which defines the 'disappearance
> of a comma'? (Helmholtz?) These are things I missed in my education.

An interval vanishes in a particular ET (or quasi-ET like a
well-temperament) if it is represented by 0 scale degrees. In face, to
specify which intervals vanish is a fairly practical way to specify a
particular system. See Fokker's writings for this.

As an example: working in the 3-5 harmonic lattice, deciding that one
wants the syntonic comma 81/80 and the lesser diesis 128/125 to vanish
commits one to working in a 12-ish system. Conveniently, one can
represent this (as Graham Breed has posted before, following Fokker)
with matrices: the 81/80 is represented by the vector (4 -1) (3 ^ 4 *
5 ^ -1, omitting 2s) and the lesser diesis by (0 3). Put the vectors
together to form a matrix, and the determinant of the matrix is (4 * 3 -
-1 * 0), or 12. This also works in higher dimensions.

Geometrically, one can imagine the unison vectors (Fokker's terminology)
specifying a way to collapse the infinite harmonic plane into a
2-manifold. The number of pitches in the ET is the area of the
parallelogram bounded by the unison vectors, or the "real" size of the
manifold--I forget what the formal mathematical term for that is. (This
also extends to higher dimensions, with parallelepipeds etc.)

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
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