source file: m1442.txt
Date: Tue, 9 Jun 1998 15:30:12 -0500 (CDT)

Subject: Re: Fokker's Matrices

From: Paul Hahn <Paul-Hahn@library.wustl.edu>

On Tue, 9 Jun 1998, Paul H. Erlich wrote:
> 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?

Here are the relevant publications.  If you can't find them any other
way they should be Interlibrary-loan-able from us.

 AUTHOR       Fokker, A. D. (Adriaan Daniel), 1887-
 TITLE        Selections from the harmonic lattice of perfect fifths and major
                thirds containing 12, 19, 22, 31, 41 or 53 notes.
 PUBLISHED    Amsterdam, Koninkl. Nederl. Akademie van Wetenschappen, 1968.   
 DESCRIPTION  251-266 p. 26 cm.
 NOTE         Caption title.
              "Reprinted from Proceedings, Series B, 71, No. 4, 1968."
              Communicated at the meeting of June 29, 1968.
 SUBJECTS     Musical intervals and scales.
              Musical temperament.

 AUTHOR       Fokker, A. D. (Adriaan Daniel), 1887-
 TITLE        Unison vectors and periodicity blocks in the three-dimensional
                (3-5-7) harmonic lattice of notes.
 PUBLISHED    Amsterdam, Koninkl. Nederl. Akademie van Wetenschappen, 1968. 
 DESCRIPTION  153-158 p. 26 cm.
 NOTE         Caption title.
              "Reprinted from Proceedings, Series B., 72, No. 3, 1969."
              Communicated at the meeting of February 22, 1969.
 SUBJECTS     Musical intervals and scales.

>            My first question would be which ETs can and cannot be
> generated by this procedure

You can generate any damn ET you want this way, although sometimes the
unisons/commas chosen to vanish aren't very small.

>                             and by analagous procedures with axes 5/3
> and 3, or 5/3 and 5,

Since the space spanned by 5/3 and 3 or 5/3 and 5 is the same one
spanned by 3 and 5, any ET generable by one should be generable by the
others, with the appropriate matrix transformations.

>                      and analagously in higher dimensions.

Yes, as the above title implies, it works in 3 dimensions, I have used
it in 4 dimensions at least, and there's no reason why it wouldn't work
in arbitrarily high dimensions.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
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