Quadratic Corners

Franck: Last year I started writing some simple BASIC programs
to factor and plot scales on various types of tonal lattices as I
needed to make some new graphics for some talks I intended to give,
and I am a slow and not very competent draftsman.

The lattices included both Cartesian types (powers of 3 on the
horizontal, 5 on the vertical, 7 on a 30 degree diagonal, etc.)
and polygonal ones similar to those invented by Ervin Wilson.
Once I had the programs running on my old monochrome Mac, I started
generating scales which had interesting visual appearances.

A Quadratic Corner is an incomplete Euler-Fokker genus consisting
for the factors 3.5.7 of 1 3 3^2 5 5^2 3x5 7 7^2 3x7 and 5x7. One
can add the inversion of each of these terms and reduce
the whole to a common octave. The concept may be extended to include
higher prime factors, such as 11 and 13.

In the 3.5.7 rectilinear lattice, this collection has a nice spikey
appearance as my program connects only nearest neighbors (tones
differing by plus or minus 1 step along any axis). Some versions of
the polygonal lattice plotters allowed diagonally related notes to be
connected as in the familiar triangular lattice where major and minor
triads appear as equilateral triangles.

I make no claims for the musical utility of these scales nor of the
related Cubic Corners, but visually they are interesting in some
representations.

--John


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