Comment on Kami's post

Kami is correct about the 3)6 [ 1 1 1 3 5 7 ] 20-any being
equivalent to the 3.5.7 Euler-Fokker Genus. I hadn't noticed
this relation before and I don't know if Erv Wilson has either
as he seldom used repeated factors.

One can generalize these relations even further by considering
the general 2)4 [x.y.x.w] hexany as the 6 vertices with 2 non-zero
and 2 zero coordinates of the 16 vertices of the 4-D hypercube
(tesseract). The 3)6 [x.y.z.w.v.u] 20-any is the 20 of the 64
vertices of the 6-D hypercube with 3 non-zero coordinates, and the
4)8 70-any is the analogous set of the 256 vertices of the 8-D
"measure polytope," in Coxeter's terminology. One can write the
vertices as N-dimensional vectors and embed them in a N-dimensional
space where N is the number of prime factors in the tone space.
Whenever 1 or 2 appears as a factor, the dimensionality may be
lowered by the assumption of octave equivalence.

Transposition thus becomes translation and other musical operations
may be replaced by their corresponding vector operations. Different
species of CPS (or any other kind of JI structure) may be obtained
by rotating around other axes, planes, or hyperplanes.

I have found such interpretations fascinating over the years.
BTW, to measure distances between intervals use the Minkowski metric
(Tenney's harmonic distance) rather than the Euclidean distance formula
as the lattice is non-continuous. Enjoy your linear algebra class!

--John


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