### Lattice Orientation

A reaction from Paul Erlich about my 'expressibility' measure made me think about the
relation of these measures and the orientation of a lattice.

Here are Paul's observations.

Now the 'translation' from an equidistant lattice to Paul's triangular lattice can be accomplished with a matrix
like this:

transforming the lattice:

to

The contour of a complete block within a certain limit of my measure looks like:

and after Paul's transformation like this:

So indeed, it's still a skewed shape, indicating the discrepancy with Paul's measure.

To create a symmetrical hexagon we would need the transformation matrix:

which would result in:

A lattice after this transform would look like:

Now we seem to hit a paradox, because the measure of a 5/4 (blue) and a 5/3 (green) should be the same, just as
in Paul's isosceles triangle case. This is caused by the fact that the boundary of my block should be tranformed
to a circle, not a regular hexagon. Performing such a transform will make the lattice look like:

Of course, since this is a non-linear transform, this only represents the metric relative to the center, not necessarily
from any other point in the lattice.

Back to intervals and periodicity blocks

Back to music

Kees home