Do any of you know of passages of music that represent 2-dimensional projections of 3- or higher-dimensional parallelotopes if one plots time against the log of frequency? A simple way to generate such a passage is to start with a two-note figure. Create a four-note figure consisting of the original two-note figure followed by a transposed version of same. Now create an eight-note figure consisting of that four-note figure followed by a transposed version of same. The eight notes can be conceived of as the vertices of a musical parallelohedron (projected into a 2D space). Apply the construction again and you’ve got a sixteen-note projection of the vertices of a 4D parallelotope. (More interesting variants exist, but that’s the basic idea.) Has this sort of geometrically-inspired scheme for organizing pitches in time been exploited by any composers to create music worth listening to? I ask because I’m working on an essay about the hypercube, so musical embodiments would fit in nicely. For that matter, has anyone built a 3D jungle gym based on 4D structures? (As I’ll mention in the article, jungle gyms were invented by the same guy who coined the term “tesseract”, and the idea of making higher dimensions more kinesthetically accessible was a goal of his.) Thanks, Jim Propp