Unfolding Toroidal Space
The spectral essence matrices produced by the HRT software are gathered up within a manifold known as "toroidal space" - so named because of its torus (doughnut) shape. A pure, coherent harmonic source is fed through the "portacle" - the modulating filter generated from the individual's voicing - and focused to a 0-dimensional point at the vortex thread (or "origin") of the flowing torus. As the torus rotates and folds through itself - each point passing through the central thread - its framework is populated by the partials of the spectral essence.
The torus, however, is not the most practical format for presentation, as the partials populating the interior or back-facing surfaces would be obscured from view. Unfolding the native toroidal space and projecting it into a 2-dimensional matrix allows us to see every partial in a uniform grid. The illustration above shows the dynamics of this process, and the characters to the right of the diagram is the actual unfolding algorithm. After unfolding is complete, the partials in the matrix are shifted by 1/2, thus restoring the origin to the center.
