Spirolab is a project I've been working on for the past few years that creates an intuitive environment for playing around with spirographs (also known as physiograms) and sound. As it turns out, most of those “laser spirographs” that you see online are really just showing the effects of phasors adding together in the time-domain and displaying the “phase plot” of your signal.
Anyways, I created an executable using Processing that you can DEMO! at http://www.nolanhergert.com/share/spirolab
Also, I've played around with making a web version of this code, although it's a little trickier and I've yet to add Web Audio API functionality (it's coming though). It's currently parked at http://spirolab.org
A very good and visual explanatory book to all things musical and mathematical. Harmonograph: A Visual Guide To The Mathematics of Music
More notes at Spirolab Notes