Very happy that Journal of Mathematics and Music has published a peer-reviewed paper on the theory that underlies the algorithmic approach used by the software described on this site.
The paper describes a self-similar map of rhythmic coherence and syncopation, across all nested combinations of:
- repetition of anticipation
- anticipation of repetition
This map collapses a seemingly large number of branching possibilities onto a concise set, a fractal-like space of interrelated (non-fractal) rhythms. Anticipation figures, and repetitions of those figures, are mapped onto the odd binomial coefficients as arranged on Pascal’s triangle, and from there onto the Sierpinski gasket.
With that, one can enumerate/characterize an axis of rhythmic structure, forming a new starting point for algorithmic analysis/composition.
The article is here.
Many thanks to the journal's Editors-in-Chief Thomas Fiore and Clifton Callender for their guidance and engagement, and to Dr. Callender for proposing the above depiction of the rhythmic components.