Exploring the space of perfectly balanced rhythms and scales

Periodic scales and meters typically embody "organizational principles"" their pitches and onset times are not randomly distributed, but structured by rules or constraints. Identifying such principles is useful for understanding existing music and for generating novel music. In this paper, we identify and discuss a novel organizational principle for scales and rhythms that we feel is of both theoretical interest and of practical utility: perfect balance. When distributed around the circle, perfectly balanced rhythms and scales have their "centre of gravity" at the centre of the circle. The present paper serves as a repository of the theorems and definitions crucial to perfect balance. It also further explores its mathematical ramifications by linking the existing theorems to algebraic number theory and computational optimizations. On the accompanying webpage, http://www.dynamictonality.com/perfect_balance_files/, we provide audio samples of perfectly balanced rhythmic loops and microtonal scales, computational routines, and video demonstrations of some of the concepts.