Infinite Canons: Maximally Self-Similar Melodic Lines and Canons with Infinite Solutions

Infinite Canons is an ongoing series of canons with infinite solutions. More specifically, each canon is based on a melodic line that can be combined in any number of voices, at any tempo ratios (rational or irrational), and with each voice moving either forward or in retrograde inversion, while maintaining harmonic consistency. This paper describes the structure of these maximally self-similar melodic lines based on two different constructions: 1) a discrete prime-factorization approach yielding self-similarity under all rational ratios, and 2) a continuous logarithmic approach extending this to irrational ratios. In both cases the vertical interval between voices in a tempo ratio of $\lambda_i / \lambda_j$ is given by a homomorphism $\phi(\lambda_i / \lambda_j)$, which is a constant independent of time. Furthermore, under these constructions retrograde inversion collapses to transposition, allowing for all manner of table canons. These structures are demonstrated with suggestive realizations of several different infinite canons. Future work includes a more complete mathematical treatment, musical applications, and an interactive program that allows users to explore an unlimited number of realizations of these pieces.