Renaissance canons with asymmetric schemes

TL;DR

This paper introduces a mathematical model and algorithm to analyze the flexibility of asymmetric canonic schemes in Renaissance music, revealing Palestrina's preference for highly flexible schemes and exploring new compositions within this style.

Contribution

It develops a first-species theoretical model and an algorithm to quantify scheme flexibility, and applies this to Renaissance canons, including new compositions and computational tools.

Findings

Palestrina favored highly flexible schemes.

The model's flexibility measure is an algebraic integer.

New compositions demonstrate unexplored canonic schemes.

Abstract

By a "scheme" of a musical canon, we mean the time and pitch displacement of each entering voice. When the time displacements are unequal, achieving consonant sonorities is especially challenging. Using a first-species theoretical model, we quantify the flexibility of schemes that Renaissance composers used or could have used. We craft an algorithm to compute this flexibility value precisely (finding in the process that it is an algebraic integer). We find that Palestrina consistently selected some of the most flexible schemes, more so than his predecessors, but that he by no means exhausted all feasible schemes. To add support to the model, we present two new compositions within the limits of the style utilizing unexplored canonic schemes. In the Online Supplement (attached via Papers with Code), we provide MIDI realizations of the musical examples and Sage code used in the numerical computations.