Renaissance canons with asymmetric schemes

By a "scheme" of a musical canon, we mean the order of voice entry with 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 the flexibility value precisely (finding in the process that it is an $h$-th root of a Pisot number). We find that Palestrina consistently selected some of the most flexible schemes, more so than his predecessors, but that he by no means exhausted the feasible schemes. We close by presenting a new composition within the limits of the style utilizing an unexplored canonic scheme.