Renaissance canons with asymmetric 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, we provide MIDI realizations of the musical examples and Sage code used in the numerical computations.

Keywords:

• Renaissance
• canon
• Palestrina
• flexibility
• Markov chain

2010 Mathematics Subject Classifications:

• 00A65
• 68R10
• 05C38
• 90B15
• 11R06

Acknowledgments

I thank Peter Smith for helpful guidance. I thank the three anonymous reviewers for patiently reading earlier drafts and catching many typos.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed online at https://github.com/emo916math/Renaissance-canons.

Notes

1 Accompanied canons where the bass is one of the canonic parts are rare; Palestrina writes them only in the case of 2-voice canon, and we will not consider them here.

2 We could equivalently define $\lambda \left(\mathcal{S}\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{V_{n+1}\left(\mathcal{S}\right)}{V_n\left(\mathcal{S}\right)}$ to be the limiting ratio of successive values (or 0 if $V_n$ is eventually 0), with only a bit more analysis needed to prove that the limit exists.

3 Unlike Palestrina, the composers of the Josquin generation often wrote three separate Agnus Dei movements in their masses, the middle Agnus Dei typically using reduced forces to prepare for a grand Agnus Dei III.