Making Rhythmic Canons
Résumé
A canon is a polyphonic piece whose voices lead the same melody with different delays. A rhythmic canon is the one whose tone onsets result in a regular pulse train with no simultaneous tone onsets at a time. Tiling in mathematics is covering an area (e.g., a square) by disjoint equal £gures. In that sense, a rhythmic canon tiles the time, providing a covering of a regular pulse train by disjoint equal rhythmic patterns. We consider the general case of tiling the time by rhythmic patterns of a few shapes. In particular, we construct a rhythmic canon from a theme, its augmentation (the theme with all durations doubled) and its double augmentation. For this purpose we develop a tiling algorithm which resembles the sieve of Eratos-thene (284-192 BC) for £nding prime numbers. Besides , we approach to an analytical solution by refor-mulating the problem in terms of polynomials and reducing it to Diophantine equations (supposedly 325-409). Finally, we describe the application of the methods developed to making a composition "Eine kleine Mathmusik".
Domaines
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|---|
| Licence |