The talk will propose an introduction to the musical problem of the construction of canons, which are polyphonic musical pieces whose voices lead the same melody with different delays. In particular, a tiling rhythmic canon is the one whose tone onsets result in a regular pulse train with no simultaneous voices at a time. Tiling in mathematics means covering an area by disjoint equal figures and in that sense, a rhythmic canon tiles the time, providing a covering with a regular pulse train by disjoint equal rhythmic patterns.
Tiling rhythmic canons can be formalized algebraically through two equivalent models: factorization of cyclic groups and products of polynomials with coefficients 0-1 The latter, in particular, underlies the Coven-Meyerowitz conditions (T1) and (T2), which are sufficient for the existence of rhythmic canons. The Vuza canons will then be identified, i.e. canons aperiodic both in the inner and in the outer rhythm. We will see how the existence of such canons depends on the order of the factorized cyclic group: it follows a division of the class of finite cyclic groups into two disjoint subclasses, explicitly defined.
Finally, the Fuglede conjecture, still open, will be presented, and we will see how the tiling rhythmic theory can present a possible approach to the solution.
The zoom link for the seminar is https://univr.zoom.us/j/97604696910
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Matteo Frigo, Chairman of Associazione Alumni Matematica Verona