Type: Article
Publication Date: 2009-06-20
Citations: 15
DOI: https://doi.org/10.1080/17459730903040709
Abstract The musical notion of rhythmic canons has proved to be relevant to some non-trivial mathematical problems. After a survey of the main concepts of tiling rhythmic canons, we discuss recent developments that enable to make, or expect, definite progress on several open mathematical conjectures. Keywords: rhythmic canonsVuzaFuglede's conjecturespectral conjecturealgorithmtiling Notes http://www.research.att.com/~njas/sequences/ Harald Fripertinger established formulas for the enumeration of RC, see Citation4. Like the more famous Langlands program, or Taniyama–Weil's conjecture. Izabella Łaba stated some results when the size of the group is not much larger than the width of the tile; but as Kolountzakis has shown, this cannot be assumed in general. This must be the case whenever all roots of A(X) are on, or inside, the unit circle, from a well-known result on polynomials. For larger numbers, a recursive construction can be envisaged. Related to the associated polynomial by . After Kolountzakis exposed his algorithm during a MaMux session at IRCAM, we managed to produce VC in the comparatively small group with his method; unfortunately they were all previously known, as members of the cyclotomic class, S A ={2, 3}. For instance, for n=120 most partitions in S A , S B yield factors that are known to satisfy (T 2). An exception is S A ={3, 5, 8}: it took extra care to check that this case yields no VC without assuming that any RC satisfied (T 2), starting from the other side S B ={2, 4}. Luckily, it also made for a shorter computation. A personal computer was unable to give the aperiodic complements of a CM Universal Complement in in a fortnight.