The catalogue of all possible factorisations of cyclic non-Hajós groups into two non-periodic subsets has been taken by some theoretically-inclined composers as the starting point for further speculations on tiling problems on music. 20| | The pieces using some of these solutions, althought in a very different way, are Coincïdences for orchestra by Fabien Lévy and a piece for small ensemble by Georges Bloch called Fondation Beyeler: une empreinte sonore. Starting from Vuza’s formalisation of tiling rhythmic canons and from the OpenMusic implementation, composer Tom Johnson posed the problem of the construction of tiling canons by augmentation. This problem, as in the case of Minkowski’s Conjecture, turned out to be unexpectedly interesting from a mathematical point of view. One solution has been proposed by E. Amiot (Amiot, February 2002) by using the polynomial representation of rhythmic canons as initially introduced by A. Tangian (Tangian, 2001) and by applying some advanced algebraic concepts from Galois Theory to music. The solution to Johnson-Tangian Conjecture by E. Amiot together with the generalisation proposed by H. Fripertinger is available online at the following address:
http://www.ircam.fr/equipes/repmus/documents/MaMuXtiling.html
For a different perspective on Galois Theory of concepts in music see Mazzola’s contribution (Mazzola, 2002) in the Fourth Diderot Mathematical Forum (Assayag and al., 2002). |
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