CONSTRUCTING AND FORMALIZING TILING RHYTHMIC CANONS: A HISTORICAL SURVEY OF A “MATHEMUSICAL” PROBLEM MORENO ANDREATTA N THIS PAPER we trace back the history of the construction and algebraic formalization of Tiling Rhythmic Canons as a “mathemusical ” problem. This neologism, which is now commonly used in mathematical music theory, was introduced by the author in the early nineties with the aim of suggesting that, although the history of relations between music and mathematics offers many examples of applications of mathematical models to music, there are also several examples showing that music can be sometimes a very important source of inspirational ideas for mathematicians. One may thus balance the usual Leibnizian perspective of music as an exercitium arithmeticae by proposing that the reverse hypothesis also holds, according to I 34 Perspectives of New Music which mathematics can be considered, in some special cases, as an exercitium musicae.1 A “mathemusical” problem is characterized by a dynamic movement between a given musical problem, the mathematical statements and general theorems obtained respectively by the formalization and generalization processes, and the musical application of these results in music theory, analysis, and/or composition. In other words, setting the originally musical problem in an appropriate mathematical framework not only gives rise, eventually, to new mathematical results, but also paves the way to new music-theoretical, analytical, or compositional constructions that would have been impossible to conceive without the process of mathematization. Although generalized mathematical results can be applied to the music domain with a focus on music theory, analysis, or composition, in the case of the algebraic approach, it is often very difficult to distinguish between these three domains, as we discussed in a previous study (see Andreatta 2003). It is this double movement, from music to mathematics and back, which makes a “mathemusical” problem so intriguing to both mathematicians and musicians. Example 1 shows the dynamical movement underlying a “mathemusical” activity. In this paper I will discuss some historical aspects of the construction of Tiling Rhythmic Canons from the previously sketched “mathemusical ” perspective. I will present two independent developments of EXAMPLE 1: THE DOUBLE MOVEMENT OF A “MATHEMUSICAL” ACTIVITY, FROM A MUSICAL PROBLEM TO ITS FORMALIZATION, GENERALIZATION, AND FINAL APPLICATION TO THE MUSIC DOMAIN Constructing and Formalizing Tiling Rhythmic Canons 35 music-theoretical constructions connected with tiling rhythmic canon structures: the first arising from Messiaen’s practice as “rythmicien,” and the second exploring the rhythmic analogies of Anatol Vieru’s modal theory. Olivier Messiaen (1908–1992) made undoubtedly one of the most significant efforts to study canons by abstracting the pitch content and focusing on the underlying rhythmic structure. A brief description of Messiaen’s compositional practice shows that the starting point is a genuine compositional problem related to some apparently very different theoretical constructions, such as Anatol Vieru’s composition of modal classes or Pierre Boulez’s chord multiplication. The history of tiling rhythmic canons is particularly interesting because it intersects a parallel history of mathematics, from number theory and the geometry of tiling to operator theory in functional analysis. We will discuss some aspects of these developments by sketching the Minkowski-Hajós problem, from the original formulation by Hermann Minkowski (1896), concerning the approximation of real numbers by means of rational numbers, through its successive interpretation from a geometric perspective again by Minkowski (1907), and up to the definitive solution provided by Hajós in the forties (Hajós 1941). Although the Minkowski-Hajós problem is totally solved, there are some weak versions of the Minkowski conjecture that are still open, and whose musical application seems potentially interesting. We will briefly present some of these conjectures before focusing on a second parallel development of the theory of tiling that originated in a problem raised by Bent Fuglede in functional analysis (Fuglede 1974), and whose solution is deeply related to a special class of tiling rhythmic canons: Vuza canons. 1. THE MUSICAL PROBLEM OF THE CONSTRUCTION OF TILING RHYTHMIC CANONS AS SEEN FROM DIFFERENT COMPOSITIONAL PERSPECTIVES There is probably no need to explain in depth the relevance of canons to music. There are few musical concepts that have been used as extensively, well beyond the boundaries of the Western classical music tradition...