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STRUCTURES, ALGORITHMS, AND ALGEBRAIC TOOLS FOR RHYTHMIC CANONS EMMANUEL AMIOT 1. WHAT IS A RHYTHMIC CANON? HE AMBITIOUS PURPOSE of this paper is to fill in the gap between theory and practice of rhythmic canons. There is indeed quite a distance between musical canons, even rather intellectual ones like Bach's in the Goldberg Variations or The Art of Fugue, and Vuza canons such as they are used by some modern composers. The basic idea of a canon is that some recognizable pattern is repeated with different offsets (usually with different instruments, or at least different voices). Sometimes this pattern (henceforth called the motif) is modified (say, augmented or retrograded). For rhythmic canons, we need only consider the occurrences of the musical events (notes, for instance), regardless of pitch, timbre, or dynamics. Hence, T 94 Perspectives of New Music with a motif A and transformations τi , i∈I (that may be only offsettings —i.e., translation in time), a canon will be the union C of the transforms ∪ i∈I τi (A). Say, for simplicity's sake, that the fundamental beats are modeled by integers; e.g., some subset D⊂ℤ. If a motif is identified with its characteristic function 1A :D→{0,1}, then the superposition of all its copies appears as a sum C=∑ i 1τi(A) . For instance, in the common case when all transformations are just different offsets in time—i.e., τi =T i =(t ↦t+i)—we get ∑ i 1A+i , which is, in general, the characteristic function of a multiset, not a set. ILLUSTRATION 1. Let M be the tango or habanera rhythm {0,3,4,6}. An infinite canon can be made by offsetting M by -2 and 0 and repeating the sequence with period 8 (Example 1). Notice that C(4)=C(6)=2, while C(5)=0, for instance. A neater canon can be made, without gaps or coincidences, by also using retrogradation (Example 2). EXAMPLE 1: A CANON WITH A TANGO MOTIF EXAMPLE 2: A MOSAIC WITH A TANGO MOTIF M AND ITS RETROGRADATION R Structures, Algorithms, and Algebraic Tools for Rhythmic Canons 95 Obviously, if one chooses to consider as possible beats all possible quavers or semiquavers (say) in a definite time-span, then most musical canons in the classical sense will feature from nil to several notes on each beat. We can try to specialize more (theoretically) interesting kinds of canons: we will call them (as suggested to this author by Jon Wild) coverings and packings. A covering is a canon where every available beat features at least one note, maybe more. One gets a trivial covering by starting a new copy of the motif on every beat. In other words, with the above notation, ∀t∈D , C (t )≥1. A packing, conversely , is a canon where there is never more than one note on every possible beat. With the above notation, ∀t∈D , C (t )≤1. A trivial packing is made of only one (or even nil!) copy of the motif. Covering is not only trivial, but lumpy (several notes on the same beat); packing, conversely, leaves many gaps. From a mathematical point of view, the obvious way to get a welldefined and interesting problem1 is to demand one and only note per beat, as in Example 2. This is called a tiling (i.e., a mosaic with copies of one motif, maybe allowing some deformations—retrogradation and augmentation among other possibilities, apart from translation in time). Hence: mosaics/tilings = coverings ∩ packings: ∀t∈D , C(t)=1. Most studies (especially on the pure mathematics side) have been devoted to the simplest case of tiling with just one tile and some of its translates (i.e., mosaic rhythmic canons by translation). In the onedimensional case (filling every beat with one and only one note) it is equivalent to the problem of tiling ℤ (see Lagarias and Wang 1996; Vuza 1991–1993); if the tile is finite, it is equivalent to the tiling of a cyclic group ℤn , as we will develop below in Theorem 2. A simple case is shown in Example 3. EXAMPLE 3: A MOSAIC RHYTHMIC CANON DESIGNED BY GEORGES BLOCH FOR A GREETING CARD 96...
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