Classification of rhythmical canons

A canon is a subset K⊆ Zn together with a covering of K by pairwise different subsets Vi ≠ ∅ for 1≤ i≤ t, the voices, where t≥ 1 is the number of voices of K, in other words
 K=.. t ∪i=1 Vi,..
such that for all i,j∈ {1,...,t}
1. the set Vi can be obtained from Vj by a translation of Zn,
2. there is only the identity translation which maps Vi to Vi,
3. the set of differences in K generates Zn, i.e. ⟨K-K⟩ := ⟨k-l | k,l∈ K⟩ = Zn.

We prefer to write a canon K as a set of its subsets Vi.

Two canons K={V1,..., Vt} and L={W1,...,Ws} are called isomorphic if s=t and if there exists a translation T of Zn and a permutation π in the symmetric group St such that T(Vi)=Wπ(i) for 1≤ i≤ t. Then obviously T(K)=L.

The canon {V1,..., Vt} can be described as a pair (V1,f), where V1 is the inner and f the outer rhythm of the canon. The inner rhythm describes the rhythm of any voice. The outer rhythm determines how the different voices are distributed over the n beats of a canon.

For example consider V1=(10011010), V2=(01010011), and V3=(11010100). We get a score of the form

 10011010 01010011 11010100
Hence the outer rhythm of this canon is f=(10010100).

A canon is called a rhythmic tiling canon if its voices are pairwise disjoint and cover entirely Zn. The canon (L,f), described by its inner L and outer rhythm f, is a tiling canon if and only if L+f=Zn and |L||f|=n, thus Zn is the direct sum of L and f.

For example ((00000101),(00110011)) is the canon

 01000001 10100000 00010100 00001010
.

A rhythmic tiling canon described by (L,f) is a regular complementary canon of maximal category (RCMC-canon) if both L and f are acyclic.

Instead of 0,1-vectors in order to describe a rhythm we also use the following representation: The vector [14,8,1,5,4,4,9,9,4,6,4,9,9,4,4,5,1,8] indicates that after 13 zeros there is the first one, then after 7 zeros the next one, then immediately again a one etc.

harald.fripertinger "at" uni-graz.at, August 1, 2017