Classification of rhythmical canons

A canon is a subset K⊆ Z_n together with a covering of K by pairwise different subsets V_i ≠ ∅ for 1≤ i≤ t, the voices, where t≥ 1 is the number of voices of K, in other words

K=^._. t
∪
i=1 V_i,^._.

such that for all i,j∈ {1,...,t}
1. the set V_i can be obtained from V_j by a translation of Z_n,
2. there is only the identity translation which maps V_i to V_i,
3. the set of differences in K generates Z_n, i.e. ⟨K-K⟩ := ⟨k-l | k,l∈ K⟩ = Z_n.

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

Two canons K={V₁,..., V_t} and L={W₁,...,W_s} are called isomorphic if s=t and if there exists a translation T of Z_n and a permutation π in the symmetric group S_t such that T(V_i)=W_π(i) for 1≤ i≤ t. Then obviously T(K)=L.

The canon {V₁,..., V_t} 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 Z_n. The canon (L,f), described by its inner L and outer rhythm f, is a tiling canon if and only if L+f=Z_n and |L||f|=n, thus Z_n 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