# 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={V_{1},..., V_{t}} and
L={W_{1},...,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 {V_{1},..., V_{t}} can be described as
a pair (V_{1},f), where V_{1} 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 V_{1}=(10011010),
V_{2}=(01010011), and V_{3}=(11010100). We get a
score of the form

__1__0011010 |

010__1__0011 |

11010__1__00 |

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