Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Making use of the standard intervallic notation for a rhythmic class, we find that the rhythmic classes associated with the couple of non periodic supplementary sets $M$ and $N$ are, respectively:

S = [12,12,3,12,12,57] R = [2,2,2,4,1,9,16,4,2,4,10,9,7,2,2,2,4,26]

If we define the power of a canon as the number $p$ of its voices, $p$ obviously is equal to $#S$ . We observe that the power $p$ alone is far from being the most important parameter. For example, take the smallest cyclic group $Z/ 72$ that does not have the Hajós property. This case has been taken as an example of remarquable number by F. Le Lionnais (Lionnais, 1989) who quoted the following decomposition of $Z/72$ in two non-periodic subsets by L. Fuchs (Fuchs, 1960, p. 316):

M = {0,8,16,18,26,34} N = {0,1,5,6,12,25,29,36,42,48,49,53}

These non periodic subsets of $/Z72$ are isomorphically associated with the following rhythmic classes

S = [8,8,2,8,8,38] R = [1,4,1,6,13,4,7,6,6,1,4,19]

Note that the power of this canon is still 6, as in the case of $Z/108$ and that in both cases the set $S$ has a stronger symmetrical character than the set $R$ .
Looking at the metric class $S$ of all these regular complementary canons of maximal category, we observe that it has a property which could be called of »partial non invertibility«. Writing $S$ as $[s1,s2,...,sn]$ , either $[s1,s2,...,sn-1]$ or $[s2,s3,...,sn]$ is a non- invertible rhythmic class (in the sense of Messiaen). By definition we say that the rhythmic class $[s1,s2,...,sn]$ is non invertible if it coincides with its inverse $[sn,sn-1,...,s1]$ . The sense of the adjective »partial« in the definition of the previous property is clearly intuitive: $S$ is non invertible with the exception of the first or the last temporal interval. The fact that every metric class of a given regular complementary canon of maximal category has the property of »partial non invertibility« seems not to be a direct consequence of the theory here under discussion. It seems appropriate, therefore, to class this fact as a conjecture which, because of its analogy with some of Messiaen’s ideas, has been called the »M-Conjecture« (Andreatta, 1997). Moreover, a comparison between this theory and some of Messiaen’s music- theoretical ideas, such as those contained in Messiaen’s recent Traité (Messiaen, 1992) shows that the problem of formalisation of canons was very central to the French composer. An interesting example is given by the piece Harawi (part no. 7, Adieu).

Figure 6:

A Messiaen’s three-voices canon in the piece Harawi

From a rhythmic point of view, the previous example realises a canon in three voices, each voice being the concatenation of three non-retrogradable rhythms, as it is shown in figure 7:

Figure 7:

Rhythmic pattern of Harawi

In Messiaen’s words, this global musical structure is an example of an »organised chaos« (Messiaen, 1992, p. 105), for the attacks of the three voices seem to be almost complementary. This is only partially true, as it is clear from the following representation of the canon