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

$f0 (- {0,1}Zn$ is the canonical representative of $Cn(f )$ if and only if $Yd(f0)$ is the canonical representative of $Cnd(Yd(f))$ .
Assume that $n > 1$ . The mapping $Z f (- {0,1} n$ is acyclic if and only if $Yd(f)$ is acyclic.

As we will see in the next theorem, these two functions $yd$ and $Yd$ can be used in order to show that there exist regular complementary canons of maximal category which are not Vuza constructible canons.

Theorem 12. Let $d > 1$ , and assume that $(L,A)$ is a regular complementary canon of maximal category in $Zn$ . Then $(Yd(L),yd(A))$ is a regular complementary canon of maximal category in $Znd$ .

The proof is straightforward and so it is omitted. As with several combinatorial constructions, sometimes when trying to find your own proof, you finally understand the problem much better, than by just following the proof of the author.

Among the $432$ regular complementary canons of maximal category of length
[4] $2.72 = 144$ with $|L|= 12$ we did not find a canon which was constructed in this way for $d = 2$ from the $18$ canons of length $72$ with $|L |= 6$ .

In conclusion we know now that there exist complementary canons of maximal category

which are Vuza constructible canons, or
which are not Vuza constructible but can be constructed from Vuza constructible canons by applying Theorem 12.

This motivates the following question: Do there exist complementary canons of maximal category which cannot be constructed with these two methods?

References

ALEGANT, B. (1992). The Seventy-Seven Partitions of the Aggregate: Analytical and Theoretical Implications. Ph.D. thesis, Eastman School of Music, University of Rochester.

ANDREATTA, M. (1997). Group theoretical methods applied to music. Visiting Student Dissertation, University of Sussex.

ANDREATTA, M.; NOLL, TH.; AGON, C.; and ASSAYAG, G. (2001). The Geometrical Groove: rhythmic canons between Theory, Implementation and Musical Experiment. In Les Actes des 8e Journées d’Informatique Musicale, Bourges 7-9 juin 2001, pp. 93-97.

DE BRUIJN, N.G. (1964). Pólya’s Theory of Counting. In BECKENBACH, E.F. (ed.), Applied Combinatorial Mathematics, chap. 5, pp. 144 - 184. Wiley, New York.