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

which are non-Hajós groups. He proved that each pair $(L,A)$ computed along the following algorithm is a regular complementary canon of maximal category.

Let $Zn$ be a non-Hajós group, whence $n = p1p2n1n2n3$ as described in Corollary 10. Vuza presents an algorithm for constructing two aperiodic subsets $L$ and $A$ of $Zn$ such that $|L |= n1n2$ , $|A|= p1p2n3$ , and $L + A = Zn$ . In order to construct $L$ , for $i = 1,2$ let $Li$ be a nonperiodic set of representatives of $-n- piniZn$ modulo its subgroup $n- piZn$ . Then set $L := L1 + L2$ . In order to determine $A$ , for $i = 1,2$ choose $-n- n- xi (- piniZn \ piZn$ and let

n ( n ) A1 = --Zn + --Zn \{0} U {x1} , p1 p2

( ) n- n- A2 = p2Zn + p1Zn \{0} U {x2} .

Choose a set $R$ of representatives of $Zn$ modulo $n3Zn$ , let $A3 := R \n3Zn$ , and put $A := A1 U (A2 + A3)$ .

Consequently, both $L$ or $A$ can serve as the inner or outer rhythm of a regular complementary canon of maximal category. Moreover, as we saw there is some freedom for constructing these two sets, and each of these two sets can be constructed independently from the other one. Vuza also proves that when the pair $(L,A)$ satisfies $L o+ A = Zn$ , then also $(kL,A)$ , $(kL,kA)$ have this property for all $k (- Z*n$ .

A regular complementary canon of maximal category which can be constructed by Vuza’s algorithm will be called Vuza constructible canon. The following table shows the numbers of isomorphism classes of Vuza constructible canons for some values of $n$ : (With $#L$ , $#A$ , and $#(L, A)$ we denote the number of possibilities to determine essentially different aperiodic sets $L$ and aperiodic sets $A$ , and non-isomorphic canons $(L,A)$ by using Vuza’s algorithm.)

|| | | | | | | | --n-||p1|p2-|n1-|n2-|n3-|#L--|--#A--|#(L,A)-- 72 || 2| 3 | 2 | 3 | 2 | 3 | 6 | 18 108 || 2| 3 | 2 | 3 | 3 | 3 | 180 | 540 120 || 2| 3 | 2 | 5 | 2 | 16 | 20 | 320 120 || 2| 5 | 2 | 3 | 2 | 8 | 6 | 48 144 || 2| 3 | 4 | 3 | 2 | 6 | 36 | 216 144 || 2| 3 | 2 | 3 | 4 | 3 | 2808 | 8424 168 || 2| 3 | 2 | 7 | 2 |104 | 42 | 4368 168 || 2| 7 | 2 | 3 | 2 | 16 | 6 | 96 180 || 2| 3 | 2 | 3 | 5 | 3 |45360 | 136080 180 || 2| 3 | 2 | 5 | 3 | 16 | 1000 | 16000 180 || 2| 5 | 2 | 3 | 3 | 8 | 252 | 2016 180 || 2| 3 | 5 | 3 | 2 | 9 | 60 | 540 180 || 3| 5 | 3 | 2 | 2 | 6 | 12 | 72 200 || 2| 5 | 2 | 5 | 2 |125 | 20 | 2500 240 || 2| 3 | 4 | 5 | 2 | 32 | 120 | 3840 240 || 2| 5 | 4 | 3 | 2 | 16 | 36 | 576 240 || 2| 3 | 2 | 5 | 4 | 16 |26000 | 416000 240 2 5 2 3 4 8 6264 50112

Table 3:

Number of isomorphism classes of Vuza ¿constructible canons in $Z n$

In order to determine the complete number of isomorphism classes of Vuza constructible canons in $Zn$ for given $n$ , we have to determine all possibilities to decompose $n$ as in Corollary 10 and sum up the number of isomorphism classes of Vuza constructible canons for these parameters. For instance, for $n = 72$ we have $18$ isomorphism classes of canons with $|L|= 6$ , and by interchanging $L$ and $A$ also $18$ isomorphism classes of canons with $|L|= 12$ , whence $36$ isomorphism classes of Vuza constructible canons.