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In order to determine the number of Vuza constructible canons of length Finally, we want to deal with the question whether there exist regular complementary canons of maximal category which are not Vuza constructible canons. In Fripertinger (2002) for an integer  If we apply
The mapping  Again the value |
with 
, we first realize that Vuza’s algorithm yields 
different sets 
and 
different possibilities for 
, which also consists of 
elements. So, when exchanging the role of 
and 
, we have to take care to count only non-isomorphic canons. As a matter of fact, here in this case the process of exchanging 
and 
yields new canons, so that in addition to the previous 
canons 
we have another 
canons of the form 
. Hence, we end up with 
Vuza constructible canons of length 
with 
. Moreover, there exist 
Vuza constructible canons with 
and 
Vuza constructible canons with 
, so that in conclusion there are 
Vuza constructible canons of length 
. 
we introduced the function 
defined on 
such that 
is the vector 
consisting of 
entries of 
, and 
is a vector consisting of 
entries of 
and 
in the last position. We write the values of 
in the form 
to each component of a vector 
we get a vector 
by concatenating all the vectors 
. Among other properties we showed that 
is the canonical representative of 
if and only if 
is the canonical representative of 
. 
is acyclic if and only if 
is acyclic.
can be considered as an augmentation, mapping a rhythm in 
to an augmented rhythm in 
. In correspondence with 
, we define the mapping 
which can be interpreted as 
-fold subdivision. It also maps a rhythm in 
to a rhythm in 
. It is defined on 
by 
for 
is obtained by concatenation of the vectors 
. It is easy to prove that the mapping 
satisfies