and described in more detail in Vuza (1991-93). By definition, a partitioning class (or Parkettierer) in is a translation class with the property that »there is a partition of the set of all twelve pitch classes into subsets belonging to that class« (Vuza, 1991-93). Obviously the order of a partitioning class must be an integer such that , i.e. . Note that, by convention, the unison is a partioning class (even if trivial) while the chromatic total does not have such a property. It can be shown (see Vuza, 1982-83) that all 2-chords are partitioning classes with the exception of the dyad (8,4). The concept of partitioning classes, together with that of »supplementary sets« (see Vuza, 1991-93) is one of the most remarkable aspects of Vuza’s theory, as is clear in relation to the Minkowski-Hajós problem outlined earlier.
Partition problems in relation to music have been posed by different authors in a number of different ways. Some are more »combinatorial« (like Milton Babbitt’s partition problem described in Bazelow and Brickle (1976)), or Halsey and Hewitt’s algorithmic strategy (which enables to compute the number of partitioning classes once it is given a group and a positive integer dividing the cardinality of ). Others are more »structural«, in the sense that they deal with the abstract, or categorical, background out of which combinatorial problems emerge, »illuminat[ing] the setting within which one wishes to deal with more concrete compositional and theoretical issues« (Bazelow and Brickle, 1976, p. 281). David Lewin’s problem of interval function, as posed in Lewin (1987), is an example of an essentially structural problem. Using concepts such as convolution and Fourier Transform, the problem »may be generalized to questions about the interrelation, in a locally compact group, among the characteristic functions of compact subsets« (Lewin, 1987, p. 103).

Before showing how partitioning classes are related to Messiaen’s transposition limited concept in the formalisation and construction of tiling rhythmic canons, we need a further preliminary definition:
Definition: Given a finite group , two intervallic classes are supplementary if there exists such that the group can be written in a unique way as , i.e. can be factored in the direct sum of the subsets and . By putting together Vuza (1991-93) and Halsey and Hewitt (1978) we have the following theorem of a characterisation of supplementary classes:
Theorem: let be a finite abelian group and let , be subsets of . The following statements are thus equivalent:

1. and are supplementary
2. and
3. and