Concrete examples of Trans₃ChordClass-Denotators are written as: Note that any representative of a Trans₃ChordClass provides suitable coordinates of the Trans₃ChordClass-Denotator, i.e., one may alternatively write:

Usually one classifies TwelveToneChords with respect to the Fifth-Transposition t₇, because by recursion one reaches all twelve transpositions. The Colimit of the corresponding M₇-Diagram yields a coarser classification than the M₃-Diagram:

In order to obtain a full chord classification with respect to the 48-elemented symmetry group of the TwelveToneSystem, one has to add two suitable arrows to the M₇-Diagram, loaded with the inversion m₁₁ (multiplication of the Cofactor indices by -1 mod 12) and fifth circle transformation m₇ (multiplication of the of the Cofactor indices by 7 mod 12):

At this point we stop working within the pointing ontology, in order to compare it with Mazzola’s one. Readers who are not familiar with category theory may skip the rest of this section and may continue with section 4, such as if

1. the Forms and Denotators would still have the prefix
2. the Form PiMod₁₂ would still be the Form TwelveTone

While we have been dealing with the category Sets (having sets as its objects and set-maps as its morphisms), there is another category of major importance for Mathematical Music Theory, representing musical parameters and their transformations. The category of modules Mod, which suits for this purposes, shows a different behaviour with respect to the universal constructions Limit and Colimit. The Power-construction does not work at all in this category. A natural way out of this problem is the consideration of functors F : Mod Sets yielding