In many practical cases, AmbientSet and FrameSet coincide, and the Identifier is the Identity map. The FrameSet is defined through one of five construction types that apply to a Coordinator, i.e., to the source structure to be used in the construction. The definition of a Form looks like this:

The five construction types are labeled as follows:

It is useful to discuss these construction types and the role of the involved coordinators in some detail and to illustrate them by musical examples. We skip the Syn-type, which stands for “Synonymy”.¹⁰

The coordinator of a Simple Form is an arbitrary one-elemented set. Pointing at some isolated object is the basic activity that motivates simple Denotation. In the pointing ontology “simple” is equivalent to “single”. Hence, the Coordinator of a Simple Form is the pointer ^. itself. The FrameSet FS(S) of any simple Form S is always the same, namely = {^.}, while its AmbientSet AS(S) can be any one-elemented set. The definition a SingleTone-Form might look like this:

Power, Limit and Colimit are known as universal constructions in category theory. For readers not familiar with that we recall the three strategies of knowledge management mentioned in the first section: hermeneutics, dogma and model. A closer inspection of the ways how to manipulate denotators following these strategies gives a suitable heuristics for these constructions.

We start with those principles of Denotator construction that are necessary for hermeneutic activity. A fundamental modality to go beyond mere denotation of single objects is called Coproduct. It is the free or “unlimited” special case of the more general Colimit type. The hermeneutician makes use of it, when refers to Denotators of two or more Forms as if they were of the same Form. One may even build Coproduct of multiple copies of a given Form. This is a suitable way to define a TwelveTone-Form:

The FrameSet FS(TwelveTone) = {₀,...,₁₁} of this Form consits of 12 copies of the element of AS(SingleTone) = {}, indexed by the positions of the 12 identical Cofactors in the Coordinator of this Form. The structure of such Coordinators is explained below.