Enders, Bernd / Stange-Elbe, Joachim (Hrsg.): Global Village - Global Brain - Global Music

The figure below shows the resulting

Form-Diagram D:

PICT

In order to understand the construction of a Limit-type Form L from this diagram we first recall specific hermeneutic activity that likely motivates such a diagram. In order to describe dependencies between Denotators, that are formally expressed in the arrows of the diagram D, a hermeneutician would base his observations on the free product of F₁, F₂ and F₃, i.e., on a diagram D₀ having three nodes loaded with F₁, F₂ and F₃, but not having arrows.¹¹

11: This is reflected in Mazzola’s convention for the notation of Coordinators of Limit and Colimit type: Form Diagrams without arrows are written as a lists of Forms. Their positions in the list represent the nodes of the diagram. The 12 Cofactors of the TwelveTone-Form correspond to nodes of a diagram without arrows, each being loaded with the SingleTone-Form.

“F123” ---> Limit(F1,F2,F3) = Limit(D0) Identity

In his discourse the hermeneutician refers to specific denotators

“d” : F123(f1,f2,f3)

which attracted his interest because of an observation, that is expressed in the following predicate P:

P(x1,x2,x3) := f(x1) = x1 /\ g(x1) = x1 /\ h(x1) = x2 /\ i(x3) = x2 /\ j(x2) = x3

The idealized dogmatist would therefore limit the scope of his interest from all possible F₁₂₃-Denotators to those for which the predicate P is true and therefore aquires the ability to filter them out – even those that practically would never have been observed by a hermeneutician. He turns the predicate P into a system of equations for Forms

PICT

The variable Form X of this system of equations involves three variable set-maps p_i : FS(X) --> A_i i = 1,2,3 from the FrameSet FS(X) of X into the AmbientSets of F₁, F₂ and F₃ and the equations read as follows: