Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

is equivalent to selecting any monomorphism $I : G >-> _O_G$ , and setting $fun(F ) = G$ . To elaborate canonical monomorpisms, consider a set $S < A@G$ for a presheaf $G$ . This defines a subfunctor $S@ < @A × F$ which in the morphism $f : B --> A$ takes the value $f@S@ = {f}× S.f$ . Since we have $IdM@S@ = {IdM} × S$ , $S$ is recovered by $S@$ . This defines a presheaf monomorphism

?@ : 2G >-> _O_G

on the presheaf $G 2$ of all subsets $A@G 2$ at address $A$ . When combined with the singleton monomorphism $sing : G >-> F in(G) : x '--> {x}$ with the codomain presheaf $G F in(G) < 2$ of all finite subsets (per address), we have this chain

G G G >-> Fin(G) >-> 2 >-> _O_

of monomorphisms. A number of common circular forms can be constructed by use of the following proposition (c.f. Montiel, 1999):

Proposition 1 Let $H$ be a presheaf in $M od@$ . Then there are presheaves X and Y in $M od@$ such that

	X Fin(H × X) and
	Y H × Fin(Y ).

Example 2 It is common to consider sound events which share a specific grouping behavior, for example when dealing with arpeggios, trills or larger groupings such as they are considered in Schenker or in Jackendoff-Lerdahl theory (Lerdahl and Ray Jackendoff, 1983). We want to deal with this phenomenon in defining MakroEvent forms. Put generically, let Basic be a form which describes a sound event type, for example the above event type $Basic = OnP iM od m,n$ . We then set

	Makro_Basic Power(Knot_Basic)
	with F = fun(Makro_Basic),FK = fun(Knot_Basic)
	and the limit form
	Knot_Basic Limit(Basic,Makro_Basic),

a form definition which by the above proposition yields existing forms.

The typical situation here is an existing form semiotic $sem$ and a bunch of >equations< $EF1,F2,...Fn(F )$ which contain the form names $F1,F2,...Fn$ already covered by $sem$ , and the new form name $F$ . The equations are just form definitions, using different types and other ingredients which specify forms. The existence of an extended semiotics $' sem$ which fits with these equations is a kind of algebraic field extension which solves the equations $E$ . This type of conceptual Galois theory should answer the question about all possible solutions and their symmetry group, i.e., the automorphisms of $' sem$ over $sem$ . No systematic account of these problems has been given to the date, but in view of the central role of circular forms in any field of non-trivial knowledge bases (Barwise and Etchemendy, 1987), the topic asks for serious research. But see Mazzola (2004) in this volume.