Given a form semiology, we shall denote a form by the symbol
and omit the identifier if it is the identity functor. We also write
if the diagram reduces to the discrete set of forms 
.
Given two forms 
in a semiotic of forms 
, a morphism 
is just a natural transformation 
. Hence every semiotic of forms defines its category 
of forms.
2.3 Conceptual Galois Theory
The general problem of existence and size of form semiotics, i.e., the extent of the 
set, maximal candidates of such sets, gluing such sets together along compatible intersections, etc., is far from being settled. We shall not pursue this interesting and logically essential branch for reasons of space. The least one should say is that regular forms, i.e., those forms which are built from simple forms by transfinite recursion, may be supposed to be included in a form semiotics without further danger concerning logical consistency.
Example 1 For non-negative integers 
, consider the forms
| OnModm Simple( m) | | |
| PiModn Simple(n) | | |
| OnPiModm,n Limit(OnModm,PiModn) | | |
| IntModm,n Limit( ) | | |
with
-modules
and
as coordinators, and with the diagram
associated with the canonical projection
onto the form
. The name
»OnMod
« symbolizes
»onset modulo...
« whereas
»PiMod
« symbolizes
»pitch modulo...
«, i.e., ordinary pitch classes. We see that the last form’s diagram is just the condition that we should take the fiber product over onset, i.e., the simultaneity of two events in pitch and onset; this is a way to encode an interval of simultaneous note events.
But circular, i.e., non-regular forms do not exist automatically, nor are they uniquely defined. For example, defining a form
