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

2.4 Denotators

The level of forms is still not the substance we are looking for. The substance is what is called a denotator. More precisely, given an address $A$ and a form $F$ , a denotator is a quatruple $N ame : A~>F (c)$ , consisting of a string $D$ (in $UN ICODE$ ), its name, its address $A$ , its form $F$ , and its coordinates $c (- A@f un(F )$ . So a denotator is a kind of substance point, sitting in its form-space, and fixed on a determined address. This approach is really a restatement of Aristotelian principles according to which the real thing is a substance plus its »instanciation« in a determined form space. Restating the above coordinates as a morphism $c : @A --> fun(F)$ on the representable contravariant functor $@A$ of address $A$ by the Yoneda lemma, the »pure substance« concept crystallizes on the representable functor $@A$ , the »pure form« on the functor $fun(F )$ , and the »real thing« on the morphism between pure substance and pure form.

In classical Mathematical Music Theory (Mazzola, 1991), denotators were always special zero-addressed objects in the following sense: If $M$ is a non-empty $R$ -module, and if $0 = 0Z$ is the zero module over the integers, we have the well-known bijection $0@M -->~ M$ , and the elements of $M$ may be identified with zero-addressed points of $M$ . Therefore, a local composition from classical mathematical music theory, i.e., a finite set $K < M$ , is identified with a denotator $K* : 0~>Loc(M )(K)$ , with form

Loc(M ) - --> Power([M ]) F in([M])>->_O_[M]

and $[M ]---> Simple(M )$ .

Evidently, this approach relates to approaches to set theory, such as Aczel’s hyperset theory (Aczel, 1988) which reconsiders the set theory as developed and published by Finsler⁶

⁶: It is not clear whether Aczel is aware of this pioneer who is more known for his works in differential geometry (»Finsler spaces«).

in the early twenties of the last century (Finsler, 1926, 1975). The present setup is a generalization on two levels (besides the functorial setup): It includes circularity on the level of forms and circularity on the level of denotators. For instance, the above circular form named $M akroBasic$ enables denotators which have infinite descent in their knot sets. Similar constructs intervene for frequency modulation denotators, see Mazzola (2001b, 1998).

The denotator approach evidently fails to cover more connotative strata of the complex musical sign system. But it is shown in music semiotics (Mazzola, 1998, section 1.2.2) that the highly connotative Hjelmslev stratification of music can be construed by successive connotational enrichment around the core system of denotators. This is the reason why the naming »denotator« was chosen: Denotators are the denotative kernel objects. For the generalization of denotators to »connotators« see Mazzola (2001b), chapter G.5.3.3.

2.5 The RUBATO Enterprise

So far, the language of forms and denotators seems to live exclusively in the esotheric universe of mathematics. Fortunately, this is not true: On the contrary,