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

3.1 Transformational Logics

In Section 3 of Noll (2003) we argued that a purely set-theoretical description of musical structures can be reconstructed as an ontology of pointing. Mazzola’s general proposal for musical forms and denotators can instead be understood as a transformational ontology and may roughly be paraphrased in the following way: Instead of accessing sets of denotators through arbitrary isolated acts of pointing one may replace the single pointer by suitable repertoires of transformations, called categories, and access denotators through coordinated families of varying perspectives, called functors.

Remark 2 An action $g : G × S --> S$ of a group $G$ on a musical space $S$ -- as studied by Lewin -- can be seen as a functor $[g] : [G] --> Sets$ from the group $G$ (seen as a category [G] with one object $G$ and iso-arrows representing the group elements $g (- G$ ) to the category of sets, such that $[g](G) = S$ and $[g](g) = mg$ for all $g (- G$ .

As a central example Mazzola (Mazzola, 2002) studies the category $M od$ of modules with affine transformations as its arrows. The choice of this category is motivated by the observation that many musical parameter spaces can be suitably described in terms of $Z$ -modules, $Q -$ or $R$ -vector spaces etc. This category includes transformations between different modules. In Noll (2003) we proposed the notion of a Mazzola ontology with reference to the associated world of forms and denotators. Likewise, it would be appropriate to speak of Lewin-ontologies for the analogous constructions within in the categories of G-actions for the groups studied by Lewin.

An important consequence of the transformational approach is a particular richness of its internal logics whenever non-invertible transformations occur. This is exemplified by an exotic property called non-wellpointedness. This property says that one cannot point at all elements of the truthvalue-object from outside. It needs a considerably high amount of mathematical machinery to fully explain the contents and techniques of topos theory. In the following subsection we instead explore the most simple case of such a non-classical transformational logics. In the final subsection we go a little step further in order to present a small music-analytical application.

3.2 The Logics of Projecting

Consider a keyboard with numbered keys $K = {0,1,...,n}$ as well as a subset of its keys $X = {k1,...,km}$ being pressed down. In this very simple example we have in mind that the two facts of a key $k$ being pressed and a tone $t$ being played are linked in a direct way such that they become logically indistinguishable: Whenever $k$ is pressed down a unique tone $t0(k)$ is played and vice versa. A logical description of a key set $X < K$ being pressed down is the characteristic function $IsP ressedX : {0,1,...,n} --> {True,F alse}$ taking the value $True$ for those keys being pressed down and the value $False$ for the others.