Local and global limit denotators and the classification of global compositions.

*(English)*Zbl 1144.18005In his tremendous book [G. Mazzola, The topos of music. Geometric logic of concepts, theory, and performance. Basel: BirkhĂ¤user (2002; Zbl 1104.00003)], the author gave a comprehensive account of his ambitious work on mathematical music theory. Within his theory he considers so-called local compositions, consisting, e.g., of collections of points in the pitch class space \(\mathbb Z_{12}\), and so-called global compositions that are spaces of local compositions glued together in a similar way as differentiable manifolds. In his theory, using a heavy machinery of category theory, particularly topoi and sheaves, he developed a classification theory of global compositions.

The present paper gives a number of technical results within this theory. In particular, he shows that the module complexes used in the classification theory are denotators of limit type (for denotator theory see [loc. cit.]). In this context he reveals some connections with the network theory of D. Lewin and H. Klumpenhouwer. Finally, he sketches a theory of global limit denotators and shows that there is a canonical functor into the category of global compositions. In this way he gets new invariants for the classification of global limit denotators.

The present paper gives a number of technical results within this theory. In particular, he shows that the module complexes used in the classification theory are denotators of limit type (for denotator theory see [loc. cit.]). In this context he reveals some connections with the network theory of D. Lewin and H. Klumpenhouwer. Finally, he sketches a theory of global limit denotators and shows that there is a canonical functor into the category of global compositions. In this way he gets new invariants for the classification of global limit denotators.

Reviewer: Klaus D. Kiermeier (Berlin)