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

perhaps unusally, the string module over these numerical modules. From these one-dimensional “scalar” modules, free modules may be contructed such as real vector spaces. Between any two of these modules morphisms may be defined, which map elements from a domain module to elements from a codomain module.

To construct forms and denotators and to manipulate them, helper classes are available. The following example will use just these helper classes to construct a simple denotator. Let’s take the common example of a note consisting of onset (O), duration (D), pitch (P), and loudness (L). Values are reals in the case of O and D (seconds), integers in the case of P (MIDI key) and strings in the case of L (pp,p,mp,mf,f,ff ).

The forms are described in DenoteX:¹

¹: The notation :. is a short form indicating an identity morphism. Likewise :@ indicates a zero address i.e. an address in the null module $0Z$ . This is the simplest case, and for the examples in this text, we don’t need the more sophisticated ones.

O:.Simple(R);
D:.Simple(R);
P:.Simple(Z);
L:.Simple(ZString);
Note:.Limit[O,D,P,L];