A Boolean theory of signatures for tonal scales

We explore the concept of tonal signatures developed and put into musical practice by one of us (Mezzadri). A tonal signature of a scale SS is a minimal subset of notes within SS that is not contained in any scale S′S′ different from SS. We present a set covering model to find a smallest signature. We also show that the signatures of a scale are the prime implicants of a suitable monotone Boolean function represented by a Conjunctive Normal Form. On this ground, we introduce a more general notion of Boolean signature, depending on a Boolean operator. The computational machinery for generating Boolean signatures remains essentially the same. The richness and variety of Boolean signatures has a great potential for the development of new paradigms in polytonal harmony.