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 S is a minimal subset of notes within S that is not contained in any scale S' different from S. 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 CNF. 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. 1. Basic definitions and notation Let us denote by N = {C, C#, ..., B , B} the set of the 12 notes, ordered in the stated order. A musical scale will always be regarded as a subset of N . All musical scales consist of 7 notes. Every scale S can be represented by a binary vector with 12 components, the characteristic vector ch(S) of S, defined as follows: 12 ,..., 1 , , 0 , 1 ) ( = ! " # $ % = i S i if S i if S chi The characteristic vectors of the three scales in C (major, minor, (minor) harmonic) are displayed in Table 1. Those of the other major, minor, harmonic scales may be obtained from the characteristic vectors of the C, Cm, Ch scale, respectively, by cyclic permutations of their components. Thus, all in all, there are 36 scales. Ital Engl Do Major Do minor Do (minor) harmonic C C m C h Do C 1 1 1 Do # C # 0 0 0 Re D 1 1 1 Mi E 0 1 1 Mi E 1 0 0 Fa F 1 1 1 Fa # F # 0 0 0 Sol G 1 1 1 La A 0 0 1 La A 1 1 0 Si B 0 0 0 Si B 1 1 1 Table 1