Proportions

In this chapter (Proportions) we present a foundation of arithmetic with proportions. We begin with a classification of diverse mathematical-musical notions oriented to some ancient notions. Then we develop the notion of proportions from these notions and introduce an axiomatic of the laws of arithmetic – called “the theory of proportions” – which is mainly centered in the Theorems 1.1 and 1.2. In doing so, we anchor the theory of proportions on basic rules, which leads to the catalogue of ancient arithmetic rules. Finally, we address the multiplication of proportions by viewing it as a process of merging (fusioning) two (or more) proportions – which is musically equivalent to layering intervals to form a new interval. This gives rise to an algebra for proportion equations, which is extremely helpful in dealing with calculations of musical intervals and scales.