Subsets of Scales in Compositions Constructed by Similarity

Abstract

We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale.