Paths on Stern-Brocot Tree and Winding numbers of modes

The aim of this paper is to propose a natural definition of a winding number for a m-note mode, generalizing the concept of well formedness (proposed by Carey and Clampitt) providing and discussing some aspects regarding the Stern-Brocot trees. We start by giving an algebraic definition of a m-notes mode in a chromatic set of n elements as a composed map between Zm and Zn. Then we introduce the winding number of a mode, as a sort of topological index related to how a certain mode is placed around the circle of fifths, and it will be discussed some connections with the concept of well formedness. Finally we shall focus on Stern-Brocot trees, discussing an interesting relation existing between winding numbers and some particular paths on the tree. Paths which can be viewed as the result of a dynamical evolution, driven by the winding numbers, from the root of the tree to the leaf corresponding to a well-formed mode. 1. MODES WITHIN A CHROMATIC SET AS A COMPOSED MAP We shall start by proposing a definition of a m-elements mode in a chromatic set of n elements (m < n) as a map from Zm to Zn: