Paths on Stern-Brocot Tree and Winding numbers of modes
نویسنده
چکیده
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:
منابع مشابه
A calculational approach to path-based properties of the Eisenstein-Stern and Stern-Brocot trees via matrix algebra
This paper proposes a calculational approach to prove properties of two well-known binary trees used to enumerate the rational numbers: the Stern-Brocot tree and the Eisenstein-Stern tree (also known as CalkinWilf tree). The calculational style of reasoning is enabled by a matrix formulation that is well-suited to naturally formulate path-based properties, since it provides a natural way to ref...
متن کاملExact arithmetic on the Stern-Brocot tree
In this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic ...
متن کاملProportionally modular diophantine inequalities and the Stern-Brocot tree
Given positive integers a, b and c to compute a generating system for the numerical semigroup whose elements are all positive integer solutions of the inequality axmod b ≤ cx is equivalent to computing a Bézout sequence connecting two reduced fractions. We prove that a proper Bézout sequence is completely determined by its ends and we give an algorithm to compute the unique proper Bézout sequen...
متن کاملAdmissible Digit Sets and a Modified Stern–Brocot Representation
We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set” yields an admissible representation of [0,+∞]. Furthermore we establish the productivity ...
متن کاملAn arborist’s guide to the rationals
There are two well-known ways to enumerate the positive rational numbers in an infinite binary tree: the Farey/Stern-Brocot tree and the Calkin-Wilf tree. In this brief note, we describe these two trees as ‘transpose shadows’ of a tree of matrices (a result due to Backhouse and Ferreira) via a new proof using yet another famous tree of rationals: the topograph of Conway and Fung.
متن کامل