# Necklaces and Bracelets: Enumeration, Algebraic Properties, and Their Relationship to Music Theory

12-2013

## Level of Access Assigned by Author

Campus-Only Thesis

## Degree Name

Master of Arts (MA)

Mathematics

George Markowsky

Robert Franzosa

Andrew Knightly

## Abstract

In music theory, a chord is a collection of k notes within an n-note scale and can be represented as a ring of n equally spaced nodes, each colored black or white. The black colored nodes represent the k notes that form the chord while the white nodes represent the n — k notes that are not part of the chord. Transposing a chord corresponds to a clockwise rotation while inverting a chord corresponds to a rotation followed by a reflection across the vertical axis of symmetry. A set class is the equivalence class of all transpositions and inversions of a chord (for example, the set of all major and minor triads). In combinatorics, a set class is identical to a binary bracelet, which is the equivalence class of all rotations and reversals (a rotation followed by a backwards reading starting from the same initial character) of a binary string of length n with k ones. It is of musical interest to know the number of set classes partitioning the set of all k-note chords within an n-note chromatic universe. Finding the solution to this problem is equivalent to finding the number of bracelets partitioning the set of binary strings of length n with k ones (called fixed- density binary bracelets). This problem motivates our introductory investigation of bracelets and their relatives. While a bracelet is the equivalence class of all rotations and reversals of a string, a necklace is the equivalence class consisting of only the rotations. Unlike necklaces and bracelets, a Lyndon word is a string representative rather than an equivalence class. More precisely, it is an aperiodic string that is lexically-least in its necklace equivalence class. In this thesis, we offer a compilation of previous results concerning enumeration of bracelets, necklaces, and Lyndon words. Additionally, we explore some of the well-known factorization properties exhibited by Lyndon words. Counting tables for binary necklaces and bracelets are provided, including tables that count the number of necklaces and bracelets partitioning the set of binary strings of length n for n up to 1000. In this work, we also compile all required preliminary lemmas and theorems and provide proofs for nearly all of these results.