Where $A$ is a topological space, let $f: [0,1] \to A$. Define the Horizontal Chord Set $D_f := {\ell \in \mathbb{R} \,|\, \exists x \in [0,1 - \ell], f(x) = f(x + \ell)}$. Let $\mathcal{L}A$ be the loop space of $A$. It has been previously proven that $\bigcap_{f \in \mathcal{L}\mathbb{R}}D_f = {\frac{1}{n} \,|\, n \in \mathbb{\mathbb{N}}} \cup {0}$. It has also previously been proven that the Lebesgue measure $\lambda(D_f) \geq \frac{1}{2}$ for $f \in \mathcal{L}\mathbb{R}$. For a topological space $A$, denote the constant $k_A = \inf {\lambda(D_f) : f \in \mathcal{L}A}$.

In this paper we characterize $k_A$ for triangulable spaces, proving $k_A = 1$ when $A$ is 0-dimensional, and $k_A = 0$ when $A$ is more than 2-dimensional, or is 1-dimensional and has a cycle. When $A$ is 1-dimensional and has no cycles it is a tree, we prove that $k_A \leq \frac{1}{n}$ where $n$ is the number of leaves, and conjecture that $k_A = \frac{1}{n}$. We show that the map $f \mapsto \lambda(D_f)$ is not continuous, making the proof of this conjecture difficult. We finally generalize Paul Levy’s Universal Chord Theorem by showing that for any tree $A$ with a vertex of degree three or more, $\bigcap_{f \in \mathcal{L}\mathbb{A}}D_f = {0,1}$.