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