Summary: A Riemannian or Finsler metric on a compact manifold \(M\) gives rise to a length function on the free loop space \(\Lambda M\), whose critical points are the closed geodesics in the given metric. If \(X\) is a homology class on \(\Lambda M\), the “minimax” critical level \(\mathsf{cr}(X)\) is a critical value. Let \(M\) be a sphere of dimension \(>2\), and fix a metric \(g\) and a coefficient field \(G\). We prove that the limit as \(\deg(X)\) goes to infinity of \(\mathsf{cr}(X)/ \deg(X)\) exists. We call this limit \(\overline{\alpha}=\overline\alpha(M, g,G)\) the global mean frequency of \(M\). As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on \(\Lambda\) of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) \(\overline{\alpha}\), or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to \(\overline{\alpha}\). The proof uses the Chas-Sullivan product and results of M. Goresky and N. Hingston [Duke Math. J. 150, No. 1, 117–209 (2009; Zbl 1181.53036)].