A Riemannian or Finsler metric on a compact manifoldM gives rise to a length function on the free loop space M, whose critical
points are the closed geodesics in the given metric. If X is a homology class on M, the “minimax” critical level 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 cr(X)/ deg(X) exists. We call this limit = (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 of sufficiently high degreee
lies hanging on closed geodesics of mean frequency (length/average index) , or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to . The proof uses the ChasSullivan product and results of Goresky-Hingston [7].