Even and odd pendulum-ring modes (${\omega }_j^2=g/L+(4k/m){\sin }^2(\pi j/N)$)

For a large ring of equal pendulums with tangential displacements and nearest-neighbour spring coupling, cyclic translation symmetry gives Fourier modes $x_s\propto e^{2\pi ijs/N}$. The locally straight-chain approximation gives the displayed spectrum. The lowest mode is uniform. For even $N$, the single highest mode alternates signs and has ${\omega }_{\max }^2=g/L+4k/m$. For odd $N$, perfect alternation cannot close; the highest pair $j=(N\pm 1)/2$ is degenerate and has ${\omega }_{\max }^2=g/L+(4k/m){\cos }^2(\pi /(2N))$. For an exact finite regular polygon with straight chord springs, multiply the spring contribution by ${\cos }^2(\pi /N)$; it tends to one in the stated large-$N$ limit.