  The spectral curve correspondence for finite-type solutions of the
sinh-Gordon equation describes how they arise from and give rise to
hyperelliptic curves with a real structure. Constant mean curvature (CMC)
2-tori in $ S ^ 3 $ result when these spectral curves satisfy periodicity
conditions. We prove that the spectral curves of CMC tori are dense in the
space of smooth spectral curves of finite-type solutions of the sinh-Gordon
equation. One consequence of this is the existence of countably many real $ n
$-dimensional families of CMC tori in $ S ^ 3 $ for each positive integer $ n
$.
