  This paper examines the relation between the spin-wave instabilities of
collinear magnetic phases and the resulting non-collinear phases for a
geometrically-frustrated triangular-lattice antiferromagnet in the high spin
limit. Using a combination of phenomenological and Monte-Carlo techniques, we
demonstrate that the instability wave-vector with the strongest intensity in
the collinear phase determines the wave-vector of a cycloid or the dominant
elastic peak of a more complex non-collinear phase. Our results are related to
the observed multi-ferroic phase of Al-doped CuFeO$_2$.
