  We generalize the cosmological duality between inflation and cyclic
contraction under the interchange $a \leftrightarrow H$ to the case of
non-canonical scalar field theories with varying speed of sound. The single
duality in the canonical case generalizes to a family of three dualities
constructed to leave the cosmological acoustic horizon invariant. We find three
classes of models: (I) DBI inflation, (II) the non-canonical generalization of
cyclic contraction, and (III) a new cosmological solution with rapidly
decreasing speed of sound and relatively slowly growing scale factor, which we
dub {\it stalled} cosmology. We construct dual analogs to the inflationary slow
roll approximation, and solve for the curvature perturbation in all three
cases. Both cyclic contraction and stalled cosmology predict a strongly blue
spectrum for the curvature perturbations inconsistent with observations.
