I think the Duke either killed his wife or had someone else do it. Either way, I think she was poisoned or "accidentally" got hurt. First, the Duke said about his wife's affections for everything, "This grew; I gave commands; then all smiles stopped altogether. There she stands as if alive." He then changes the subject by saying, "Will't you please rise? We'll meet the company below." and "Notice Neptune, though…" he acted jealous of the smiles, said his wife died, then changed the subject. This sounds suspicious, like he was guilty of killing her or having her killed.