The Duke is talking to an emissary of a count. The Duke is trying to marry the count's daughter, so he is trying to impress the emissary. In describing the painting of his most recent duchess, he communicates more about himself than he does of his late wife. He notes that the Duchess's look of "joy" in the painting is not just because he was present during the painting, her joy being caused by his presence. He claims that she had a heart "too soon made glad." In other words, he claims that she liked to be looked at and liked looking at others. This is his coy way of saying she was flirtatious. But what really comes across here is his jealousy. He would not even leave her alone with a friar (Fra Pandold).
In lines 45-47, he seems to respond to a question posed by the count's emissary, asking if the Duchess ever smiled. The Duke replies that she did smile but that as he became more oppressive (commanding), she stopped smiling. Here, he quite blatantly admits that as he became more domineering, she became less and less happy. He says, "I gave commands; / Then all smiles stopped together." The Duke ends his dramatic monologue to the emissary by calling his attention to a statue of the God Neptune taming a seahorse. The Duke unwittingly illustrates his own behavior towards his late wife in showing his admiration for a God controlling a "lesser" creature. This mirrors his controlling nature.