Art--particularly, the portrait of the Duchess and the sculpture of Neptune taming the sea horse--has a critical role in "My Last Duchess."
The comments made by the Duke on the portrait of the Duchess, for example, allow us to understand both the Duke and the Duchess (to a lesser extent), and the conclusions we ultimately reach about the Duke are the result, in part, of the artist's ability to catch important aspects of the Duchess's character in the portrait.
The artist has been able to capture the Duchess's "earnest glance," and the Duke's visitor has apparently asked a question about it: "How such a glance came there; so not the first/Are you to turn and ask thus." This question, which was undoubtedly asked to please the Duke and compliment the Duchess, misfires in that the Duke begins his criticism of the Duchess's grace, genuine politeness, and openness with everyone. After noting that the artist's comment about the difficulty of producing the "'Half-blush that dies along her throat,'" drew a blush from the Duchess, he begins his real criticism of her--"A heart--how shall I say--too soon made glad,/Too easily impressed. . . ." Most of us would find joy in a person with such a light heart who could find pleasure in the smallest of things, but the Duke begins to show an aversion to the Duchess's ability to be joyful, not an expected reaction.
We learn in the next several lines that the Duke's resentment, for that what it is, goes much deeper than one would expect. He recites a litany of things that gave her joy, and then, in a paroxysm of jealousy, gets to the root of his resentment--she likes everything equally "as if she ranked/My gift of a nine-hundred-years-old name with anybody's gift." He continues to berate her for not understanding his hurt pride and inability to "be lessoned," to be taught what she should value rather than what she does value. Because he is so stiff with pride, he chose "never to stoop," in other words, never to treat her as an equal and explain his feelings so that she could understand his hurt pride.
His answer to her inability to understand his feelings--keep in mind the fact that he didn't explain himself to her adequately--was to command her not be so easily pleased, command which only caused her sadness. And, without any explanation, the Duchess disappears from the discussion.
The second piece of art--Neptune taming the sea horse--is a emblem of the Duke's controlling personality--just as he tried to control the Duchess and failed, perhaps resulting in her death, the statue of Neptune controlling the sea horse, which was made specifically for the Duke, exemplifies the Duke's view of proper domination and subordination.
Woe to the prospective duchess the Duke and his visitor go downstairs to discuss with her father.