The speaker of "My Last Duchess" is, of course, the Duke of Ferrara. But it’s important to think about him, not only as a character, but as a speaker. We need to consider his rhetoric, and syntax, and speech patterns. We know what kind of a man the Duke is, but what kind of an orator is he?
First of all, the Duke’s speech is highly formalized, using strict rhyme and meter to organize itself into couplets (AABBCC etc.). He’s a man who appreciates control, and he takes pains to control his own statements. But the syntax, or sentence structure, of the poem pulls against its rhyme scheme. The lines are paired in rhymed couplets, but these couplets are "open" – that is, the sentences don’t finish at the same time the lines do. For example:
I repeat, The Count your master’s known munificence Is ample warrant that no just pretence Of mine for dowry will be disallowed; Though his fair daughter’s self, as I avowed At starting, is my object. (48-53)
This statement is one sentence and contains two couplets, but the sense of the lines continually spills past the rhyming words. The Duke can shape his speech into couplets, but his thoughts strain against that structure and try to break it. There’s a sense of struggle in his lines, as though he’s just barely managing to rein things in and about to lose it at any moment. Given what happened to his "last Duchess," we’re frightened of what will happen when he finally loses control.