In broadest terms, a tragedy ends in the death of a major character or characters; in a comedy, the major characters survive. Antony and Cleopatra's suicidal deaths are main tragic elements of this play.
Underlying these deaths is the tragic current implying (from the start) that both are foredoomed. The play hints that fate or fortune is on the side of Rome, Octavius, and all the virtues of self-discipline, reason, and sternness that Rome represents. Just as in a Greek tragedy like Oedipus Rex, there is a sense that forces bigger even than the lives of monarchs rule the universe, using humans as pawns. As Antony says near the end:
Tis paltry to be Caesar; / Not being Fortune, he's but Fortune's knave, / A minister of her will
In other words, Octavius is simply the tool of bigger powers, without free will, bringing the action to a fore-ordained end that he can't control any more than Antony can.
Antony's great personal tragedy is that he is so besotted with Cleopatra that he can't muster...
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