The narrator believes his tone to be persuasive and calm, as he is trying to convince someone that he is not crazy. In the first line, he asks,
True!—nervous—very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses—not destroyed—not dulled them.
He describes how perceptive his senses have become, especially his hearing, saying, "I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad?" He believes that this acuteness of his senses is evidence of how healthy he is, and he believes that he can tell his story "calmly." However, the fact that he believes he can hear what's happening in heaven and hell, and even everything on the earth itself, is pretty crazy-sounding, and the sheer number of exclamation points used in the story—forty-three—makes it seem as though "calm" is not exactly the right word to describe his tone!
Instead, the narrator's tone is somewhat aggressive, excitable, and even manic. His many, many references to his auditor make him sound aggressive, especially in the first three paragraphs, where he addresses "you" seven times. His mania and aggression are both apparent in the third paragraph, which begins,
Now this is the point. You fancy me mad. Madmen know nothing. But you should have seen me. You should have seen how wisely I proceeded—with what caution—with what foresight—with what dissimulation I went to work!
In these lines, he is obviously trying to sound convincing, addressing his auditor directly three times, though this makes him sound aggressive. The abbreviated structure of the first three sentences also makes him seem manic. He is very excitable.