Thanks so much, @Som and @Carlos Luna, for your encouragement and advice.
And, thanks for the info and the link, Abdullah. Unfortunately, in my personal case it's not really a matter of my hearing: I scored 97% on the test you linked to. But there's a considerable problem with how my brain processes (or fails to process) the signals my ears deliver to it: I got completely flustered over one of the questions, and started to panic until I realized it was actually an example, and the answer was <em>right there</em>. I can tell the difference between a melody and a rhythm, but can't carry the one or reliably keep the other. And I've encountered more than a little household contention over being unable to tell whether a toilet tank is filling, or if the toilet is running and someone needs to go jiggle the handle, by whether the sound of it is rising or not.
So maybe there are two questions here: one, whether a tone-deaf person could learn Mandarin; and two, a rather different question of whether <em>I</em> could learn Mandarin.