One known technology for producing a musical tone with the human body is to use the vocal chords and hum or sing.
Another is to use the lips and whistle.
Since these methods are partially independent, it seemed to me as if I ought to be able to produce two-part harmony on my own.
Step one was to be able to whistle and sing, separately, which I've been able to do since I was little. (If anyone wants help learning how to whistle, feel free to ask; I can at least try to help you).
Step two was to learn how to, while making one tone with my voice, whistle. That wasn't much harder. I just sang a note, and while I was doing that, moved my lips into whistling position. It didn't take long to produce two distinct sounds at the same time - although there is some interference.
Step three was to figure out how to change the pitch of the whistle while sustaining the vocal tone. This was also pretty easy - if you can change the pitch of your whistle, you can change the pitch of your whistle while singing at a single pitch. It feels pretty much the same.
Step four was to be able to alter the vocal pitch while keeping the whistle constant. This is actually hard, because the whistle tone and the vocal tone seem to interact somehow. So I'd have to change my lip position just to keep the whistle at the same pitch as I altered my vocal pitch. Eventually I got it, after a day or two of annoying everyone around me.
Step five was to be able to alter them simultaneously, so as to sistle in two part harmony. This was really, really hard. I think what makes this the hardest step is that it's not about learning the physical positions - it's about the cognitive ability to track the two melodies simultaneously.
The way you sing or whistle of course is not by consciously consulting a giant lookup table between pitches and physical behaviors. Instead, you learn to associate a tone with a bodily behavior, so that when you think of the tone, your body prepares to produce that tone automatically. I think the same tone-memory in my mind is linked to both a whistling and singing behavior. So when I tried to keep track of two pitches, I didn't have the cognitive skill of remembering which tone went with which part of my behavior. Instead I got leakage - I'd try to move my vocal pitch and move my whistling pitch instead, or vice versa. (This is also part of what made step four a little hard.)
On top of that there are range problems - it's harder to whistle below than above your vocal pitch, and you're stuck with your maximum comfortable vocal and whistling ranges - minus a little bit since you have less room to for example help your whistle by changing your mouth shape.
However, after having a bunch of fun with that, eventually I got to the point where I could do some mediocre two-part harmony. Here are some examples I recorded today - they sound pretty terrible, but the point is that I can do it at all: