Why Does Inhaling Helium Make Your Voice Squeaky?

Sucking a lungful of helium out of a balloon makes your voice sound hilarious. But contrary to popular belief, the switch from air to helium gas doesn't actually increase the pitch of your voice (at least not very much).

Instead, it affects a much more mysterious property of the sound, called 'timbre.' Rather than chirping high notes like Tweety Bird, you start quacking words like Donald Duck. Why does your voice get that reedy tone?