Hmmm... I've been giving this more thought...
it makes sense that a reed's vibrations is dependent on the density of both objects, and likewise the air density NOT having much of an effect on a string...
What about a speaker? It seems more like it would be analogous to a string... A speaker's movement is controlled by the signal driving it (instead of the tension and mass of the string), and the "rate" would be independent of air density, but the displacement with each travel obviously would change depending on air density and the voltage amplitude driving it, at least in theory I think a "thick air" would require more power to push the driver the same displacement...
Suppose I have a breathing source and I was in a room full of another gas, such as helium since we've probably all inhaled helium (and I'd guess not many of us have inhaled sulfur hexafluoride - but as an FYI, inhaling N2O, which is also denser than air (but I don't think nearly as much as sulfur hexafluoride) also results in an audible "pitch lowering" effect). Then, suppose we sent a 100 Hz sine wave to the speaker. Regardless of the density of the medium, as long as there is "a" medium to transfer the pressure changes and the amplitude was enough to move the medium such that it could move my tympanic membrane (and thus everything else in my ear) by the time it gets to me, then wouldn't I still hear 100 "pressure changes" per second, which my ear would detect as the same pitch as if the speaker and I were in a "normal" atmosphere? Better yet, to eliminate any of the differences in passing from one medium to another regarding the path in our ears, suppose we hooked a microphone to an o'scope... Wouldn't it still read 100 Hz regardless whether it was air, helium, sulfur hexafluoride, N2O, Butane, etc.? And if our ear (pitch) works on "pressure changes per second" then wouldn't we hear it at the same pitch whether it was helium vs. air between the speaker and your ear? The sound may travel faster in different mediums resulting in each pressure change getting to our ear from the source in a shorter time, but the number of pressure changes per second at your ear would be the same as they would be at the speaker, which is the same frequency as the voltage going into the speaker. Right? I still feel like I'm missing something...
Does the apparent change in pitch have anything to do with the fact that there are two different mediums at play, first through helium in the persons lungs transitioning to air? OR, is it all related to what you said about vocal cords being more like a reed than a string or "cord" ?
I know I am somewhat contradicting my previous post...