F1 cars make "weird" noises when gears are being shifted for a couple of reasons. First, the act of shifting gears on a transmission that's linked to an engine revving at 20,000 RPM is pretty violent. I'm not sure on the actual time figures, but I think the computer-controlled upshifts and downshifts happen in about 0.015 seconds. That's an amazingly small amount of time for an engine to make a 10,000+ RPM change. That fact is made even more astounding when you realize that the force of acceleration on the engine's pistons is about 9,000 Gs. So it's not a quiet, delicate procedure.
And second, shifting gears is done electro-hydraulically so that the clutch, transmission, and engine all work in concert with each other. Some of that "weird" noise can be attributed to the ECU making micro-adjustments to all of those components, especially the throttle which gives the shift that "popping" sound, so that shifts happen as fast and efficently as possible.
To add to the perception of sounds, there is one source that is relevant. We hear it in high performance jet fighters, we hear it in musical instruments. When fast moving mass of airflow comes into contact with a slower moving mass of air, there is friction between the two interfaces. That is how a flute works, as well as a jet fighter. It's almost a tearing noise, the violent interaction between two masses of air colliding with each other. So how the exhaust finally exits the exhaust pipes can be a source (or depression) of the sounds. If you can get the exhaust airflow to move much slower on exit, to be as close to the velocity of the outside airstream, then the air interaction is less violent, less noise.
In technical terms, to reduce disparate flow velocities.
To continue Dave's train of thought with regards to the shearing of gases at different velocities, you can add the factor of ignition cutting during shifts. The ignition is cut for a small amount of time in order to temporarily reduce the power being transmitted through the drivetrain. I wouldn't be surprised if the fuelling during this period remains constant, and consequently unburnt fuel ends up in a hot exhaust. Here it will ignite and cause the loud 'pop' heard during shifts. This probably increases the velocity briefly in the exhaust and causes increased shear with the air.
I believe traction control works on a similar principle, but perhaps by cutting a few cylinders rather than all 8. That's why you get the harsh-sounding crackle out of corners.
...sorry for the double post. My CPU controls me sometimes 
