Umm, not exactly. There are two separate questions here. (1) How does the pitch change with increasing speed of sound in the gas? (2) Why is the speed of sound in helium faster than in air?
(1) The sound travels by compression waves (molecules bunching up and getting farther apart), not by the movement of the gas. The vocal cords actually vibrate at the same rate, air or helium, but the sound waves produced by resonance in the throat/neck/mouth are shorter wavelength (thus higher frequency) in a higher sound-speed gas. The helium molecules are smaller and lighter so are easier to move around more quickly. If you made the throat smaller, you'd get a higher pitch too. Thus, women and children tend to have higher pitched voices.
(2) The sound speed is inversely proportional to the compressibility of the gas, and to its density. A "spongy" gas (highly compressible) will have a lower sound speed; the compression waves are more sluggish. A denser gas will have a lower sound speed; the molecules are harder to move around thus compression waves move around more sluggishly.
On balance, air and helium are similar in compressibiity, but are greatly different in density, so the less dense helium has a higher sound speed than air, by a factor of about 2.9.
By the way, the difference in the speed of sound in N2, O2 is the basis for a gas analyzer that someone built. If you know the combined speed of sound in the mixture, you can calculate the fraction of each gas. You can make this even more complicated by adding in a third gas, like He, and measuring one of the gases, like O2. Then you can calculate how much N2 and He are there by knowing the combined sound speed of the whole mixture. You get the sound speed by measuring the pitch of the sound produced by resonating in a chamber of known dimensions.