If the speed of sound depended on the frequency (pitch) of the sound, then there could be no such thing as a band or a choir performance. Everybody on the stage would play or sing together, but by the time the sound got out to the customers in the audience, the flutes and tubas would be way apart, and so would the sopranos and the baritones.
The density of the medium determines the speed of sound through it, and the speed is the same for all sound frequencies.
