How is it possible to change the pitch and the tempo of an audio track independently of each other?

Question

If you slow down a turntable or cassette-player, both pitch and tempo are decreased. How is it possible to change one without affecting the other?

Answer 1

Here's how I'd think about it, but I'm not an authority, so I'm making my answer CW in the hope that if I've made an error, others who know better would be happy to edit my answer to correct it.

Think of the audio stream as a series of notes, each note a signal $f_n$. Each signal is a superimposition of frequencies, so $f_n = \int \hat{f}_n(\omega)e_\omega d\omega$.

To change tempo, you should alter the rate at which it emits signals. To change frequency, you should alter the frequency decomposition of each signal: $\hat{f}_n(\omega)\to \hat{f}_n(c\omega)$.