I want to write an app to transpose the key a wav file plays in (for fun, I know there are apps that already do this)... my main understanding of how this might be accomplished is to

1) chop the audio file into very small blocks (say 1/10 a second)

2) run an FFT on each block

3) phase shift the frequency space up or down depending on what key I want

4) use an inverse FFT to return each block to the time domain

5) glue all the blocks together

But now I'm wondering if the transformed blocks would no longer be continuous when I try to glue them back together. Are there ideas how I should do this to guarantee continuity, or am I just worrying about nothing?