Increasing the FFT length with the book implementation to a large number like 65,536 results in really echo-y sound, even a small bump up to 8,192 doesn't sound right. I guess I'm trying to figure out if this is due to some result of what a phase vocoder does, or due to a limitation in the implementation that would need to be adjusted.
There is no limitation in the implementation regarding length of FFT (you have all the code, you can check that for yourself).
Yes, increasing the size of the FFT will result in a more echo-y sound. In addition, it makes the phase vocoder unusable for realtime effects (think pitch shifting stomp box - your pitch shifted sound would take more than a second at 44.1kHz to arrive at the output).
To understand the ramifications, implement the robot effect and change the FFT length. You will hear what is happening directly.
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