Polyphonic pitch detection is much more involved and requires more advanced algorithms which are computationally intensive and thus hard to perform in real-time.

Commercial closed-source software like Melodyne can do this, at least in off-line processing. AFAICT, the latest Melodyne versions also do some real-time processing, but I haven't used Melodyne for some time and so I don't know how well that works.

I'm not sure either whether there are any good open-source codes for this available yet, maybe others can provide corresponding links. But here are some relevant answers from Stackoverflow and Stackexchange:

http://stackoverflow.com/questions/9613768/multiple-pitch-detection-fft-or-other/9626849#9626849

http://dsp.stackexchange.com/questions/11433/polyphonic-detection-mulit-pitch-detection-chord-recognition

Also, here's an interesting recent DAFx paper on doing polyphonic pitch detection using autocorrelation:

http://www.dafx14.fau.de/papers/dafx14_sebastian_kraft_polyphonic_pitch_detectio.pdf

And then there's the work of Anssi Klapuri and others at Tampere University:

http://www.cs.tut.fi/~klap/iiro/

Also, there are algorithms for doing spectrum estimation such as filter diagonalization methods (FDM) and the classical Prony algorithm, but due to their complexity these probably aren't well-suited for real-time processing either (the Prony algorithm also suffers from numerical instabilities IIRC), and you still have to do the partitioning of the overtone series afterwards.

There's surely more, but that's what I could find with a quick Google search or remember from the top of my head.

Hope this helps,

Albert