A number of alternatives that may be used.

Use notes. Let each note represent a wav file.
Write code that sums multiple wav files (remember that many instruments have sound that slowly declines).
Send the sum of the currently sounding tones to a 16-bit DAC. If low quality is acceptable, look for an ulaw/alow codec solution.

With a fast enough processor, you can play quite a large number of concurrent tones. With 10-20 concurrent tones you get a reasonably good result playing new chords while still playing the remaining sound from the previous chords. Best is to perform the math using more than 16 bits - 24 bits is a good choice.

Another alternative. Use a large SPI flash chip and store full music recordings. 1GB of flash is about 1.5 times larger than a CD, so it gives an indication how much music you can store.

A third alternative. If you need huge amounts of sound - use a MP3 chip or get a processor with MP3 decoder or a fast enough processor to run the decoding in software. The third alternative isn't so fun unless you have other reasons for a quite fast processor.

But as noted - it is not easy to convert sound into tones, since many instruments produces very rich sounds with a lot of frequencies involved. And you normally don't play a single tone at a time either.