Multiple synthesizer projects have been done for the Arduino, but few have been able to utilize the full power of the Arduino processor. DZL from GeekPhysical wrote a 4 voice wavetable synthesizer that is one of the more advanced software based synths for the Arduino.
It has wavetables included (sine, saw, square and triangle) and envelopes to create beats.This instructable will show you how to make a simple sequencer that plays a little melody. You can hear how it sounds here:
It is up to you to modify the code to do many other things. Possible examples could be a midi controlled synth (sample code included), it could be a strange noise machine like this one:
For more detail: Minimal Arduino-based wavetable synth