The point here is that you have to determine exactly what you wish to synchronise with what.
Merely having a pulse generator - such as a NE555 - is of no value as we have been enthusiastic to point out. The Arduino can do that in a far more useful manner than the NE555; it can generate pulses (with considerable accuracy) and programattically adjust them to varying specifications.
If your thought was to have (what is in fact, a metronome) the pulse generator provide synchronising pulses to both the Arduino and the music system, then the correct way to do this is to program the Arduino to generate the pulses as it performs its part, and output those same pulses to the audio system.
If on the other hand, it is the audio system which actually provides such synchronisation signals, then they can be interfaced acordingly (an optocoupler would be appropriate, as in the conventional MIDI interface) to the Arduino which as Mike explains, as part of the many functions which continuously cycle in the loop, checks this input to enable the appropriate response.
However it happens, there is no need for a timebase as a third component.