In fact, it is better to think of an electronic musical instrument that is involved in a performance as an Event-Driven system. In such a system, what drives transitions from one state to the next is not the transitions of a clock signal, but the occurence of an Event, usually generated externally to the system. For example, events can be created by the pressing of a key on the keyboard of a synthesizer, or the striking of a drumstick on an electronic drum pad. As you are no doubt aware, even when playing music such as Bach, where a very precise periodic rhythm is desired, the events produced by human performers are by no means periodic. Some players are much better than others, of course, but then there is the drummer from your neighborhood garage band.
Some music is arrhythmic.
Some music has the timing of its events derived from the environment. In all of these cases, musical events essentially occur asynchronously , that is to say, they occur independently of a periodic clock. The main problem is that the order in which notes are played in an arpeggiaton is usually in order of pitch, rather than in order of being played. Instead one must scan the notes that are being played, in pitch order. This scanning must be done continually, as notes can be played or released at any time.