Consider the possibility that the Page (having a tin ear) has quite misunderstood Touchstone's comment and believes that simply having 'kept time' means that he was also 'in tune'. That makes the exchange funny, and provides the shift of meaning needed for Touchstone's rejoinder to follow naturally from it. It would also require that the Elizabethan audience would understand much the same distinctions as we do now; though, of course, a good many, as now, would not.

My suspicion would be that we aren't so much looking at different attitudes to timing or intonation, as to (slightly) different assumptions about the meaning of words and the continuity of meaning within an exchange. Remember that one of Shakespeare's most individual characteristics was the way in which he could shift meaning through a whole spectrum in the space of just a few lines. You need to follow that shift rather than try to pin down one single meaning out of the continuum in order to make sense of the exchange.