It is strange the way that a motivational or business-success-related concept can translate into something musical. Consider Rajesh Setty's Focus on the Edges on his Life Beyond Code blog.
The edge for me is the millisecond when sound begins. The clarity and confidence of that millisecond determines nearly everything about the note that will follow. Every bow stroke, every stroke of a wind-player's tongue or a percussionist's stick (or hands or fingers) is an interaction with "the edge."