I think something is missing! What changes the duty cycle? For example say you adjust the duty cycle by turning a POT, then the time could be seconds/minuets/hours depending on how fast you turn it. Is it adjusted by some kind of interrupt? What I am trying to say is we need to know what controls the duty cycle so we know the time frame. Personally unless this was a quick 20%-70% I would be using a deep memory Logic analyzer. That way you just run it and then scroll back to 20% place a cursor there, then scroll forward and place other cursor at 70%.
Can you give details of the system? Otherwise say for example you have a interrupt that makes the duty cycle increase every 10 seconds by 5%, your never going to capture that on a scope.
Does your scope have a post trigger setting? and a delay setting? Do you have access to a Logic analyzer? do you have say a PK2 or PK3?