The Last Minute Before the Hour

You have two ropes and a supply of matches. Each rope takes exactly 60 minutes to burn from one end to the other, but neither burns at a steady rate: one section might vanish quickly while another smolders for much longer. The ropes may also differ from each other. You may light either end of either rope. Once burning, a rope cannot be extinguished, cut, marked, or measured. You have no clock, so after lighting the first match you can act only when you observe a rope finish burning. Using both ropes, what is the longest interval of less than one hour that you can measure exactly?