The log periodic antenna is the ultimate broadband beam. It can provide high gain on a single band or moderate gain on multiple bands. On vhf you can easily design a single feed 2 and 6 meter antenna. On hf the size and element diameters are usually a drawback .Wire versions are usually the answer. Although 4 bands is usually stretching it, don't be afraid to try more. I had an inverted 'T' loaded 13 element designed for 160 to 6 meters and it worked. It covered an acre ,was 120 feet wide ( 40 meters), 350 feet long (105 meters) and was suspended from 3 towers. The 1/2 mile of wire looked like a suspension bridge. The neighbours are still cautious to talk to me and the XYL doesn't want to discuss it