Sesquiterpene cyclase catalyzing the production of sixfold more beta-farnesene than alpha-bergamotene from farnesyl diphosphate. Involved in indirect defense by producing volatile signals attracting natural enemies of herbivores. (2E,6E)-farnesyl diphosphate = (E)-beta-farnesene + diphosphate Secondary metabolite biosynthesis; terpenoid biosynthesis. The Asp-Asp-Xaa-Xaa-Asp/Glu (DDXXD/E) motif is important for the catalytic activity, presumably through binding to Mg(2+). Belongs to the terpene synthase family.