Based on the generalized cyclotomy of order two with respect to n=p1e1+1p2e2+1…ptet+1, where p1, p2, …,pt are pairwise distinct odd primes and e1, e2,…, et are non-negative integers satisfying gcd (piei (pi-1), pjej (pj-1)) = 2 for all i ≠ j, this paper constructs a new family of generalized cyclotomic sequences of order two with length n and investigates their linear complexity. In the view of cascade theory, this paper obtains the linear complexity of a representative sequence.