We construct a family of purely infinite C¤-algebras, Q¸ for ¸ 2 (0, 1) that are classified by their K-groups. There is an action of the circle T with a unique KMS state Ã on each Q¸. For ¸ = 1/n, Q1/n »= On, with its usual T action and KMS state. For ¸ = p/q, rational in lowest terms, Q¸ »= On (n = q − p + 1) with UHF fixed point algebra of type (pq)1. For any n > 1, Q¸ »= On for infinitely many ¸ with distinct KMS states and UHF fixed-point algebras. For any ¸ 2 (0, 1), Q¸ 6= O1. For ¸ irrational the fixed point algebras, are NOT AF and the Q¸ are usually NOT Cuntz algebras. For ¸ transcendental, K1(Q¸) »= K0(Q¸) »= Z1, so that Q¸ is Cuntz' QN, [Cu1]. If ¸ and ¸−1 are both algebraic integers, the only On which appear are those for which n ´ 3(mod 4). For each ¸, the representation of Q¸ defined by the KMS state Ã generates a type III¸ factor. These algebras fit into the framework of modular index theory / twisted cyclic theory of [CPR2, CRT] and [CNNR].