We construct a family of purely infinite C ⁎ -algebras, Q λ for λ ∈ ( 0 , 1 ) that are classified by their K-groups. There is an action of the circle T with a unique KMS state ψ on each Q λ . For λ = 1 / n , Q 1 / n ≅ O n , with its usual T action and KMS state. For λ = p / q , rational in lowest terms, Q λ ≅ O n ( n = q − p + 1 ) with UHF fixed point algebra of type ( p q ) ∞ . For any n > 1 , Q λ ≅ O n for infinitely many λ with distinct KMS states and UHF fixed-point algebras. For any λ ∈ ( 0 , 1 ) , Q λ ≠ O ∞ . For λ irrational the fixed point algebras, are NOT AF and the Q λ are usually NOT Cuntz algebras. For λ transcendental, K 1 ( Q λ ) ≅ K 0 ( Q λ ) ≅ Z ∞ , so that Q λ is Cuntz' Q N [Cuntz (2008) [16]]. If λ and λ − 1 are both algebraic integers, the only O n which appear are those for which n ≡ 3 ( mod 4 ) . For each λ, the representation of Q λ defined by the KMS state ψ generates a type III λ factor. These algebras fit into the framework of modular index theory/twisted cyclic theory of Carey et al. (2010) [8], Carey et al. (2009) [12], Carey et al. (in press) [5].