  We construct a family of purely infinite $C^*$-algebras,
$\mathcal{Q}^\lambda$ for $\lambda\in (0,1)$ that are classified by their
$K$-groups.
  There is an action of the circle
  $\T$ with a unique ${\rm KMS}$ state $\psi$ on each $\mathcal{Q}^\lambda.$
For $\lambda=1/n,$ $\mathcal{Q}^{1/n}\cong O_n$, with its usual $\T$ action and
${\rm KMS}$ state.
  For $\lambda=p/q,$ rational in lowest terms, $\mathcal{Q}^\lambda\cong O_n$
($n=q-p+1$) with UHF fixed point algebra of type $(pq)^\infty.$ For any $n>0,$
$\mathcal{Q}^\lambda\cong O_n$ for infinitely many $\lambda$ with distinct KMS
states and UHF fixed-point algebras. For any $\lambda\in (0,1),$
$\mathcal{Q}^\lambda\neq O_\infty.$ For $\lambda$ irrational the fixed point
algebras, are NOT AF and the $\mathcal{Q}^\lambda$ are usually NOT Cuntz
algebras. For $\lambda$ transcendental, $K_1\cong K_0\cong\Z^\infty$, so that
$\mathcal{Q}^\lambda$ is Cuntz' $\mathcal Q_{\N}$, \cite{Cu1}. If $\lambda^{\pm
1}$ are both algebraic integers, the {\bf only} $O_n$ which appear satisfy
$n\equiv 3(mod 4).$ For each $\lambda$, the representation of
$\mathcal{Q}^\lambda$ defined by the KMS state $\psi$ generates a type ${\rm
III}_\lambda$ factor. These algebras fit into the framework of modular index
(twisted cyclic) theory of \cite{CPR2,CRT} and \cite{CNNR}.
