  Recently, frame multipliers, pair frames, and controlled frames have been
investigated to improve the numerical efficiency of iterative algorithms for
inverting the frame operator and other applications of frames. In this paper,
the concept of biframe is introduced for a Hilbert space. A biframe is a pair
of sequences in a Hilbert space that applies to an inequality similar to frame
inequality. Also, it can be regarded as a generalization of controlled frames
and a special kind of pair frames. The basic properties of biframes are
investigated based on the biframe operator. Then, biframes are classified based
on the type of their constituent sequences. In particular, biframes for which
one of the constituent sequences is an orthonormal basis $\{e_k\}_{k=1}^\infty$
are studied. Then, a new class of Riesz bases denoted by $[\{e_k\}]$, is
introduced and is called b-Riesz bases. An interesting result is also proved,
showing that the set of all b-Riesz bases is a proper subset of the set of all
Riesz bases. More precisely, b-Riesz bases induce an equivalence relation on
$[\{e_k\}]$.
