For any connected space
X
{\mathbf {X}}
and ring
R
R
, we describe a first-quadrant spectral sequence converging to
H
~
∗
(
X
;
R
)
{\tilde H_*}({\bf {X}};R)
, whose
E
2
{E^2}
-term depends only on the homotopy groups of
X
{\mathbf {X}}
and the action of the primary homotopy operations on them. We show that (for simply connected
X
{\mathbf {X}}
) the
E
2
{E^2}
-term vanishes below a line of slope
1
/
2
1/2
; computing part of the
E
2
{E^2}
-term just above this line, we find a certain periodicity, which shows, in particular, that this vanishing line is best possible. We also show how the differentials in this spectral sequence can be used to compute certain Toda brackets.