We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane R² . The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane R² is decomposed into an infinite union of the translates of the rectangular periodicity cell Ω⁰, and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of Ω⁰ consist of a neighborhood of the boundary of the cell of the width h and thus has an area comparable to h, where h > 0 is a small parameter. Using the methods of asymptotic analysis we study the position of the spectral bands as h → 0 and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided h is small enough.