Two well-known methods of spectrum estimation are the classical windowed-periodogram method described by Blackman and Tukey and the maximum entropy method (MEM) proposed by Burg. Comparisons of the two methods are diflicult, in part because their motivations are based on differing sets of assumptions. My development, aimed at unification, rests on two new but simple formulas that extend a classical concept, thereby exhibiting an element common to both methods: the spectral window. The first formula expresses the MEM estimate as a convolution of the spectrum with a data-dependent and spectrum-dependentequivalent spectral window. The second formula gives the MEM estimate as a convolution of the periodogram with a data-dependentperiodogram window. These formulas provide a new outlook on MEM that may be quite useful for a variety of purposes, including (a) the development of heuristic interpretations of line splitting and peak shifting and (b) the motivation of natural definitions of bandwidth for MEM estimates.