By treating a small cluster embedded in an effective medium (described here by the coherent-potential approximation) we can reproduce the "exact" numerical frequency-distribution spectra of the vibrating-linear-chain alloy AcB1-c with mass disorder. The theory is especially applicable to concentrations 0.05c0.95 throughout the alloy regime, where other theories are quantitatively unreliable. Unlike purely numerical calculations, the present method can be practically applied to real three-dimensional alloys. The theory is valid for all concentrations c and all mass ratios mBmA; it satisfies the oscillator strength sum rule, and it reduces to the exactly soluble single-defect theory in the limits c0 and c1. Its greatest virtue is that it is computationally efficient, because it does not require large clusters.