A formula to compute the index of stellar scintillation detected with a finite spectral bandpass and with arbitrary aperture is derived in the limit of weak perturbations. It also applies to differential scintillation (relative fluctuations of light intensity in a pair of apertures), where the effect of finite bandpass turns out to be significant. The new formula is used for measurements of free-atmosphere seeing and low-resolution turbulence profiles with concentric-ring apertures.