The temperature dependencies of the fundamental energy gaps of group-IV, III±V, and II±VI materials are fitted by means of a relatively simple analytical four-parameter expression. This is shown here to be capable of providing fine numerical fittings in combination with physically reasonable estimations of basic parameters. The majority of the materials under study are found to pertain to the regime of intermediate dispersion, where neither Varshni's formula nor the Bose-Einstein-related model function are capable of producing adequate fits. The effective phonon temperatures are estimated to amount as a rule to fractions of about 0.5 to 0.7 of the corresponding Debye temperatures. The availability of experimental E(T) data up to the vicinity of the Debye temperature in a given material is found to be a necessary condition for trustworthy determinations of the limiting slopes of the E(T) curves in the high-temperature region. This is achieved here for many materials by simultaneous fittings of fractional data sets owing to different authors and/or different experimental methods. The general cause of break-down of Varshni's formula, which has been noticed particularly in wide bandgap material studies, is clearly shown here by an inspection of higher-order derivatives to be due to its largely inadequate (ad hoc imputed) analytical form.