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Exponential band edges have been observed in a variety of materials, both crystalline and amorphous. In this Letter, we infer the structural origins of these tails in amorphous and defective crystalline Si by direct calculation with current ab initio methods. We find that exponential tails appear in relaxed models of diamond silicon with suitable extended defects that emerge from relaxing point defects. In amorphous silicon (-Si), we find that structural filaments of short bonds and long bonds exist in the network, and that the tail states near the extreme edges of both band tails are also filamentary, with much localization on the structural filaments. We connect the existence of both filament systems to structural relaxation in the presence of defects and of topological disorder.