Human speech production obeys the same acoustic principles as vocal production in other animals but has distinctive features: A stable vocal source is filtered by rapidly changing formant frequencies. To understand speech evolution, we examined a wide range of primates, combining observations of phonation with mathematical modeling. We found that source stability relies upon simplifications in laryngeal anatomy, specifically the loss of air sacs and vocal membranes. We conclude that the evolutionary loss of vocal membranes allows human speech to mostly avoid the spontaneous nonlinear phenomena and acoustic chaos common in other primate vocalizations. This loss allows our larynx to produce stable, harmonic-rich phonation, ideally highlighting formant changes that convey most phonetic information. Paradoxically, the increased complexity of human spoken language thus followed simplification of our laryngeal anatomy.