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Put another way, is pitch purely a function of how the vocal tract is constricted, or do the vocal folds modulate the frequency of how they vibrate as well?

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Pitch is the perceptual correlate of fundamental frequency. The fundamental frequency of a semi-periodic waveform is the frequency with which the vocal folds vibrate, e.g. 150 times per second. The rate of vibration of the vocal folds is not determined by the degree of constriction in the vocal tract, until the constriction becomes so significant that air-flow is blocked and the vocal folds cease vibrating entirely.

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