A method for tracking fundamental frequency variations in speech is proposed, based on multitaper analysis. Using the multitaper technique, a statistical test is developed for detecting the presence of harmonic components at multiples of a fundamental frequency, embedded in coloured noise. It is shown that this can be applied to speech to estimate the fundamental frequency, when present, as well as the amplitude and phase of each harmonic. The method is validated on synthetic data, to determine accuracy and robustness, and evaluated on a small corpus of real speech data, comparing simultaneous acoustic and electroglottographic measurements to assess performance. Acoustic measurements are marginally less accurate than electroglottographic measurements, but often continue to provide useful fundamental frequency estimates in situations where electroglottography fails