An optimum processor theory for the central formation of the pitch of complex tones

A comprehensive theory is formulated for the central formation of the pitch of complex tones, i.e., periodicity pitch [Schouten, Ritsma, and Cardozo, J. Acoust. Soc. Amer. 34, 1418–1424 (1962)]. This theory is a logical deduction from statistical estimation theory of the optimal estimate for fundamental frequency, when this estimate is constrained in ways inferred from empirical phenomena. The basic constraints are (i) the estimator receives noisy information on the frequencies, but not amplitudes and phases, of aurally resolvable simple tones from the stimulus and its aural combination tones, and (ii) the estimator presumes all stimuli are periodic with spectra comprising successive harmonics. The stochastic signals representing the frequencies of resolved tones are characterized by independent Gaussian distributions with mean equal to the frequency represented and a variance that serves as free parameter. The theory is applicable whether frequency is coded by place or time. Optimum estimates of fundamental frequency and harmonic numbers are calculated upon each stimulus presentation. Multimodal probability distributions for the estimated fundamental are predicted in consequence of variability in the estimated harmonic numbers. Quantification of the variance parameter from musical intelligibility data in Houtsma and Goldstein [J. Acoust. Soc. Amer. 51, 520–529 (1972)] shows it to be dependent upon the frequency represented and not upon other stimulus frequencies. The quantified optimum processor theory consolidates known data on pitch of complex tones.