Optimum Processor Theory Research Articles

Pitch identification of simultaneous dichotic two-tone complexes.

The optimum processor theory of Goldstein can, in principle, account for pitch perception phenomena involving simultaneous dichotic complex tones. The frequency-coding noise function, which is the only free parameter of the model, was estimated with pitch identification data of two simultaneous two-tone complexes presented to different ears. This "sigma" function was found to have a shape similar to that of the function derived from data on identification performance for single pitches. The sigmas in the simultaneous pitch identification experiment are larger by an amount that differs from subject to subject. By using different methods of data analysis it was found that the pitch estimation processes for the two tones are independent for most subjects. This allows a simple extension of Goldstein's optimum processor theory.

Theoretical and experimental analysis of a central optimal processor for pitch of multicomponent inharmonic tones.

This paper describes some experiments on the pitch produced by multicomponent tones. Stimuli used were the sum of a pure tone at frequency f = 1 kHz and of the first six successive subharmonics at frequencies psi i = f/i (i = 2, 3, 4, 5, 6, 7) or, in separate experiments, the sum of six (i = 2, 3, 4, 5, 6, 7), five (i = 3, 4, 5, 6, 7) or four (i = 4, 5, 6, 7) subharmonics, respectively. Pitch data produced by these stimuli have been compared with the predictions of the optimum processor theory developed by Goldstein and coworkers (Goldstein, J.L. (1973): J. Acoust. Soc. Am. 54, 1496-1516; Goldstein, J.L. et al. (1978): J. Acoust. Soc. Am. 63, 486-497; Gerson, A. and Goldstein, J.L. (1978): J. Acoust. Soc. Am. 63, 498-510). Model predictions are shown to deviate significantly from the experimental results. By assuming the existence of an additional transformation to be included into the original mathematical model, without any other modification, predictions are in complete agreement with pitch data.

Frequency and intensity difference limens for harmonics within complex tones.

A two-interval, two-alternative forced choice task was used to estimate frequency difference limens (DLs) for individual harmonics within complex tones, and DLs for the periodicity (i.e., number of periods per s) of the whole complexes. For complex tones with equal-amplitude harmonics, the DLs for the lowest harmonics were small (less than one percent). The DLs increased rather abruptly around the fifth to seventh harmonic. The highest harmonic in each complex was also well discriminated, and the discriminability of a single high harmonic was markedly improved by increasing its level relative to the other components. The DL for a complex tone was generally smaller than the frequency DL of its most discriminable component. The DL for a complex was found to be predictable from the DLs of the harmonics comprising the complex, using a formula derived by Goldstein [J. Acoust. Soc. Am. 54, 1496-1516 (1973)] from his optimum processor theory for the formation of the pitch of complex tones. The DL for a complex is sometimes primarily determined by high harmonics, such as the highest harmonic, or a harmonic whose level exceeds that of adjacent harmonics. We also measured intensity DLs for individual harmonics within complex tones. The intensity DLs were smallest for low harmonic numbers, and for the highest harmonic in a complex. An excitation-pattern model was used to determine whether the frequency DLs of harmonics within complex tones could be explained in terms of place mechanisms, i.e., in terms of changes in the amount of excitation at appropriate frequency places. We conclude that place mechanisms are not adequate, and that information about the frequencies of individual harmonics is probably carried in the time patterning of neural impulses.

Simulation of auditory analysis of pitch: an elaboration on the DWS pitch meter.

A model was developed for estimating the pitch of complex sounds that are partially masked by background sound. Our ultimate aim is to obtain a model that can separate two simultaneous sounds on the basis of the harmonic structure of at least one of the sounds. The MDWS model is an extension of the Duifhuis, Willems, and Sluyter pitch meter (DWS) [J. Acoust. Soc. Am. 71, 1568-1580 (1982)] which is a practical implementation of Goldstein's optimum processor theory of pitch perception [J. Acoust. Soc. Am. 54, 1496-1516 (1973)]. The main modifications incorporated in MDWS consist of a more faithful modeling of auditory frequency analysis and of an alteration to the criterion used to decide which fundamental best fits a set of resolved components. Effects of the latter modification were investigated in a comparison between model estimates of the pitch of inharmonic complex signals and results obtained for humans. Furthermore, the accuracy of model estimates of the pitch of periodic signals (among which were synthesized vowel sounds), partially masked by noise, was compared with the just noticeable difference of fundamental frequency of these sounds for human observers. The results of these two tests show that the model estimates come close to human perception.

Pitch of harmonic two-tone complexes of unequal amplitudes

Melodic interval identification experiments were performed using dichotic two-tone complexes of successive random harmonics with intensity ratios of lower to higher tone of 20, 10, 0, −10, and −20 dB. Experimental confusion matrices were correlated with theoretical matrices computed from Goldstein's optimal processor theory, Wightman's pattern transformation theory, and from an analytic pitch theory which assumes that a two-tone complex has one of two pitches, corresponding to the pure tone frequencies, with the likelihood of each pitch dependent on the relative loudness of the tone components. Results indicate that the best predictions are achieved by a combination of the optimum processor mode and the analytic mode. At low harmonic numbers, the optimum processor theory alone accounts very well for the data, while at higher harmonic numbers a combination of optimum processor and analytic theories yields the best correlation with observed data. At harmonic numbers higher than about six, correlation between data and any theory or combination of theories is clearly less than perfect, suggesting that pitch behavior at higher harmonics has some randomness which is not accounted for by any of the current pitch theories. [Work supported by NIH, Grant NS11680-03.]

Musical pitch of two-tone complexes and predictions by modern pitch theories.

Most studies of the musical pitch of harmonic tone complexes have utilized signals comparing two or more successive harmonics. The present study provides systematic data on melodic interval recognition by three musically experienced subjects with sounds whose missing fundamentals were represented by two nonsuccessive harmonics nf0,(n + m)f0, delivered to separate ears. Data were obtained in the ranges 1 less than or equal to n less than or equal to 9, 2 less than or equal to m less than or equal to 4, and 200 Hz less than or equal to f0 less than or equal to 1000 Hz. The data are interpreted in the light of three theories, the "optimum processor theory," the "virtual pitch theory," and the "pattern transformation theory." For each theory, a constraint on preformance is proposed based on interference between the "analytic" and "synthetic" pitch perception modes. The former is obtained with large spacings between harmonics, where listeners are more likely to perceive harmonics as individual tones, each having their own pitch. This degrades the listener's ability to hear the fundamental pitch.

Pitch of two‐tone complexes comprising nonsuccessive harmonics

Most studies of the musical pitch of harmonic tone complexes have utilized signals comprising two or more successive harmonics. It is also easily observed that tones from a clarinet, whose even harmonics are heavily suppressed, evoke a strong and unambiguous pitch. The present study provides systematic data on melodic interval recognition by three musically experienced subjects, with sounds made up of two nonsuccessive harmonics. If we describe such a two‐tone complex as nf0, (n + m)f0, data were obtained for the ranges 1 ⩽ n ⩽ 10, 2 ⩽ m ⩽ 4, 200 ⩽ f0 ⩽ 1000 Hz. Trends in the data are interpreted in the light of three popular models, the “optimum processor theory” (Goldstein), the “pattern transformation theory” (Wightman), and the “learning matrix theory” (Terhardt). A constraint on performance is proposed which is based on the “analytic” and “synthetic” perception modes of complex tones (Helmholtz). Apparently, when the harmonic spacing increases, the listener is more likely to operate in the analytic mode, perceiving harmonics as individual tones each having its own pitch. This degrades the tracking ability for the missing fundamental. [Work partially supported by NIH.]

Evidence for a general template in central optimal processing for pitch of complex tones.

Optimum processor theory successfully accounts for earlier pitch data by including the constraint that component tones in a complex stimulus are estimated as successive harmonics. This constraint gives the paradoxical prediction that a periodic complex tone comprising nonsuccessive harmonics cannot evoke periodicity pitch corresponding to its period. Most published data from pitch-shift experiments imply the necessity for this constraint. New periodicity pitch experiments on pitch shift and musical interval recognition were performed which prove that the theoretical constraint is not generally true. New and old data are reconciled by replacing the maximum likelihood estimation of the theory with maximum posterior probability estimation and removing the successive harmonic constraint. Periodicity pitch is estimated by optimizing the match between the aurally measured frequencies of stimulus components and a general harmonic template over some a priori expected pitch range. The new, more general, formulation reduces in many experimental situations to the successive harmonic constraint as a special case.

Verification of the optimal probabilistic basis of aural processing in pitch of complex tones.

Periodicity pitch for complex tones has been quantitatively accounted for by a two-stage process of Fourier-frequency analysis subject to random errors and significant nonlinearities, followed by an harmonic pattern recognizer that makes an optimum probabilistic estimate of the fundamental period of musical and speech sounds. The theory predicts that periodicity pitch is a multimodal probabilistic function of a given stimulus. A clear and empirically supported distinction is made between limitations on the pitch mechanism caused by the stochastic nature of aural frequency representation and by the deterministic resolution bandwidths of aural frequency analysis. This model was developed earlier [J. L. Goldstein, J. Acoust. Soc. Am 54, 1496-1516 (1973)] to account for probabilistic data on pitch errors [A. J. M. Houtsma and J. L. Goldstein, J. Acoust. Soc. Am. 51, 520 (1972)] measured with periodic stimuli comprising two successive harmonics. This paper presents new predictions by the theory that were calculated, with computer simulation where needed, for known probabilistic pitch data from stimuli comprising three to six successive harmonics. Predicted pitch errors increase with increasing errors in estimating the frequencies of stimulus harmonics and decrease as more harmonics are added to the stimulus. Optimum processor theory fully accounts for the multicomponent pitch data on the basis of similar errors in estimating component stimulus frequencies as reported earlier, thus providing further evidence for the optimum probabilistic basis of aural signal processing in pitch of complex tones.

Pitch of dichotically presented amplitude modulated sinusoids

Periodicity pitch resulting from dichotic presentation of harmonically related tonal complexes generated by amplitude modulation of pure tones was investigated. In the first experiment, AM signals were presented dichotically in such a way that if the individual components from each ear were combined and rank ordered, a single pitch was heard. Otherwise, separate pitches were heard at each ear. Pitch fusion was investigated as a function of frequency spacing between components and frequency region. In the second experiment, loudness balances were attempted at each ear for the periodicity pitch percept as a function of the intensity of tonal complexes. (identical to those used in the first experiment) at the other ear. The purpose of this experiment was to determine if pitch strength of tonal complexes could be increased at one ear by the simultaneous presentation os a complementary complex at the other ear. The results of the experiments are discussed in terms of the time fine-structure theories, the pattern-transformation theory, and the optimum processor theory.

Pitch confusions and the optimum processor theory

Goldstein's optimum processor model for complex tones [J.L. Goldstein, J. Acoust. Soc. Am. 54, 1496–1516 (1973)] allows one to compute the multimodal probability density function of the estimated fundamental frequency for any two-tone complex with only one variance parameter which depends on stimulus frequency and subject. From a given set of complex tone stimuli a set of receiver-operating characteristics can be computed which have a staircase shape because of the discrete nature of the underlying density functions. Data from musical interval identification experiments using dichotically presented harmonic two-tone complexes are processed and compared with computed ROCs. Bias effects in the criteria and other bias caused by the particular musical context of the stimulus set are discussed. [Work supported by the National Institutes of Health, grant 1RO1 NS 11680-01.]

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.

An Optimum Processor Theory for the Central Formation of the Pitch of Complex Tones

A theory was formulated for the central formation of the pitch of complex tones, i.e., periodicity pitch. 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. These constraints are (1) 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 (2) 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 indifferent to 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 existing data on pitch of complex tones. [Work supported by grants from the National Institutes of Health.]

Pitch Ambiguities of Harmonic Two-Tone Complexes

A tone complex comprising two arbitrary successive harmonics of low order, presented monotically or dichotically, evokes a pitch sensation which corresponds to the missing fundamental frequency. When the harmonic numbers are higher, the pitch becomes increasingly more ambiguous. These ambiguities are the major cause of incorrect responses when subjects are asked to identify musical intervals played with harmonic tone complexes of relatively high harmonic number. A detailed investigation was made of the particular confusions a subject makes when he responds incorrectly in a musical interval identification test by obtaining confusion matrices for each harmonic number. The results imply a multimodal distribution for pitch with the locations of the modes approximated by the so called “first effect” [J. F. Schouten, R. J. Ritsma, and B. L. Cardozo, J. Acoust. Soc. Amer. 34, 1418 ( 1962)]. Results are compared with theoretical performance predicted by Goldstein's Optimum Processor Theory. [Work supported by grants from the National Institutes of Health.]