The perception of frequency peaks and troughs in wide frequency modulations. IV. Effects of modulation waveform Laurent Demany and Sylvain Cle´ment Laboratoire d’Audiologie Expe´rimentale et Clinique, Universite´ Bordeaux 2, BP 63, 146 rue Leo-Saignat, F-33076 Bordeaux Cedex, France

~Received 11 February 1997; revised 21 July 1997; accepted 21 July 1997! This work extends previous studies on the perceptual asymmetry between the local maxima and minima of wide frequency modulations ~FMs! @L. Demany and K. I. McAnally, J. Acoust. Soc. Am. 96, 706–715 ~1994!; L. Demany and S. Cle´ment, J. Acoust. Soc. Am. 97, 2454–2459 ~1995!; L. Demany and S. Cle´ment, J. Acoust. Soc. Am. 98, 2515–2523 ~1995!#. In experiment 1, subjects had to discriminate frequency shifts in the temporally central vertex of V- and L-shaped FMs imposed on 200-ms sinusoidal tone bursts. The precise shapes of these FMs varied in eight steps from quasi-triangles ~with a durationless central vertex! to quasi-squares ~with a long-duration central vertex!. The central vertex was either a minimum or a maximum, but in each case the corresponding frequency was near 1000 Hz and the FM span was about 0.5 oct. For each FM shape, the discrimination threshold was lower when the vertex was a maximum than when it was a minimum, but ~in four subjects out of five! this difference decreased monotonically as the FM became less and less triangular. FM shape had a remarkably small effect on the discrimination of the maxima, and the thresholds measured for the sharpest maxima were unexpectedly low. In subsequent experiments, subjects had to discriminate frequency shifts in the starting point or the final point of unidirectional FMs ~tone glides! that spanned about 0.5 oct in 100 ms. The relevant frequency extremum was near 1000 Hz in each condition. At the final point of the glides, discrimination was better for rising glides than for falling glides. At the starting point of the glides, discrimination was better for falling glides than for rising glides. Thus discrimination was always better when the relevant frequency extremum was a maximum than when it was a minimum, and this effect was produced both ‘‘forward’’ and ‘‘backward.’’ The latter fact suggests that the perceptual asymmetry of FM originates at least partly from central factors. © 1997 Acoustical Society of America. @S0001-4966~97!02211-X# PACS numbers: 43.66.Fe, 43.66.Hg, 43.66.Mk @WJ#

INTRODUCTION

This is the fourth in a series of papers devoted to the perception of ‘‘instantaneous pitch’’ in widely frequency modulated sounds ~Demany and McAnally, 1994; Demany and Cle´ment, 1995a, 1995b; henceforth, these three previous articles will be referred to as papers 1, 2, and 3!. The whole series is focused on an intriguing perceptual asymmetry that may be termed ‘‘the peak/trough effect.’’ Our first experiment displaying this effect used periodic modulators consisting of the sum of a few sinusoids ~with a maximum frequency of 10.5 Hz!. These complex FMs were imposed on a sinusoidal carrier of about 1 kHz. Within such stimuli, most listeners appeared to hear a repetitive sequence of tones with precise pitches. As could be expected, the heard tones corresponded to local frequency extrema of the FM waveforms. However, although the FM waveforms were symmetric on the dimensions of time and log frequency, we found that tones were much more often heard at local maxima ~FM peaks! than at local minima ~FM troughs!. This occurred even for maxima and minima at the same frequency. In subsequent experiments, we simplified the FM waveforms. They were reduced to a single cycle of a cosine function on a logarithmic frequency scale. The subjects’ task was then to discriminate stimuli that differed only by the frequency vertex occurring in their temporal centers ~cf. Fig. 3 2935

of paper 1!. The cosine cycle of the FM started and ended either at 0° ~in the ‘‘trough’’ condition! or at 180° ~in the ‘‘peak’’ condition!. We found that frequency shifts of a given vertex were typically much better detected in the ‘‘peak’’ condition than in the ‘‘trough’’ condition. This was the case for FM carriers consisting of sinusoids at low or medium frequencies, various kinds of harmonic complexes, and even amplitude-modulated noise. As assessed in this way, the peak/trough effect appeared to be a robust phenomenon: It did not diminish with practice in the discrimination task. The goal of the present experiments was to investigate more systematically the effect of modulation waveform on the perception of FM peaks and troughs. We thus measured again, for new FM waveforms, just-noticeable shifts in frequency extrema. The FMs used in experiment 1, like the cosine modulators that we had previously employed, produced a single ‘‘there and back’’ pitch motion. However, instead of being cosinusoidal, their waveforms ranged from quasi-triangles ~without steady vertex! to quasi-rectangles ~with a long-duration steady vertex!. In experiments 2 and 3, the FMs of primary interest were unidirectional rather than bidirectional. The frequency extrema under study were then located at the starting point or the end point of stimuli con-

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FIG. 1. The family of FM waveforms used in the ‘‘trough’’ condition of experiment 1. The vertical separations of the waveforms are arbitrary. In the standard stimuli, the initial and final instantaneous frequency was always 0.5 oct away from the temporally central frequency vertex.

sisting of a constantly rising or constantly falling frequency glide. I. EXPERIMENT 1 A. Stimuli and rationale

In each trial of this experiment, the subject had to discriminate between two slightly different FMs imposed on sinusoidal tone bursts lasting 200 ms ~complete envelope! and gated on and off with 10-ms cosinusoidal amplitude ramps. Each of the two FMs produced a temporally symmetric ‘‘there and back’’ pitch motion. For the ‘‘standard’’ stimulus, the frequency vertex reached after 100 ms (F vertex) was at 1000 Hz and the initial and final value of instantaneous frequency ( f inst) was 0.5 oct away from F vertex . The ‘‘target’’ stimulus, to be discriminated from the standard, differed from it by a slight shift of F vertex , the initial and final value of f inst being unchanged; this shift of F vertex always increased the overall span of f inst . Figure 1 displays the family of standard FM waveforms used in the ‘‘trough’’ condition. ~The vertical separations of the curves are merely intended to facilitate visual comparisons.! Upside down, this figure shows instead the standard FM waveforms of the ‘‘peak’’ condition. Within each stimulus, f inst varied according to the formula: h

f inst~ t ! 5F vertex exp~ a u t20.1u ! ,

~1!

where t represents time in seconds (0
J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997

the frequency vertex occurring at t50.1. The vertex was extremely sharp for h 51.25 ~highest curve in Fig. 1!. On the contrary, for h 516 ~lowest curve!, f inst was very close to F vertex during more than half of the stimulus. Variable a ~positive in the ‘‘trough’’ condition and negative in the ‘‘peak’’ condition! was adjusted as a function of h and F vertex in order to obtain the required value of f inst for t50 and t50.2, that is 707.1 Hz in the ‘‘peak’’ condition and 1414.2 Hz in the ‘‘trough’’ condition. For each value of h, therefore, discrimination thresholds for F vertex were measured in the ‘‘peak’’ and ‘‘trough’’ conditions. Precise predictions about the effect of h on thresholds could be drawn from a model presented by Horst ~1989!. We wished to compare our data to these predictions. For the largest value of h, given that f inst was very close to F vertex for a long time, it could be expected that thresholds would be lowest and that no clear peak/trough effect would be apparent: Obviously, it should become impossible to observe a peak/trough effect when h is so large that the standard stimuli of the ‘‘peak’’ and ‘‘trough’’ conditions differ from each other only during a few ms, at their very beginning and end. An interesting question was: As h decreases and gets closer and closer to 1, does the peak/trough effect always increase, or is there some intermediate value of h for which this effect is maximum? The second alternative, a nonmonotonic effect of h on the magnitude of the peak/ trough effect, could seem more likely than the first one. To see why, note first that shifting F vertex produced a concomitant shift in the FM slopes before and after t50.1. Shifts in an FM slope are much less detectable than frequency shifts in a steady tone ~cf., e.g., Dooley and Moore, 1988!. Thus, when h was large or very large, one could reasonably expect that subjects would perform the discrimination task by detecting shifts in the frequency plateau centered on t50.1; it was unreasonable to hypothesize instead that subjects would use an FM slope cue. When h was close to 1, however, the latter hypothesis became quite plausible because: ~1! the FM waveforms did not contain any plateau centered on t50.1; ~2! within each stimulus, the FM slopes were approximately constant for a relatively long time. Assume that FM slope becomes the only discrimination cue when h is close to 1. In this case, similar results are expected in the ‘‘peak’’ and ‘‘trough’’ conditions since the corresponding standard stimuli do not differ from each other with respect to FM slope ~in oct/s!. Even if one-half of the stimuli perceptually dominates the other half, the results should still be similar in the two conditions, because just-noticeable shifts in the slope of frequency glides appear to be quite similar for rising versus falling glides ~Dooley and Moore, 1988!.

B. Procedure

The stimuli were digitally generated in real time, at a sampling rate of 20 kHz, with the equipment already described in paper 1. Each FM waveform was defined in a table containing 1024 frequency samples ~i.e., 5.12 frequency samples per ms!. The output of the DAC was low-pass filtered at 8 kHz and presented binaurally, at 55 dB SPL, via L. Demany and S. Clement: Frequency peaks and troughs

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TDH 39 earphones. Five subjects with normal hearing, including the two authors, were tested individually in a doublewalled soundproof booth. On each trial, the standard and the target ~presented in this order or the opposite order, at random! were separated by a 500-ms silent interval. The subject’s identification of the target’s position was given by pressing one of two keys, and was immediately followed by visual feedback. During each trial block, a threshold was measured for a fixed value of h. The adaptive procedure used to measure thresholds ~for details, see paper 1, Sec. II A1! estimated the 75% correct point of the psychometric function. Within the experimental sessions, as well as preliminary training sessions, the ‘‘peak’’ and ‘‘trough’’ conditions were always presented in alternation. The training period was short ~1–4 h, depending on the subject!, but each subject had previously taken part in at least one closely related experiment. Two ‘‘sub-experiments’’ were run. In the first one, h did not exceed 4; we did not use larger h values because we guessed that the peak/trough effect would already be quasiinexistent for h 54. As the results showed that this was not the case, we performed ~on only four of the five subjects! a second sub-experiment in which h was equal to 4, 8, or 16. C. Results

The individual results of the five subjects are displayed in five panels of Fig. 2. In these panels, each symbol represents the mean of ten threshold measurements. The results obtained for h 54 in the two sub-experiments are shown separately. Note that the two axes of the figure are logarithmic. Four of the five subjects ~see the four upper panels of Fig. 2! behaved similarly. For these subjects, thresholds were essentially a monotonic function of h, in both the ‘‘peak’’ and ‘‘trough’’ conditions. However, the effect of h on thresholds was larger in the ‘‘trough’’ condition than in the ‘‘peak’’ condition. For the largest value of h, thresholds were lowest and there was only a weak peak/trough effect. The peak/trough effect increased monotonically as h decreased, and became quite pronounced for h 51.25. For subject JJ, by contrast, the thresholds measured in the first subexperiment did not strongly depend on h and the peak/trough effect was always weak. This subject was not tested in the second sub-experiment. D. Discussion

Overall, the results did not confirm our expectation that the peak/trough effect would be a nonmonotonic function of h. Since this expectation was based on the assumption that, in both the ‘‘peak’’ and ‘‘trough’’ conditions, the subjects’ discrimination cue for the lowest value~s! of h would be FM slope rather than F vertex , one is led to think that this assumption was wrong. Indeed, as subjects of the experiment, the authors felt that the discrimination cue that they used in the ‘‘peak’’ condition was the pitch corresponding to F vertex for each value of h, even the lowest one. ~It was more difficult to specify from introspection the discrimination cue, or cues, used in the ‘‘trough’’ condition.! Moreover, all the thresh2937

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FIG. 2. Results of experiment 1. The discrimination thresholds of the five subjects ~LD, SC, JL, MP, JJ! are displayed in separate panels, as a function of the nature of the frequency vertex ~peak or trough! and the value of exponent h in Eq. ~1!. The axes are logarithmic. Thresholds are expressed in ‘‘cents’’ rather than in Hz for the sake of consistency with papers 1–3. One cent51/100 semitone51/1200 oct. For a reference frequency of 1000 Hz ~the standard value of F vertex!, 10 cents correspond to a frequency difference of about 5.8 Hz. In the bottom-right panel ~mean results for subjects LD, SC, JL, and MP!, the two dashed curves represent predictions of a model described by Horst ~1989; see the text for details!.

olds measured in the ‘‘peak’’ condition were too low to be consistent with the FM slope hypothesis: According to Dooley and Moore ~1988!, the Weber fraction for the detection of shifts in FM slope is about 0.06; this would yield a threshold of 36 cents in the present experiment. Note that although all the thresholds measured in the ‘‘peak’’ condition were much lower, this was not the case in the ‘‘trough’’ condition: In the latter condition, for h 51.25, the mean threshold of the five subjects was 33 cents. Thus FM slope may have been the dominant discrimination cue in the ‘‘trough’’ condition when h was very small. The bottom-right panel of Fig. 2 displays the average results of the four subjects who behaved similarly. The two dashed curves are predictions from the model proposed by Horst ~1989! to account for the results of his own study on the discrimination of FM waveforms. In Horst’s study, the standard and target stimuli also consisted of a ‘‘there and back’’ frequency movement, but the central frequency vertex was always a peak. Horst assumed that subjects detect shifts in F vertex by making measurements of its value within a rectL. Demany and S. Clement: Frequency peaks and troughs

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TABLE I. Width ~2t! of the temporal window minimizing S in Eq. ~2!, as a function of h ; a516.

h

1.25

1.5

2t ~ms!

8.0

10.2

1.75 12.8

2.0

2.5

3.0

4.0

8.0

16.0

15.4

20.8

26.4

37.2

72.8

112.4

angular temporal window centered on the vertex. The accuracy of these frequency measurements is limited by two factors: ~1! the variation of f inst within the window; ~2! a reciprocal relation between frequency accuracy and time, reflecting the well-known ‘‘uncertainty principle’’ ~see Hartmann, 1997, Chap. 13!. The first factor favors the choice of a narrow window, but the second factor favors instead wide windows. Subjects are supposed to adopt the best compromise, that is to select the window width ~2t! that minimizes the sum of the imprecisions due to the two factors. This sum (S) can then be considered as an approximation of the detection threshold for shifts in F vertex . For our stimuli, following Horst’s model, S was given ~in Hz! by the following equation: S5F vertex@ exp~ a • t h ! 21 # 1 ~ 2 t a ! 21 ,

~2!

where F vertex51000 and a is a constant which can be assessed from psychophysical data concerning the effect of duration on the frequency discrimination of tone bursts near 1000 Hz. In his own equation for S, Horst set a at 2.5. Taking the same value of a in Eq. ~2! leads to the predictions corresponding to the upper dashed curve in the bottom-right panel of Fig. 2. However, the psychophysical data reported by Moore ~1973! suggest that it is much more realistic to set a at 16: For a 1000-Hz tone burst of 25 ms, Moore measured a frequency discrimination threshold of about 2.5 Hz, and this threshold was multiplied by about 2 and 4, respectively, when stimulus duration was divided by 2 and 4. Setting a at 16 in Eq. ~2! gives predictions corresponding to the lower dashed curve in Fig. 2. It was a priori obvious that Horst’s model could not account for all of our data since this model predicted identical thresholds in the ‘‘peak’’ and ‘‘trough’’ conditions. Yet, it could be expected that the model would yield correct predictions in the ‘‘peak’’ condition, because its basic assumption was that subjects perform the discrimination task by making measurements of F vertex : This assumption seemed to be valid for ‘‘peak’’ stimuli, whatever h. In fact, the model largely failed to predict the effect of h on thresholds for ‘‘peak’’ stimuli; paradoxically, the model was rather more successful for ‘‘trough’’ stimuli. A comparison of the results obtained in the ‘‘peak’’ condition with the lower dashed curve ~giving, in principle, more reasonable predictions than the upper dashed curve! indicates that the measured thresholds were higher than predicted for h .3, but lower than predicted for the smallest values of h. For h .3, it could be expected that the predictions would be too low, and that the error would increase with h, because: ~1! according to the model, the width of the temporal window used by the subjects exceeded 25 ms ~see Table I!; ~2! for durations exceeding about 25 ms, the ‘‘uncertainty principle’’ reflected by the last term of Eq. ~2! is no longer verified psychophysically 2938

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~Chih-an and Chistovich, 1960; Moore, 1973!. When h did not exceed 3, however, the temporal windows minimizing S were always short enough to justify the last term of Eq. ~2!. Thus, for h <3, the discrepancy between the predicted and measured effects of h is remarkable. Actually, a related discrepancy is apparent in Horst’s paper. In his experiment, he varied the depth of the standard FM while keeping constant the standard value of F vertex ~always a peak!. His model predicted that thresholds would increase with the depth of the standard FM. However, as long as this parameter was larger than 0, the measured thresholds were approximately constant. This discrepancy, and the one found here, suggest that one cause of the peak/ trough effect is some auditory mechanism permitting an ‘‘abnormally accurate’’ perception of FM peaks. One of the experiments reported in paper 3 suggested instead that the perception of FM troughs is impaired by a deleterious factor. It is of course possible that one and the same mechanism, unidentified as yet, has a beneficial effect on the perception of peaks and a deleterious effect on the perception of troughs. II. EXPERIMENT 2 A. Rationale and method

In experiment 1, as well as those reported in papers 1–3, we were concerned with the perception of frequency extrema that are both preceded and followed by a frequency glide. An obvious question to ask about the perceptual asymmetry found in these experiments is then: Does it originate mainly from a ‘‘forward’’ effect of the preceding glide’s direction, or on the contrary from a ‘‘backward’’ effect of the following glide’s direction? It is rather easy to imagine how a perceptual advantage of FM peaks over FM troughs could be produced by a forward effect involving neural inhibition or neural facilitation. The necessary physiological ingredients can be found, for instance, in the cochlear nucleus and the inferior colliculus of the cat ~see, e.g., Ehret, 1992!. So, a mechanism acting forward might exist below the auditory cortex. On the other hand, a mechanism acting backward would probably be located in the auditory cortex itself, as the effect should be understood as a kind of mnemonic interference or ‘‘recognition masking’’ ~Massaro and Idson, 1977!. In studies using rapid sequences of tone bursts, Watson et al. ~1975; see also Divenyi and Hirsh, 1975! found that the perceptual encoding of the frequency of one tone can be markedly impaired by the subsequent presentation of higherfrequency tones. It is tempting to think that a similar frequency-asymmetric backward interference effect plays a role in FM perception ~even though stimulus uncertainty appears to have opposite effects on the perceptual asymmetry of FM and the phenomenon observed by Watson et al., as shown in paper 2!. A simple way to determine the relative weights of forward and backward effects in the perceptual asymmetry of FM is to use unidirectional FMs, i.e., constantly rising or constantly falling frequency glides, and to measure discrimination thresholds for the frequency extremum reached at each temporal extremity. A source of perceptual asymmetry L. Demany and S. Clement: Frequency peaks and troughs

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acting forward is demonstrated if thresholds are lower at the end of a rising glide than at the end of a falling glide. A source acting backward is demonstrated if thresholds are lower at the start of a falling glide than at the start of a rising glide. For fair comparisons, the standard frequency extremum should be the same in the four experimental conditions ~2 directions32 temporal extremities!. Some recent studies on the discrimination of stimuli imitating speech formant transitions ~Porter et al., 1991; Collins et al., 1990; van Wieringen and Pols, 1995! suggested that a perceptual advantage of FM peaks over FM troughs can be produced both forward and backward. The study by Porter et al. was already described in paper 1. For the starting point of formant transitions lasting 60 or 120 ms, these authors measured lower thresholds with falling transitions than with rising transitions. However, their standard rising transitions started at 1500 Hz, whereas the standard falling transitions started at 2100 Hz, and they ascribed the perceptual advantage of falling transitions to this difference in frequency register rather than to an intrinsic advantage of frequency peaks over frequency troughs. For the discrimination between the ends of formant transitions, Collins et al. ~1990! found, on the opposite, a perceptual advantage of rising transitions over falling transitions; but the factor of transition direction was again confounded with a frequency register factor. In experiments similar to those of Porter et al. and Collins et al., van Wieringen and Pols ~1995! obtained results from which they concluded that there was no significant effect of transition direction. However, this negative conclusion appears to be based on the fact that their discrimination thresholds were measured in Hz, that is in terms of absolute frequency differences: If the data displayed in their Table III are converted into relative frequency differences, the logical conclusion becomes that frequency peaks have a marked perceptual advantage over frequency troughs, both at the beginning and at the end of formant transitions. The unidirectional FMs used in experiment 2 had the shape of one-half of a cosine cycle on a log frequency scale. They took place in the central 100 ms of stimuli which had a total duration of 110 ms and were gated on and off with 10-ms amplitude ramps. The amplitude ramps were cosinusoidal ~same shape as the FMs!. During the first 5 and last 5 milliseconds of the stimuli, f inst was kept on a plateau corresponding to the extremum reached by the FM.1 In the standard stimuli, the critical frequency extremum was always at 1000 Hz. The standard span of f inst was equal to 0, 600, or 1800 cents. When this standard span was 600 cents, the stimuli could be described as the first-half or second-half of stimuli that we previously employed ~see especially paper 2!. For each type of critical frequency extremum ~end of rise, end of fall, start of rise, or start of fall!, Fig. 3 depicts the standard stimuli ~thick lines! and target stimuli ~thin lines! corresponding to a standard span of 0 cent ~left part of the panels! and a larger standard span ~right part of the panels!. On each trial, as before, the span of f inst was larger in the target stimulus than in the standard stimulus. Six listeners with normal hearing took part in the experiment. Two of them ~AS and YB! were psychoacoustically naive at the outset. For these subjects, in addition to sounds 2939

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FIG. 3. Schematic spectrograms of the stimuli used in the four main conditions of experiment 2. Standard and target stimuli are, respectively, represented by thick and thin lines. The standard stimuli were either steady tones or frequency modulated tones, as shown, respectively, in the left and right parts of each panel.

containing a unidirectional FM, the stimulus set included 200-ms ‘‘control’’ stimuli in which the FM consisted of one full cycle of a 5-Hz cosine function and the standard span of f inst was 600 cents. With these stimuli, as in experiment 1, we measured discrimination thresholds for the temporally central frequency vertex ~standard value: 1000 Hz! in a ‘‘control peak’’ condition and a ‘‘control trough’’ condition. For the other four subjects, including the two authors, such data had already been collected in other studies. All the stimuli were binaurally presented at 70 dB SPL.2 They were generated as in experiment 1 ~except that f inst was now sampled at an even higher rate!. The adaptive procedure used to measure thresholds was the same as before. In the conditions involving unidirectional FMs, only three and two subjects were, respectively, tested with standard spans of 0 and 1800 cents; for a given subject, extremum type, and standard span, at least 20 threshold measurements were made. For both unidirectional and bidirectional FMs, the thresholds reported below are ~arithmetic! means of the last ten measurements made in each condition. B. Results and discussion

Table II presents the thresholds obtained from each subject in each condition. The standard errors of the measurements ~in parentheses! amount to about 10% of the means. In order to quantify the peak/trough asymmetries which could be ascribed to ~1! a purely ‘‘forward’’ action, ~2! a purely ‘‘backward’’ action, ~3! the sum of ~1! and ~2!, we computed the ratios of the thresholds obtained in the conditions: ~1! ‘‘end of rise’’ and ‘‘end of fall’’ ~EF/ER!; ~2! ‘‘start of fall’’ and ‘‘start of rise’’ ~SR/SF!; ~3! ‘‘control peak’’ and ‘‘control trough’’ ~CT/CP!. The obtained ratios are displayed in Fig. 4. Let us consider first the ratios obtained when the standard span of f inst was 600 cents ~middle panel of Fig. 4!. The geometric means of the EF/ER and SR/SF ratios were, reL. Demany and S. Clement: Frequency peaks and troughs

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TABLE II. Discrimination thresholds measured in experiments 2 and 3. These thresholds are expressed in cents. The standard errors of the threshold measures ~10 measures for each cell! are given in parentheses. The means across subjects are geometric. Numbers in italics represent data collected in the course of other studies.

Subject

End of rise

End of fall

Start of fall

Start of rise

Control peak

Control trough

Experiment 2 Standard span of f inst : 0 cent SC 7.6 ~1.0! 6.5 ~0.7! LD 4.3 ~0.6! 7.4 ~0.9! JJ 14.2 ~3.2! 17.9 ~1.5! Mean 7.7 9.5

17.1 ~2.8! 11.7 ~1.1! 28.2 ~3.1! 17.8

Standard span of f inst : 600 cents SC 10.5 ~1.1! 23.4 ~3.4! LD 7.1 ~0.6! 24.9 ~1.6! JJ 22.8 ~2.1! 29.2 ~3.5! AS 12.5 ~1.1! 19.6 ~3.2! MP 11.8 ~1.5! 47.2 ~4.4! YB 16.2 ~1.5! 22.2 ~2.2! Mean 12.6 26.5

16.1 24.2 44.7 88.7 46.5 52.3 39.4

Standard span of f inst : 1800 cents SC 11.2 ~1.3! 24.8 ~2.4! LD 11.1 ~1.5! 34.8 ~4.7! Mean 11.1 29.4

18.6 ~1.6! 35.8 ~6.2! 25.8

23.5 ~1.1! 18.5 ~1.8! 26.3 ~2.4! 22.5

~1.2! ~1.8! ~4.8! ~8.2! ~5.0! ~6.7!

93.0 53.0 43.3 55.3 103.7 63.2 65.3

~9.5! ~6.5! ~3.7! ~4.1! ~10.6! ~6.9!

9.4 6.3 11.6 14.6 12.4 23.8 12.0

(0.9) (1.1) (1.2) ~1.3! (0.8) ~2.3!

81.3 28.0 32.9 77.2 64.7 53.8 52.2

(7.0) (2.2) (3.6) ~8.9! (4.7) ~4.3!

5.5 11.6 9.7 20.1 10.6

~0.6! ~1.2! ~1.0! ~2.8!

25.9 32.9 29.2 49.7 33.3

~1.5! ~3.6! ~2.6! ~5.6!

76.8 ~7.0! 48.1 ~5.3! 60.8

Experiment 3 Standard span of f inst : 600 cents LD 17.2 ~1.4! 58.2 ~6.2! JJ 43.7 ~1.5! 88.4 ~6.9! JL 25.0 ~1.7! 65.3 ~5.4! TV 70.7 ~5.3! 203.1 ~15.7! Mean 34.0 90.9

17.8 48.0 24.1 46.6 31.3

spectively, 2.10 and 1.65. These two values are larger than 1 and rather similar, which suggests that the peak/trough effect can be produced both forward and backward, with approximately equal strengths. However, looking at the individual data, one notes that EF/ER was larger than SR/SF in five subjects out of six and that the SR/SF ratios were not always larger than 1. Thus, the evidence for a retroactive creation of the peak/trough effect is not spectacular. Overall, both the SR/SF and EF/ER ratios were rather small. For each subject, the CT/CP ratio was larger. The geometric mean of its individual values was 4.35. When the standard span of f inst was 0 ~upper panel of Fig. 4!, there was no reason to expect that the EF/ER and SR/SF ratios would differ markedly from 1; they were indeed close to 1. On the other hand, when the standard span of f inst was 1800 cents ~lower panel!, ratios larger than 1 were again expected, and obtained. It is interesting to consider, for the two subjects who were tested at each standard span ~SC and LD!, the effect of this variable on the measured thresholds. For each extremum type, as shown in Table II, thresholds increased notably when the standard span varied from 0 to 600 cents ~except for subject SC in the SF condition!. However, a further widening of the standard span, up to 1800 cents, had little effect. This pattern of results is similar to that observed by Horst ~1989! for sounds resembling our CP stimuli ~cf. Sec. I D of the present paper!. Table II also shows that for each standard span, frequency extrema in final position were better discriminated than frequency extrema in initial position. Similar end/start 2940

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~2.3! ~3.2! ~2.8! ~4.0!

113.8 81.8 59.3 154.8 96.2

~8.2! ~6.0! ~5.6! ~22.2!

asymmetries in frequency discrimination were previously reported in a number of papers concerning mostly the perception of formant transitions ~Mattingly et al., 1971; Collins, 1984; Cosgrove et al., 1989; Elliott et al., 1992; van Wieringen and Pols, 1995!. This phenomenon is probably related to some observations on the pitch of stimuli consisting of short frequency glides: Brady et al. ~1961! and Na´beˇlek et al. ~1970! found that the final frequency content of such stimuli is very heavily weighted by the pitch extractor. It was also found by the same authors that this ‘‘pitch bias’’ is stronger for rising glides than for falling glides. The latter finding is consistent with the fact that we obtained lower thresholds in the ER condition than in the EF condition. To some extent, the CP condition could be considered as a mere juxtaposition of the ER and SF conditions ~for a standard span of 600 cents!. Thus a crude model of discrimination in the CP condition predicted that, for each subject, the CP threshold would be equal to either the ER threshold or the SF threshold, whichever was the lowest ~and it was always the ER threshold!. Similarly, it could be predicted that the CT threshold would be equal to the EF threshold, which was always lower than the SR threshold. An examination of Table II reveals that the first of these two predictions worked well; its only serious error was an overestimation of JJ’s CP threshold by a factor of 2. However, the second prediction did not work at all for three of the six subjects ~SC, AS, and YB!, who performed much worse than predicted in the CT condition. The success of the first prediction, as well as the failure of the second one, tally with L. Demany and S. Clement: Frequency peaks and troughs

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FIG. 5. Schematic spectrograms of the stimuli used in the four main conditions of experiment 3. Standard and target stimuli are, respectively, represented by thick and thin lines. The filled rectangles represent noise bursts.

FIG. 4. Threshold ratios computed from the data displayed in the upper part of Table II ~results of experiment 2!. Each subject is represented by a specific symbol. Ratios larger than 1 reflect a perceptual advantage of frequency maxima over frequency minima. This perceptual advantage could be produced by the previous acoustic context ~EF/ER!, the following acoustic context ~SR/SF!, or both contexts ~CT/CP!.

subjects’ conscious impressions about the stimuli: In the CP stimuli, the initial rising glide is perceptually much more salient than the following falling glide; introspectively, therefore, the CP condition was similar to the ER condition; by contrast, the CT stimuli did not sound similar to either purely rising or purely falling glides. III. EXPERIMENT 3 A. Rationale and method

In experiment 2, we obtained lower thresholds when the critical frequency extremum was at the final point of the stimuli than when it was at the starting point. For this reason, comparing the EF/ER and SR/SF ratios was a somewhat problematic way to assess the relative weights of ‘‘forward’’ and ‘‘backward’’ actions in the production of a perceptual asymmetry between FM peaks and troughs. In experiment 3, we wished to assess again the relative weights of forward 2941

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and backward actions, but using stimuli in which the critical frequency extremum would always have a temporally central position, as in most of our previous experiments. The solution was to re-employ the 200-ms CP and CT stimuli of experiment 2 and to mix them with a 100-ms noise burst coinciding with, and masking, either their second-half ~for the assessment of forward actions! or their first-half ~for the assessment of backward actions!. In doing so, as shown in Fig. 5, we produced four experimental conditions which were labeled again as ‘‘end of rise’’ ~ER!, ‘‘end of fall’’ ~EF!, ‘‘start of rise’’ ~SR!, and ‘‘start of fall’’ ~SF!. Essentially, these four conditions differed from the corresponding conditions of experiment 2 only in that the critical frequency extrema were adjacent to a noise ~with a large bandwidth! instead of a silence. It was especially interesting to compare, after this addition of noise, the SR and SF thresholds. In the absence of noise, when the critical frequency extrema of conditions SR and SF coincided with a stimulus onset, the neural encoding of these frequency extrema was probably affected by ‘‘adaptation’’ phenomena ~see, e.g., Palmer, 1995!. By contrast, when we introduced noise just before a critical frequency extremum, the neural response to this frequency extremum could be considered as a ‘‘post-adaptation’’ response. In each of the four experimental conditions, frequency discrimination was of course likely to be impaired by the noise bursts preceding or following the critical extremum. However, this was not, per se, a methodological weakness. The essential point was that, due to their physical properties ~see below!, the noise bursts were not liable to produce a larger forward masking effect in condition SR than in condition SF, or a larger backward masking effect in condition EF than in condition ER. More generally, there was no reason to think that the noise bursts could give an artificial advantage to the perception of frequency maxima. The standard span of f inst was fixed at 600 cents. As in experiment 2, the standard value of each critical frequency extremum was 1000 Hz. However, the ~partially masked! CP and CT stimuli were presented at 55 instead of 70 dB. We did so in order to avoid having to present the concurrent noise bursts at a high SPL. The noise bursts were obtained L. Demany and S. Clement: Frequency peaks and troughs

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by bandpass filtering the output of a pink noise source between 500 and 2000 Hz. They were gated on and off abruptly. The rejection slopes of the bandpass filter ~Kemo VBF 8! were about 75 dB/oct. For each subject, the levels of the noise just necessary to mask the standard CP stimulus, and the standard CT stimulus, were determined in a 2I-2AFC paradigm with an adaptive procedure which was similar to that used for the measurement of the frequency discrimination thresholds.3 In each trial, the two listening intervals contained a 200-ms noise burst. The two noise bursts were independent of each other but had the same SPL. In one of the intervals, to be identified by the subject, the noise was mixed with the 55-dB CP or CT stimulus. At the outset of a trial block, the noise SPL was always small enough to make the subject’s task easy. Following a correct response, the noise SPL was increased by 3 dB ~initially! or 1.5 dB ~after the second reversal!. Following a wrong response, the noise SPL was decreased by 9 dB ~initially! or 4.5 dB ~after the second reversal!. The SPL for which 75% of responses were correct was estimated as the arithmetic mean of the SPLs used on the 12 trials corresponding to reversals 3–14. For each subject, more than 20 measurements of this ‘‘threshold SPL’’ were made with the CP stimulus and ~in separate blocks of trials! the CT stimulus. The grand means of the measured threshold SPLs were 64.1 dB for the CP stimulus and 65.8 dB for the CT stimulus. A slight advantage for the CT stimulus was observed in each subject. In the main part of the experiment, the SPL of the 100-ms noise bursts was set at the threshold SPL for the CT stimulus; it was fixed across conditions, but varied slightly across subjects, in a 2-dB range. In conditions ER, EF, SF, and SR, at least 30 measurements of the frequency discrimination threshold were made for a given subject. The mean threshold values reported in the next section were computed from the last ten measurements. Frequency discrimination was also assessed in the absence of noise, in a ‘‘CP’’ and a ‘‘CT’’ condition. These two conditions replicated those labeled identically in experiment 2, at 55 instead of 70 dB SPL. In each of them, again, the formal data were collected only after several practice sessions and consisted of ten threshold measurements. The experiment was performed on four listeners with normal hearing. Two of them ~LD and JJ! were also subjects in experiment 2. The other two listeners had no previous experience with psychoacoustic tasks. B. Results and discussion

The results are displayed in the lower part of Table II and in Fig. 6. They were simpler and more clear-cut than those obtained in experiment 2 for the same standard span of f inst ~600 cents!. Looking first at the EF/ER and SR/SF ratios ~Fig. 6!, one notes that they were globally somewhat larger than in experiment 2, and less variable from subject to subject. The mean values of EF/ER ~2.67! and SR/SF ~3.07! were similar to each other, and also similar to the mean value of CT/CP ~3.16!. Another important fact was that, this time, very similar thresholds were obtained in conditions ER and SF, as well as in conditions EF and SR ~see Table II!. Thus, in the four main experimental conditions, thresholds did not 2942

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FIG. 6. Threshold ratios computed from the data displayed in the lower part of Table II ~results of experiment 3!.

depend on whether the critical frequency extremum was at the starting point or the final point of the audible FM. What only mattered was the nature of the extremum in the frequency domain, i.e., whether it was a maximum frequency or a minimum frequency. Clearly, these results provide strong evidence for both forward-acting and backward-acting sources of the peak/trough effect. Note that the mean thresholds measured in conditions EF and SR were very large ~90.9 and 96.2 cents!. The corresponding frequency shifts produced a change in FM slope of about 15%. Apparently, smaller changes in FM slope could not be detected and used to perform the task. This is rather surprising in view of the suggestion by Dooley and Moore ~1988! that listeners can detect 6% changes in a FM slope. However, the stimuli used by Dooley and Moore differed in many respects from those used in the present experiment, so that there is no flagrant discrepancy between the two sets of results. From the finding that the peak/trough effect can be produced both forward and backward, with approximately equal strengths, what do we learn about the mechanisms of this perceptual phenomenon and their location in the auditory system? The existence of retroactive factors argues against the idea that the peak/trough effect would completely originate from cochlear mechanisms and thus be reflected in some aspect of the eighth nerve’s responses to FM stimuli ~see, in this respect, Duifhuis, 1973!. In paper 3 ~Sec. III!, we hypothesized that the peak/trough effect reflects properties specific to a temporal coding of instantaneous frequency by the auditory system. The present data do not support this hypothesis, as they suggest that at least part of the peak/trough effect has a central origin: Due to neural phase-locking, there is a temporal representation of instantaneous frequency in the auditory nerve ~Sinex and Geisler, 1981!; beyond the cochlear nuclei, however, it seems that neural phase-locking rapidly disappears ~de Ribaupierre et al., 1972; Steinschneider et al., 1980!. To account for a retroactive production of the peak/ trough effect, it seems necessary to admit that within the neural network that stores auditory information concerning frequency extrema—a network probably located in the auditory cortex—the representation of a local frequency miniL. Demany and S. Clement: Frequency peaks and troughs

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mum tends to be degraded by a subsequent frequency rise. The production of the peak/trough effect in a forward direction might rest on completely different mechanisms, unrelated to auditory memory per se and located subcortically. Note, however, that the cochlear nonlinearity termed as ‘‘two-tone suppression’’ cannot play a crucial role since this process does not significantly persist after the cessation of the suppressor ~Arthur et al., 1971; Stanny and Elfner, 1980!. A parsimonious hypothesis is that the neural interactions producing the peak/trough effect forward are actually the same as those acting backward. Cle´ment ~1996! designed a formal neural network intended to model crudely the perception of instantaneous frequency. The network consists of two layers of tonotopically organized cells with asymmetric inhibitory connections ~in the first layer! and temporal integration properties. When its input is a continuous FM, the network essentially identifies the local frequency maxima. It is less sensitive to local frequency minima, and what we wish to stress here is that this asymmetry occurs both at the starting point and the final point of stimuli consisting of a unidirectional glide. In the model, therefore, the peak/trough effect is produced both forward and backward, by identical ~and physiologically plausible! mechanisms. It remains to be seen, among other things, if the model correctly simulates the influence of glide duration. The results of Porter et al. ~1991!, as well as pilot data collected in our laboratory, suggest that the peak/trough effect is no longer observed at very short stimulus durations. Further studies on this issue are needed. IV. CONCLUSIONS

Overall, the present experiments show that the perceptual asymmetry between FM peaks and troughs ~the ‘‘peak/ trough effect’’! does not have a simple origin. In experiment 1, using a family of modulation waveforms ranging from quasi-triangles to quasi-squares, we found that the peak/ trough effect was maximum for the most triangular modulations, with essentially durationless vertices. The detection threshold of shifts in a durationless frequency maximum appeared to be surprisingly small ~about 10 cents, i.e., 0.6%!. This suggests that some unknown auditory mechanism permits a hyperacute perception of frequency maxima, and thus that the peak/trough effect is not reducible to the effect of a deleterious factor on the perception of frequency minima. By showing that the peak/trough effect can be produced retroactively, experiments 2 and 3 made clear that its origin is at least partly central ~certainly post-cochlear and maybe cortical!. This does not support our previous hypothesis that the peak/trough effect reflects properties specific to a temporal coding of instantaneous frequency by the auditory system. ACKNOWLEDGMENTS

We thank Alain de Cheveigne´, J. Wiebe Horst, and John P. Madden for comments on a previous version of this paper. We are also grateful to Jean-Jacques Sierra for his active participation in the experiments. 1

Thus the amplitude ramps were twice as long as the frequency plateaux.

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Another methodological option was to use equally long amplitude ramps and frequency plateaux. However, the frequency plateaux had to be short, for obvious reasons, whereas the amplitude ramps had to be relatively long in order to minimize spectral splatter. 2 For subject JJ, in the ‘‘control peak’’ and ‘‘control trough’’ conditions, the SPL was 55 dB instead of 70 dB. The corresponding data were actually collected during experiment 3. 3 ´ ˇ Nabelek ~1978; see also Collins and Cullen, 1984! measured the detection thresholds, in a wideband masking noise, of gliding tones that were comparable to the first-half and the second-half of our CP and CT stimuli. For tone durations of 50–200 ms, thresholds appeared to be independent of glide direction. Here, therefore, we could assume that when the noise just masked an entire CP or CT stimulus, both the first-half and the second-half of this stimulus were just masked. This is why we did not measure masking thresholds for the two halves separately. Arthur, R. M., Pfeiffer, R. R., and Suga, N. ~1971!. ‘‘Properties of ‘two-tone inhibition’ in primary auditory neurones,’’ J. Physiol. ~London! 212, 593– 609. Brady, P. T., House, A. S., and Stevens, K. N. ~1961!. ‘‘Perception of sounds characterized by a rapidly changing resonant frequency,’’ J. Acoust. Soc. Am. 33, 1357–1362. Chih-an, L., and Chistovich, L. A. ~1960!. ‘‘Frequency-difference limens as a function of tonal duration,’’ Sov. Phys. Acoust. 6, 75–80. Cle´ment, S. ~1996!. ‘‘Mode´lisation de la perception auditive des modulations de fre´quence,’’Master’s thesis in Cognitive Science, Universite´ Bordeaux 2, France. Collins, M. J. ~1984!. ‘‘Tone glide discrimination: Normal and hearingimpaired listeners,’’ J. Speech Hear. Res. 27, 403–412. Collins, M. J., and Cullen, J. K. ~1984!. ‘‘Effects of background noise level on detection of tone glides,’’ J. Acoust. Soc. Am. 76, 1696–1698. Collins, M. J., Cullen, J. K., Jackson, D. F., and Porter, R. J. ~1990!. ‘‘Discrimination of formant-like frequency transitions preceded or followed by steady states,’’ Abstracts of the Thirteenth Midwinter Meeting of the Association for Research in Otolaryngology, pp. 181–182. Cosgrove, P., Wilson, J. P., and Patterson, R. D. ~1989!. ‘‘Formant transition detection in isolated vowels with transitions in initial and final position,’’ Proc. IEEE ICASSP 1, 278–281 ~Bell & Brian, Glasgow!. Demany, L., and McAnally, K. I. ~1994!. ‘‘The perception of frequency peaks and troughs in wide frequency modulations,’’ J. Acoust. Soc. Am. 96, 706–715. Demany, L., and Cle´ment, S. ~1995a!. ‘‘The perception of frequency peaks and troughs in wide frequency modulations. II. Effects of frequency register, stimulus uncertainty, and intensity,’’ J. Acoust. Soc. Am. 97, 2454– 2459. Demany, L., and Cle´ment, S. ~1995b!. ‘‘The perception of frequency peaks and troughs in wide frequency modulations. III. Complex carriers,’’ J. Acoust. Soc. Am. 98, 2515–2523. Divenyi, P. L., and Hirsh, I. J. ~1975!. ‘‘The effect of blanking on the identification of temporal order in three-tone sequences,’’ Percept. Psychophys. 17, 246–252. Dooley, G. J., and Moore, B. C. J. ~1988!. ‘‘Duration discrimination of steady and gliding tones: A new method for estimating sensitivity to rate of change,’’ J. Acoust. Soc. Am. 84, 1332–1337. Duifhuis, H. ~1973!. ‘‘Consequences of peripheral frequency selectivity for nonsimultaneous masking,’’ J. Acoust. Soc. Am. 54, 1471–1488. Ehret, G. ~1992!. ‘‘Le me´sence´phale auditif, une ‘gare de triage’ du traitement de l’information acoustique,’’ in Le Syste`me Auditif Central, edited by R. Romand ~INSERM, Paris!. Elliott, L. L., Hammer, M. A., and Carrell, T. ~1992!. ‘‘Discrimination of converging and diverging frequency transition,’’ in Auditory Physiology and Perception, edited by Y. Cazals, L. Demany, and K. Horner ~Pergamon, Oxford!. Hartmann, W. M. ~1997!. Signals, Sound, and Sensation ~AIP, Woodbury, NY!. Horst, J. W. ~1989!. ‘‘Detection and discrimination of frequency modulation of complex signals,’’ J. Acoust. Soc. Am. 85, 2022–2030. Massaro, D. W., and Idson, W. L. ~1977!. ‘‘Backward recognition masking in relative pitch judgments,’’ Percept. Mot. Skills 45, 87–97. Mattingly, I. G., Liberman, A. M., Syrdal, A. K., and Halwes, T. ~1971!. ‘‘Discrimination in speech and nonspeech modes,’’ Cogn. Psychol. 2, 131–157. Moore, B. C. J. ~1973!. ‘‘Frequency difference limens for short-duration tones,’’ J. Acoust. Soc. Am. 54, 610–619. L. Demany and S. Clement: Frequency peaks and troughs

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Na´be˘lek, I. V. ~1978!. ‘‘Temporal summation of constant and gliding tones at masked auditory threshold,’’ J. Acoust. Soc. Am. 64, 751–763. Na´beˇlek, I. V., Na´beˇlek, A. K., and Hirsh, I. J. ~1970!. ‘‘Pitch of tone bursts of changing frequency,’’ J. Acoust. Soc. Am. 48, 536–553. Palmer, A. R. ~1995!. ‘‘Neural signal processing,’’ in Hearing, edited by B. C. J. Moore ~Academic, New York!. Porter, R. J., Cullen, J. K., Collins, M. J., and Jackson, D. F. ~1991!. ‘‘Discrimination of formant transition onset frequency: Psychoacoustic cues at short, moderate, and long durations,’’ J. Acoust. Soc. Am. 90, 1298–1308. Ribaupierre, F. de, Goldstein, Jr., M. H., and Yeni-Komshian, G. ~1972!. ‘‘Cortical coding of repetitive acoustic pulses,’’ Brain Res. 48, 205–225. Sinex, D. G., and Geisler, C. D. ~1981!. ‘‘Auditory-nerve fiber responses to frequency-modulated tones,’’ Hearing Res. 4, 127–148.

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Stanny, R. R., and Elfner, L. F. ~1980!. ‘‘An ‘inhibitory’ influence on brainstem population responses,’’ Science 208, 418–419. Steinschneider, M., Arezzo, J., and Vaughan, Jr., H. G. ~1980!. ‘‘Phaselocked cortical responses to a human speech sound and low-frequency tones in the monkey,’’ Brain Res. 198, 75–84. Watson, C. S., Wroton, H. W., Kelly, W. J., and Benbassat, C. A. ~1975!. ‘‘Factors in the discrimination of tonal patterns. I. Component frequency, temporal position, and silent intervals,’’ J. Acoust. Soc. Am. 57, 1175– 1185. Wieringen, A. van, and Pols, L. C. W. ~1995!. ‘‘Discrimination of single and complex consonant–vowel- and vowel–consonant-like formant transitions,’’ J. Acoust. Soc. Am. 98, 1304–1312.

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