Perception of acoustically presented time series with varied intervals

Acta Psychologica 147 (2014) 105–110

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Perception of acoustically presented time series with varied intervals Jiří Wackermann ⁎, Jakob Pacer, Marc Wittmann Dept. of Empirical and Analytical Psychophysics, Institute for Frontier Areas of Psychology and Mental Health, Freiburg, Germany

a r t i c l e

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Article history: Received 18 January 2013 Received in revised form 20 August 2013 Accepted 27 September 2013 Available online 5 November 2013 PsycINFO classification: 2340 2360 Keywords: Duration discrimination Interval of subjective uniformity Perspectival contraction of time Subjective shortening Time perception Uniformity illusion

a b s t r a c t Data from three experiments on serial perception of temporal intervals in the supra-second domain are reported. Sequences of short acoustic signals (“pips”) separated by periods of silence were presented to the observers. Two types of time series, geometric or alternating, were used, where the modulus 1 + δ of the inter-pip series and the base duration Tb (range from 1.1 to 6 s) were varied as independent parameters. The observers had to judge whether the series were accelerating, decelerating, or uniform (3 paradigm), or to distinguish regular from irregular sequences (2 paradigm). “Intervals of subjective uniformity” (ISUs) were obtained by fitting Gaussian psychometric functions to individual subjects' responses. Progression towards longer base durations (Tb = 4.4 or 6 s) shifts the ISUs towards negative δs, i.e., accelerating series. This finding is compatible with the phenomenon of “subjective shortening” of past temporal intervals, which is naturally accounted for by the lossy integration model of internal time representation. The opposite effect observed for short durations (Tb = 1.1 or 1.5 s) remains unexplained by the lossy integration model, and presents a challenge for further research. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Standard methods of experimental research in time perception usually operate on single time intervals (production, estimation), or pairs of successively presented time intervals (reproduction, discrimination), which are marked by sensorially perceivable events (e.g. acoustic or visual). However, in real-world settings we often perceive series of events forming a temporal pattern, e.g. musical rhythms, sound sequences produced by mechanical devices, and movement patterns in sports or physical exercises. Identification of a global temporal pattern must somehow rely upon elementary discrimination of its constituting intervals, but how these two levels of perception are related is still an open question. The relevance of musical rhythms for understanding time perception was early recognized by the now “classic” authors in the field (Fraisse, 1958; Mach, 1922), and perception of musical rhythms has become a special topic of its own (Boltz, 1989; Handel, 1992; Hirsh, Monahan, Grant, & Singh, 1990; Povel, 1977, 1981). Musical rhythms are acoustic patterns consisting of integer multiples or simple fractions of a base duration, which is specified by the tempo and lies in the sub-second or circa-second domain. Musical rhythms are thus only very special instances of serial events. In other situations, intervals between single events may vary continuously—e.g. perception of “acceleration” of a bouncing ball losing its kinetic energy. ⁎ Corresponding author at: Institut für Grenzgebiete der Psychologie, Wilhelmstr. 3a, D-79098 Freiburg, Germany. E-mail address: [email protected] (J. Wackermann). 0001-6918/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.actpsy.2013.09.015

The study of serial events with longer base durations may provide further insights on specific characteristics of time perception in the supra-second region. Perception of durations longer than 2–3 s shows a remarkable “subjective shortening” (Wearden & Ferrara, 1993) of past intervals, which is revealed (i) by an asymmetry of the discrimination function in pairwise comparisons of time intervals (Hellström, 1977, 1985; Wackermann & Späti, 2006), and (ii) by progressive shortening of the response with longer standards in the reproduction task (Eisler & Eisler, 1992; Ulbrich, Churan, Fink, & Wittmann, 2007; Wackermann & Ehm, 2006; Wackermann, Späti, & Ehm, 2005). However, empirical evidence should be mentioned suggesting that the “subjective shortening” effect is variable and partly dependent on the range of presented durations (Lejeune & Wearden, 2009; Noulhiane, Pouthas, & Samson, 2009). In any case, we may expect the “subjective shortening” also to affect perception of series of events. In fact, such an effect was described succinctly by Ernst Mach in his Analysis of Sensations (1922, chap. XII, §9): “The phenomenon is perfectly analogous to that which we observe in the province of the space-sense […]. In walking forwards, we have a distinct sensation that we are moving away from a starting-point, but the physiological measure of this removal is not proportional to the geometrical. In the same manner, elapsed physiological time is subject to perspectival contraction, its single elements [i.e., events] becoming less and less distinguishable.” In other words, as multiple events (“elements” in Mach's parlance) recede to the past, the intervals separating the singular events seem subjectively to shrink, thus creating an impression of “perspectival contraction”.

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Fig. 1. Monotonic (left) and alternating (right) pips sequences for different values of δ.

These observations motivated our study of serial discrimination of acoustically presented intervals in the supra-second domain using two types of stimuli: (i) series of monotonically increasing or decreasing intervals, resulting in a global judgment of accelerating, decelerating, or uniform sequence, and (ii) series of alternating intervals, resulting in a global judgment of regular or irregular sequence.

2. Methods 2.1. Apparatus and stimuli Series of acoustic events (“pips”) were generated by a program running under BSD Unix on a portable iBook G4 (Apple Inc.) computer, using the system clock for the interval timing and the console beeper as a sound source. The acoustic signal from the computer was fed via an amplifier (Sony TA-FE310R) to a pair of headphones (Sennheiser HD 201). Pips of a nominal frequency of 2000 Hz and a duration of 20 ms were used in all reported experiments. The inter-onset intervals between subsequent pips were varied to form either geometric series (mode mono), or alternating series (mode mach1), using the formulae: n

ð1 þ δÞ T n ¼ Tb


n

T n ¼ T b 1 þ ð−1Þ

δ 2



ðmonoÞ ðmachÞ

where n = 0,1,2,… is an ordinal index of the interval, Tb denotes the “base duration”, and δ is a parameter modulating the series. In the mono mode, the base duration was identical to the first interval presented (T0 = Tb) while in the mach mode, the base duration was the mean value of subsequent, alternating durations (Fig. 1). Tb and δ were two control parameters determining the temporal structure of the stimulus which were varied according to experimental designs described below.

2.2. Participants Participants were students from the University of Freiburg recruited by advertisement. All participants were reportedly of good health and with no known neurological or psychiatric problems. The participants signed an informed consent before the beginning of the session and received a moderate financial compensation when the session was completed. Twenty-four observers, 12 women and 12 men in the age range 21–33 years (mean age 25.8 years) participated in Experiment 1. Eighteen observers, 9 women and 9 men in the age range 21–29 years (mean age 24.4 years) participated in Experiment 2. Twelve observers, 6 women and 6 men in the age range 21–28 years (mean age 24.7 years) participated in Experiment 3. 1 Mach (1865) was the first to use acoustic series with alternating intervals in a study of temporal discrimination.

2.3. Procedures In each trial, the observer listened to the pips series played through the headphones. Duration of the pips series was limited to a maximum of 1 min. However, the observer could interrupt the stimulus presentation by pressing a button on a pointing device (“mouse”) before the time limit was reached.2 Thereafter a list of possible judgments was displayed on the monitor, from which the observer had to choose her/his response. In the mono mode (3), the three response categories were “accelerating”, “uniform”, and “decelerating”. In the mach mode (2), the two response categories were “regular” and “irregular”. In Experiments 1 and 3 only monotonically modulated series were used. In Experiment 2 both types of stimuli, mono and mach were used. 2.3.1. Experiment 1 (N = 24 = 2 × 12 subjects.) Three blocks, each consisting of twelve trials, were run with different base durations in a fully permuted order, using Tb = 1.5, 3, 6 s for a sub-group of twelve subjects, and Tb = 1.1, 2.2, 4.4 s for the other twelve subjects. Each block consisted of two parts. In part 1, δ varied from −0.05 to +0.05 in steps of 0.02, following a simple up/down or down/up staircase scheme with two repetitions for each δ value. A point of subjective uniformity (PSU) was roughly estimated from the data, and a new range of δs distributed symmetrically around the PSU with halved steps of 0.01 was used in part 2, following the same scheme. This two-phase procedure was designed to adapt the δ-sampling scheme to the subject's individual performance. 2.3.2. Experiment 2 (N = 18 = 2 × 9 subjects.) Each session consisted of two blocks, one block with monotonic series (mode mono) and the other block with alternating series (mode mono). Two base durations were used in each block, Tb = 2.2 or 4.4 s with nine subjects, and Tb = 3 or 6 s with the other nine subjects, in a permuted order. A mono block consisted of 26 trials, with δ that varied from −0.06 to +0.06 in steps of 0.01, and two repetitions for each δ value. A mach block consisted of 22 trials, with δ that varied from 0 to 0.25 in steps of 0.025 for base durations 2.2s and 3 s, or a doubled range up to 0.5, and doubled steps of 0.05 for base durations 4.4 s and 6 s. These settings were based on a series of pilot experiments showing that the discrimination in the mach mode was definitely inferior to that in the mono mode. An interleaved up/down or down/up staircase scheme was used, in which the full δ-range was traversed four times with different starting points and interleave factor four. This resulted in a pseudo-random, from the participants' point of view unpredictable sequence.

2 This allowed the observer to give her/his response as soon as s/he arrived at a firm judgment, thus speeding up the experiment and reducing the participants' boredom. Only in Experiment 3, where the number of presented intervals nP was fixed for each trial (and varied across blocks of trials) the observer had to listen to the entire sequence, and the stopping option was disabled.

J. Wackermann et al. / Acta Psychologica 147 (2014) 105–110

A

1

E

.5

107

0

D −.06

0

+.06

−.06

0

+.06

Fig. 2. Transformation of a 3AFC dataset to two PMFs. Left: Individual responses (×) for δ varying from −0.06 to +0.06: A = accelerating, E = uniform, D = decelerating, dashed broken curve = central response tendency. Right: Relative frequencies of responses ‘accelerating’ (△) and ‘decelerating’ (∇), to which psychometric functions ψA(δ) (red) and ψD(δ) (blue) are fitted; the two curves cross the dashed line P = 0.5 at points δ = θA and δ = θD, respectively. (Illustrative data from Exp. 2, mode mono, Tb = 3 s.).

2.3.3. Experiment 3 (N = 12 subjects.) Unlike Exps. 1 and 2, the number of presented intervals (nP) was limited to 2, 5, or 10. The observer had to listen to the entire sequence of nP + 1 pips before giving a response.3 Combining the three nP values with two base durations (Tb = 3 and 6 s) resulted in 2 × 3 = 6 blocks, each block consisting of 18 trials. For nP = 5 and 10, δ varied from −0.1 to +0.1 in steps of 0.025; for nP = 2, the δ range was adjusted to [−0.4,+0.4], and the steps increased accordingly to 0.1. Again, these settings were based on exploratory pilot experiments. An interleaved staircase δ-sampling scheme similar to Exp. 2 was used. 2.4. Data reduction and analysis Separately for each subject and base duration Tb, data were sorted by δ, relative frequencies of response categories were calculated, and Gaussian psychometric functions (PMFs) were fitted to the response frequencies, using the standard maximum-likelihood method (Bush, 1963). For the 3AFC data acquired with monotonically modulated time series two PMFs were fitted, namely,
 θ −δ ψA ðδÞ ¼ Φ A ; σA


 δ−θD ψD ðδÞ ¼ Φ ; σD

specifying probabilities of responses “accelerated” and “decelerated”, respectively; Ф denotes the normal cumulative distribution function (CDF). The discrimination threshold θA thus determines the value of δ for which the probabilities of responses “accelerated” vs. “nonaccelerated” are equally 1/2; and similarly for θD and responses “decelerated” vs. “non-decelerated” (Fig. 2). For the 2AFC data acquired with alternating time series (Exp. 2, mode mach), the probability of response “regular” is given by the function ψR ðδÞ ¼ Φ


 θR −jδj σR

where Ф again denotes the normal CDF, and θR is the value of δ for which the probabilities of responses “regular” and “irregular” are equally 1/2. In this way, the 3AFC data were reduced to a pair [θA,θD], delimiting an “interval of subjective uniformity” (ISU).4 The 2AFC data were reduced, as usual, to a single value, θR = “point of subjective indifference” (PSI) between judgments “regular” and “irregular.” The slopes of the PMFs at −1 −1 the thresholds, σ−1 A , σD , and σR , specifying acuity of the respective discrimination judgment, were not of interest. Our analyses focused on dependences of ISUs (and PSIs, where applicable) on base duration

3 For nP = 2 the observer's task consisted in a comparison between two empty intervals marked by three pips. Response alternatives “the second interval was shorter” or “the second interval was longer” were used and coded in data as “accelerating” or “decelerating,” respectively. 4 Note that the ISU is an interval in the mathematical sense—i.e., a contiguous subset of real numbers—whereas in time perception research the word “interval” is often used for a duration of an interval, i.e., a single value.

Tb. For a comprehensive presentation, the their respective midpoints and widths, 1 e θ ¼ ðθA þ θD Þ; 2

ISUs

were also described by

Δθ ¼ θD −θA :

Parameter e θ naturally characterizes a bias of the subjective uniformity judgment w. r. t. the physical uniformity (δ = 0), while Δθ is a singlevalue measure of subjective uncertainty of uniformity detection. 3. Results Of interest were primarily dependences of the ISUs on experimental conditions, i.e., base durations Tb, presentation modes (mono, mach) and number of presented intervals nP. Note that in Experiments 1 and 2, participants were subdivided into two sub-groups using different sets of base durations Tb, to obtain a finer picture of Tb-dependent effects. 3.1. Experiment 1 The ISU width Δθ varied from ~0.023 at shortest base durations, 1.1 and 1.5 s, to 0.052 for the longest, Tb = 6 s, whereas the ISU midpoints e θ shifted from +0.005 to −0.004 (Fig. 3a). A closer look at the data reveals that these effects are due to a systematic and mostly significant shift of the acceleration threshold towards negative values with increasing Tb (Fig. 3b), verified by intraindividual pairwise t tests (all with df = 11; here and in the following, uncorrected P values are reported): 1.1 vs. 2.2 s: t = 1.94 (P = 0.082), 1.5 vs. 3 s: t = 5.00 (P b .005), 2.2 vs. 4.4 s: t = 3.57 (P b .02), and 3 vs. 6 s: t = 2.62 (P b .05). On the other hand, the deceleration threshold θD did not change significantly with Tb, only varied around ~0.018. This “dissociation” between acceleration and deceleration threshold is also supported by insignificant pairwise correlations between θA and θD. Interestingly, three subjects never gave the response “accelerating”, in spite of varying δ throughout the entire range specified by the experimental design. This failure to detect acceleration was observed in one subject at Tb = 4.4 s, and two subjects at Tb = 6.0 s. These cases can be interpreted as extreme manifestations of the “θA traveling” effect.5 A survey of actually perceived intervals nP reveals, above all, extreme inter-individual variability: some participants arrived at their judgments quite quickly and utilized the stopping option regularly, while others listened to presented pips series in full length. Fig. 4 shows common patterns of response behavior in terms of grand means nP , displayed as functions of δ for different base durations Tb. For short Tb = 1.1 or 1.5 s, these patterns have generally an inverted V form, with a maximum at δ = 0 and quickly decreasing with increasing |δ|. This tendency flattens for Tb = 2.2 and 3 s, and completely disappears for long Tb = 4.4 and 6 s. 5 To make further statistical evaluation possible, in these three cases the missing θA values were replaced by the group mean across all remaining subjects for the same base duration. This is a super-conservative estimate, in fact somewhat counter-acting the observed effect which nonetheless remains significant.

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0.04 +.010 2.2

1.1

0.03

ISU midpoint θ˜

+.005 0.02 0

threshold θD

4.4

0.01

-.005 6.0

3.0

-.03

-.02

1.5

0

-.010 1.11.5

2.2

3.0

4.4

6.0

-.04

-.01

0

threshold θA

base duration Tb [s]

θ plotted as a function of base duration Tb. Right: Interrelation between mean thresholds θA and θD across Fig. 3. Dependence of ISU on base duration Tb in Exp. 1. Left: Mean ISU midpoint e different base durations Tb (indicated by tiny numbers and dotted lines). Group means ± 1 SEM are shown (N = 12 for each data point). Two subsets of different Tb are distinguished by graphic symbols and ○.

3.2. Experiment 2 The results for the mono mode only partially confirm those from Exp. 1. Comparisons between Tb = 2.2 and 4.4 s show a negative shift of both θA (t = 2.15, df = 8, P = .067) and θD (t = 2.60, df = 8, P b .05), resulting in a significant shift of the midpoint e θ of the entire ISU (t = 3.34, df = 8, P = .01). Comparisons between Tb = 3 and 6 s show also negative but non-significant shifts of the ISU. The midpoints e θ were 0.0078 and − 0.0063 for Tb = 2.2 and 4.4 s, and 0.0003 and − 0.0055 for Tb = 3 and 6 s, respectively. As for the mach mode, mean regularity thresholds θR were 0.221 for Tb = 2.2 s, and slightly lower for longer base durations: 0.135 (Tb = 3 s), 0.188 (Tb = 4.4 s), and 0.164 (Tb = 6 s). These values, which are by almost one order of magnitude higher than mean θA (in the range from −0.017 to 0.031) or θD (in the range from +0.018 to +0.033), indicate that recognition of deviation from uniformity is considerably more difficult for non-directed than directed (monotonic) interval series. Intraindividual pairwise comparisons between shorter and

longer base durations (2.2 vs. 4.4 s, or 3 vs. 6 s) did not reveal any significant differences. No significant correlations were found between θA and θR, or θD and θR. There seems to be no strong relation between duration discrimination in the two modes, mono and mach. 3.3. Experiment 3 The mean ISU width Δθ decreased from 0.388 for nP = 2 via 0.118 for nP = 5 to 0.084 for nP = 10 (average across both base durations, 3 and 6 s).6 This effect obviously reflects a reduction of the observer's uncertainty with increasing number of perceived intervals. For all three nP conditions, the ISU midpoints e θ shifted from positive values at Tb = 3 s to negative values at Tb = 6 s, reflecting simultaneous shifts of both θA and θD. In spite of this common pattern, ISU shifts were statistically significant only for nP = 5: for θA, t = 2.80 (P b .02), for θD, t = 3.16 (P b .01), for e θ, t = 3.48 (P b .01) (all t tests have 11 df). The ISU width Δθ did not change significantly with base duration. 4. Discussion

¯P 30

The background assumption in this study was that the detection of a global property of a “pips” series results from serial comparisons between subsequent between-pips intervals. Therefore two types of stimuli were used, where the intervals form a geometric series (mono) or an alternating series (mach). It is easy to see that for the two types of time series the following relations hold:

20

T nþ1 −T n ¼ δ ðmonoÞ; Tn T nþ1 −T n ≈  δ ðmachÞ: Tn

10

That is, the Weber ratios between subsequent intervals7 are constant (mono), or constant in the absolute value (mach) with alternating sign. We may thus assume, for a given δ, a more-or-less uniform

δ

0 -.05

0

+.05

Fig. 4. Graphic summary of average numbers of intervals perceived (nP) until judgment in Exp. 1. Shown are grand means nP taken across subjects and repeated trials, as functions of the stimulus control parameter δ, sorted by base duration Tb. Base durations are distinguished by graphic symbols ⋄ (1.1, 2.2, 4.4 s) and ○ (1.5, 3.0, 6.0 s) and line widths (thin: short, medium: medium, thick: long), respectively.

6 Two subjects yielded extremely flat PMFs with Tb = 6 s and nP = 2 or 5, respectively, so that the ISU could not be estimated reliably for these cases. Similarly as in Exp. 1 (see footnote 5), the missing θA and θD values were replaced by the group means across all remaining subjects for the same conditions. 7 The term “Weber ratio” is here used simply as a synonym for an increment of duration relative to the preceding duration. This does not imply that we postulate that duration discrimination is strictly Weberian, which is still a controversial issue in time perception research (cf. (Bizo, Chu, Sanabria, & Killeen, 2006; Grondin, 2003; Masin, 2009)).

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accumulation of cognitive evidence resulting in the observer's response.8 Estimates of δ for which a property of interest is detected with probability 1/2 can be so far considered as a generalization of the concept of Weber ratio for such sequential stimuli. The observers' task was, essentially, the detection of deviation from uniformity, which could be directional (“accelerating” or “decelerating”) with monotonic series, or direction-less (“irregular”) with alternating series. Our use of the 3AFC paradigm in experiments with the monotonic time series was based mainly on observations from early pilot experiments: in some subjects there was quite a broad region of subjective uncertainty between distinct perception of “accelerating” and “decelerating” series, so that a forced choice between the two alternatives would not make sense to them. Since we preferred not to force the participants to give responses that do not match their subjective perceptual experience, a neutral response (“neither–nor”, that is, “uniform” or “uniform-like”) was included. This turned out to be a fortunate choice, as it allowed us to analyze temporal discrimination in terms of two parameters, θA and θD, delimiting the interval of subjective uniformity (ISU), and derived parameters e θ and Δθ.9 The most important finding in this study is the systematic shift of the ISU with increasing base duration Tb in the monotonic time series. In Exp. 1, a “dissociation” between θA and θD was observed: the acceleration threshold θA moved to more negative values with increasing Tb, while θD did not change significantly; whereas in Exps. 2 and 3 the ISU shifted as a whole. The source of this discrepancy is at present unclear. Provisionally, the term “ISU traveling effect” used in this paper covers both cases. Importantly, results from Exp. 1 suggest that the ISU shift is not a linear or linear-like function of Tb, but rather a steep transition occurring between Tb = 2.2 s and 3 s (cf. Fig. 3a). This finding may be related to other evidence suggesting that temporal processing of durations up to 2 to 3 s qualitatively differs from that of longer intervals. Integration of events within a temporal horizon up to 2–3 s has been reported in various experiments in perception and action (Fraisse, 1984; Grondin, 2010; Pöppel, 1997; Wittmann, 2011). This can be illustrated by phenomena such as perceptual grouping of metronome beats: at a moderate tempo, perceptual gestalts are formed with an accent on every nth beat (1-2, 1-2, or 1-2-3, 1-2-3), whereas for inter-beat intervals longer than 2–3 s only singular beats are perceived (1,1,1, etc.) (Szelag, von Steinbüchel, Reiser, Gilles de Langen, & Pöppel, 1996; von Steinbüchel, Wittmann, & Szelag, 1999). Also, in a task to synchronize motor actions (finger taps) to a sequence of presented tones, effortless timing of behavior—i.e., the precise anticipation of tones—breaks down with inter-stimulus intervals N2 s (Mates, Müller, Radil, & Pöppel, 1994). Our results thus seem to point out the significance of a critical duration boundary, 2–3 s, in a different experimental setting. This qualitative observation, however, leaves some questions unanswered. Our results indicate not only a shift towards “subjective shortening” of subsequent intervals for Tb ≥ 3 s but also an opposite effect—not just zero, i.e. “veridical perception”—for short inter-pips

8 This assumption should be substantiated by a mathematical model, predictions of which would be compared with empirical data. Specifically, the model would have to predict (1) distributions of numbers of intervals needed for a decision and their dependence on the stimulus parameters δ and Tb (cf. Fig. 4) or, inversely, (2) dependence of detection thresholds on the presented number of intervals. This is clearly an enterprise beyond the aims of the present study, and should be left as a task for future elaboration of this paradigm. 9 For 2 paradigms, the concept of a “point of subjective indifference” is uniquely defined: the value of the stimulus for which perception and non-perception of a property   of interest, say X, are equi-probable, PrfX g ¼ Pr X ¼ 12. By contrast, 3 paradigms offer a richer spectrum of possible parametrizations (cf. (Burro, Sartori, & Vidotto, 2011)). For example, taking PMF for the neutral response “uniform”, ψU = 1 − (ψA + ψD), we could determine the region of subjective uniformity in terms of points equalizing ψU = ψA and ψU = ψD, and take the local maximum of ψU as a point-wise estimate of its position. However, the definition of the ISU used in our study seems to be the most natural choice for the given purpose.

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intervals at Tb = 1.1 and 1.5 s (Fig. 3). Interpretation of this finding is at present unclear. Effects of opposite signs at opposite ends of a control parameter continuum make one first think of an “adaptation level” (AL) effect (Helson, 1964). In our experiment stimuli of different base durations Tb were used in separate blocks, not inter-mixed within a single run of trials. Thus we find it unlikely that the reversal at the critical duration is merely an AL effect10; we are rather inclined to assume that it reflects an intrinsic property of the internal duration representation: transient effects prevailing at short durations and its “lossy” character prevailing at longer durations. Worth mentioning in the present context is also an earlier study by Halpern and Darwin (1982), using patterns of four clicks with different base durations between the initial three clicks, and an adjustable period between the 3rd and 4th clicks, in a duration discrimination paradigm. They reported a significant shift of the PSI to negative values with increasing base duration. The authors operated in a much shorter range of base durations, from 0.4 to 1.45 s, which does not overlap with the Tb range in our experiments, the magnitude of the perceptual bias was much smaller, but the direction of the effect was the same as in our experiments. Results of Experiments 2 and 3 shed additional light on the findings of Experiment 1. Experiment 2 demonstrated that detection of uniformity was more difficult with an alternating direction of difference between subsequent intervals (mode mach) than with monotonically accelerating or decelerating time series (mode mono). This observation may be of importance for identification of mechanisms involved in assessment of those different perceptual categories. Yet one could expect that subjects performing better in one type of the task would also score better in the other task, regardless of their specific levels of difficulty. Thus the lack of intra-individual correlation between θA,D (mono) and θR (mach) in Exp. 2 appears somewhat surprising. However, the analyzed data subsets may be too small to allow for reliable conclusions. The progressive narrowing of the ISU with increasing number of presented intervals nP (Exp. 3) supports the heuristic hypothesis of the observer's judgment being a result from an ongoing accumulation of sequential comparisons. These results are in agreement with studies of sequential perception of uniform interval sequences, which found increasing sensitivity (lowered Weber ratio) for higher nP (Grondin, 2008; Miller & McAuley, 2005). Also, a comparison with Exp. 1, where the numbers of intervals perceived (nP) were under the participants' control, is instructive. As seen from Fig. 4, mean nP until response were on the average higher than fixed nP used in Exp. 3. We may thus conjecture that, in Exp. 3, in most trials the subjects had to give a response before they could arrive at a firm judgment. Even in such a situation of “forced guessing” the ISU width, i.e., the uncertainty of perceptual discrimination, decreased with increasing nP. Coming back to the main finding of our study: how can we explain the ISU “traveling effect”, and what are its consequences for time perception in real world settings? With longer durations (Tb N 3 s), the threshold for acceleration detection (and thus also the midpoint of the entire ISU) shifts to more negative values, corresponding in our experimental paradigm to more rapid physical shortening between subsequent intervals. This is in line with findings on “subjective shortening” of past temporal intervals known from duration reproduction or discrimination experiments (cf. Section 1). This part of the effect can be accounted for by the lossy integration model of internal time representation (Sysoeva, Wittmann, & Wackermann, 2011; Wackermann, 2011), also known as a “dual klepsydra model” (DKM) (Wackermann & Ehm, 2006). According to this model, the iterated reproduction of a given duration unit (so-called “klepsydraic clock”: (Wackermann, 2008)) results in a series of progressively shortening intervals. Although this is not an exact equivalent of the monotonic

10 AL-like effects may also occur in blockwise experimental designs, although with a smaller effect size (Hellström, 2000).

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series used in our study,11 the qualitative agreement between our experimental findings and the DKM-based prediction is encouraging. It seems that in the supra-second domain we are constantly subject of a subtle, yet psychophysically measurable, illusion in time perception: event series making a subjective impression of “uniformity” must be in fact slightly accelerating, while event series which are really (physically) uniform are more likely to be perceived as slowing down. The “uniformity illusion” and the “perspectival contraction” described by Mach (see Section 1) thus appear as two sides of the same coin or, in other words, two manifestations of the same principle underlying the neural representation of duration. Admittedly, we have no similarly convenient explanation for the opposite part of the “ISU traveling effect” observed in the circa-second domain (Tb ≤ 1.5 s). The lossy integration model in its present form predicts no deviation from veridical reproduction/discrimination for such short intervals. So far the bidirectional nature of the effect remains a major challenge for further research. We cannot exclude even a possibility of a combination of two effects: a unidirectional shift due to the lossy character of internal duration representation, and a bidirectional effect on the adaptation level basis.12 A next promising step would be an extension of the DKM to a “cascaded klepsydrae model”, combining integrators of different time constants. A “fast” (short time constant) front-end integrator may be responsible for transient effects, while a “slow” (long time constant) back-end integrator acts serves the “lossy” duration representation. It seems that such a model would be able to account for effects in both directions, i.e. subjective shortening or lengthening (work in progress). List of abbreviations nAFC AL CDF DKM ISU PMF PSI PSU SEM

n alternatives forced-choice adaptation level cumulative distribution function dual klepsydra model interval of subjective uniformity psychometric function point of subjective indifference point of subjective uniformity standard error of the mean

Acknowledgments We wish to thank two reviewers, Å̊ke Hellström and Tsuyoshi Kuroda, for their careful reading of and commenting on an earlier version of the manuscript. Their critical remarks and suggestions greatly helped us to improve the paper. Thanks are also due to Oksana Gutina for conducting part of the experiments and for general assistance. References Bizo, L. A., Chu, J. Y., Sanabria, F., & Killeen, P. R. (2006). The failure of Weber's law in time perception and production. Behavioral Processes, 71, 201–210. Boltz, M. (1989). Tempo discrimination of musical patterns: Effects due to pitch and rhythmic structure. Perception & Psychophysics, 60, 1357–1373. Burro, R., Sartori, R., & Vidotto, G. (2011). The method of constant stimuli with three rating categories and the use of Rasch models. Quality and Quantity, 45, 43–58. Bush, R. R. (1963). Estimation and evaluation. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology, Vol. I. (pp. 429–469). Eisler, H., & Eisler, A.D. (1992). Time perception: Effects of sex and sound intensity on scales of subjective duration. Scandinavian Journal of Psychology, 33, 339–358. Fraisse, P. (1958). Les structures rythmiques. Paris: Érasme.

11 In the present study, the subsequent intervals in the mono mode were related by a linear function, while the “klepsydraic reproduction function” is generally non-linear (Wackermann, 2006). 12 Thanks to Å̊ke Hellström for suggesting this interpretation.

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