Perception & Psychophysics 1979, Vol. 26 (5),340-354 The perception of time

LORRAINE G. ALLAN McMaster University, Hamilton, Ontario L8S 4KJ, Canada The present paper organizes and evaluates selected portions of the time perception literature. Emphasis is on data and theory concernedprimarily with judgments of brie/temporal intervals. Research concerning the psychophysical law for time, Weber's law, the time-order error, and the role of nontemporal information is evaluated. This is followed by a consideration of current, quantitatively oriented, theoretical formulations for time perception.

Five years ago, Allan and Kristofferson (1974a) the experimenter (E) presents a temporal interval and could write: "There are few quantitative theories of the subject (S) gives a verbal estimate of its duration duration discrimination and few established empirical in clock time. In production, E states the duration phenomena to guide theorizing" (p. 26). Since that of the interval in clock time and S produces that time, there has been a dramatic change. Many interval. In reproduction, E presents the temporal articles on the discrimination of brief temporal inter­ interval and S reproduces it. In the method of com­ vals, and several new quantitative models, have parison, E presents two temporal intervals in succes­ appeared in a period of a few years. The last major sion and S makes a judgment of relative duration. general review and evaluation of the literature con­ Many studies asked (1) which method produced cerned primarily with the perception of brieftemporal judgments of greatest accuracy (least difference from intervals was published almost 30 years ago by the presented temporal interval) and consistency Woodrow (1951). Recent reviews by Doob (1971), (least intra- and intersubject variability), and (2) Fraisse (1963), and Peppel (1978) emphasize the per­ whether there were significant correlations among ception of relatively long temporal intervals. The the basic methods (e.g., Carlson & Feinberg, 1968a, present paper organizes and evaluates those portions 1968b, 1970; Danziger & Du Preez, 1963; Doehring, of the more recent time perception literature con­ 1961; Du Preez, 1963; Gilliland & Humphreys, 1943; cerned primarily with judgments of brief temporal Gilliland & Martin, 1940; Goldfarb & Goldstone, intervals, generally less than 10 sec, by normal 1963b; Goodfellow, 1934; Hawkes, Bailey, & Warm, human adults. First, the more common methods 1961; Hornstein & Rotter, 1969; Kruup, 1961; used are described. Second{ the research relevant to McConchie & Rutschmann, 1970, 1971; Ochberg, four central issues is evaluated. Finally, current Pollack, & Meyer, 1965; Siegman, 1962; Spivak and quantitatively oriented theoretical formulations & Levine, 1964; Treisman, 1963). No single method for time perception are considered. can claim consistent superiority. Furthermore, what Even within these limits, no attempt at covering emerges clearly is a lack of correlation among the all the relevant literature was made; the literature methods, which prompted Carlson and Feinberg considered is best described as representative rather (1970) to conclude that "present methods of assessing than comprehensive. (A fairly comprehensive recent time judgment are of uncertain validity and gener­ bibliography of the time perception literature is alizability" (p. 171). Zelkind and Sprug, 1974.) Most early research on time perception was non­ theoretical: The subject's responses were treated as METHODOLOGY direct estimates of perceived duration. Recently, several quantitative models have been generated, and Past attempts to impose order on the methodology new methods, dictated by the models, have become of time perception (e.g., Bindra & Waksberg, 1956; popular. Those frequently used are listed in Table 1. Clausen, 1950; Woodrow, 1951) usually agreed that There are two basic research methodologies, dura­ there were four basic methods. In verbal estimation, tion scalingand duration discrimination. In a duration scaling task, S is asked about the perceived durations Preparation of this manuscript was supported by the Natural of a set of easily discriminable temporal intervals; Sciences and Engineering Research Council of Canada. I would in a duration discrimination task, S is asked to dis­ like to thank A. B. Kristofferson and Cern Kaner for reading tinguish among a set of highly confusable intervals. and commenting upon an earlier draft, and Martha Teghtsoonian for her helpful editorial comments. Requests for reprints should be sent to Lorraine G. Allan, Department of Psychology, Duration Scaling McMaster University, Hamilton, Ontario L8S 4KI, Canada. A variant of the method of verbal estimation is

Copyright 1979 Psychonomic Society, Inc. 340 0031-5117/79/110340-15$01.75/0 THE PERCEPTION OF TIME 341

Table I Methods Used in Time Perception Studies Dura tion Scaling Duration Discrimination

(I) Verbal Estimation (I) Method of Comparison (2) Magnitude Estimation (a) forced-choice with a fixed standard (3) Category Rating (b) forced-choice with a roving standard (4) Production (2) Single Stimulus (5) Ratio-Setting* (3) Many-to-Few (6) Synchronization (4) Identification "Reproduction, fractionation, multiplication. that of magnitude estimation. Here the S's verbal FOUR CENTRAL ISSUES response is not in temporal units. Rather, S assigns a number to represent the magnitude of the per­ The following four questions have been addressed ceived duration. A standard duration may be pre­ by the various theoretical accounts of time percep­ sented on each trial, or just at the beginning of the tion: (1) What is the psychophysical law for time session, or no standard may be designated. A similar perception? (2) Does Weber's law hold for time per­ scaling task is category-rating. E presents the tem­ ception? (3) What is the source of the time-order poral interval and S locates its perceived duration in error? (4) What is the influence of nontemporal one of m ordered categories. In category rating, the information on temporal judgments? boundaries and subdivisions of the response scale are defined by E; in magnitude estimation, they are The Psychophysical Law for Time Perception defined by S. At first glance, there appears to be no consensus Reproduction is a subcategory of the method of regarding the nature of the relationship between ratio-setting. In a ratio-setting task, E presents the stimulus time and perceived time. Early studies temporal interval and S generates a specified propor­ (see Woodrow, 1951) often described the data in tion of that interval. When the proportion is 1.00, terms of Vierordt's law-that short intervals are the task is referred to as reproduction; less than 1.00, overestimated and long ones underestimated. The fractionation; greater than 1.00, multiplication. indifference interval is that which is neither over­ A variant'of the method of reproduction is syn­ estimated nor underestimated. Woodrow (1930, 1933, chronization. In one version, E presents a fixed dura­ 1934) demonstrated that there was no universal tion standard and S responds in synchrony with its tendency for Vierordt's law to hold, and concluded termination. In another version, E presents a series that, even when an indifference interval was found, of brief events once per unit time and S reproduces there was considerable variablility in the value a sequence of responses at the same rate. across laboratories and that "even under fixed experimental conditions, there is no single indiffer­ Duration Discrimination ence interval valid for all subjects" (Woodrow, In the method of comparison, two duration values 1951, p. 1226). are presented sequentially on each trial. Usually one More recent studies have tried to determine the value is constant within a block of trials and is functional form of the relationship between stimulus referred to as the standard; the other value varies time and perceived time. The terms perceived, from trial to trial and is referred to as the compari­ internal, subjective, psychological, and apparent son or the variable. In thejorced-choice (FC) version duration (time) are used interchangeably; generally of the method of comparison, the standard duration speaking, they refer to the temporal value used by can occur either first or second and S is required the subject in making his judgment. Efron (1970, to indicate which. In the roving-standard version, the 1973) has argued that duration of a percept is not standard value also varies from trial to trial. necessarily the same as perceived duration. He con­ In the method of single stimulus (SS), one of cluded, based on simultaneity judgments, that for two possible duration values is presented on each stimuli briefer than a critical duration (about trial and S indicates whether it was the shorter or 150 msec), perceptual offset latency increases as the longer value. In the many-to-jew modification, stimulus duration is decreased, the result being a con­ the number of possible stimulus values is increased stant minimum perceptual duration, while for stimuli but the binary response maintained. In the identifi­ longer than 150 msec the duration of the percept cation modification, both the number of possible increases linearly with stimulus duration. However, stimulus values and of response alternatives is in­ new data from simultaneity tasks similar to Efron's creased. Identification differs from category-rating (Allan, 1976; Penner, 1975, 1978) do not confirm his in that the duration values are highly confusable. basic finding. Penner (1975) suggested that, although 342 ALLAN there may be a constant minimum perceptual duration, Michon, 1967; Painton, Cullinan, & Mencke, 1977; her subjects adopted different criterion values than Steiner, 1968; Stevens & Galanter, 1957; Stevens & did Efron's. Later Allan (1976) and Penner (1978) Greenbaum, 1966). In a number of these studies argued that Efron's resultscould be attributed entirely (Bobko et aI., 1977; Jones & MacLean, 1966;Stevens to criterion changes, and concluded that a strong case & Galanter, 1957), the estimate of the power func­ for a constant minimum perceptual duration had not tion exponent was very close to unity, but in none been made. of the studies was there any attempt to compare Support for a power relation and for a direct the goodness-of-fit of the power function with that linear relation as the psychophysical law has been of a linear function. Kaner and Allan (Note 3) have claimed. The main support for a power function has made this comparison. They obtained magnitude been derived from ratio-setting data and from magni­ estimation data from 11 well-practiced subjects for tude and verbal estimation data. Bjorkman and three ranges of duration values. All subjects generated Holmkvist (1960), Eisler (1975), and Ekman (Note 1; estimates for stimulus durations between 20 and Ekman & Frankenhauser, Note 2) developed models 300 msec (short range) and for durations between for the analysis of ratio-setting temporal data. These 400 and 8,100 msec (long range); 10 of the subjects models provide discrepant estimates of the power generated estimates for durations between 20 and function exponent. For example, the average expo­ 8,100 msec (full range). nent reported by Bjorkman and Holmkvist (1960), Kaner and Allan estimated for each subject the by Ekman and Frankenhaeuser (Note 2), and by exponent n of the power function by determining the Frankenhaueser (1960), is considerably larger than slope n of the best fitting linear function in log­ 1.00, whereas the average exponent reported by log coordinates. For 29 of the 32 functions, the Eisler (1975) is about .84. While the average exponent exponent n was less than 1.00, with a median val­ over many studies derived from Eisler's model is ue of .767. Kaner and Allan also determined the about .90 (Eisler, 1976), there is considerable between­ . best fitting linear function, and found that for 23 subject (Allan, 1978; Eisler, 1975) and between­ of the 32 cases the coefficient of determination, 2 experiment (Eisler, 1976) variability. Further, r , was larger for the linear function than for the Allan (1978) has shown that the same exponent value log-log linear function (p < .05). That is, a sim­ is not derived from reproduction, half-setting, and ple linear function often fit the data better than double-setting data. While Eisler (1975) does report a power function, even in cases where the size of some high correlations among ratio-setting exponents, the exponent appears to rule out the linear function. he found that the correlation between exponents These results indicate that arguments against linear estimated from reproduction data and from half­ models for time based solely upon the size of the setting data was only .14. power function exponent should be regarded with Allan (1978) and Blankenship and Anderson caution. (1976) have pointed out that, even though the Eisler (1975) and Michon (1967) found that their ratio-setting models for duration have two parameters, data, when plotted in log-log coordinates, appeared it is not possible to obtain estimates for both param­ curved, and they fit their data with two linear, log­ eters from conventional ratio-setting data. The log segments. Michon allowed both the slope and the parameter of interest is the exponent of the power intercept of the two segments to be different, while function. The other parameter represents the propor­ Eisler allowed only the intercept and the location of tional relationship between the duration values the point of intersection, the breakpoint, to vary. generated by S and presented by E. In the ratio­ Kaner and Allan also found that two linear, log-log setting models, it is assumed that the experimenter­ segments provided a better description of their data defined proportion (P) and the value adopted by S than the simple linear function. However, since the (p) are identical. Only if the relationship between P linear function accounted for most of the variance and p is assumed, can an estimate of the exponent (mean r2 = .99), the improvement was negligible, be obtained. The conclusion derived from ratio­ especially since the two linear, log-log segments setting data, that time perception is not veridical, is allowed for five parameters (two intercepts, two . as valid as this assumption. Based on her review of exponents, and one breakpoint value), whereas the ratio-setting models, Allan (1978) suggested that, linear function allows for only two parameters. Also, given the current status of model development for there was no between-subject consistency in the value ratio-setting temporal data, these data should not be of the breakpoint, or any sensible relationship among the basis for any firm conclusion regarding the form the breakpoint values for the different ranges. Fur­ of the psychophysical law in time perception. thermore, it should be noted that a satisfactory theo­ A large number of studies have concluded that retical rationale for a breakpoint analysis is not verbal and magnitude estimates are a power function available. of stimulus duration (e.g., Bobko, Thompson, & In all published verbal and magnitude estimation Schiffman, 1977; Eisler; 1975, 1976; Jones & MacLean, studies of time, the empirical relationship between 1966; Kunnapas, Hallsten, & Soderberg, 1973; the subject's estimates and stimulus duration has THEPERCEPTION OF TIME 343 been interpreted as a direct reflection of the psycho­ developed. The exponent derived from magnitude physical law. It has been assumed, implicitly or ex­ estimation data is frequently close to unity, and plicitly, that the obtained function represents the Kaner and Allan (Note 3) have demonstrated that transformation of stimulus time to perceived time, even when the exponent deviates considerably from and that the subject's response is a simple linear 1.00, a two-parameter linear function often provides transformation of perceived time. This assumption a better description of the data than a two-parameter has been questioned. For example, Ekman (1964) power function. Category-rating and discrimination pointed out that if the internal representations of the data are consistent with a linear relationship between response value and of the stimulus value were both perceived time and stimulus time. logarithmically related to the external values, the power law would result. Weber's Law Moreover, it has been shown that for other stimulus According to Weber's law, the just noticeable dif­ dimensions, responses are not necessarily a simple erence, lJ., between two stimulus durations is a linear transformation of the internal stimulus values. constant proportion, k, of do, the shorter of the two The extensive work of Curtis and of Rule (e.g., values. In a discrimination task, the value of lJ. can Curtis, 1970; Curtis, Attneave, & Harrington, 1968; be estimated from the slope of the psychometric Rule, Curtis, & Markley, 1970) has shown that mag­ function, and one statistic that has been used is the nitude estimation data can be analyzed in terms of standard deviation, SD(d). Thus, if Weber's law at least two stages: an input stage that transforms holds, the standard deviation of the distribution of stimulus values to perceived values and an output temporal intervals generated in a scaling task should stage that transforms perceived values to responses. be directly proportional to the mean duration value Recent work on magnitude estimation of loudness being generated. (e.g., Braida & Durlach, 1972; Durlach & Braida, Allan and Kristofferson (1974a) and Woodrow 1969; Green & Luce, 1974; Green, Luce, & Duncan, (1951), who reviewed the relevant literature, con­ 1977; Jesteadt, Luce, & Green, 1977; Luce & Green, cluded that there was little support for this simple 1974) also points to the necessity of a more detailed version of Weber's law. In scaling studies, the analysis of magnitude estimation data. None of these standard deviation of the generated response distribu­ approaches have been applied to magnitude estimation tion increases monotonically with but not propor­ of temporal intervals. tionally to the mean response value (e.g., Doehring, However, a beginning in this direction has been 1961; Woodrow, 1930, 1933). In discrimination made with category-rating data. Curtis and Rule studies, the just noticeable difference is an increasing (1977) applied their two stage model to category function of do, but the relationship is not usually ratings of temporal intervals ranging from about .5 linear (e.g., Abel, 1972a, 1972b; Blakely, 1933; to 10 sec, and Blankenship and Anderson (1976) Creelman, 1962; Henry, 1948; Kinchla, 1972; made use of the functional measurement approach Small & Campbell, 1962; Stott, 1933; Tanner, Patton, (Anderson, 1977) with temporal intervals ranging & Atkinson, 1965). from 0 to 30 sec. The data from both studies are Allan and Kristofferson (1974a) summarized data consistent with a linear relationship between perceived from their laboratories which indicated that, for well­ duration and stimulus duration. practiced subjects, lJ. remained invariant over a wide The duration discrimination data also provide range of do values (Allan & Kristofferson, 1974b; support for a linear relationship. A number of Allan et al., 1971; Rousseau & Kristofferson, 1973; decision theory models specify that the expected Kristofferson, Note 4). More recently, Hopkins value of the internal duration distribution is a linear (Note 5) and Kristofferson (1976) have reported syn­ function of stimulus duration (Allan, Kristofferson, chronization data which show that the variability of & Wiens, 1971; Creelman, 1962; Kinchla, 1972). A the generated response distributions is constant for large quantity of data has been analyzed in terms of synchronization intervals from about 200 to 550 msec. these models, and in general the data are in accord Thomas and Brown (1974) have reported reproduc­ with the linearity assumption. Furthermore, tion data that also indicate constancy in the standard Kristofferson (1977) has shown that, under some deviation of the response distributions: While the circumstances, duration discrimination data are average standard deviation of reproduced intervals compatible with the real-time criterion model which was larger for a stimulus range of 4,500-5,500 msec states that, for the purpose of duration discrimination, than for one of 750-1,750 msec, there was no sys­ there is no transformation of stimulus time into tematic relationship between the average standard psychological time. According to this model, internal deviation and stimulus value within either range. time is real time. Probably the data most often cited in favor of In summary, an adequate model for the analysis Weber's law are those of Treisman (1963), generated of ratio-setting temporal data has not yet been using the methods of estimation, production, repro- 344 ALLAN duction, single stimuli, and comparison. However, and Brown (1974) could not be described by Equa­ Treisman concluded that his data did not support tion 2. Clearly, a viable theory for time perception proportionality, though they could be described by will have to account both for data that demonstrate SD(d) = k(d+a). invariance and for data that are consistent with Four recent studies (Divenyi & Danner, 1977; Getty's generalized form ofWeber's law. Getty, 1975, 1976; Thompson, Schiffman, & Bobko, 1976) have provided data that .have been interpreted The Time-Order Errorin TemporalJudgments as supportive of Weber's law. Thompson et al. In scaling and discrimination tasks involving the (1976)determined the Weber fraction for three dura­ sequentlal presentation of two or more duration tion ranges. They found the Weber fraction to be values, it has frequently been reported that the order .056, .053, and .043 for stimulus duration ranges of presentation influences performance. Most of the centered at 1, 2, and 3 sec, respectively, although an systematic work on presentation order effects has analysis of variance on the individual subject Weber used the method of comparison, in particular the fractions indicated a nonsignificant effect due to forced-choice (FC) procedure (e.g., Blakely, 1933; duration. Philip, 1947; Stott, 1933, 1935; Woodrow, 1935; If Weber's law holds, the function relating log Woodrow & Stott, 1936). In a typical FC task, there !J. to log do should have unit slope. Divenyi and Danner are two orders of presentation: either the shorter (1977) obtained a slope of .93 for data averaged over duration follows the longer by t msec (StSo) or the three subjects, and concluded that their data sup­ longer follows the shorter (SoSt). Correct judg­ ported the view that Weber's law holds for time per­ ments can be summarized by two conditional prob­ ception. However, their function is based on only abilities, P(Rto I s.s.i and P(Rot I s.s.i, where three values of do. . RIO indicates a judgment that the longer dura­ The data that most strongly support Weber's law tion was presented first, Rot the reverse. The signed are those of Getty (1975, 1976). He argued that difference between the two conditional probabilities, variability in a discrimination task arises from several P(R IO I StSo) - P(Rot I SoSt), is called the time-order sources, only some of which are dependent on error (TOE). Early studies found that the TOE was stimulus magnitude. If the variance of the observed duration-dependent. For brief durations, TOE was psychometric function is partitioned into two compo­ often positive; for longer durations, it was often nents, negative. The duration that resulted in zero TOE was referred to as the indifference interval. (1) The basic interpretation of TOE offered by Woodrow (1935)was that the perceptual duration of where VR represents the stimulus-independent vari­ the first-presented duration "gravitated" towards a ance, taking the square root of Equation 1 results in remote standard, the indifference interval. The what Getty refers to as a generalized form of Weber's greater the discrepancy between the presented law, stimulus duration and the indifference interval, the greater the gravitational effect. This explanation (2) places the source ofTOE in perceptual memory. Stott (1933, 1935), Woodrow (1935), and Woodrow Data from two well-practiced subjects were in and Stott (1936) found that a negative TOE became excellent agreement with Equation 2 for duration less negative and a positive TOE became less positive values between 50 and 2,000 msec (Getty, 1975), and with practice in a comparative judgment task involv­ data from a synchronization task were in better ing a number of variable durations. They suggested agreement with Equation 2 than with a model that that assimilationof the perceivedduration of the first­ specifies a proportional relation between the variance presented duration to the average value of all the of the produced intervals and stimulus duration duration values attenuates TOE with practice. (Getty, 1976). Allan and Kristofferson (1974a) summarized the­ In summary, the conclusion reached by Allan and data from more recent studies involving comparative Kristofferson (1974a) and by Woodrow (1951) is still judgments of brief stimulus durations (Allan, justified: Weber's law in its simple form does not Kristofferson, & Rice, 1974; Carbotte, 1973; Creelman, hold for time perception. Getty's generalized form of 1962; Small & Campbell, 1962). For a constant set Weber's law has received strong support, and it of duration values, for some subjects no TOE was would be of value to reanalyze existing data in terms observed, for some it was positive, and for others of Equation 2. However, the data reported by Allan negative. Such results suggested that the source of (Allan & Kristofferson, 1974b; Allan et al., 1971), TOE might be in the decision process rather than in by Hopkins (Note 5) , by Kristofferson (Note 4, 1976; perceptual memory. However, the investigation of Rousseau & Kristofferson, 1973), and by Thomas TOE was not the major purpose of these studies, THEPERCEPTION OF TIME 345 and it has since been demonstrated that use of process. A similar counterargument can be made for trial feedback could have influenced the subject's the instruction data reported by Jamieson and responding in such a way as to attenuate or even Petrusic (1975a, 1975b, 1976, 1978). eliminate any systematic relationship between the size Other arguments against a decision process source and sign of TOE and the stimulus duration values of TOE have also been advanced. Jamieson and being compared (Jamieson & Petrusic, 1976, 1978). Petrusic (1975a) note that, in the absence of trial A number of studies designed specifically to inves­ feedback, TOE occurs in the predicted direction for tigate TOE in duration discrimination tasks have individual subjects, and argue that such between­ recently been reported (Allan, 1977; Hellstrom, 1977; subject consistency implies the unlikely event that all Jamieson, 1977; Jamieson & Petrusic, 1975a, 1975b, subjects adopt a similar response bias or criterion. As 1975c, 1976, 1978). Hellstrom (1977) and Jamieson a counterexample, Allan (1977) suggested a model and Petrusic (1975a, 1975b, 1976, 1978) argued that, that places the source of TOE in the decision process if the source of TOE was response bias, TOE should and predicts that TOE should be positive for short be dependent on the particular mode of responding, durations and negative for long. whereas if TOE was a perceptual or memory phe­ In her experiment, Allan used a roving-standard nomenon, it should be independent of response design. Jamieson (1977) has argued that data from a instructions. Hellstrom (1977) ran four groups of roving-standard design reflect primarily assimilation subjects: The 1LS group was instructed to make a effects rather than TOEs. In assimilation, the longer/ shorter decision about the first duration memory of the first duration is influenced by the presented and the 2LS group about the second dura­ average duration value presented, so that durations tion presented; the L12 group had to decide which longer than the average decrease in magnitude while duration (first or second) was longer, and the S12 held in memory and durations briefer increase. group which duration was shorter. Hellstrom con­ Jamieson reanalyzed Allan's data in terms of three sidered a decision theory model which represented versions of Luce and Galanter's (1963) basic choice the subject as comparing the two internal duration model. To allow for presentation-order effects in the values by computing a duration difference value form of TOEs to occur, Luce and Galanter sug­ which was then compared to values of a difference gested that a parameter representing a multiplicative criterion. He correctly argued that if TOE was simply bias be associated with the second-presented stimulus. a matter of response bias or preference (longer vs. Jamieson developed an assimilation version of the shorter or first vs. second), then the various instruc­ choice model by introducing a parameter to represent tions should yield different response patterns and the memory distortion of the first-presented stimulus. therefore differential TOE results. For example, if The third version of the choice model contained both the subject has a preference for making a longer the TOE and assimilation parameters. All three ver­ response over a shorter response, then under the sions provided a better account of Allan's roving­ 1LS condition this would lead to more correct standard data than did the model that she suggested. responses to S,So than to SoS, and a positive TOE, Jamieson (1977) concluded that there were at least while under the 2LS condition it would lead to more two presentation order effects, TOE and assimilation. correct responses to SoS, than to S,So and a negative What is puzzling about his reanalysis of Allan's TOE. Hellstrom used two duration ranges, one (1977) data is that while he (Jamieson & Petrusic, centered at I sec, the other at 2 sec, within the same 1975a, 1975b, 1976, 1978) has consistently argued experimental session. He found a positive TOE for that TOE is not a decision criterion or response bias 1 sec and a negative TOE for 2 sec for all four groups phenomenon, by using the Luce and Galanter bias of subjects, and concluded that his data ruled out the version of the choice model, he treated TOE as a decision process as the source of TOE. response bias phenomenon. Furthermore, if TOE is Hellstrom's (1977) conclusion is not justified if the neither assimilation nor bias, what is it? decisionprocess is conceptualized in a slightly different In retrospect, it could be argued that the approach manner. For example, suppose the subject sets a taken in recent research on TOE is counter-productive. criterion difference value and labels as long-short An argument over whether the source of TOE is in (RIO) all differences larger than, and as short-long perceptual memory or in the decision process, out­ (Ro,) all differences smaller than, the criterion value. side the framework of a general theory, is likely to RIO is translated to longer by the 1LS group, to shorter generate many data but few firm conclusions. by the 2LS group, to first by the L12 group, and to Jamieson and Petrusic (1975a, 1975b, 1975c, 1976, second by the S12 group. There is no reason to 1978) have not provided a theoretical account for suppose that the different instructions would influence their provocative data. Hellstrom's (1977) adaptation/ the placement of the criterion. Hellstrom's data are weighting model is by far the most systematic theo­ incompatible with the particular decision process that retical account of TOE. Each of the two internal he describes, but not necessarily with any decision duration values generated on a trial is modified by 346 ALLAN the adaptation level or mean subjective duration, and exist. Tanner et al. (1965) report that for duration differential weights are then applied to the modified values around .5 sec a visual filled interval was internal values. The response is based on a compari­ judged as longer than an auditory filled interval, and son of these weighted internal values with a criterion that for longer durations judged duration was not internal value. The model predicts that the sign affected by modality. Hawkes et al. (1961), Bobko and magnitude of TOE should be dependent upon et al. (1977), and Brown and Hitchcock (1965) did the duration values being compared, and Hellstrom not find a modality effect. presents various versions of the model which, in A filled interval is judged as longer than an empty general, provide a good description of his data. interval of the same stimulus duration, the filled Further, Hellstrom (Note 6) identifies the differential duration illusion (Craig, 1973; Goldfarb & Goldstone, weights associated with the first and second duration 1963a; Goldstone & Goldfarb, 1963; Steiner, 1968), stimuli with retroactive and proactive interference ef­ and an empty interval containing intervening discrete fects, respectively. His interpretation of the weight elements is judged as longer than a completely empty parameters is consistent with the findings of Jamieson interval, judged duration increasing with the number and Petrusic (1975a, 1975b, 1976, 1978) that TOE, of interveningelements (Adams, 1977; Buffardi, 1971; whether positive or negative, tends toward zero as Israeli, 1930; Jones & MacLean, 1966; Schiffman & the interduration interval, t, is increased. Interference Bobko, 1977; Thomas & Brown, 1974). Brown and effects should decrease with increasing t, and there­ Hitchcock (1965) found that visual filled intervals fore so should TOE. alternating between two intensities, and auditory At its present stage of development, Hellstrom's filled intervals alternating between two intensities model is directed towards providing an account of and frequencies, were judged as longer than constant TOE and does not address other aspects of perfor­ filled intervals. mancein time perceptiontasks. The only generaltheory Other features of the signal marking the temporal for time perception that incorporates TOE is Eisler's interval havebeen shown to influencejudged duration. (1975). As Eisler himself notes, his TOE expression It has usually been found that judged duration is has many degrees of freedom. More important, an directly related to the energy levelof the filled interval evaluation of the TOE aspect of Eisler's model has (Berglund, Berglund, Ekman, & Frankenhaeuser, not been reported. Future research should place 1969; Goldstone & Lhamon, 1974; Goldstone, greater emphasis on incorporating TOE into general Lhamon, & Sechzer, 1978; Steiner, 1968; Zelkind, theories of time perception. 1973); however, Treisman (1963) reports a decrease in produced and reproduced intervals with increasing TheRole of Nontemporal Information tone intensity. Woodrow (1928) showed that the It is welldocumented that intervals having identical duration of the marker of an empty interval in­ stimulus durations are not always judged as equal in fluenced its perceived duration: Lengthening either perceived duration. Rather, the judgment is influenced the onset marker or the offset marker results in an by such nontemporal characteristics of the marker increase in judged duration, with the duration of the as its modality, nature (filled vs. empty), energy, onset marker having the larger effect. Schiffman and and complexity. In this section, the effect of these Bobko (1974) found that the more ordered and variables on the judgment of temporal intervals simple the flash pattern of eight lights marking a presented in the visual and auditory modalities will temporal interval, the shorter the reproduced dura­ be considered. tion. Thomas and Cantor (1975, 1976) found that Goldstone and Goldfarb (Goldfarb & Goldstone, judged duration was directly related to visual area 1964; Goldstone, 1968; Goldstone, Boardman, & of circles marking temporal duration. In a later Lhamon, 1959; Goldstone & Goldfarb, 1963, 1964a, study, Cantor and Thomas (1977) found that judged 1964b; Goldstone, Jernigan, Lhamon, & Boardman, duration was directly related to the area of a visual 1959; Goldstone & Lhamon, 1974) have reported data form and inversely related to the perimeter. showingthat a filled auditory interval is judged longer "Cognitive" variables have also been shown to in­ than a filled visual interval of the same stimulus fluence judged duration. Thomas and Weaver (1975) duration. This modality difference has been demon­ found that a brief tachistoscopic presentation of a strated using a number of different tasks, over a wide word or of a "nonsense" word is judged as longer range of duration values, for filled visual intervals of than the presentation of a blank field. Avant and various intensities and wave lengths, and for filled Lyman (1975) and Avant, Lyman, and Antes (1975) auditory intervals of various intensities and frequen­ report that the brief presentation of a three-letter cies. Other investigators have also reported that a nonword was judged as longer than that of a three­ filled auditory interval appears longer than a filled letter word, which in turn was judged as longer than visual interval (e.g., Behar & Bevan, 1961; Stevens that of a single letter. That is, the more "familiar" & Greenbaum, 1966). However, conflicting data do the stimulus, the shorter the judged duration. Devane THE PERCEPTION OF TIME 347

(1974), Warm, Greenberg, and Dube (1964), and 1973; Kristofferson, 1977; Kristofferson & Allan, Warm and McCray (1969), also found an effect of 1973; Rousseau & Kristofferson, 1973; Kristofferson, stimulus familiarity on judged duration. However, Note 4) makes use of filled and empty visual and their data indicate that judged exposure time for auditory intervals. These experiments show that familiar words was longer than that of unfamiliar discriminability is not constant across the vari­ words. ous types of intervals. Thus, while discrimina­ Clearly, judged duration is very much influenced bility of filled and unfilled intervals is independent by the way in which the temporal interval is defined of the energy values of the signals, discriminability or marked. This might suggest that in a duration appears to be dependent upon modality and upon discrimination task the subject bases his judgment on whether the interval is filled or empty. A clearer some aspect of the presented interval other than its picture of this dependency is needed. temporal extent. Allan and Kristofferson (l974a) In summary, there is an abundance of data on the reviewed a large number of duration discrimination role of nontemporal information in the perception of studies over a wide range of duration values. They time, and a fairly clear empirical picture emerges. concluded that, for filled visual or auditory intervals, However, little effort has been directed at explaining discrimination performance was independent of the the data and research directed at providing a theo­ intensity, frequency, or bandwidth characteristics of retical base for the empirical relationships is needed. the signal as long as the signal was readily detectable Of particular importance would be studies designed (Abel, 1972a; Allan et aI., 1971; Creelman, 1962; to determine why certain variations in the marker Henry, 1948; Small & Campbell, 1962). Similarly, have a dramatic effect on scaling performance but discrimination of empty intervals was independent of leave discrimination performance unaffected. the temporal or intensity values of the visual or auditory signals that bound the interval to be judged (Abel, 1970; Carbotte & Kristofferson, 1973; Nilsson, MODELS FOR TIME PERCEPTION 1969, 1979; Rousseau & Kristofferson, 1973). The only possible exception were Abel's (1972b) data, Only those quantitative models concerned primarily which indicated better discrimination of empty audi­ with the perception of brief temporal intervals are tory intervals as the intensity of the noise markers considered in this section. The interesting and influ­ was increased. ential theorizing of investigators concerned mainly Recent data from discrimination studies of empty with relatively long temporal intervals (e.g., Doob, auditory intervals support the conclusions of Allan 1971; Fraisse, 1963; Michon, 1972, 1978; Ornstein, and Kristofferson (l974a). Divenyi and Danner (1977) 1969) and models which are concerned with one par­ conclude that for empty auditory intervals longer ticular issue (e.g., Bjorkman & Holmkvist, 1960; than 80 msec "temporal discrimination performance Hellstrom, 1977; Ekman & Frankenhaeuser, Note 2) was little affected even by large changes in the acous­ are not discussed. tic properties of markers" (p. 139). Penner (1976) The models to be considered assume, implicitly found that discrimination performance for empty or explicitly, that a common mechanism underlies auditory intervals deteriorated only when the tem­ time perception in the visual and auditory modalities. poral and intensity values of noise markers were This assumption is supported by a number of empiri­ chosen randomly and independently for onset and cal findings. Loeb, Behar, and Warm (1966) found offset from trial to trial, compared to a control that intermodal correlations of category-ratings of condition of fixed intensity and duration markers. durations were of about the same magnitude as intra­ It is somewhat surprising that there are no systematic modal correlations. Warm, Stutz, and Vassolo (1975) comparisons of the discriminability of filled vs. demonstrated transfer effects in the reproductions of empty intervals, or of visual vs. auditory signals. The temporal intervals between the two modalities. little evidence that does exist suggests that discrimin­ Eijkman and Vendrik (1965) showed that detection ability may be affected. Abel (1972a, 1972b) found of an increment in the duration of visual and audi­ the discrimination of filled auditory intervals to be tory signals is completely correlated, and argued that better than that of empty ones. Tanner et al. (1965) this correlation indicates the existence of a common used a FC task and found superior discriminability of "duration" center. filled intervals for auditory than for visual signals. Creelman (1962) developed the first quantitative Accuracy was worst when one filled intervals was model for duration discrimination, and a number of auditory and the other visual. Goodfellow (1934) the more recent models derive from his counting reports that the auditory is less than the visual dif­ model. These models are primarily concerned with ference threshold. The research of Allan and of the psychophysical law and with Weber's law. In the Kristofferson (Allan et al., 1971, 1974; Allan & Creelman model, the subject bases his judgment on Kristofferson, 1974a, 1974b; Carbotte & Kristofferson, the number of pulses that are accumulated during a 348 ALLAN temporal interval. The source of these pulses consists dard deviation of the internal distribution. A specific of a large number of independent elements, each with model that predicts proportionality between the a fixed probability of firing at any moment.The ex­ standard deviation of the internal distribution and pected number of pulsesaccumulated during a d-msec stimulus duration is Gibbon's (1977) scalar expectancy interval and the variance in the number of accumu­ theory, which has had considerable success in the lated pulses are both equal to Ad, where Ais a con­ animal time literature. However, Gibbon's model is stant representing the rate of firing of the pulse difficult to apply to the human duration discrimina­ source. The distribution of the number of accumulated tion task, sinceits underlying process is reinforcement­ pulses is Poisson, but for large Ad this distribution dependent. is approximately normal. Allan and Kristofferson (1974a) summarized dura­ Most data analyzed in terms of the Creelman tion discrimination data from their laboratories, model have found that Ais not constant across the indicating a linear relationship between the expected d values used (e.g., Abel, 1972a, 1972b; Allan et aI., value of the internal duration distribution and stimu­ 1971). These data have resulted in modifications of lus duration, and a standard deviation invariant over the basic Creelman model. For example, Divenyi and a considerable range of duration values. According Danner (1977) explored the possibility that A was to the onset-offset model proposed by Allan et al. related to d by a negative exponential function. I (1971), all the variability in the internal duration The expansion of Creelman's one rate parameter, values generated by a fixed stimulus duration is the A, to three parameters plus the inclusion of an addi­ result of variation in the time, with respect to stimu­ tional parameter results in a four-parameter model. lus time, at which the internal duration begins and Divenyi and Danner found that their model provided ends. For any stimulus duration, d, the perceptual a reasonable description of their data. However, onset and offset latencies are each uniformly and observed performance was consistently superior to independently distributed over a range of' q msec, predicted. resulting in a triangular distribution of internal dura­ Kinchla (1972) proposed a model similar to tions spanning 2q msec, with an expected value pro­ Creelman's in predicting that the variance in the portional to stimulus duration and a variance, q2/6, number of accumulated pulses was proportional to independent of stimulus duration. stimulus duration, but different in that the expected The onset-offset model specifies that the variability number of pulses was not equal to the variance but in the onset and offset perceptual latencies is inde­ to stimulus duration. Kinchla did not propose a pendent of stimulus duration, and this independence process which would generate these relationships. appears to be true over a considerable range of Also, the fit between the theory and her data is not stimulus durations. However, the variability does particularly good. change betweenstimulusranges (Allan & Kristofferson, Getty (1975, 1976) questioned the linear relationship 1974b; Kristofferson, Note 4). This implies that the between variance and stimulus duration predicted by variability in the onset latencies is a function of the Creelman model and proposed, as an alternative, stimulus duration, an unlikely possibility. Allan and his Weber's law model, for which he presents strong Kristofferson (1974a) suggested, as an alternative, supporting data. According to Getty, the standard that the variability represented by the triangular dis­ deviation of the psychometric density function obeys tribution is part of the criterion mechanism, and Weber's law. There are two classes of models­ Kristofferson (1977) has developed this alternative as logarithmic and proportionality-which are consis­ the real-time criterion model. He showed that under tent with this relationship, and he did not choose some circumstances duration discrimination can be between them. Logarithmic models assume that interpreted as temporal order discrimination. Ac­ stimulus duration is transformed logarithmically into cording to the model, the internal onset of the dura­ internal duration, resulting in an internal distribution tion stimulus triggers an internally timed interval, the with a constant standard deviation and an expected criterion. If the criterion interval ends before the value that increases logarithmically with stimulus internal offset of the duration stimulus, the subject duration. The human time-perception literature responds long; if the duration stimulus ends before provides no support for a logarithmic psychophysical the criterion interval, the subject responds short. law. For example, Kaner and Allan (Note 3) found According to this model, errors in duration dis­ that in only 1 of 32 cases did a logarithmic func­ crimination are due entirely to variability in the tion between magnitude estimates and stimulus dura­ criterion, and the variance of the triangular dis­ tion provide a better fit to the data than either a tribution of criterion values depends upon the range power function or a linear function. Proportionality and the values of the stimulus durations used in the models assume that both the expected value and the discrimination task. Thus, for the data summarized standard deviation are proportional to stimulus dura­ by Allan and Kristofferson (l974a), a range of tion. In this case, Weber's law applies to the stan- stimulus duration values appears to have the same THE PERCEPTION OF TIME 349 variance merely because the variance arises from the need for both an input and an output function. The internal criterion interval, a source which is the same Eisler model does not contain an output function. for all the values. Although the real-time criterion Furthermore, Curtis and Rule (1977) found that per­ model has not been extensively tested, it does provide formance in a category-rating task involving the an excellent fit to the data reported by Kristofferson simultaneous presentation of two temporal intervals (1977). was very different from performance when the two The models discussed above were developed to intervals were presented sequentially. According to account for data from discrimination tasks and are the Eisler model, the monitoring is supposed to be concerned primarily with the psychophysical law and simultaneous even when the intervals are presented with Weber's law. At about the same time that sequentially, and therefore it is difficult to reconcile Creelman (1962) proposed his counting model for the Curtis and Rule data with this aspect of the duration discrimination, Treisman (1963) introduced Eisler model. a general theory for time perception. His internal Thomas and Brown (1974) have presented a general clock model was proposed to account for both framework that provides a schema for the interrela­ scaling and discrimination data. According to the tion of data from different time perception tasks. model, a pacemaker produces a regular sequence of Stimulus input is encoded as a vector, one component pulses at a constant rate which can be affected by being an encoded, and the other the decoded, dura­ the arousal level of the subject. These pulses are tion. They show that if the encoding function is of counted and are either read into the "store" for a particular form, and if decoding is the inverse later retrieval by the comparator or directly entered of encoding, then a number of specific predictions into the comparator. The postulated characteristics of about d I in the SS task can be made. In particular, the pacemaker, the counter, the store, and the com­ d I will be (1) directly related to the absolute differ­ parator yield predictions about Weber's law, the in­ ence between presented and reproduced intervals in difference interval, and other findings in the time­ a reproduction task, (2) inversely related to the vari­ perception literature. Although the Treisman paper is ability of the reproduced intervals, and (3) unrelated frequently cited, there are no models which derive to the slope of the curve relating reproduced and from it and little if any data have been analyzed presented intervals. They also show that a reversible in terms of it. encoding function predicts average reproduced inter­ Eisler (1975) has proposed a general model for vals that are linear with clock time, accounts for the time perception which was developed to account for overestimation of short intervals and the underesti­ ratio-setting, magnitude estimation, and discrimina­ mation of long ones, and accounts for the filled dura­ tion data. The model is based on the power law as tion illusion. As they note, the appropriate data for the psychophysical law. The comparison on each trial testing many of the predictions concerning the inter­ is based on the simultaneous monitoring or parallel relation among various time-perception tasks do not processing of two internal duration values-the exist. internal value of the total duration (standard dura­ In later papers, Cantor and Thomas (1977), tion plus variable duration) and the internal value of Thomas and Cantor (1975, 1976, 1978) and Thomas the variable duration. The internal standard dura­ and Weaver (1975) have examined the role of non­ tion, which is the difference between the internal temporal information in the processing of temporal value of the total duration and the internal value of information. Thomas and Weaver (1975) suggested the variable duration, is continuously calculated and that perceived duration was influenced by the time compared to the internal value of the variable. The spent processing nontemporal information. Their overt response is based on this comparison. Eisler model assumes that a stimulus of duration d, consist­ argues that the subject always makes a comparative ing of nontemporal information, I, is analyzed by judgment, even when only one stimulus duration is a timer and by an information processor. The output presented on a trial. In the SS task, it is assumed of the timer is a temporal encoding which is directly that the stimulus value on trial n is compared with related to the stimulus duration, d; the output of the the stimulus value on trial n - 1. processor contains encodings of the nontemporal Eisler has not yet applied his model to discrimina­ stimulus features and an encoding or memory of the tion data. We have already noted one difficulty with time spent processing I. Attention is shared between Eisler's model in its application to ratio-setting data. the timer and the processor, and the values and the In order to apply the model, it is necessary to as­ reliability of the two outputs are a function of the sume either a value for the exponent of the power amount of directed attention. Perceived duration is function or that the subject's ratio is identical to the a weighted average of the output of the timer and the experimenter's ratio. When the model is applied to encoding of the time spent processing I. Thus far, magnitude estimation data and category-rating data, the application of the model has been restricted to the work of Curtis and Rule (1977) emphasizes the d< 100 msec. 350 ALLAN

Thomas and Weaver (1975) demonstrated that test stimuli which differ in duration is followed, after manipulations to increase the time spent processing I a variable interval, by a second stimulus, the mask. and to increase the amount of attention devoted In the simplest version of the task, the subject is to the outcome of the processor led to an increase required to identify the test stimulus as long or short. in the perceived duration of a fixed-value stimulus Performance is influenced by the duration of the duration. Cantor and Thomas (1977) and Thomas interval between the test stimulus and the mask and Cantor (1975, 1976) have presented data relating (lSI) and by the duration of the mask. Idson and judgments of perceived duration and perceived size Massaro (1977) and Massaro and Idson (1976, 1978) which are in accord with the Thomas and Weaver have shown that increasing lSI increased the correct proposal that perceivedduration is related to the time identification of the long test stimulus, but decreased spent processing the nontemporal features of the that of the short test stimulus. Furthermore, correct stimulus. identification on no-mask relative to mask trials was Gomez and Robertson (1979) report that the size high for the short stimulus and low for the long of the visual stimulus did not influence perceived stimulus. duration when size was a between- rather than a Massaro and Idson (1976)proposed a time percep­ within-subject variable. They appear to think that tion model based on Massaro's (1975) theory for this result is incompatible with the model described auditory recognition. An auditory stimulus is stored by Thomas and Weaver (1975). However, it is not. in preperceptual memory, and information in preper­ The model specifies that perceived duration is influ­ ceptual store is read out continuously for approxi­ enced by the time spent processing nontemporal mately 250 msec. If a second auditory stimulus, the information if attention is directed to the information mask, is presentedbefore processing is completed, the processor. When the task does not require the sub­ representation of the test stimulus is erased and its ject to process size information, it is entirely plausi­ processing terminated. Thus the perceived 'duration ble that no attention is directed at the information of the test stimulus will increase with lSI. If suf­ processor and, therefore, that physical size should ficient time is allowed for complete processing of the have no influence on perceivedduration. test stimulus, its perceived duration will be directly Thomas and Cantor (1978) present a more detailed related to the test stimulus duration; if less than com­ description of the timer. Stimulus onset, after a plete processing time is available, the test stimulus is variable delay, activates the timer, and stimulus off­ perceived as shorter than its asymptotic value. At set, after a variable delay, stops the timer. In a SS short lSI values, the long stimulus will tend to be task, the response is made after a criterion interval, inaccurately identified as short, while the short C, or at timer offset, whichever occurs first. If C stimulus will be identified accurately. With increas­ occurs first, the response is long; if the timer offset ing lSI, a greater proportion of the test stimuli occurs first, the response is short. This viewis similar will be identified as long, simultaneously increasing to the onset-offset model proposed by Allan et al. accuracy on the long, and decreasing accuracy on the (1971) in that errors are due to onset and offset short, stimuli. It is also assumed that the perceived latencies. It also incorporates Kristofferson's (1977) duration of the mask is incorporated, in an additive proposal that duration discrimination can be viewed manner, into the judgment of the perceived duration as temporal order discrimination. of the test stimulus. Since the contribution of the It should be noted that in most conceptions of time mask is to lengthen the perceived duration of the test perception, the response is dependent upon obtaining stimulus, both test stimuli will tend to be classified a measure of the temporal extent of the stimulus as short on no-mask trials, yielding excellent perfor­ duration. Two exceptions are Kristofferson's (1977) mance on the short test stimulus and poor perfor­ real-time criterion model and the Thomas and mance on the long one. Cantor (1978) description of the timer. In both cases, In the original version of the time perception the response is determined by the outcome of a race model (Massaro & Idson, 1976), the rate of growth between the termination of the criterion interval and of perceived duration was assumed to be the same the internal registration of the termination of the during lSI as during the test stimulus itself. However, stimulus. The two views differ in the source of the rate was assumed to be dependent upon the dura­ errors. For Kristofferson (1977), all of the variability tion of the test stimulus, resulting in two rate para­ is in the value of the internal criterion; for Thomas meters, Os for the short stimulus and 0L for the long. and Cantor (1978), all of the variability is in the Allan and Rousseau (1977) and Kristofferson (1977) onset and offset latencies. pointed out that since short and long test stimuli are Cantor and Thomas (1976), Idson and Massaro identical for a major portion of their extent, the rate (1977), and Massaro and Idson (1976, 1978) inves­ of growth of the perceived duration has to be identical tigated the relationship between perceived duration over this span, and Os = 0L. Consequently, Idson and processing time in a different way, by using a and Massaro (1977) and Massaro and Idson (1978) SS backward-masking task. On each trial, one of two modified the original model to include three rate THE PERCEPTION OF TIME 351 parameters: The rate of growth during the test duration values being judged. Many features of this stimulus, eD, is independent of the duration of the model need elaboration, especially the determination test stimulus, and the rates of growth during lSI, of the relationship between criterion variance and es and eL, depend upon test stimulus duration. stimulus range. Idson and Massaro (1977) and Massaro and Idson The real-time criterion model specifies that, under (1976, 1978) have had considerable success in account­ some circumstances, order information is sufficient ing for auditory temporal data. However, the varia­ for discrimination among stimuli of varying durations. bility among the parameter values estimated from the The response does not depend upon obtaining a various studies is surprising considering that the test measure of the temporal extent of the stimulus dura­ durations and the mask durations were very similar tion. It is possible that some of the discrepancies in in all the studies. For example, the estimated value the literature concerning the relation between stimulus for es was 102.36 in Massaro and Idson (1976; cited duration and discriminability arise from alternative by Massaro & Idson, 1978), 40.51 in Massaro and ways of performing in a time judgment task, specifi­ Idson (1978), and 25.00 in Idson and Massaro cally on whether the response is based on an order (1977). judgment or on the temporal extent of the duration The Massaro-Idson model falls into the category stimulus. of erasure or interruption models for backward masking. Cantor and Thomas (1976) offer an inte­ SUMMARY gration model for their backward masking data. They suggest that the mask and the test stimulus are There are now a number of fairly well established processed as an integrated configuration, whose pro­ phenomena to guide theorizing about time perception: cessing time is dependent upon lSI and on the simi­ (1) There are considerable data that are consistent larity between the test stimulus and the mask, and with a linear relationship between perceived time and influences perceived duration. Cantor and Thomas stimulus time. Data that have been interpreted as argue that an integration model and an interruption supportive of a power function as the psychophysical model make different predictions concerning the rela­ law should be regarded with caution. tionship between perceived duration and lSI. An (2) Neither discrimination data nor scaling data interruption model predicts that perceived duration are in agreement with Weber's law in its simple form. increases monotonically with lSI; an integration model Some of the existing data lend strong support to allows perceived duration to be monotonic with lSI, Getty's generalized form of Weber's law. Other data constant, or nonmonotonic, depending upon the simi­ indicate that the standard deviation of the internal larity between the test stimuli and the mask. Cantor duration distribution is invariant over a considerable and Thomas found that the perceived duration of range of duration values. one set of stimuli (two different forms) increased (3) In the absence of feedback, a strong and sys­ with lSI, whereas the perceived duration of another tematic TOE is usually observed. set of stimuli (two different-sized circles) was rela­ (4) Perceived duration is influenced by nontemporal tively constant over lSI. They suggest that experi­ characteristics of the marker. Discrimination per­ ments designed specifically to manipulate similarity formance is also dependent upon the modality and might allow a distinction between an integration and the nature of the marker; however, it is generally an interruption view of visual masking to be made. unaffected by large variations in the intensive, fre­ Theory is considerably richer now than when Allan quency, and bandwidth characteristics of the marker. and Kristofferson (1974a) summarized the duration The relationship between the effects of nontemporal discrimination literature only a few years ago. Of the variables on perceived duration and on discrimination general theories for the perception of brief temporal performance has not been studied. intervals, the work of Thomas and of Cantor is There is no theory for time perception that at­ especially exciting. Assets of their theory are its flexi­ tempts to encompass all these empirical phenomena. bility, its ability to predict relationships between Some models have been concerned primarily with scaling tasks and duration discrimination tasks, and the psychophysical law and/or Weber's law, some the emphasis placed on the influence of the type of primarily with interrelations among different kinds material and of processing time on perceived dura­ of time-judgment tasks, and some primarily with the tion. role of nontemporal variables and/or processing time Of the specific models for duration discrimination, on temporal judgments. the real-time criterion model is the only one that can address the observed invariance in discriminability. The novel feature in this model is that it is the in­ REFERENCE NOTES ternal criterion interval that is variable rather than 1. Ekman, G. Subjective power functions and the method of the internal value of the stimulus duration. The cri­ fractionation. 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