ON THE ROLE OF IMPUTED VELOCITY IN THE AUDITORY KAPPA EFFECT. Molly J. Henry. A Thesis

Transcription

1 ON THE ROLE OF IMPUTED VELOCITY IN THE AUDITORY KAPPA EFFECT Molly J. Henry A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS May 2007 Committee: J. Devin McAuley, Advisor Dale Klopfer Verner Bingman

2 ii ABSTRACT J. Devin McAuley, Advisor The auditory kappa effect is a is a systematic effect of manipulations to pitch on judgments about auditory sequence timing. Two experiments were conducted to examine the role of imputed pitch velocity in the auditory kappa effect. In both experiments, participants judged the timing of a target tone embedded in a three tone sequence (a kappa cell), while ignoring manipulations of the target s pitch. Experiment 1 examined the effects of the presence/absence of a context sequence that reinforced the pitch velocity imputed to the kappa cell on the magnitude of the auditory kappa effect. Presence of a context sequence tended to weaken the kappa effect. Experiment 2 varied the pitch velocity of sequences between trials. Generalizing findings from vision, the magnitude of the auditory kappa effect increased as pitch velocity increased. Findings are discussed with respect to the perceptual interdependence of space and time.

3 iii ACKNOWLEDGMENTS I would like to acknowledge the help and guidance provided by the Timing Research Group, specifically that of my advisor, J. Devin McAuley, and Laura Dilley. Thanks to Danielle Champney for a tremendous help with running participants and to Noah MacKenzie for answering all of my naïve questions.

4 iv TABLE OF CONTENTS Page INTRODUCTION. 1 Auditory Kappa Effect... 3 Imputed Velocity Hypothesis... 4 Local vs. Global Imputed Velocity... 7 Overview... 9 EXPERIMENT Method Design Participants Apparatus Stimuli Procedure Data Analysis Results Application of the Imputed Velocity Model Discussion EXPERIMENT Method Design Participants Apparatus... 21

5 v Stimuli Procedure Data Analysis Results Application of the Imputed Velocity Model Discussion GENERAL DISCUSSION Perceptual Interdependence Tau Effect Auditory Streaming Extrapolations Based on Constant Velocity and Acceleration/Deceleration Representational Momentum Auditory Pattern Perception Music and Speech Perceptual Independence Music Perception Amusia Conclusions REFERENCES FIGURE CAPTIONS APPENDIX A: TRAINING PROCEDURES... 65

6 vi LIST OF FIGURES Figure Page

7 1 INTRODUCTION Time and space are abstract dimensions for which there exist neither stimuli nor receptors. The perceptual experience of movement requires the extraction and integration of information about transformations along both dimensions simultaneously. In general, transformations along the dimensions of time and space are coherently and lawfully related. The tendency towards regularity in space-time transformations undergone during movement allows for the mental extrapolation of the coherent space-time trajectory to an expected location of a future event. It has been proposed that the ability to dynamically predict the future location of a moving stimulus is the mechanism that allows a person to cast out attentional thrusts forward to when in time and where in space an event may be expected (Jones, 1981). A question can then be raised as to the nature of the connections between spatial and temporal information that yields a unified, holistic perceptual experience. An intriguing possibility is that time and space are inextricably linked, and thus perception along each dimension is dependent on the other. An example of such perceptual interdependence is the kappa effect, which is the focus of this thesis. The kappa effect is the dependence of the perceived timing of events on their relative spatial orientation. The kappa effect has been primarily studied in the visual modality (e.g. Cohen, et. al., 1953, 1955; Price-Williams, 1954; Matsuda & Matsuda, 1979, 1981; Jones & Huang, 1982; Huang & Jones, 1982; Sarrazin, et. al., 2004). The canonical method is as follows: Three stimuli (a kappa cell) are flashed sequentially; the first and third (bounding) stimuli remain fixed in space and in time across trials. The spatiotemporal position of the middle, target stimulus is manipulated. The observer is asked to judge when the target occurs, ignoring where it is located. The kappa effect is a tendency for judgments of timing to be systematically influenced by the spatial position of the target. When the spatial extent delimited by two stimuli is

8 2 decreased, the corresponding time interval is often underestimated. Conversely, if the spatial extent between two stimuli is increased, the corresponding temporal interval is likely to be overestimated. Thus, duration judgments are dependent on the magnitude of the spatial separation between two stimuli that define one spatiotemporal interval. The visual kappa effect is robust to manipulations of the number of stimuli making up the sequence, the spatial and temporal separation between individual stimuli, and to variations in the task used to examine the effect. The visual kappa effect has been demonstrated using sequences made up of between two (Price-Williams, 1954) and eight (Sarrazin, et. al., 2004) stimuli, delimiting between one and seven spatiotemporal intervals, respectively. The spatial separation between individual stimuli has been varied between 3 cm (Sarrazin, et. al., 2004) and 32 in (Price-Williams, 1954). The durations between individual stimuli have been varied between 150 ms (Jones & Huang, 1982) and 11 s (Price-Williams, 1954). The visual kappa effect has been demonstrated using both perception (Cohen, et. al., 1955; Jones & Huang, 1982) and production tasks (Cohen, et. al., 1953; Price-Williams, 1954; Matsuda & Matsuda, 1979, 1981; Sarrazin, et. al., 2004). With respect to the nature of responses made by observers, variability is large between studies. A question emerges from visual kappa effect research as to whether the same dependence of time on space can be shown to hold for the auditory modality. In general, less is known about the kappa effect in the auditory versus the visual modality. Studies of the auditory kappa effect have primarily manipulated pitch space to elicit distortions of temporal judgments (Cohen, et. al., 1954; Shigeno, 1986, 1993; Crowder & Neath, 1994; MacKenzie & Jones, 2005). The general aim of this thesis is to extend previous research on the auditory kappa effect.

9 3 The remainder of this introduction is organized into four subsections. First, research on the auditory kappa effect is reviewed. Second, an imputed velocity hypothesis is considered as an explanation for the kappa effect in both the visual and auditory modalities. Third, potential roles of local and global velocity are discussed. Finally, an overview of the aims of the current thesis is provided. Auditory Kappa Effect The canonical auditory kappa task is comparable to that used in vision. A set of three tones (a kappa cell) is presented sequentially, and can be either ascending or descending in pitch. The spatiotemporal position of the middle, target tone is manipulated. The listener is asked to make judgments about when the target tone occurs, ignoring where in pitch space it is located. The auditory kappa effect is a tendency for the perceived duration of the to-be-judged time interval to be dependent on the relative magnitude of the corresponding pitch interval. Specifically, when the target tone is closer to either bounding tone in pitch space, decreasing the pitch distance between the tones, the corresponding temporal interval is generally underestimated. Conversely, when the target tone is moved farther from either bounding tone in pitch space, increasing the pitch space spanned by the interval, the corresponding duration is likely to be overestimated. The auditory kappa effect has been elicited using both perception (Shigeno, 1986, 1993; Crowder & Neath, 1994; MacKenzie & Jones, 2005) and production (Cohen, et. al., 1954) tasks. The ranges of durations, pitch intervals, and number of individual elements making up stimulus sequences has not been examined as thoroughly in audition as in vision, however. Cohen, et. al. (1954) was the first to examine the kappa effect in the auditory modality. Listeners were presented with sets of three sequentially presented tones. The location of the

10 4 middle, target tone was manipulated in pitch and in time. Listeners adjusted the timing of the target tone so that the interval between the bounding tones was temporally bisected by the target. When the target tone was closer in pitch to either of the bounding tones, the corresponding time interval was often underestimated. Conversely, moving the target tone away from either bounding tone increased the likelihood that the corresponding time interval was overestimated. Shigeno (1986) presented to listeners ascending auditory kappa cells, the middle tone of which had been shifted in pitch space and time. Listeners indicated whether the target tone was positioned closer in time to the first or third bounding tone of the kappa cell. The findings corroborated those found previously. When the middle, target tone was closer to the either bounding tone in pitch, it was perceived as being closer in time to that same tone. Conversely, when the target tone was moved farther in pitch space from either bounding tone, the perceived temporal location of the target tone was farther from that same tone. In one of few auditory kappa experiments to manipulate characteristics of the sequences within which the to-be-judged interval was embedded, MacKenzie and Jones (2005) preceded the three tone kappa cell by an isochronous, three tone context sequence that reinforced the rhythmicity implied by the bounding tones of the three tone kappa cell. The context sequence was not extended in pitch space; all tones took on the same pitch level. This manipulation was shown to eliminate the auditory kappa effect. Imputed Velocity Hypothesis What is the mechanism responsible for the perceptual dependence of time on visual or auditory space? The dominant explanation of the visual kappa effect is an imputed velocity hypothesis, which rests on the assumption that general principles applicable to real-world, moving stimuli are being applied to static stimulus displays that only imply motion (Price-

11 5 Williams, 1954; Cohen, et. al., 1955). Successive stimuli, of the sort used to study the kappa effect, take on the qualities of a unified, moving stimulus, though they are not seen or heard as moving per se, because of their simultaneous and coherent transformations in space and time. Experience of observers with natural relationships of changes in space to changes in time yields an imputation of constant velocity to the array when stimuli are regularly spaced and unfolding in time at a regular rate (Cohen et. al., 1955). Observers are able to generate expectancies about where and when an element of a sequence would occur by extrapolating the space-time trajectory implied by the fixed spatiotemporal positions of the first and third stimuli making up the kappa cell. Violations of these expectancies along one dimension, then, are perceptually compensated for by systematic distortions along the other dimension. Responses to manipulations of space and time in judgments of duration are assumed to be predictable based on the actual duration of the time interval preceding the target stimulus, and the expected duration of the same interval formed based on imputed constant velocity and the given spatial position of the target. From this perspective, a duration judgment can be considered to be a weighted combination of the observed, objective duration of the single interval preceding the target stimulus, and an expectation about the amount of time needed to traverse the corresponding spatial interval at a constant velocity (Anderson, 1974; Jones and Huang, 1982). This has been quantified by Jones & Huang (1982): R = f [ wt + (1 w) E( t)]. (1) Here, t is the actual duration of the time interval preceding the target, and E(t) is the expected duration of the same time interval based on the imputed velocity of the sequence and the given spatial extent. This follows from algebraic manipulation of the equation for velocity: V = s/ t becomes E(t) = s/v. The function f describes how the combination of the observed time interval

12 6 t, and the expected time interval E(t), is mapped onto a probability of giving a particular response. The weight parameter, w, [0,1], then quantifies the relative contributions of t and E(t) to perceived duration, and the resulting magnitude of the kappa effect. When w = 1, only the actual duration of the time interval preceding the target is used to calculate perceived duration. No perceptual distortion results, and no kappa effect obtains. Perception of the to-be-judged duration is veridical. When w < 1, however, the perceived duration of the time interval preceding the target is a weighted combination of actual time and expected time based on constant velocity and given spatial extent. A kappa effect would present under this latter set of circumstances. A smaller value of w would be associated with a larger kappa effect. Huang and Jones (1981; as cited in Jones & Huang, 1982) demonstrated that the magnitude of the visual kappa effect, as quantified by w, increased as the imputed velocity of the sequence was increased. This increase in the magnitude of the kappa effect corresponded to a decrease in w. In vision, as imputed velocity increases, responses are based more on the expected duration of the interval preceding the target, E(t), and less on objective time, t. Decreasing the imputed velocity of the sequence, on the other hand, was found to weaken the visual kappa effect. The weight parameter, w, approaches 1 with decreasing velocity. The response is then based more on objective time, t, than on expected time, E(t). The imputed velocity hypothesis is equally applicable to the kappa effect in the auditory domain. Pitch velocity is quantified as change in the distance in pitch space per change in time, V p = p/ t. The same general principles discussed with respect to the application of the imputed velocity hypothesis to vision hold here. The expected duration of the time interval preceding the target tone is now based on imputed pitch velocity and given extent in pitch space, E(t) = p/v p.

13 7 Equation 1 has been used to account for performance in auditory kappa effect tasks (MacKenzie & Jones, 2005) in addition to visual kappa effect tasks (Jones & Huang, 1982). It is hypothesized that an effect of imputed velocity similar to that demonstrated by Huang and Jones (1981; as cited in Jones & Huang, 1982) should be demonstrable in audition. If the effects in the two modalities are comparable, it should be the case that increasing imputed pitch velocity will correspond to an increase in the observed magnitude of the auditory kappa effect. Conversely, decreasing imputed pitch velocity should decrease the magnitude of the auditory kappa effect. The modulation of the magnitude of the kappa effect by imputed velocity should be apparent in changing values of w. Local vs. Global Imputed Velocity The visual kappa effect has been demonstrated to emerge in conditions much different from those used in the canonical task (Price-Williams, 1954; Matsuda & Matsuda, 1979, 1981; Sarrazin, Giraudo, Pailhous, & Bootsma, 2004). The visual kappa effect has been demonstrated when trials are composed of only two stimuli delimiting one spatiotemporal interval, or as many as eight stimuli delimiting seven spatiotemporal intervals. The kappa effect also obtains whether velocity is held constant across all trials, or is randomized between trials. The question raised, then, is whether imputed velocity is derived from a single source, or is an aggregate comprised of velocity information from multiple sources. Price-Williams (1954) made use of single intervals delimited by two flashes of light, and the durations of those intervals were reproduced by observers. By relying on two lamps, denoted A and B, it was possible to form four spatial intervals: AA, AB, BA, and BB. The target stimulus, in this case the second lamp to flash, was manipulated spatially through the assignment of either the same or different positions than the fixed stimulus. Neither the space between the

14 8 lamps nor the duration of the interval was held constant across trials. When participants reproduced the temporal intervals between the flashes, those durations marked by the fixed and target lights in different spatial positions (AB and BA) were overestimated relative to intervals marked by the fixed and target lights in the same spatial position (AA and BB). Systematic overestimation of the durations between the two stimuli demarcating the largest of the spatial intervals additionally emerged (Price-Williams, 1954). On what basis can constant velocity be imputed to single, isolated intervals? Matsuda and Matsuda (1979, 1981) presented observers with visual kappa cells, the third stimulus of which had been manipulated in space. Observers indicated by a button press the appropriate time-of-arrival of the third stimulus at a fixed spatial position that would cause the two durations delimited by the kappa cell to be equivalent. When space-time pairings of the first spatiotemporal interval were randomized, a visual kappa effect emerged. It appeared that observers were making their judgments based on given spatial extent and imputed velocity averaged over trials. However, when durations of the first spatiotemporal interval were randomized between trials, and spatial extent was held constant, no kappa effect was present. In this latter condition, a constant spatial extent was randomly paired with widely varying durations. Thus, it appeared that observers were capable of using imputed velocity averaged over trials in their judgments of duration. However, there appeared to be limits on the range of velocities over which an average could be calculated online (Matsuda & Matsuda, 1979, 1981). The use of up to seven spatiotemporal intervals can induce a visual kappa effect in a single trial (Jones and Huang, 1982; Sarrazin, et. al., 2004). Sarrazin, et. al. studied the visual kappa effect by presenting subjects with eight horizontally arranged points on a computer screen. Successive presentation of these points created seven spatiotemporal intervals. Observers

15 9 recreated the perceived pattern of durations delimited by these points. When the physical separation between any two points was manipulated, the perceived duration of the corresponding time interval was systematically biased in the direction of the spatial manipulation. Research related to the auditory kappa effect has used almost exclusively the standard three tone kappa cell presented across all trials at a constant velocity. Thus, the potentially separable influences of velocity imputed locally from single trials, and globally, averaged across trials, have not been demonstrated in the auditory modality in a kappa effect study. However, in a task that required participants to rate a comparison sequence as being Shorter, Longer, or Same relative to a standard that was part of a context sequence, Jones and McAuley (2005) found that the base rates of individual sequences and the average of rates presented throughout the session contributed jointly to patterns of listeners errors. One possibility is that similar contributions are being made to duration judgments by local and global velocity. It can be hypothesized that the expectation that produces the distortions in time judgments is a combination of velocity imputed globally, an average of velocities experienced throughout the experiment or block, and locally, from a single trial (MacKenzie & Jones, 2005). It is unknown, however, to what extent distorted perceptions such as those generated by the kappa effect are shaped separately by global and local velocity, and what stimulus conditions change the relative contribution of each. Overview Two experiments examined the auditory kappa effect. For both experiments, participants judged the timing of a target tone in a three-tone sequence (a kappa cell) while ignoring pitch. Experiment 1 examined the effects of the presence/absence of an isochronous, ascending context sequence that preceded the kappa cell on the strength of the kappa effect. If the presence of a context sequence reinforces the coherent pitch-time trajectory of the sequence and therefore the

16 10 representation of velocity of the sequence, then the presence of a context sequence should potentially increase the magnitude of the kappa effect. Experiment 2 varied pitch velocity by randomizing the rates of sequences between trials, holding pitch changes constant, in an attempt to generalize findings from visual kappa effect studies to the auditory modality. A secondary goal of this manipulation was to potentially distinguish between the contributions of local and global imputed velocity to the auditory kappa effect.

17 11 EXPERIMENT 1 Of interest in Experiment 1 was whether the presence of a context sequence preceding a kappa cell would facilitate the imputation of a constant pitch velocity to the sequences and thus potentially increase the magnitude of the kappa effect. To address this issue, participants judged the timing of the penultimate target tone in a three-tone kappa cell that was presented in isolation (context absent), or in a kappa cell that was preceded by a three-tone context sequence (context present) that reinforced the constant pitch velocity implied by the fixed spatiotemporal positions of the bounding kappa cell tones. Both the pitch level and timing of the target tone varied from trial to trial. If reinforcing the imputed velocity strengthens the auditory kappa effect, then a larger kappa effect would be expected to be found in the context present condition than in the context absent condition. Method Design Experiment 1 employed a 2 x 5 mixed factorial design. The two context sequence conditions (context present versus context absent) were crossed with five pitch levels of the target tone. Pitch of the target tone was varied by -3, -1, 0, +1, or +3 semitones (ST) relative to the expected pitch level of the target tone based on a constant imputed pitch velocity. Context sequence condition was a between-subjects factor, while pitch level of the target tone was a within-subject factor. Participants Thirty-three individuals with a range of musical training (M = 4.18 yrs, SD = 4.32) were randomly assigned to either the context absent or context present condition. Participants were students enrolled in summer classes at Bowling Green State University or members of the

18 12 Bowling Green community. All participants were adults between the ages of 18 and 65 (M = 22.79, SD = 3.62), with self-reported normal hearing. Participation was in exchange for a cash payment of $5. Data for five participants in the context present condition were excluded due to instability of their CE and/or JND estimates (see Data Analysis). Final participant numbers for the context absent and context present conditions were n = 15 and n = 13, respectively. Apparatus Stimulus generation and response collection were controlled by an IBM PC compatible computer running the MIDILAB software package, with a time resolution of 1 ms (Todd, Boltz, & Jones, 1989). Auditory sequences were presented to participants at a comfortable listening level through Grado SR-80 headphones attached to a Yamaha PSR-70 MIDI keyboard set to a piano voice. Stimuli Kappa cells were three-tone ascending sequences. All tones had onset to offset durations of 100 ms. The first and third (bounding) tones of kappa cells were always separated in pitch and time by 8 ST and 1000 ms, respectively. The pitch of the first tone was always E4 ( Hz) and the pitch of third tone was always C5 ( Hz). Across the three-tone kappa cell, there was an implied constant pitch velocity of 4 ST / 500 ms. In line with previous kappa studies, the spatiotemporal location of the target tone was varied between trials. There were five possible pitch values for the target tone: F4 ( Hz), G4 (329 Hz), A-flat4 (415.3 Hz), A4 (440 Hz) or B4 ( Hz). Relative to the expected 4ST pitch distance based on the implied constant velocity of 4ST / 500, these target tone pitch levels corresponded to deviations of -3, -1, 0, +1, and +3 ST. The timing of the target tone was manipulated so that the interval between the first bounding tone and the target tone was 500 ms ± 4%, 8%, 12%, or 16%. These deviations

19 13 correspond to absolute intervals of 420, 440, 460, 480, 520, 540, 560, or 580 ms. At no time did the target tone temporally bisect the interval between bounding tones. In the context absent condition, stimulus sequences consisted of an isolated kappa cell (see Figure 1), while in the context present condition, stimulus sequences consisted of a kappa cell preceded by an isochronous three-tone context sequence that ascended in pitch and reinforced a 4ST/500 ms velocity (see Figure 2). Thus, in the context present condition, the pitch and time distances between each context tone were fixed at 4 ST and 500 ms, respectively. The resulting pitch progression of context sequences was E3 ( Hz), A-flat3 ( Hz), and C4 ( Hz). Procedure Participants were randomly assigned to either the context absent or context present condition. Prior to testing, participants completed a single training session that familiarized them with the task. See Appendix A for a detailed description of the training procedures for the context present and context absent conditions. Participants in the context absent condition were shown a diagram and explained that they would listen to three tone sequences. They were told that the middle tone had been moved around in time, changing the durations of the time intervals delimited by the three tones. Participants were asked to judge whether the target tone was closer in time to the first tone than the third tone (by responding short-long ) or was closer in time to the third tone than the first tone (by responding long-short ). Thus, for each threetone sequence, participants were asked to make a short-long or long-short judgment about the pattern of durations delimited by the tones. Participants in the context present condition were similarly shown series of diagrams, but were told that they would listen to sets of six tones. They were told that the fifth tone had been

20 14 moved around in time, changing the durations of the time intervals delimited by the final set of three tones. Participants were asked to make short-long and long-short responses about the pattern of durations delimited by the latter set of three tones. Critically, participants were instructed to ignore pitch and to focus only on the timing of the target tone. Responses were made by pressing one of two labeled buttons on a response box. No feedback was provided. All participants were tested on two separate days. Training was completed on the first day. Participants completed ten test blocks (5 blocks per day) with 40 trials per block. Within a test block, participants heard each combination of the five pitch levels and eight time levels of the target tone once. There were 200 trials per session and each session lasted approximately 30 minutes. In total, ten observations were obtained for each of the 40 conditions (5 target pitch levels x 8 time levels). Data Analysis Estimates of Point of Subjective Equality (PSE) and Just Noticeable Difference (JND) thresholds were determined for each participant at each of the five target tone pitch levels using three methods. Methods 1 and 2 involved first calculating the proportions of long-short responses, collapsed across the ten test blocks, for each of the eight temporal deviations and five target tone pitch levels of the target tone. This yielded five psychometric curves for each participant, one for each relative target tone pitch level. Each of these curves approximated a cumulative normal distribution. Calculating PSE using Method 1 required using the z-transforms of these curves, which were straight lines with x-intercepts equal to the time at which the proportion of long-short answers was Solving for this number yielded the PSE for each participant at each pitch level (as described by Macmillan and Creelman, 1991, pp ).

21 15 JND, using Method 1, was quantified as half of the stimulus distance between the 25 th and 75 th percentiles, and was calculated from the same straight line z-transform of each participant s cumulative response curve. JNDs were calculated by first setting the transformed equation equal to the z-scores corresponding to proportions of long-short responses of 0.25 and 0.75, and then solving for the corresponding times at which these proportions were produced. The distance between these two numbers was then halved, yielding JNDs for each participant at each pitch level (MacMillan & Creelman, 1991). For Method 2, estimates of PSE and JND were obtained using each participant s raw cumulative response curve for each pitch level. The following equation was used to fit each participant s data: 1 f ( x) =. (2) 1+ e^ ( γ ( x +θ )) Equation 2 is used to convert a perceived duration into a probability of making either a shortlong or a long-short response. Working backwards, proportions of long-short responses given by each participant were used to fit the parameters of Equation 2. The values of θ and γ that provided the best fit to the data based on root mean square error were converted to estimates of PSE and JND respectively. Specifically, θ = -PSE and γ = 1/JND. Method 3 was a bootstrapping method, which involves using weighted linear regression to fit a cumulative Gaussian psychometric function to a set of binary data. A threshold is calculated according to a criterion set equal to This value corresponds to PSE. The inverse of the slope of the function corresponds to JND. The program BOOTPROB (Foster & Bischof, 1991) was used to carry out the bootstrap analysis. Standard deviations across the three methods were calculated for each subject for both PSE and JND estimates at each relative target tone pitch level. Participant data were discarded if

22 16 the maximum standard deviation across the three methods exceeded 30% of the 500 ms IOI on which constant velocity was based for any pitch level. Data for five participants was discarded based on this criterion. Raw PSE scores for each participant were used to calculate relative constant error (CE) estimates. Absolute CE was calculated by subtracting PSE from the base rate of the sequence. This yielded under- or overestimations of the time interval preceding the target relative to the base rate of the sequence in milliseconds. Raw CEs were then converted to relative CE by calculating the percent under- or overestimation based on a 500 ms base IOI. Negative CEs are indicative of an underestimation of the time interval preceding the target, and positive CEs indicate an overestimation of the same time interval. All statistics were performed on CEs (derived from PSEs) and JNDs obtained using Method 1. Results Figure 3 shows relative CEs for each of the five relative pitch levels of the target tone (-3, -1, 0, +1, +3 ST) for the context present and context absent conditions. Consistent with the presence of an auditory kappa effect, duration judgments were systematically affected by the relative pitch level of the target tone. When the relative pitch level of the target tone was negative (i.e., smaller than expected based on constant imputed pitch velocity) then CEs were generally negative, indicating that the time interval preceding the target tone tended to be underestimated. When relative target tone pitch level was negative, listeners were more likely to respond short-long. Conversely, when the relative pitch level of the target tone was positive, (i.e., the pitch distance of the target tone was larger than expected based on constant imputed velocity), then CEs were generally positive, indicating that the time interval preceding the target tone tended to be overestimated. Listeners were more likely to respond long-short. With

23 17 respect to the effect of context, Figure 3 shows that the magnitude of the auditory kappa effect was weaker in the context present condition than the context absent condition. These findings were supported by separate 2 x 5 mixed-measures ANOVAs performed for relative CE and JND. With respect to CE, a main effect of relative target tone pitch level obtained, (F(4, 104) = 13.45, MSE = , p <.05), as did a target tone pitch level x context interaction (F(4, 104) = 5.46, MSE = , p <.05). A trend analysis revealed that both the linear and cubic trends were significant for both context conditions (context absent: F lin (1, 14) = 12.26, MSE = , p <.05; F cubic (1, 14) = 24.33, MSE = 55.45, p <.05; context present: F lin (1, 12) = 13.61, MSE = 33.28, p <.05; F cubic (1, 12) = 11.18, MSE = 2.59, p <.05). This implies that though the kappa effect was weakened in the context present condition, it was not entirely eliminated. Figure 4 shows relative JNDs for the context present and context absent conditions for each of the five relative pitch levels of the target tone. The benefit of presence of the context sequence in terms of sensitivity to deviations from isochrony is evident in the nonsignificant trend for relative JND to be lower in the context present condition (M = 9.85 ± 2.24) 2 than in the context absent condition at all relative target tone pitch levels (M = ± 2.15), F(1, 26) = 2.23, MSE = , p =.15. No significant differences in JND were found as a function of any independent variable or interaction (p >.05 in all cases). Application of the Imputed Velocity Model Finally, the imputed velocity model (Equation 1) was used to quantify the magnitude of the kappa effect (Jones & Huang, 1982). The model was used to estimate a weight parameter, w, for each participant. In this model, w refers to the weight placed on the actual duration of the interval preceding the target tone by the observer when making duration judgments. A weight 2 All means are reported with standard error of the mean.

24 18 parameter equal to 1 would indicate that the duration judgment had been made based entirely on the objective time interval and therefore no kappa effect was present. A weight parameter less than 1 would indicate that a weighted combination of the actual duration of the interval preceding the target and the expected duration of the same time interval based on imputation of constant velocity was used by the observer in making a duration judgment. A weight parameter less than 1 indicates the presence of a kappa effect. A smaller weight parameter is associated with a larger the kappa effect. An alternative means to examine the effect of the presence of a context sequence, then, is by investigating the effects of the manipulation on values of w. A one-way between-subjects ANOVA was performed on w to examine whether the presence of a context sequence influenced w estimates. With respect to w, the effect of sequence length was marginally significant, F(1, 26) = 3.95, MSE = 0.01, p = The context absent condition was associated with a smaller w, than was the context present condition (M =.88 ±.03 versus M =.95 ±.03, respectively). Finally, Pearson s r was calculated for w and years of musical training assess possible effects of musical training on the magnitude of the kappa effect. The correlation between w and years of musical training was not significant, r (26) = -.13, p >.05. Discussion In general, an auditory kappa effect was found in Experiment 1 for both the context present and context absent conditions. However, the presence of a context sequence reinforcing the constant velocity implied by the kappa cell tended to decrease the magnitude of the kappa effect rather than increase it as was hypothesized. This finding is supported by a significant relative target tone pitch level x context interaction, and additionally by the increased w value that obtained for the context present condition relative to the context absent condition.

25 19 It was assumed that the addition of a context sequence preceding the kappa cell strengthened the expectations listeners had generated for the expected location of the target tone in pitch-time. This was expected to strengthen the perceptual distortions resulting from incongruent transformations of the stimuli in pitch space and time, and therefore to increase the magnitude of the kappa effect. However, the opposite was found to be true. The presence of an isochronous context sequence presumably reinforces not only the constant velocity of the sequence, but also the rhythm at which individual tones are presented. This may have afforded listeners an opportunity to perceptually segregate spatial and temporal properties of the sequence. Shorter sequences, made up of only a kappa cell, do not afford this same information, as individual sequences never contained an instance of a 500 ms IOI. The shorter sequences therefore reinforced velocity over the course of a session, but did not reinforce the isochronous rhythm implied by the velocity.

26 20 EXPERIMENT 2 The primary goal of Experiment 2 was to investigate whether the strength of the auditory kappa effect is modulated by sequence velocity, as has previously been shown to be the case for the visual kappa effect. Huang and Jones (1981, as cited in Jones & Huang, 1982) found that the magnitude of the visual kappa effect increased as velocity increased. If the visual and auditory kappa effects are similar, then increasing pitch velocity should increase the magnitude of the auditory kappa effect. Of additional interest were the relative contributions of local sequence velocity and global session velocity to expectations about the timing of the target tone. In Experiment 1, local and global velocities were confounded. In order to separate the potential differential effects of the two, Experiment 2 varied the local context sequence velocity from trial to trail, but held the average (global) velocity constant throughout. Musical experience was additionally included as a factor in Experiment 2 in order to test the hypothesis that musical training might enable individuals to more easily ignore pitch when making judgments about the timing of the target tone. Method Design Experiment 2 employed a 3 x 5 x 2 mixed factorial design. Three sequence velocities (4 ST / 400 ms, 4 ST / 500 ms, and 4 ST / 600 ms) were crossed with five relative target tone pitch level and two levels of musical training (musicians versus non-musicians) 3. Pitch of the target tone was -3, -1, 0, +1, or +3 ST relative to the expected pitch level of the target tone based on a constant imputed pitch velocity. Relative target tone pitch level and pitch velocity were within subjects factors, and musical training was a between subjects factor. 3 Musicians (n=7) were defined as having three or more years of training on at least one instrument or voice (M = 8.57 years). Non-musicians (n=7) were defined as having two or fewer years of musical training on any instrument or voice (M = 0.07 years).

27 21 Participants Eighteen individuals participated in Experiment 2. Participants were students enrolled in summer courses at Bowling Green State University, or members of the Bowling Green community. All participants were adults between the ages of 18 and 65 (M = 28.15, SD = 7.13), with self-reported normal hearing. They received $10 for their participation. Data for four participants was excluded due to instability of CE and/or JND estimates (see Data Analysis). Data for fourteen subjects were used in the analyses (musicians, n = 7; non-musicians, n = 7). Apparatus Stimulus generation and response collection was controlled by an IBM PC compatible computer running the MIDILAB software package, with a time resolution of 1 millisecond (Todd, Boltz, & Jones, 1989). Auditory sequences were presented to participants at a comfortable listening level through Grado SR-80 headphones attached to a Yamaha PSR-70 MIDI keyboard set to a piano voice. Stimuli Stimulus sequences were of the same format as those used in the context present condition of Experiment 1 (see Figure 2). All sequences consisted of an ascending three-tone kappa cell preceded by an isochronous three-tone context sequence that also ascended in pitch and reinforced the constant velocity implied by the fixed spatiotemporal positions of the bounding tones of the kappa cell. Again, all tones had onset to offset durations of 100 ms. The critical manipulation in Experiment 2 was the varying of pitch velocity between trials. Each sequence was assigned one of the following local sequence velocities: 4 ST / 400 ms, 4 ST / 500 ms, or 4 ST / 600 ms. Over the experimental session, the average (global) context velocity remained constant at 4 ST / 500 ms.

28 22 As in Experiment 1, the first and third bounding tones of the kappa cell were fixed in pitch space and in time. Bounding tones were always separated in pitch by 8 ST (pitches were fixed at E4 ( Hz) and C5 ( Hz)) in all velocity conditions; temporal separation of bounding tones was dependent on velocity condition. The IOI separating bounding tones was equally often 800 ms, 1000 ms, or 1200 ms corresponding respectively to implied constant velocities of 4 ST / 400 ms, 4 ST / 500 ms, and 4 ST / 600 ms. The spatiotemporal location of the target tone was varied between trials. There were five possible pitch values for the target tone: F4 ( Hz), G4 (329 Hz), A-flat4 (415.3 Hz), A4 (440 Hz) or B4 ( Hz). Relative to the expected 4ST pitch distance based on the implied constant velocity, these target tone pitch levels corresponded to deviations of -3, -1, 0, +1, and +3 ST. The timing of the target tone was manipulated so that the interval between the first bounding tone and the target tone was ± 4%, 8%, 12%, and 16% of the base IOI of the sequence, i.e., 400 ms, 500 ms, or 600 ms. These deviations correspond to the raw target tone times of 336, 352, 368, 384, 416, 432, 448, or 464 ms in duration for the 4 ST / 400 ms condition, 420, 440, 460, 480, 520, 540, 560, or 580 ms for the 4 ST / 500 ms condition, and 504, 528, 552, 576, 624, 648, 672, or 696 ms for the 4 ST / 600 ms condition. At no time did the target tone temporally bisect the interval between bounding tones. In all velocity conditions, kappa cells were preceded by isochronous, ascending context sequences that reinforced the constant velocity of each sequence. The pitch distance between each context tone was 4 ST (pitches were fixed at E3 ( Hz), A-flat3 ( Hz), and C4 ( Hz)) for all velocity conditions. Tones making up context sequences were separated by 400 ms, 500 ms, or 600 ms, dependent upon the velocity of the individual sequence.

29 23 Procedure Each participant completed one training session to familiarize them with the task. See Appendix A for details related to training for Experiment 2. As in Experiment 1, context present condition, participants were shown a series diagrams and told that they would listen to sets of six tones. They were told that the fifth tone had been moved around in time, changing the durations of the time intervals delimited by the final set of three tones. Participants were asked to judge whether the target tone was closer in time to the fourth tone than the sixth tone (responding short-long ) or was closer in time to the sixth tone than the fourth tone (responding longshort ). Thus, for the second set of three tones, participants were asked to make a short-long or long-short judgment about the pattern of durations delimited by the kappa cell. Critically, participants were instructed to ignore pitch and to focus only on the timing of the target tone. Responses were made by pressing one of two labeled buttons on a response box. No feedback was provided. All participants were tested over three days. Training was completed on the first day. Participants responded to blocks of 60 trials, hearing half of the possible pitch x velocity x time combinations in each block. Participants completed six experimental blocks during the first session, and seven blocks during the second and third sessions, for a total of 20 blocks. Participants were given a break halfway through each session. Each participant responded to 360 trials during the first session, and 420 trials during both the second and third sessions. Each session lasted approximately 45 minutes. Across sessions, 10 observations were obtained per each of the 120 conditions (5 pitches x 8 timing manipulations x 3 velocities).

30 24 Data Analysis Estimates of Point of Subjective Equality (PSE) and Just Noticeable Difference (JND) thresholds were determined for each participant at each of the fifteen velocity x target tone pitch level combinations using each of the three methods outlined in Experiment 1, Data Analysis. Proportions of long-short responses, collapsed across the 20 test blocks, were calculated for each of the eight temporal deviations at each of the relative target tone pitch level x velocity combinations. Experimenter error resulted in data for the 4 ST / 500 ms x 0 ST x -8% condition being discarded. This was remedied differently for each of the three methods. Using Methods 1 and 2, the proportion of long-short responses for this combination of factors was simply removed from the analysis. PSEs and JNDs were calculated normally for all pitch level conditions in both the 4 ST / 400 ms and 4 ST / 600 ms conditions, as well as for the -3, -1, +1, and +3 ST pitch levels for the 4 ST / 500 ms velocity condition. PSEs and JNDs for the 4 ST / 500 ms x 0 ST condition were computed as if participants had responded to only seven factorial combinations of pitch and time. In order to implement Method 3 sans this point, the flawed pattern file had to be specified as a training pattern. This allowed the raw data to be analyzed, ignoring this pattern. This method produced all of the information necessary to calculate PSEs and JNDs for the 4 ST / 400 ms and 4 ST / 600 ms conditions, as well as for the -3, -1, +1, and +3 ST pitch levels for the 4 ST / 500 ms velocity condition using the BOOTPROB program. For the 4 ST / 500 ms x + ST condition, bootstrapping files were generated that had to be opened in a word processor to be edited. This problem point was removed and bootstrap files were changed so that BOOTPROB would read them as having been generated using only seven factorial pitch-time combinations. This allowed

31 25 for estimation of PSE and JND for each participant at all relative target tone pitch x velocity levels. Standard deviations across the three measures were calculated for each participant s PSE and JND estimates at each pitch x velocity level. Participant data was discarded if the maximum standard deviation across measures exceeded 30% of the IOI on which constant velocity was based. Data for three participants were discarded based on this criterion. Data for one additional participant was discarded because of a negative estimate of JND. Raw PSE scores for each participant were used to calculate relative CE. Absolute CE was again calculated by subtracting PSE from the base rate of the sequence, 400 ms, 500 ms, or 600 ms. This yielded the under- or overestimations for each condition in milliseconds. These raw CE values were converted to relative CE by calculating the percent under- or overestimation relative to the IOI of the sequence. Negative CEs are indicative of an underestimation of the time interval, and positive CEs indicate an overestimation of the time interval. All statistics were performed on CEs (derived from PSEs) and JNDs obtained using Method 1. Results Figure 5 shows relative CE as a function of relative target tone pitch level for each of the three pitch velocity conditions. Consistent with the presence of a kappa effect, duration judgments were systematically affected by the relative pitch level of the target tone. When the relative target tone pitch level was negative, indicating a smaller increase in pitch than expected based on constant velocity, the time interval preceding the target tone was generally underestimated, as evidenced by negative values of CE. At negative relative target tone pitch levels, listeners were more likely respond short-long. Conversely, when values of relative target tone pitch level were positive, positive CEs obtained. At pitch distances that were larger

32 26 than expected based on imputed constant velocity, time intervals preceding the target tone were overestimated. Listeners were more likely to respond long-short. Figure 5 also shows that the largest kappa effect obtained at the highest pitch velocity (4 ST /400 ms) and the smallest kappa effect obtained at the lowest pitch velocity (4 ST / 600 ms). These findings were supported by the results of separate 3 x 5 x 2 mixed-measures ANOVAs conducted for relative CE and JND. With respect to CE, the main effect of pitch level was significant, F(4, 48) = 16.38, MSE = 69.65, p <.05, indicating the presence of a statistically reliable auditory kappa effect. The presence of a target tone pitch level x velocity interaction indicated that velocity had a modulatory effect on the magnitude of the kappa effect, F(8, 96) = 2.91, MSE = 16.90, p < With respect to the fast, 4 ST / 400 ms, velocity condition, a trend analysis revealed that both the linear and cubic contrasts reached significance at this velocity, F lin (1, 13) = 16.26, MSE = , p <.05; F cubic (1, 13) = 12.71, MSE = 15.88, p <.05. Linear and cubic trends also reached significance for the medium, 4 ST / 500 ms, condition, F lin (1, 13) = 16.42, MSE = 67.63, p <.05; F cubic (1, 13) = 17.23, MSE = 15.76, p <.05. However, for the slow, 4 ST / 600 ms, velocity condition, only the linear trend was significant, indicating a decrease in the strength of the auditory kappa effect at this velocity, F lin (1, 13) = 8.26, MSE = 78.92, p <.05; F cubic (1, 13) = 3.81, MSE = 17.90, p =.073. Findings from vision regarding the role of increasing velocity in the magnitude of the kappa effect, therefore, generalize here to the auditory modality. Increasing the imputed pitch velocity of sequences increased the magnitude of the auditory kappa effect. Musical training was assessed as a between subjects factor. Figure 6 shows that the kappa effect was weakened for musicians relative to non-musicians. The target tone pitch level x musical training interaction was found to be significant, F(4, 48) = 3.97, MSE = 69.65, p <.05.