Run vs. Gap for each Session cross=musician, square=non-musician

Transcription

1 training are plotted as a plus sign and the subjects without formal musical training are plotted as squares. The musically untrained subjects are clustered in the right portion of the graph whereas the musically trained subjects are spread across the left and middle of the graph. The clustering of the parameter estimates for the musically untrained subjects farthest from the origin of the graph suggests that their data is more predictable by the run{gap rules than are the data from the musically trained subjects. Gap Parameter Value Run vs. Gap for each Session cross=musician, square=non-musician Run Parameter Value Figure 7: Run vs Gap parameter values for musically trained (plus signs) and musically untrained (squares) subjects. References [Boker, 1995] Steven M. Boker. Predicting the grouping of rhythmic sequences using local estimators of information content. In Proceedings of the Fourteenth International Joint Conference on Articial Intelligence, San Mateo, CA, Morgan-Kauman. [Garner and Gottwald, 1968] W. R. Garner and R. L. Gottwald. The perception and learning of temporal patterns. Quarterly Journal of Experimental Psychology, 20:97{109, [Garner, 1974] Wendell R. Garner. The Processing of Information and Structure. Lawrence Erlbaum Associates, Hillsdale, NJ, [Handel, 1989] Stephen Handel. Listening: An Introduction to the Perception of Auditory Events. MIT Press, Cambridge, MA, [Lashley, 1952] K. S. Lashley. The problem of serial order in behavior. In L. Jeress, editor, Cerebral Mechanisms in Behavior, pages 112{136. Wiley, New York, [Royer and Garner, 1966] F. L. Royer and W. R. Garner. Response uncertainty and perceptual diculty of auditory temporal patterns. Perception & Psychophysics, 1:41{47, [Shannon and Weaver, 1949] C. E. Shannon and W. Weaver. The Mathematical Theory of Communication. The University of Illinois Press, Urbana, Discussion These analyses suggest that there exist large, stable individual dierences in the perception of rhythmic structure in isochronous auditory sequences. This result indicates that rule{based deterministic algorithms such as the run{gap rules are unlikely to be able to predict the perception of musical rhythmic structure. The apparent negative covariance between musical training and adherence to the run{gap rule system suggests that an articial musical generation program which adheres to such a rule{based system may produce music which is perceived as being similar to that produced by musically untrained individuals. We argue that a parameterized, probability{based system which constructs structure from local information theoretic estimates may provide a better approximation to the variety of individual dierences in music produced by trained individuals.

2 consistency in the predictions from the run{gap rules. The histogram of responses shown in Figure 4{A is predicted by the run{gap principles, whereas the pattern in Figure 4{B is almost identical and yet the responses have now shifted to a point which is not predicted by the run{gap rules. Any theory of organization of rhythmic perception must take into account this type of shift; a number of these dichotomies appear in the response data from this experiment Distribution of Beats for Subject 2 Length = 6 Histogram of number of taps on each beat Distribution of Beats for Subject 6 Length = 6 Histogram of number of taps on each beat Figure 5: Histograms showing response distributions from two subjects for all stimulus patterns of length 6. Note the dierence between the subject labeled A whose responses are well predicted by the run{gap rules and the subject labeled B who's responses that are highly patterned but are not structured in the way that the run{gap rules would predict. 3.2 Individual Dierences in Response Distributions. Figure 5 shows two histograms which represent the responses to all patterns of length two for two individual subjects. The responses of the subject shown in Figure 5{A are highly predictable from the run{gap rules. Subjects showing this type of pattern indicated that they had little or no formal musical training and didn't presently play a musical instrument. The responses of the subject shown in Figure 5{B are quite dierent from the previous subject. A large proportion of the responses from this subjects were not predicted by the run{gap rules. The responses from subjects indicating that they had musical training tended to show patterns of response unpredicted by the run{gap rules. Gap Parameter Value Run vs. Gap for each Session number=subjectid Run Parameter Value Figure 6: Run vs Gap parameter estimates for each individual session. The number in the scatterplot represents the subject ID number. 3.3 Model Fitting Results The histograms from Figure 5 indicate that there may be large, stable individual dierences in these data. In order to examine both interindividual dierences and intraindividual variability a structural equation model was t to the data from each experimental session. In this way, the parameter estimates are not aggregated either across time or across subjects. The Run{Gap Model shown in Figure 1 was separately t to each session for each individual using the SAS Proc Calis structural model tting procedure. The parameter estimates for P run, the eect of the Run predictor variable on S the latent perceived structure variable, and for P gap, the eect of the Gap variable on S are summarized in Figures 6 and 7. Figure 6 plots the parameter estimates for P run against P gap. The numbers in the scatterplot represent the subject ID number from the experimental data. There is evident clustering of the subject ID numbers, which suggests that the intraindividual variation is smaller than the intraindividual variability. Figure 7 again plots the parameter estimates for P run against P gap, but now the subjects with formal musical

3 edge of the histogram. Since the stimulus pattern is always repeated, responses which occur at the end of the last beat can also be thought of as occurring just before the rst beat in the pattern. Therefore, Figure 2{A can be read as showing a distribution of responses with a mode centered 25{50 ms before the predicted starting point of the measure. Similarly, each of the other histograms in Figure 2-B show distributions which closely follow the pattern predicted by the run{gap principles. Note that when the gap is long, the distribution of responses around the predicted starting point spreads out. But when every beat is lled except one, as in Figure 2{B, the distribution of responses is more precise. This is not at all surprising when one considers the task as being the prediction of the rst sounded beat; the longer the gap before the sounded beat, the more dicult the task of synchronizing one's response with the occurrence of that beat Distribution of Beats for [ ] space6-90(1) Run=0 Gap=0 [ ] Distribution of Beats for [ ] space6-90(2) Run=0 Gap=0 responses that the run{gap principles did not predict particularly well. Figure 3{A shows an example of one of the stimuli for which a sizable percentage of responses occur on the beat following the predicted starting point. The distributions of quite a few of the stimuli show this characteristic distribution of responses onto the beat following the predicted starting point even when this second beat is in the middle of a run of 1's Distribution of Beats for [ ] space8-90(4) Run=0 Gap=0 7 8 [ ] Distribution of Beats for [ ] space8-90(6) Run=0 Gap=0 7 8 [ ] Figure 4: Histogram showing response distributions for one stimulus pattern which elicited responses predicted by the run{gap rules. Histogram showing response distributions for a nearly identical stimulus pattern which elicited responses which were not predicted by the run{gap rules. [ ] Figure 3: Histograms showing response distributions from all subjects for two selected stimulus patterns which elicited responses showing lower degrees of agreement on the starting point for the stimulus pattern and substantial variation from the starting point predicted by the run{gap rules. Figure 3 shows two histograms of the distributions of responses to two more selected rhythmic stimuli. These histograms represent some of the stimuli which elicited Figure 3{B shows an example of another phenomenon which was observed to occur in response to a number of stimuli. The distribution of responses is split almost equally between the predicted starting point and a solitary 1 following a small gap following the longest run of 1's. There is nothing in the run{gap rules which accounts for this large, reliable shift in responses. A theory of the perception of rhythmic organization must be able to account for this type of shift which has been observed to occur in pairs of nearly identical stimuli. Figure 4 shows two histograms which highlight the in-

4 2.2 Experimental Procedure Each subject was asked to complete the experimental procedure once on each of ve occasions, where the occasions were separated by as little as 24 hours and by as much as three weeks. In each trial, subjects were presented with one repeating rhythmic auditory pattern and were asked to respond by striking a key on a synthesizer keyboard synchronous with the perceived starting point of the pattern. Subjects were asked to continue to strike the key at the beginning of each repetition until condent that they had perceived the starting point. Once subjects were condent of their response, they were asked to press a mouse button ending the trial. A single rhythmic stimulus was composed of a xed number of beats: equal intervals of time which could either be empty or be lled with a percussive sound at the beginning of the interval. The set of stimuli for Experiment 1 consisted of the 69 unique rhythmic patterns of length 8 or less with oversampling applied to the most ambiguous patterns, thus creating 115 trials per session. A more complete description of the experimental procedure and the structural modeling analysis is given elsewhere in this volume [Boker, 1995]. Run Prun Pgap Gap 3 Results 3.1 Response Histograms Figure 2 presents two histograms of the distributions of responses to two selected rhythmic stimuli from all subjects on all occasions of measurement. Figure 2{A shows the distribution of responses to a simple stimulus, a measure of four beats in which one beat is sounded and the others are silent. The label for each histogram contains a binary representation of the stimulus pattern which elicited the distribution of responses. For instance, Figure 2{A is labeled with [ ]. Elapsed time is always increasing from left to right, and each number on the X{ axis represents a duration of 250 ms. The run{gap rules predict a starting point for the rhythmic stimulus aligned with the far left side of each histogram and consequently also with the rst binary digit in the label Distribution of Beats for [ ] space4-100(0) Run=0 Gap=0 RB e R S V e V Figure 1: Path diagram showing a structural model of Garner's basic run{gap theory. A e A [ ] Distribution of Beats for [ ] space4-100(2) Run=0 Gap= [ ] 2.3 Latent Path Model A latent variable structural equation model was constructed which t Garner's \run{gap" heuristic predictions to the data gathered from this experiment. Figure 1 shows a path model of Garner's run{gap heuristics. The predictor variables are Run, the run principle, and Gap, the gap principle. The latent variable is S, the perceived structure of the rhythmic pattern. The measured outcome variables are RB, the response within the beat; A, the accuracy of the response; and V, the velocity of the response. Figure 2: Histograms showing response distributions from all subjects for two selected stimulus patterns which elicited responses showing high degrees of agreement on the starting point for the stimulus pattern and agreement with the run{gap rules. Responses which occurred just after the predicted starting point appear at the far left edge of the histogram, and the responses which occurred at the end of the duration of the last beat appear at the far right

5 Individual Dierences in the Perceptual Segmentation of Auditory Rhythmic Sequences. Steven M. Boker y and Michael Kubovy Department of Psychology The University of Virginia Charlottesville, Virginia boker@virginia.edu boker/ Abstract Musical rhythm is inherently structured in such a way that it is perceived to be partitioned into segments. A repeating rhythmic stream has many possible segmentations. We describe an experiment which explores the individual variability in the perception of the segmentation of repeating auditory rhythms. Histograms of responses to this experiment show large individual dierences which are not predicted by Garner's run{gap rules [Garner, 1974]. Analysis of individual dierences in parameters of a latent variable model of the run{gap predictors indicates that predictability by the run{gap rules inversely covaries with musical training. 1 Introduction Music presents the auditory system with a continuous stream of sensory data which is perceived to have a regular, self{referential and sometimes repeating structure. The perceptual segmentation of this auditory stream into a sequence of events which have further relationships with each other seems so automatic that it forms an often unstated assumption at the basis of musical theory. But upon closer inspection this task is not as automatic nor as rule{based as it rst appears. The problem of segmentation of continuous streams of sensory data into temporally ordered sequences has posed a longstanding problem for cognitive psychologists [Lashley, 1952]. There is more ambiguity in the temporal structure of musical sequences than it might appear. Musical structure is disambiguated through a variety of devices such as stress and temporal anticipation of critical segmentation points [Handel, 1989]. By studying the nature of the ambiguity in unstressed isochronous rhythmic sequences we can understand and predict the organization Presented to the International Joint Conference on Articial Intelligence Workshop on Music and Articial Intelligence, August 21, 1995 y Supported by the Institute for Developmental and Health Research Methodology which will be perceived to be inherent in the temporal structure of the sequence. A simple repeating rhythmic pattern which contains no stressed elements may be perceived as having a variety of starting points. Figure 1 shows a repeating sequence which could be perceived as having one of three potential starting points. Some starting points have a higher probability of being perceived than others, but each of these probabilities is greater than zero : : : : : : : : : Figure 1. A repeating sequence of length three has potential starting points in three dierent positions. Garner and his colleagues [Royer and Garner, 1966; Garner and Gottwald, 1968; Garner, 1974] studied these types of rhythmic patterns and devised rules which they named the run principle and gap principle by which predictions could be made regarding the organization that would be perceived by individual subjects. Our models replace Garner's rule{based system with an information theoretic [Shannon and Weaver, 1949] estimation of the probability of perceiving any starting point as a segmentation boundary. The data which we present suggests that not only do dierent individuals perceive the same rhythmic stream dierently, but that the same individual will perceive the same rhythmic stream dierently on dierent occasions. This result creates a problem for rule{based systems, but is consistent with a model which predicts behavior based on a distribution of probabilities generated from characteristics of the input. 2 Methods 2.1 Subjects Eleven subjects participated in the experiment, 8 males and 3 females. Age of the subjects ranged from 18 to 41. Six subjects reported having received 4 years of training in playing a musical instrument, while the remaining ve subjects reported no formal training in playing a musical instrument.