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Psychol Res (1984) 45:315-337

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A Theoretical Frameworkfor Rhythm Perception Dirk-Jan Povel University of Nij megen, Department of Experimental Psychology, P.O, Box 9104, 6500 HE Nijmegen, The Netherlands

Summary. This study relates to the perception of simple rhythmical patterns. A theoretical framework is presented that aims at predicting the perceived organization, the judged complexity, and the experienced rhythmical value of temporal sequences. Two simple assumptions form the basis of the framework. The notion of the 'temporal grid' is proposed to specify the temporal structure of a sequence. Such a grid is a time scale consisting of isochronic intervals. Since a rhythmical pattern generally allows for several different possible grids, an 'economy principle' is employed for selecting the most efficient grid. Economy of description is determined by the number of tones covered by the grid and the ease of specifying the noncovered tones. Since this basic model cannot explain all relevant phenomena, it is extended in order to incorporate three additional factors, namely the starting point of a sequence, subjective accents, and tempo.

The tones that are the constituents of tone sequences have a dual function. On the one hand they are bearers of characteristics like pitch, loudness, timbre and duration. On the other hand the tones, or more precisely the onset of the tones, mark points in time thus dividing the time continuum in a sequence of intervals. By their first function the tones produce structures in time, by their second they produce structures o f time. Similarly, in the perception of tone sequences, two different aspects can be distinguished. One process concerns the coding of the characteristics of the tones, while the other process is concerned with the coding of the temporal structure formed by the intervals between the onsets of the tones. Melody perception seems more directly related to the first process, rhythm perception more to the second. It should be realized, however, that this is too simple a view. Melody perception cannot adequately be studied apart from the temporal structure of the tones occurOffprint requests to: D.-J. Povel


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ring in the sequence. Likewise, the perceived rhythmical aspects in a tone sequence do not solely depend on the temporal structure of that sequence. In fact, the experienced rhythmical value will be the result of the interaction of the temporal structure formed by the tone onsets and by the structural characteristics of the elements. Accordingly, a complete understanding of rhythm perception should take into account both aspects. However, for the problem to remain manageable, it is necessary to reduce the object of research. Therefore, this article confines itself to the study of the perception of rhythms composed of tones that are identical in all respects: frequency, intensity, spectral composition and duration. The elements in the resulting so called equitone sequences have primarily the function of marking time. Equitone sequences may therefore be said to be pure temporal sequences. Later, it will be shown that this view is not completely correct. Here it is important to note that equitone sequences, which result when in an actual rhythm all differences between the tones are eliminated, still possess most of the rhythmical characteristics of the original pattern. To illustrate, an actual rhythm can be reproduced by tapping with a pencil on a table, showing that a basic rhythmical aspect is formed by the sequence of onsets of the tones in the rhythm. At the turn of the century rhythm perception was intensively studied, major issues being: the nature of the rhythmical experience, the objective conditions of auditory rhythms, and the role of kinesthesis. For most of the investigators of that time the central aspect of rhythm perception consisted in the mental activity by which the elements in a rhythmical pattern were transformed into an understandable percept. The temporal structure and the occurring accents were considered the most important psychological characteristics of a rhythmical pattern. The main subjects studied during this period were the phenomenon of subjective rhythm that may arise when listening to a uniform series of tones (Bolton 1894; Mach 1868; Meumann 1894), the role of accents in rhythm perception (Bolton 1894; MacDougall 1902; Meumann 1894; Woodrow 1909; Wundt 1911), the range of intertone intervals at which perceptual organization is manifest (Bolton 1894; Meumann 1894), the interaction between temporal and intensity factors (Bolton 1894; MacDougall 1902 ; Meumann 1894; Woodrow 1909; Wundt 1911), the estimation of short durations and the just noticeable difference of duration (Hall & Jastrow 1886; Mach 1868), the 'indifference' interval (the interval that is neither over or under estimated) (Vierordt 1868; Woodrow 1930; Wundt 1911), the concomitant movements accompanying the listening to rhythms (Bolton 1894; Koffka 1909; MacDougall 1902; Miner 1903; Ruckmick 1913; Stetson 1905; Wundt 1911), and characteristics of rhythm performance (Meumann 1894; Sears 1902). After this period of research activity followed a period of relative silence with the exception of some attention from the Gestalt psychologists K/Shler (1929) and Koffka (1935) who again emphasized the importance of organization for the understanding of rhythm perception. The last two decades have again shown an increasing interest in the research of rhythm and of music in general; most of the relevant work of this period will be discussed below. In this paper an attempt has been made to develop a conceptual framework that incorporates some of the earlier developed notions in an elaborated form. The ulti-


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mate goal is to present an explanation for the major phenomena in the field on the basis of a specification of the internal representation of rhythmical patterns.

The Basic Model

Introduction What is it that a theory or rhythm perception should explain? A comprehensive theory should first of all consist of a model of the internal representation of rhythms and should be able to explain how this internal representation may give rise to the typical attributes that subjects ascribe to rhythms 1 . The most outstanding characteristic of a rhythm is what may be called its rbytbmical value, which tentatively can be described as the amount of tension that accompanies the perception of a rhythm. There are definitely great differences between the rhythmical value of temporal sequences: some are dull while others may give rise to feelings of joy and excitement (cf. Seashore 1938). Another important psychological characteristic of a rhythm is its complexity. A distinction should be made here between perceptual and productive complexity, the latter being the degree of difficulty in producing the rhythm. Usually, perceptually complex rhythms will seem to be hard to produce, but this is not necessarily so. This paper only deals with perceptual complexity. For a further discussion of the present distinction, see Collard and Povel (1982) and Povel and Collard (1982). The theory proposed in this paper aims at predicting both the complexity and the rhythmical value of temporal sequences based on a model of the internal representation of rhythms. A first, perhaps rather obvious assumption states that the complexity and the rythmical value associated with a tone sequence mainly depends on the structural aspects of that sequence. More accurately, the assumption states that only those structural characteristics are important that contribute to the development of the internal representation of the rhythm. To say that only structural aspects of tone sequences are the determinants of the complexity and the rhythmical value implies that nonstructural characteristics like overall loudness and tempo are unimportant in this respect. This implication is probably correct for loudness (higher overall loudness only resulting in a higher overall arousal), but tempo appears to be a variable that definitely affects the perceptual quality of a sequence and consequently its experienced rhythmical value. The effect of tempo will be dealt with in Section III. The theoretical framework developed below is deliberately kept as simple as possible, using a minimal number of concepts. At first, I try to understand all relevant phenomena related to rhythm perception b y means of a framework that only incorporates temporal concepts, that is, concepts that describe the temporal structure formed by the onsets of the tones in the sequence. In this first approach, the notion of the 'temporal grid' is proposed, and is further elaborated in Section II. Throughout this paper the terms (tone) sequence, pattern, and rhythm are used indiscriminately. Gradually, it will become clear how the subset of rhythms differs from the set of tone sequences.


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Later, w h e n it appears t h a t this f r a m e w o r k is t o o simple t o e x p l a i n all p e r c e p t u a l p h e n o m e n a , o t h e r c o n c e p t s will b e i n t r o d u c e d stepwise. This e x t e n s i o n o f t h e basic m o d e l is d e s c r i b e d in S e c t i o n III.

The Perception of Simple Temporal Patterns Let m e s t a r t w i t h a simple d e m o n s t r a t i o n . A s u b j e c t is p r e s e n t e d w i t h an e q u i t o n e s e q u e n c e in w h i c h t h e t o n e s all are at equal intervals, t h u s f o r m i n g w h a t is k n o w n as an ' i s o c h r o n i c s e q u e n c e ' . L e t t h e t o n e s have a d u r a t i o n o f 50 m s a n d t h e o n s e t intervals b e 2 0 0 ms, see Figure 2. N o w t h e s u b j e c t m a y r e p o r t hearing t h e s e q u e n c e as divided i n t o g r o u p s o f t o n e s . This p e r c e p t u a l p h e n o m e n o n , t h e 'subjective r h y t h m ' , is n o t very s t a b l e a n d s h o w s great individual differences. T h e same s e q u e n c e m a y be perceived as consisting o f g r o u p s o f 2, 3 or 4 t o n e s , or m a y b e h e a r d w i t h o u t

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2 In depicting the tone sequences the use of music notation is deliberately avoided because in that code timing aspects are rather akwardly represented. Instead, a notation is adopted in which the tone onsets are displayed as vgrtical lines on a time axis. So, the distance between two vertical lines represents the inter-onset interval. Unless otherwise specified, the tones have a duration of 50 ms and the smallest occurring inter-onset interval, the 'grain' is 200 ms. Apart from that, in order to get a perceptual impression of the sequences discussed, the reader is urged to listen to the sequences as they sound when played by some tone generating device.


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any grouping. Bolton (1894) reported a prevailing tendency in almost all of his 50 subjects to group by four. Generally, it is found that the number of tones that are perceptually grouped depends on the tempo of the tone sequence: the higher the tempo, the more tones are grouped (Bolton 1894; Meumann 1894; Fraisse 1956; Vos 1973). Next, the subject listens to the sequence depicted in Figure l b , which consists of a part of the just described isochronic sequence alternated with two longer intervals that are precisely three times as long as the short intervals, namely 600 ms. Now the string of clustered tones will unavoidably be heard as two groups of three tones. This organization is quite stable. If the long intervals are made four times as long as the shorter one (Figure lc), the string will inevitably be heard as divided into two groups of four tones. Although this may seem a trivial demonstration, it points to a fundamental mechanism in the perception of temporal sequences. It essentially shows that an interval in a sequence may be employed to render structure to other (shorter) intervals in that same sequence. A concomitant effect of the mechanism is that subjects may report hearing accents on the tones that form the borders of the perceived groups. In effect, accents may be heard on tones 3 and 6 of pattern l b , and on tones 3 and 7 of pattern lc. The same mechanism is also shown in the following demonstration. Subjects heard sequences l d (identical to sequence l b ) and l e randomly inserted among others and were asked to imitate each sequence, which was presented repetitively, as soon as they could 3. Note that in pattern l d the group of short intervals has a duration twice that of the long interval, whereas in pattern l e this does not apply. Typically, subjects imitated sequence l d correctly, both with regard to the number of elements and to the interval structure, whereas in the reproduction of sequence l e subjects sometimes added an extra element and made timing errors, showing that the conceptualization of this pattern is more problematic. In a study of the reproduction of simple tone sequences, in which the interval ratios were systematically varied, Povel (1981) showed that subjects only correctly reproduce the temporal relation of the occurring intervals if the pattern of the intervals has such a form that one interval can be used to organize the others. An example may clarify the point. Subjects imitated a repetitively presented temporal pattern consisting of two onset intervals of 250 and 750 ms, respectively. Instead of reproducing an interval ratio of 0.33 (250/750) they produce an interval ratio of 0.39 (cf. Sternberg, Knoll & Zukofsky 1982). Moreover, in continued reproduction a drift was found in the ratio to move further and further away from 0.33, showing that the internal representation of the pattern was unstable. In another experiment subjects imitated a pattern consisting of four intervals of 250, 2 5 0 , 2 5 0 and 750 ms, respectively. Now the pattern was reproduced perfectly with an average duration ratio of the two intervals of 0.34, while no sign of a systematic drift was found. Apparently, in the first pattern the subjects are unable to detect that the second interval is three times as long as the first. The fact that the second pattern is reproduced correctly can be understood if the subjects conceive of the three short inter3 All empirical observations referred to in the text, were collected in experiments in which at least 10 subjects participated. These experiments will be reported in more detail elsewhere (see for instance Povel & Essens, submitted for publication).


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vals as a subdivision of the long interval. From these and similar results it may be concluded that subjects are essentially unable to store a series of different intervals unless the structure of the sequence (i. e., the ratio and the order of the occurring intervals) is such that it can be recoded into a conceivable form. The next section goes into the nature of this coding.

The Temporal Grid Notion It is indeed clear from research done so far that temporal sequences are not stored as a series of independent durations, as would be supposed b y an association model. Also the idea, advocated by Fraisse (1946, 1956) that in the coding of temporal sequences the occurring intervals are subdivided into two classes, namely long and short intervals that roughly relate as 1:2, has been shown to be too simple (Pore1 1981). The last two examples given above clearly illustrate that the coding applied by the subject depends on a specific combination of the occurring duration ratios and the interval pattern. The assertion that an interval in a sequence may be used to structure other (shorter) intervals in that sequences, combined with the observation that a stable representation is only arrived at if the total duration of a subsequence of neighbouring short intervals match the duration of a longer interval in the sequence, readily leads to the notion of the ~ m p o r a l grid. The temporal grid is a time scale on which the tone sequence is mapped as a first step in the specification of the temporal structure of the sequence. The temporal grid consists in a sequence of isochronic intervals. Further, it is assumed that the temporal grid is not a fixed but a flexible structure, the features of which are determined by characteristics of the sequence. The temporal grid, since it at least partly overlaps with the sequence, will fixate part of the elements (tones) of the sequence. At the same time, it creates a frame that allows for the specification of the remaining elements. (Note that if the elements are fixed, the intervals are fixed and vice versa.) With regards to the selection of an appropriate grid for a sequence, two assumptions are made: 1. Each interval (either empty or filled) occurring in a tone sequence is a potential candidate for the grid interval. 2. That grid will be selected that allows the most economical description. Figures 2, and 3, which present some sequences along with different potential grids, clearly show that grids differ considerably in the amount to which they 'cover' the sequence. Intuitively, economy of description bears upon at least three aspects. First, a grid is more economical the more elements it fixates. Second, a g r i d is less economical the more points in time it fixates at which no tones occur. Third, a grid is more economical the easier the nonfixated elements can be specified within the given grid. A remark is in order here about the relationship between the temporal grid notion and the beat concept in music theory. Listeners usually tend to beat time to rhythms by regularly tapping with hand or foot. This regular tapping may be considered a direct expression of the temporal grid a subject utilizes while perceiving a temporal sequence. We should mention here the work of Longuet-Higgins and Lee (1982) who have proposed a model of how a listener infers the beat of a tone sequence. Their model


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Fig. 2a,b. Examples of sequences with their alternative grids. The duration of the shortest interval (grain) in each sequence is 200 ms. For explanation, see text is a process model incorporating a number of optional procedures which finally lead to a 'metric interpretation' of the sequence. It cannot easily be compared with the model presented here which is not a process model and which only employs an abstract ' e c o n o m y ' principle for grid selection. The relationship between the two models is discussed by Povel and Essens (submitted for publication). With the help of the two examples in Figure 2, the temporal grid notion and the economy principle will be further clarified. Tone sequence 2a contains five different intervals from which five different grids can be formed, numbered 1 through 5 4. Grid 1 has the smallest interval corresponding with the smallest interval occurring in the sequence, whereas Grid 5 has the largest interval which corresponds to the period of the sequence. Grid 1 fixates all elements in the sequence, but moreover it fixates two points in time where n o elements occur (to be called deletions). Therefore, if this grid were used to describe the sequence, it should be added in the description that the elements at positions 2 and 6 are deleted. Grid 2 fixates all elements except element 3, which could be fixated by noting that the second grid interval is subdivided into two equally long intervals. Grid 3 fixates only two elements, which leaves the elements 2 and 4 to be specified. Since elements 2 and 4 divide the grid 4 The grid is only indicated for one complete period, but must of course be thought of as being continuous.


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intervals in which they occur into unequal intervals (forming ratios of 2:1 and 1:2, respectively), the specification of these two elements in this grid is rather complicated. Grid 4 forms a special case since it does not fit within the period. This is the reason why it is discarded as a useful grid. Grid 5, finally, with a grid interval coinciding with the period of the repeated sequence will yield a rather uneconomical description since it only fixates one element, leaving all others to be specified. In sum, after elimination of Grid 1 and 5, three candidates remain, of which Grid 2 seems to yield the most economical description. Indeed, if subjects are asked to beat time with this sequence, they will tap on d e m e n t s 1, 2 and 4; that is they will tap in accord with Grid 2. F r o m the example given it follows that a useful potential grid interval will usually be smaller than the period, and must be a divisor of the period. Moreover, it should be clear that a sequence may be specified within a grid, either by specifying deletions (locations in the sequence where no tones occur) and/or by specifying subdivisions of one or more of the grid intervals. Figure 2b presents a sequence with seven potential grids that obey the two criteria mentioned above. The grids do not only differ with respect to the size of their intervals but also with respect to the number of deletions (indicated with a small circle on t o p of the corresponding grid line) and with respect to their 'locations'. Grid 1 fixates all tones, but not less than eight deletions should be indicated. Grid 2 has a grid interval twice as long as that of Grid 1, two deletions, and two tones (2 and 4) that have to be further specified. Grid 3 has in its turn an interval twice that of Grid 2, one deletion, but five tones (2, 3 , 4 , 6 and 7) that must be specified. Apparently there is a reverse relationship between number of deletions and number of tones to be specified: the more deletions the less tones have to be specified and vice versa. Grid 4 has a location that differs from the former three: it has the same interval as Grid 2, but is shifted one 'grain' to the right with respect to Grid 2. Note that this grid has as much as six deletions, and only two grid points that actually coincide with an element. In other words, there is only one interval in the sequence (the interval between elements 2 and 4) that actually 'suggests' this grid. There remain six tones to be specified within the grid (1, 3, 5, 6, 7 and 8). Clearly, this is a most uneconomical grid. Grid 5 has again another location. Henceforth, I will indicate the location of a grid with the number of the element in the sequence where the given grid is suggested for the first time. Thus the location of Grid 5 is 3, that of Grid 4 is 2, and that of the Grids 1, 2 and 3 is 1. Note that Grid 5 has the same interval as Grid 3 and has also only one deletion, leaving the same number of notes to be specified, although different ones. Grids 6 and 7, finally, have the same interval which equals half the period duration, but different locations. In both grids 6 tones have to be specified. Two things should be clear from the examples presented so far. First, a grid has two parameters: (1) the duration of its interval and (2) its location in the sequence. Second, the formal definition of the notion 'economical description' is much more complicated than it may seem at first: even rather simple sequences may have several different potential grids that may differ in both parameters mentioned.


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Coding, Complexity, Ambiguity The temporal sequences studied in this paper all are equitone sequences. Such sequences can be conceived of as sequences of numbers formed by the durations of the successive onset intervals. What is needed is an algorithm that describes what transformations are applied to such a sequence and in what code it finally is stored in memory. A program GRID has been written that in one step determines all potential grids of a given sequence. Therefore, starting from each element in the sequence, the program determines which grids are feasible. Excluded are grids that are characterized by an interval that is longer than the period duration and grids that have an interval that does not form a divider of the period. Interestingly, for somewhat more complex sequences the program gave more solutions than was expected, demonstrating that grid selection is not a trivial problem. This is illustrated with the two sequences presented in Figure 3 for both of which the program generated nine different potential grids. As a first step towards the selection of the most economical description, the program determines for each grid the number of tones that are fixated by the grid and the number of deletions. Moreover, the program describes the location of the nonfixated tones in the respective grid intervals by specifying how these nonfixated tones subdivide the grid intervals (see Fig. 3). In order to formalize the economy assumption made above, a rule is needed that decides which of the grids yields the most economical description. This decision rule might be a weighted function of the parameters, the number of deletions, and the number and position of the nonfixated tones. Some remarks are in order regarding the way these parameters should be weighted. First, with regard to the deletions, it must be realised that if many deletions are associated with a grid, it may theoretically still be a potential candidate, but practically it may not even be considered by the perceptual system since there is too little information in the sequence that suggests this grid (see for instance Grid 4 of pattern 2b). This seems to imply that the parameter deletions should receive a relatively high negative weight. The second parameter, the number of nonfixated tones, can obviously not be entered as such in the decision rule for the simple reason that the ease of specifying a tone in an interval depends on how this tone subdivides the interval and whether other tones occur in the same interval. Consider, for instance, sequence 3b. The specification within Grid 1 of the nonfixated element 3 clearly differs from the specification of the nonfixated elements 5 and 6. Element 3 must be specified by defining how the grid interval is partitioned, whereas the elements 5 and 6 can be specified by noting that grid interval 3 is filled with intervals at one-third the length of the grid interval. The specification of the nonfixated tones of the same sequence in Grid 2 seems to be more complicated. To specify the four non-fixated elements 2, 4, 5 and 7, the subdivision of three different grid intervals has to be noted. Moreover, none of these intervals is subdivided in a simple way. In a first approach, the location of nonfixated elements within the appropriate intervals is notated by reference to a subdivision of the intervals which is based on the smallest occurring interval (grain) in the sequence. Thus, the intervals of Grid 1 are subdivided into three, of Grid 2 into four intervals. Accordingly, the four


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intervals of Grid 1 can respectively be specified as 00, 10, 11, and 00, whereby O0 indicates an empty interval and 11 an interval that is completely filled. Analogously, the three intervals of Grid 2 can be described as 001, O11 and 100. If it is accepted that e m p t y intervals at this level need not be specified (LonguetHiggins 1978) and that a completely filled interval can easier be coded than an interval that is subdivided b y an occurring element into two intervals of different length, it is reasonable to suppose that the specification of the sequence under consideration is more economical within Grid 1 than within Grid 2. For each grid corresponding to the patterns in Fig. 3, is given the location, the number of deletions and the subdivision of the intervals.


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Complexity A first attempt can now be made to define the relative complexity of a temporal sequence. First, complexity is presumably related to the number of grid intervals that are needed to describe the sequence. A more important determinant of complexity is probably the manner in which the different grid intervals are subdivided. According to Povel (1981), only three types of subdivisions of a grid interval are conceivable. The interval may either be empty, subdivided into a number of equal intervals, or subdivided into two intervals that relate as 1:2. Other subdivisions cannot be correctly conceptualised. Moreover, in the same study it was shown that the occurrence of two different subdivisions in one sequence (unless they relate as 1:2) is not conceivable. So, subjects have trouble reproducing the sequence 450, 450, 300, 300, 300, although it can be specified in a grid with a grid interval of 900, but now the first interval is subdivided into two, the second into three subintervals. Some examples were given above (pattern 2a and 3b) in which the difference in complexity seems intuitively clear. A more formal specification of complexity, however, has to be postponed until more data concerning the subjective complexity of equitone rhythms are available.

Structural Ambiguity A sequence or pattern may be said to be ambiguous if it fits two or more structural descriptions at the same time. Strictly speaking, since every pattern can be described within several grids, every pattern is ambiguous. In practice, however, ambiguity will only arise if two (or more) structural descriptions are feasible that are equally economical. Thus, pattern 2a is theoretically ambiguous because two mutually noncompatible grids can be found that describe the sequence: Grids 2 and 3. But, since Grid 3 yields a much less economical structural description than Grid 2, this ambiguity is perceptually not apparent. Figure 4 shows some sequences that are structurally ambiguous. The two grids from which ambiguity may arise, are indicated below the sequences. The grids are mutually exclusive and give rise to structural descriptions that seem to be equally efficient. To what extent this ambiguity actually plays a role in perception, has to be verified experimentally, but it is true that with these sequences, beats corresponding to both indicated grids can be tapped. In contrast, it is practically impossible to tap a beat that corresponds to Grid 3 of pattern 2a. Here, it is tentatively proposed that the rhythmical value of a sequence is related to structural ambiguity. That is, rhythmical tension will arise if the sequence fits two (or more) equally efficient structural descriptions. Below, it will be shown that this preliminary definition of complexity and ambiguity, which is based on a model that conceives a rhythm purely as a sequence of intervals with variable durations, is not complete. The model developed so far insufficiently takes into account process aspects that are involved in the perception


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of rhythms. In particular, it will prove necessary to consider the fact that a temporal sequence is presented sequentially to a system with a limited storage capacity.

Interaction with Processing Aspects It was silently assumed before that the coding of a rhythm is performed after being stored in some temporary buffer. Moreover, the line was adopted that only temporal aspects are relevant. This latter assumption, besides being parsimonious, seemed reasonable since the equitone sequences studied here are completely specified by the successive intervals between the elements. A somewhat closer investigation of the perception of these sequences, however, soon makes clear that both assumptions are incorrect.

Starting Point Several students of temporal patterns have reported that the actual starting point of a repeatedly presented temporal sequence may affect the subjective organization of the sequence. In effect, the actual starting point may influence the subjectively chosen starting point. The starting point effect can be avoided either by starting the sequence at a subliminal intensity level which is slowly increased (Vos 1973) or by beginning the sequence with a high rate that is gradually decreased (Royer & Garner 1970). Fraisse (1978) seems to see the effect purely as an experimental problem that is to be avoided, but in my opinion, the effect shows that coding of a sequence is not postponed until the complete sequence is stored (and its periodicity detected), but rather that the encoding process starts as soon as the first intervals are entered. In the case of equitone sequences, the process of perception may be seen as a hypothesis-testing activity in which the successive intervals in the sequence are used to construct temporal grids that are subsequently tested as a potential frame to specify the sequence under consideration. Only if either the location or the duration of a


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latter intervaI suggests a grid that appears to yield a more economical description will the latter grid be adopted (cf. Steedman 1977). In studying equitone sequences, the starting point effect is encountered frequently. If a sequence starts with an element that is not the finally chosen subjective starting point, perceptual reorganization will occur which may result in a response delay. Patterns 5a and 5b are examples of such sequences. An important determinant of the subjective pattern is the 'gap principle' (Garner 1974; Vos 1973). According to this principle, the longest interval occurring in the sequence will be the end of the perceived pattern (Deutsch 1980; Handel 1973; Handel 1974). So, in the sequences shown, the longest interval will be heard as the interval between the periodically repeated patterns. The element that is consequently heard as the initial element is underlined. Another possible effect of the physical starting point regards the choice of the grid. This is shown in patterns 5c and 5d which are cyclic permutations of each other, or in other words are the same pattern with different starting points. If subjects are asked to beat time to these sequences (interspersed between others) they tend to do so in accordance with the grids indicated. That is, they chose different grids for the two patterns: for pattern 5c they consistently tap on elements 1, 5 and 8, while for pattern 5d most subjects tap on elements 1, 4, 5 and 8. According to the tenqporal grid notion this implies that the sequence is coded differently, depending on the actual starting point. On the one hand, this means that the judged complexity of the two sequences should be different, showing a far reaching effect of the order of presentation. On the other hand, this very phenomenon seems to pose a problem for the economy principle proposed above.

Accents

The assumption that all psychologically relevant characteristics of rhythms made from equitone sequences can adequatley be described by only referring to the tern-


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poral characteristics of these sequences must also be abandoned. It can be shown that there are rhythms for which the selection of the grid is not solely determined by the economy principle described before. Moreover, rhythms are encountered with highly similar structural descriptions that give rise to completely different impressions. Pattern 6a is an example of a sequence having two equally efficient grids, namely the grids indicated as 1 and 2, where subjects always select Grid 2. With pattern 6b (which is identical to pattern 4b) two appropriate grids are associated, also indicated 1 and 2. If subjects are asked to beat time to this rhythm they will always do so in accordance with Grid 2. Still, this grid seems the less economical of the two, considering that in Grid 1 the elements 4, 5, 6 and 7 fall within one grid interval, making their specification easy. Patterns 6c and 6d are two rhythms that have very similar structural descriptions, but are perceptually very different (the two sequences are respectively the basic rhythm forms of tango and beguine). If subjects are asked to tap with these sequences they will do that in accordance with the grids indicated. The structural descriptions of the two sequences only differ with respect to the first grid interval, which subdivides into an interval ratio of 3:1 for pattern 6c, versus 1:3 for pattern 6d. It is not clear why this difference by itself should produce such a big perceptual difference. An explanation that not only accounts for the experienced difference between pattern 6c and 6d, but also explains the grid selection in patterns 6a and 6b, holds that some elements in these sequences bear accents and that there is a strong tendency for these accented elements to coincide with the border of a grid interval. It is known that subjects have great difficulty in tapping with nonaccented elements in temporal sequences. This has been shown for nonaccented syllables in spoken


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sentences by Allen (1972). Now, it seems reasonable to assume that subjects will preferably select a temporal grid that coincides with accented tones. Povel and Okkerman (1981) have shown that some tones in equitone sequences, though these tones are identical in all respects, are perceived as accented. The main finding can be summarized as follows: if a tone (duration 50 ms) is followed by a second tone within 2 0 0 - 3 0 0 ms that in its turn is followed b y a considerable longer interval, the second tone will be heard as being accented. The perceived accentuation is considerable as shown b y the fact that a 4 dB increase of the nonaccented tone is needed to balance the effect. If this finding is combined with the assumption that the borders of grid intervals preferably coincide with accented elements, it can be understood why Grid 2 is selected in pattern 6a and pattern 6b: in both cases that grid coincides with the accented element (respectively elements 2 and 3). If the same line of reasoning is applied to patterns 6c and 6d, it can be understood why these two patterns are perceptually so very different: in pattern 6c the accented element 3 coincides with the grid, but in pattern 6c, the accented element 2 does not coincide with the grid, thus creating a special form of ambiguity which may cause the experience of a special rhythmical tension. (Note that for pattern 6d there is no appropriate grid that coincides with the accented element 2). The introduction of accent as an indispensable explanatory factor brings the model closer to other theoretical accounts on r h y t h m (Cooper & Meyer 1960; Davies 1978; Fraisse 1956; Koffka 1909; Longuet-Higgins 1976; Perkins & Howard 1976; Seashore 1938; Sturges & Martin 1974; Woodrow 1909). The two latter factors introduced, the starting point of a sequence and the effects of accent, can be incorporated into the model if the first assumption is slightly modified. This assumption stated that every interval occurring in a sequence is a potential candidate for a grid interval. The amended assumption now goes: each interval occurring in a sequence is a potential grid interval, but marked intervals are better candidates. An interval is marked if it starts or ends with an accented element. Also, an interval occurring at the beginning of a sequence has a better chance to become the grid interval. If the initially chosen grid is not suitable, perceptual reorganization will take place. Such recognization is experienced when listening to pattern 6d. First a grid is tried out with a grid interval that coincides with the interval that starts at the accented element 2. But neither the interval between elements 2 and 3, nor the interval between elements 2 and 4 results in an acceptable grid. Therefore, the grid is finally selected that starts at element 1, which, however, is not accented. It was indeed found that subjects take longer before beginning to beat time to pattern 6d than to pattern 6c. An instructive demonstration showing the effect of accented elements is obtained when in pattern 2a the elements 1 and 3 are made louder. The accents that now appear in the sequence fit best with Grid 3, while as shown before, the temporal structure is most economically specified within Grid 2. These two grids, evoked by eompletetly different mechanisms, are not compatible. This gives rise to a highly ambiguous percept. This pattern indeed shows a high degree of rhythmical tension. If subjects are required to beat time to this sequence, they will consistently tap in accordance with Grid 2, showing that in this example a grid is chosen that yields the most efficient description of the temporal structure of the sequence. For pattern


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6b it was found that subjects consistently beat time in accordance with Grid 2, showing that in this case, the occurring accents are the decisive factor.

Tempo A parsimonious theoretical account of rhythm perception should preferably start with the assumption that tempo, being an overall variable that does not effect the temporal structure, is an unimportant variable. This assumption, however, is soon found to be incorrect when tested; changing the tempo of temporal sequences may cause dramatic changes in the perceived rhythmical characteristics. Actually, tempo modifications that can practically be brought about are rather limited because the range of interval durations that are normally used in the construction of rhythms is not very large. Roughly, this range lies between approximately 125 ms for the shortest and 1 5 0 0 - 2 0 0 0 ms for the longest interval (Bolton 1894; Fraisse 1956). A group of tones with onset intervals smaller than 125 ms is perceived as an inarticulated train of events; such a series cannot be heard as being subdivided into subgroups. If, on the other hand, the duration of an interval between two onsets is made longer than about 1500 ms, the tones tend to be perceived as isolated events. This means that a temporal sequence that contains intervals of 200 and 800 ms (a 1:4 ratio very common in musical rhythms) is speeded up or slowed down twice; it already exceeds the range of practical durations. That this range is so limited is probably related to restrictions in the intake mechanism like limited temporal resolution and storage capacity. Two effects will be mentioned that are related to tempo changes. First, tempo appears to be a factor in the selection of the grid. Figure 7 displays two simple sequences, for which the selection of the grids depends on the tempo of presenta-

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tion. If the smallest interval is 400 ms or more, subjects tend to beat according to Grid 1; if the tempo is increased, a beat corresponding to Grid 2 will be tapped. In pattern 7b the same findings are observed: Grid 1 is selected for a rather slow tempo, Grid 2, for a faster tempo. So far the findings in the two examples are identical. One might propose that what is observed here is a preference to tap at a medium speed in what essentially is the same grid: Grid 1 and 2 forming two different levels of the same (hierarchical) grid. Since this supposition introduces a new concept, I will postpone a further discussion until later. If the presentation rate of pattern 7b is further increased until the smallest interval has a duration of about 100 ms (which is about the shortest tone value occurring in music), subjects show a strong tendency to beat time in accordance with Grid 3. This seemingly strange behavior is presumably the consequence of an increased accent on element 3 which attracts attention to the grid. A similar phenomenon is shown in pattern 7c. If the smallest interval is about 400 ms long, subjects tap Grid i ; if the tempo is doubled making the smaller interval 200 ms, subjects tap Grid 2. If, subsequently, the tempo is further increased until the smallest interval is 100 ms, the r h y t h m seems to collapse; it becomes impossible to beat time with the sequence. The reason seems to be that the group of tones formed by elements 4, 5 and 6 is heard as an indifferentiated cluster with a strong accent on the last element. Note that in order to tap a beat that either conforms to Grid 1 or to Grid 2, element 5 must be isolated from the group formed by elements 4, 5 and 6. Tentatively, I would summarize the reported phenomena related to tempo by stating that changes in tempo mainly affect the perceived accentuation of tones in the sequence to the extent to which groups of proximate tones are turned into a cluster of nondistinguishable events. Therefore, tempo does not have to be introduced as a separate factor in the theory.

Discussion In the present study, I have attempted to show that such important characteristics as the complexity and the rhythmical value of temporal sequences can be understood from a perceptual model that basically assumes that the perceiver specifies the temporal structure within a temporal grid. This temporal grid is chosen so that it yields the most economical description. In this code, the number of deletions and the nature of the subdivisions of the grid intervals determine the complexity of the sequence. Rhythmical tension arises especially when the sequence 'suggests' two (or more) descriptions that are mutually exclusive. Alternative codings for one sequence may be evoked when two descriptions are structurally equivalent. A practically more important cause of rhythmical tension is that the temporal structure of the sequence suggests one grid, while some intervals that are marked in a special way (e.g., by accented border elements) suggest another. In the theoretical account presented so far, the concept of grouping was not needed. This may seem strange since according to several authors, grouping is a basic mechanism in the perception of rhythms (Bolton 1894; Fraisse 1 9 5 6 ; K o f f k a 1909; Vos 1973; Woodrow 1909, 1951; Wundt 1911). In the introduction it was pointed


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out that in the perception of tone sequences, a distinction should generally be made between the organization of the temporal structure formed by the series of consecutive intervals in the sequence on the one hand, and the perception of the structure that may he present in the characteristics of the elements (accent and pitch structure) on the other. Most research on tone sequences has concentrated on the latter aspect (Deutsch & Feroe 1981; Dowling & Fujitani 1971;Garner 1974; Jones 1974, 1978; Simon & Sumner 1968; Vitz & Todd 1969). An interesting theoretical view on the interaction between the two aspects of processing discussed here has been presented by Jones (1976). In the introduction it was also stated that a complete understanding of musical rhythms should take into account both aspects of the perception process. Later it was shown, that even for the understanding of the rhythmical aspects of equitone sequences, the concept of accent, which is of course a feature of the elements, was needed. If grouping is loosely defined as the perception that a number of events belong together, it is clear that grouping also occurs in the perception of equitone sequences: Subjects very easily detect the periodicity in a repetively presented sequence. Moreover, subgroups may be heard within the period. The law of proximity plays an important role here: It may determine the formation of subgroups and also the perceived starting point (Deutsch 1980; Garner 1974; Handel & Yoder 1975; Michon 1977, 1978; Restle 1972; Vos 1973). Note that the grouping considered here concerns the grouping of the elements, whereas the grid as a structuring device may also be said to divide the period in subgroups, in as far as it segments the sequence in a number of equal intervals. Perhaps it is more correct to say that the grid groups intervals rather than elements. The two types of grouping discerned here are: the organization of the elements and the organization of the intervals. Though presumably occurring in parallel, they may to a greater or lesser extent be incompatible. Thus for instance in pattern 3b the elements 4 through 7 may be heard as a group on the basis of the law of proximity, whereas this group is subdivided by Grid 2 so that elements 4 and 5 fall in one segment and element 7 in another and element 6 forms the border of the two segments. With the addition of the concept of grouping of the elements, the theoretical framework may now also be extended to the understanding of more complex rhythmical phenomena that are the result of the interaction of the two types of organization. Though a detailed elaboration is beyond the scope of this paper, a couple of examples will be given that may clarify the principle. Consider pattern 8a which shows an isochronic sequence consisting of alternating groups of three low (indicated C) and three high (indicated G) tones. Subjects will tap to this sequence in accordance with the grouping introduced by the pitch dimension, as indicated in Figure 8. This shows that pitch characteristics in a tone sequence may determine grid selection. Steedman (1974) has proposed a model of how melodic repetition acts upon beat selection. In actual tone sequences the two determinants (temporal structure and pitch structure) may either give rise to one and the same grid as in pattern 8b, or they may yield competing grids as in tone sequence 8c, where the temporal structure gives rise to Grid 1 and the melodic structure to Grid 2. Numerous examples can be found in music where the different determinants of the grid, temporal structure, accents, and pitch structure give rise to different grids thus creating a special


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rhythmical tension. Figure 7 presents two fragments of pieces written in respectively 3/4 and 3/8 meter, in which (part of) the melodic line suggests another meter. In the Bach fragment things are even more complex since the upper line suggests a 6/8 measure and the lower line a 3/4 measure. These examples sufficiently show how complex a phenomenon rhythm is. More importantly, they show how a framework that incorporates both components of the perception process (element and interval perception) may account for these complex rhythmical phenomena. A definitive coding language for rhythms, especially for more complex rhythms will have to face the question of whether a more hierarchical description than the one presented here should be considered. The notion of measure in music of course points to such hierarchy. Many theorists have argued that rhythms are hierarchically organized (Cooper & Meyer 1960; Jones 1974, 1978; Martin 1972; Michon 1974; Perkins & Howard 1976; Vorberg & Hambuch 1978), but so far the empirical evidence is meager (Dworak & Sullivan 1976; Povel 1981; Vorberg & Hambuch 1978; MacKenzie, Nelson & Wills 1982). Until pertinent data have been collected, the development of a definitive coding language will have to wait.


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In this paper the perception of what might be called pure, that is not actually performed tone sequences, has been studied. Every listener, however, knows that an important part of the rhythmical tension is caused by deviations brought about by the performer. Many textbooks are written about the question of how music should be performed in the temporal domain. Also, some descriptive studies are devoted to this question (Clarke 1982; Gabrielson 1974; Michon 1974; Povel 1977; Sears 1902; Seashore 1938; Shaffer 1981) which all show that performers modify the temporal structure as notated in the score considerably when .interpreting a piece of music. In my opinion, the tension created by the performer is superposed on the tension that a rhythm has in its pure form, The nature of the interpretive modifications brought about by performers may be better understood if sufficient understanding is gained of how temporal sequences are internally represented. Apart from that, it seems that some rhythms are best defined in terms of deviations from some simple interval pattern. Thus, the Viennese Waltz for instance, is characterized by the playing of the second beat slightly ahead of time. A final remark may be made with respect to the distinction between the process of perception and the result of this process: the memory code. The actual theory as presented here pertains to the result of the perceptual process: the memory code formed of the rhythm. I have shown, however, that in the case of the perception of temporal sequences, it is impossible to predict the final coding of a sequence without referring to the process of perception. Indeed, I have argued that the perception of temporal sequences is a hypothesis-testing activity that starts as soon as sufficient information about the sequence has reached the subject to form a potential hypothesis. Therefore, the study of rhythm perception may not only yield a better comprehension of the result of the perceptual process, the memory code, but also a greater insight in this process itself.

Acknowledgement. I am grateful for the comments and help of Peter Essens, Rent Collard, Joan Woodcock, Leon van Noorden, Piet Vos, Herman Kolk, Hans-Leo Teulings, and John Michon.

References Allen GD (1972) The location of rhythmic stress beats in English: an experimental study. I. Language and Speech 15:72-100 Bolton TL (1894) Rhythm. Am J Psychol 6:145-238 Clarke EF (1982) Timing in the performance of Erik Satie~s "vexations". Acta Psychol 50:1-19 Collard RFA, Povel DJ (1982) Theory of serial pattern production: tree traversals. Psychol Rev 89:693-707 Cooper G, Meyer L (1960) The Rhythmic Structure of Music. University of Chicago Press, Chicago Davies DB (1978) The Psychology of Music. London: Hutchinson & Co. Ltd. Deutsch, D (1980) The processing of structured and unstructured tonal sequences. Percept & Psychophysics 28:381-389


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Deutsch D, Feroe J (1981) The internal representation of pitch sequences in tonal music. Psychol Rev 88:503--522 Dowling WJ, Fujitani DS (1971) Contour, intervals and pitch recognition in memory for meoldies. J Acoust Soc Am 49:524-531 Dworak P, Sullivan MP (1976) Confusion and reversals in rhythm perception. Internal Report, Dep. of Music and Electronic Engineering, Carnegie Mellon University, Pittsburg Fraisse P (1946) Contribution a l'etude dy rythme en tant que forme temporelle. Journal de psychologic normale et pathologique 39:283-304 Fraisse P (1956) Les structures rythmiques. Publication Universitaires de Louvain, Louvain Fraisse P (1978) Time and rhythm perception. In: Carterette EC, Friedman MP (eds) Handbook of Perception, Vol. 8. Academic Press, New York Gabrielson A (1974) Performance of rhythm patterns. Scand J Psychol 15:63-72 Garner WR (1974) The Processing of Information and Structure. Wiley, New York Hall GS, Jastrow J (1886) Studies of rhythm. Mind 11:55-62 Handel S (1973) Temporal segmentation of repeating auditory patterns. J Exp Psychol 101:46-54 Handel S (1974) Perceiving melodic and rhythmic auditory patterns. J Exp Psychol 103:922-933 Handel S, Yoder D (1975) The effects of intensity and interval rhythms on the perception of auditory and visual temporal patterns. Q J Exp Psychol 27:111122 Jones MR (1974) Cognitive repesentations of serial patterns. In: Kantowitz BH (ed) Human Information Processing: Tutorials in Performance and Cognition. Wiley, New York Jones MR (1976) Time, our lost dimension: Toward a new theory of perception, attention, and memory. Psychol Rev 83:323-355 Jones MR (1978) Auditory patterns: Studies in the perception of structure. In: Carterette EC, Friedman MP (eds) Handbook of Perception (vol. 8) Academic Press, New York K6hler W (1929) Gestalt Psychology. Liveright, New York Koffka K (1909) Experimental-Untersuchungen zur Lehre vom Rhythmus. Zeitschrift fiir Psychologie 52:1-109 Koffka K (1935) Principles of Gestalt Psychology. Lund Humphries, London Longuet-Higgins HC (1976) Perception of melodies. Nature 263:646--653 Longuet-Higgins HC (1978) The perception of music. Interdisciplinary Science Reviews 3:148--156 Longuet-Higgins HC, Lee CS (1982) The perception of musical rhythms. Perception 11:115-128 MacDougall R (1902) Rhythm, time and number. Am J Psychol 13:88-97 MacKenzie CL, Nelson JA, Wills BL (in press) A preliminary investigation of motor programming in piano performance as a function of skill level. In: Rogers DR, Sloboda JA (eds) The Acquisition of Symbolic Skills. Plenum Press Inc. Mach E (1911) Die Analyse der Empfindungen. Jena: Fischer, 1886 (1st ed), (6th ed)


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Martin JG (1972) Rhythmic (hierarchical) versus serial structure in speech and other behavior. Psychol Rev 79:487-509 Meumann E (1894) Untersuchungen zur Psychologie und Asthetik des Rhythmus. Philosophische Studien 10:249-323,393-431 Michon JA (1974) Programs and "programs" for sequential patterns in motor behavior. Brain Res 71:413-424 Michon JA (1977) Holes in the fabric of subjective time: Figure-ground relations in event sequences. Acta Psychol 41:191-203 Michon JA (1978) The making of the present. A tutorial review. In: Requin J (ed) Attention and Performance VIII. Erlbaum, Hillsdale, New Jersey Miner J B (1903) Motor, visual and applied rhythms. Psychol Rev Monographs 5 Perkins DN, Howard VA (1976) Toward a notation for rhythm perception. Interface 5:69-86 Povel DJ (1977) Temporal structure of performed music. Some preliminary observations. Acta Psychol 41:309-320 Povel DJ (1981) Internal representation of simple temporal patterns. J Exp Psychol Human Percept PerfOrm 7:3-18 Povel DJ, Collard RFA (1982) Structural factors in patterned finger tapping. Acta Psychol 52:107-123 Povel DJ, Essens P (submitted for publication) Perception of temporal patterns. Povel DJ, Okkerman H (1981) Accents in equitone sequences. Percept & Psychophysics 30:565-572 Restle F (1972) Serial patterns: The role of phrasing. J Exp Psychol 92:385-390 Royer FL, Garner WR (1970) Perceptual organization of nine-element auditory temporal patterns. Percept & Psychophysics 7:215-220 Ruckmick CA (1913) The role of kinaesthesis in the perception of rhythm. Am J Psychol 24 Sears CH (1902) A contribution to the psychology of rhythm. Am J Psychol 13: 28-61 Seashore CE (1938) The Psychology of Music. McGraw-Hill, New York Shaffer LH (1981) Performances of Chopin, Bach, and Bartok: Studies in motor programming. Cog Psychol 13:326-.376 Simon HA, Sumner RK (1968) Pattern in music. In: Kleinmuntz B (ed) Formal Representation of Human Judgement. Wiley, New York Steedman J (1977) The perception of musical rhythms and meter. Perception 6: 555-569 Sternberg S, Knoll RL, Zukofsky P (1982) Timing by skilled musicians: Perception, production and imitation of time ratios. In: Deutsch D (ed) The Psychology of Music. Academic Press, New York Stetson RH (1905) A motor theory of rhythm and discrete succession. Psychol Rev 12:250-270,292--350 Sturges PT, Martin JG (1974) Rhythmic structure in auditory pattern perception and immediate memory. J Exp Psychol 102:377-383 Vierordt H (1968) Der Zeitsinn nach Versuchen. Laupp, T/ibingen Vitz PC, Todd TC (1969) A coded element model of the perceptual processing of sequential stimuli. Psychol Rev 76:433--449


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Vorberg D, Hambuch R (1978) On the temporal control of rhythmic performance. In: Requin J (ed) Attention and Performance VIII. Erlbaum, Hillsdale, New Jersey Vos PG (1973) Waarneming van metrische toonreeksen. Unpublished dissertation, University of Nijmegen Woodrow H (1909) A quantitative study of rhythm. Arch Psychol 14:1-66 Woodrow H (1951) Time perception. In: Stevens SS (ed) Handbook of experimental psychology. Wiley, New York Wundt W (1874) Grundziige der physiologischen Psychologie. Engelmann, Leipzig

Received November8, 1983/January 15, 1984


1984Povel