Page  125 ï~~The Detection of Rhythmic Repetition Using a Self-Organising Neural Network Simon Roberts and Mike Greenhough Department of Physics and Astronomy University of Wales College of Cardiff Abstract An artificial neural network known as SONNET [Nigrin, 1993], which is capable of classifying temporal patterns from a continuous sequential input, is described. When the network is exposed to a sequence of inter-onset-intervals, it is able to detect rhythmic repetition without being supplied with any additional information, e.g. metrical structure. Therefore, the network can be incorporated within a model of rhythm perception to assist with the determination of grouping and metrical structures. Results of network simulations are presented. 1 Introduction Repetition is an important factor when considering rhythm perception, as it contributes towards the formation of the grouping and metrical structures. When discussing grouping principles, Deutsch [1986] states that "repetition of a subsequence within a sequence induces the listener to group the elements of the subsequence together". Repetition is also a perceptual cue for metre [Palmer and Krumhansl, 1990], as a repeated rhythmic pattern is likely to occur in the same metric position. Lerdahl and Jackendoff [1983] refer to similar passages of music as being "parallel". Parallelism is included in their "Preference Rules" for grouping and metrical structure. Longuet-Higgins and Lee [1982] have developed a model for the perception of musical rhythms. They concluded that the inability of the model to take account of rhythmic repetition was a serious limitation. The current paper describes an artificial neural network which is capable of detecting rhythmic repetition. Similar research has been carried out by Rosenthal [1989], who developed a computer model that constructs a hierarchical description of a rhythm. Rosenthal's model uses information about metrical structure to assist with the segmentation of the incoming events. The motivation for the work presented in the current paper, is to develop a system which will support the determination of a rhythm's grouping and metrical structures. Therefore, segmentation is based only on the regularities inherent within the input environment. The neural network encodes information about the structure of recurring rhythmic patterns, and is also able to generate expectations. This is beneficial for a model of rhythm perception [Desain, 1992]. 2 SONNET Overview SONNET (Self-Organising Neural Network) [Nigrin, 1993] is an artificial neural network which is capable of classifying patterns from a continually changing input. When the network is exposed to an environment it self-organises, using unsupervised learning, to form stable categories for recurring patterns which exist within the environment. The network creates its own segmentations in response to a stream of incoming events, and is therefore potentially suitable for real-time operation. SONNET can recognise patterns which are surrounded by extraneous information (embedded patterns) in a context-sensitive manner. That is, when a long, previously learned pattern is presented to the network, SONNET allows the category for that pattern to mask out categories which represent constituent parts of it. (Neural networks which possess this property are known as masking fields.) Alternatively, if a sub-part of the long pattern is presented, then a corresponding smaller category is able to classify the shorter pattern. In addition, SONNET allows multiple existing categories to represent novel large patterns. SONNET also has a number of other desirable features, such as the ability to operate using different learning speeds, create arbitrarily coarse classifications, generate expectations, and represent hierarchical structures. 3 Description of SONNET The network consists of two fields of cells: F1 and F2. The input is applied at F1, which acts as a shortterm memory (STM) and converts a temporal sequence of events into a spatial pattern. The activities of the cells in F1 are fed forward to each of the cells in the F2 field via bottom-up excitatory connections. The F2 cells classify the F1 patterns. When a novel pattern exists at F1, many F2 cells will obtain low activations. After learning, a single F2 cell will activate strongly whenever its corresponding pattern exists at Fl. The F2 cell is said to be committed to the pattern. The long-term memory representation is stored by the magnitude of the excitatory weights on the bottom-up connections to each F2 cell. The network architecture described above is based on the Adaptive Resonance Theory ICMC Proceedings 1994 125 Neural Nets

Page  126 ï~~circuits developed by Carpenter and Grossberg. (See for example [Carpenter and Grossberg, 1987]). As learning progresses, the network self-organises into a masking field, where the F2 cells have different "sizes". The "size" of an F2 cell increases with the number of strong bottom-up connections associated with that cell, thus "larger" F2 cells classify longer patterns. The F2 cells compete to gain a high activation via lateral, non-uniform inhibitory connections. The inhibitory connectivity pattern is initially uniform, but as the network is exposed to an environment the inhibitory weights self-organise. Eventually, only F2 cells which respond to overlapping patterns provide mutual inhibition, i.e. patterns which have items in common. Topdown feedback connections from the F2 field to the F1 field allow expectation to be introduced. The feedback weights self-organise so that they become approximately parallel to the bottom-up weights. 4 Application of SONNET to the Detection of Rhythmic Repetition The network was implemented with each F1 cell representing a particular inter-onset-interval (101), measured in musical time, i.e. beats at some metrical level. An additional system is required to identify and track the beats of a particular metrical level, e.g. tactus, to allow the network to be tempo invariant. Also, some form of pre-processing is required to convert an 101 to a place on a spatial map, to allow the correct F1 cell to be fired. A supervised neural network, such as a multi-layer perceptron trained using back-propagation of error, could be used in the input stage. In the network simulations, the correct mapping was contrived and the Fl cells were fired accordingly. After firing, an F1 cell's activation increases. When the next IOI is detected, its corresponding F1 cell will fire and the currently active F1 cells will increase their activity. The activities increase with time to enable expectations to be correctly generated. (See Nigrin [1993] for further details.) However, in the simulations, the top-down feedback connections were disabled, so no expectations were generated. To prevent F1 overload, a number of the most active F1 cells are reset after the total activity in the F1 field has exceeded some threshold. In the simulations, the threshold was set so that F1 reset would occur after 7 lOis were held in the STM. This value is the typical STM depth for humans (Miller, 1956]. After an F2 cell has become committed to a pattern, it is able to chunk-out its pattern from the STM, thus reducing the total activity in the F1 field. A restriction is placed on the length of a pattern which can be learnt by an F2 cell. The maximum length was chosen to be commensurate with the number of items which humans can recall before order information becomes confused. This is known as the transient memory span and has a typical value of 4 items [Miller, 1956]. Rhythms often only consist of a few different IOIs, so it is necessary for multiple F1 cells to correspond to the same 101, to allow repeated lOis to be present in the STM. So, the number of required F1 cells is dependent on the number of different IOIs to be represented, and the maximum number of occurrences of a particular 101 to be held in the STM. The latter was taken to equal the STM depth, i.e. 7 F1 cells relate to the same IOI. Now, when an 101 is presented to the network, one of its associated inactive F1 cells will fire. The network architecture is shown in Figure 1. Allowing multiple F1 cells to represent a single 101 increases the complexity of the network. The reason for this is best explained using an example. Let Q1, Q2...Q7 denote 7 F1 cells, each of which represents a quarter-note IOI. If 3 of these cells are activated in the order Q1, Q2, Q3 and an F2 cell starts to learn this pattern, then this F2 cell should respond to J J J regardless of how it is stored in the STM. Therefore, the F2 cell should respond equally well to the patterns Q3Q2Q1, Q5Q6Q7 or any other permutation of 3 F1 cells representing consecutive quarter-notes. Nigrin [1993] achieves this behaviour by allowing multiple links to exist from each F1 cell to each F2 cell. Each link represents an occurrence of the F1 cell's associated 101, in a particular position in the pattern encoded by the F2 cell. The earlier the position in the pattern, the larger the excitatory weight on the link. This mechanism can selforganise assuming that an F2 cell can identify which F1 cells represent the same 101. Network Output lr {c4 c...A F2 10... JF =ZNW % - StMWi Map (P mcuo) Figure 1: SONNET architecture with multiple F1 cells representing the same IO!. Fl is fully connected to F2 via bottom-up excitatory links, with multiple links from each F1 cell to each F2 cell. The lateral inhibitory connections in the F2 field and the top-down feedback connections are not shown. Neural Nets 126 ICMC Proceedings 1994

Page  127 ï~~We developed slight modifications to enable embedded patterns to be recognised, when multiple occurrences of an 101 could be stored in the STM. For example, suppose an F2 cell has learned the pattern ))J. The modifications were necessary to allow this cell to recognise that its pattern had recently occurred, when sequences like )J ))41 )4J were held in the STM. 5 Simulations The same network architecture and network parameters were used for each of the sequences that were presented. The sequences collectively contained 8 different lOis, so 56 F1 cells were used to allow 7 occurrences of each 101 to be held in the STM. Twenty-five F2 cells were used, i.e. a maximum of 25 patterns could be encoded by the network. The cell activities and all of the weights were only permitted to change during a fixed time period after each 101 was presented. This time period will be referred to as the attention-span. An attention-span of 0.2s was chosen by considering the shortest 101, from each sequence, at a typical tempo. (NB: The simulations were run in pseudo real-time.) A parameter which greatly affects SONNET's ability to detect rhythmic repetition is known as the learning rate. A high learning rate enables the network to learn patterns very quickly, but may prevent regularities in the input from being identified. A low learning rate allows recurring patterns and embedded patterns to be learnt, but more sequence presentations are required. A low learning rate was chosen to allow the network to learn regularities from the sequences. Each sequence was presented 30 times in a continuous manner, i.e. the first 101 of the sequence followed directly on from the last 101 of the previous presentation. The input sequences were based on the rhythms displayed in Figure 2. These rhythms differ by length, complexity of rhythmic structure, and time signature. Figure 3 shows how the network segmented each sequence on the 30th presentation. Bolero lJ Ja zJ- -]IJ J --m 3 Greensleeves Kong Kristian (Danish Royal Anthem) UdlU 4 J 41 J. )JJ J 4 J U d4 IJTflJ J)). )4J J1 UJ J1 J J 1J. )J1 J11 IJJ-JJJ4 JJ4JJ IJ J1 UJ t1I IJ1J41 U.. )J41IJ. )J. )4, IJI Figure 2. Rhythms used to form input sequences Boer 5 5jrh IT) is ~5I~JJSj~~~ J) J) J1 J J1 JJ J) a: J) - J4 J771)4 J4u KM"JfJ' J. Kong Krisia (DaanhRoyal Anthem) 414 414 4.) 414 41J 41.J J]).r J J.) 4 J14 4 4.) 4J.rj jj j rj j jj ji JJ 1.) J 1.) J.) JJ 4 J.) 414 4 Figure 3: The segmentation of each sequence, performed by the network on the last sequence presentation. When SONNET was exposed to a sequence, it gradually formed categories, with the most frequently occurring patterns being encoded the earliest. The ability to recognise a recurring pattern is dependent on the number of contexts in which that pattern appears. For example, when the Bolero sequence was presented, the network failed to recognise every occurrence of flf, because SONNET could not form a category for F' (which occurs at the end of the first bar), as this pattern always occurred in the same context. Unless short patterns are presented in multiple contexts, SONNET lumps them together with the surrounding IOIs. This problem could be overcome if top-down feedback were used. A committed F2 cell would then be able to suppress other F2 cells when its pattern is only partly present at F1, and the remainder of its pattern is expected. For Bolero, no F2 cell encoded the pattern A because, after an F2 cell had become committed to f', an uncommitted F2 cell could only respond to the triplet alone when it was not preceded by an eighthnote (otherwise it received a large inhibitory signal). This only happens at the end of the second bar, so an uncommitted F2 cell could only respond to the triplet in one context. Consequently, a maximal length pattern of 4 IOIs was formed. After SONNET had classified the commonly recurring patterns, categories continued to form until a stable representation was obtained for the entire sequence. The number of presentations necessary to achieve a stable representation increased with the complexity of the sequence structure. Bolero required 6 presentations, Greensleeves required 13 and Kong Kristian required 20 presentations. For Greensleeves, SONNET only failed to detect the repetition of 4'7. This was because J. was always preceded by this pattern and so the network lumped all of these IOIs together. As Kong Kristian has a more complex structure, the repeating patterns occur in multiple contexts, and therefore SONNET was able to detect all of the ICMC Proceedings 1994 127 Neural Nets

Page  128 ï~~rhythmic repetition. During the presentation of this sequence, SONNET allowed 2 F2 cells to simultaneously classify their patterns, as there was no overlap between these patterns. This occurred when the F2 cell encodin a quarter-note combined with the F2 cell encoding Jito represent J J'. After SONNET was exposed to the Greensleeves or Kong Kristian sequences, some committed F2 cells had become redundant, i.e. these- cells never classified their patterns on later presentations. The reason for this was that the patterns which these cells encoded became partly chunked-out of the F1 field, as further regularities were classified by other F2 cells. For example, during the first presentation of Greensleeves, an F cell became committed to. After the patternsJ) and 3J had been encoded by the network, this cell became redundant. As SONNET only uses regularities in the input to form its categories, the resulting segmentations are not necessarily human-like, as humans involve many principles when grouping IOIs together. If the attention-span increased with 101,.then segmented patterns are likely to end with a long 101, thus the network would produce more human-like segmentations. This grouping organisation is known as the gap principle [Deutsch, 1986]. Music psychologists can benefit from SONNET, because factors which affect the segmentation of a continuous stream of IOIs can be investigated in isolation. For example, the attention-span can be varied to analyse to what extent longer IOIs affect grouping. 6 Further Work The above simulations served as a preliminary investigation into the performance of SONNET for the detection of rhythmic repetition. A number of alterations are required to create a more compact and elegant system. In the work discussed above, there are 2 distinct representations for time: a particular F1 cell represents a specific IOI and the F1 activity pattern encodes/the order information. The STM can be implemented so that the F1 activity pattern represents both the IOIs and the order in which these occur. This is achieved by continuously modifying the F1 cell activities at short, regular intervals in time, as opposed to only allowing modification to take place during a fixed time period after an event occurs. The firing of an F1 cell would then simply correspond to the occurrence of an event onset, thus the number of F1 cells would depend only on the required STM depth. Fewer F1 cells is advantageous for the simulations, because the computation time increases dramatically with the number of cells in the network. Also, with this STM implementation, no pre-processing is required to convert an 101 to a place on a spatial map. The absence of a pre-processing system overcomes the need to deal with expressive timing in the input stage. Expressive timing can now be processed directly by the SONNET network, as a vigilance parameter controls the coarseness of the classifications. Multiple SONNET networks can be lumped together to form a hierarchical structure. This property is desirable for the classification of patterns which are inherently hierarchical, such as musical rhythms. Future work will investigate the performance of lumped SONNET networks. 7 Summary An artificial neural network, known as SONNET, which can classify temporal patterns from a continuous sequential input, has been described. SONNET's ability to detect rhythmic repetition has been demonstrated, by exposing it to 3 different sequences of IOIs. The network is useful for rhythm perception models because grouping structure and metrical structure are dependent on repetition. References [Carpenter and Grossberg, 1987] Gail Carpenter and Stephen Grossberg. A massively parallel architecture for a self-organizing neural pattern recognition machine. Computer Vision, Graphics and Image Processing, 37, pp.54-115, 1987. [Desain, 1992] Peter Desain. A (de)composable theory of rhythm perception. Music Perception, 9(4): pp.439-454, 1992. [Deutsch, 1986] Diana Deutsch. Auditory pattern recognition. In K. R. Boff, L. Kaufman and J. P. Thomas (Eds.) Handbook of Perception and Human Performance, Volume 2: Cognitive Processes and Performance, John Wiley and Sons, New York, 1986. [Lerdahl and Jackendoff, 1983] Fred Lerdahl and Ray Jackendoff. A Generative Theory of Tonal Music, The MIT Press, ISBN 0-262-62049-9, 1983. [Longuet-Higgins and Lee, 1982] H. C. LonguetHiggins and Christopher Lee. The perception of musical rhythms. Perception 11,pp.115-128, 1982. [Miller, 1956] George A. Miller. The magical number seven, plus or minus two: some limits on our capacity for processing information. The Psychological Review, 63(2), pp.81-97, 1956. [Nigrin, 1993] Albert Nigrin. Neural Networks for Pattern Recognition, The MIT Press, ISBN 0-262 -14054-3, 1993. [Palmer and Krumhansl, 1990] Caroline Palmer and Carol Krumhansl. Mental representations for musical meter. Journal of Experimental Psychology: Human Perception and Performance, 16(4), pp.728-74l, 1990. [Rosenthal, 1989] David Rosenthal. A model of the process of listening to simple rhythms. Music Perception, 6(3), pp.315-328, 1989. Neural Nets 128 ICMC Proceedings 1994