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Page 00000001 A Neural Network Model of Metric Perception and Cognition in the Audition of Functional Tonal Music. Jonathan Berger CCRMA Stanford University Stanford, CA 94305, U.S.A email@example.com Dan Gang * Institute of Computer Science Hebrew University Jerusalem 91904, Israel firstname.lastname@example.org Abstract In our previous work we proposed a theory of cognition of tonal music based on control of expectations and created a model to test the theory using a hierarchical sequential neural network. The net learns metered and rhythmecized functional tonal harmonic progressions allowing us to measure fluctuations in the degree of realized expectation (DRE). Preliminary results demonstrated the necessity of including metric information in the model in order to obtain more realistic results for the model of the DRE. This was achieved by adding two units representing periodic index of meter to the input layer. In this paper we describe significant extensions to the architecture. Specifically, our goal was to represent more general meter tracking strategies and consider their implications as cognitive models. The output layer of the sub-net for metric information is fully connected to the hidden layer of sequential net. This output layer includes pools of three and four units representing duple and triple metric indices. Thus the sub-net was able to influence the resulting DRE, that was expected by the net. Moreover, by including multiple metric parsings in the output layer the net reflects conflicts between parallel possible interpretations of meter. This output was fed back into the sub-net to influence the next predictions of the DRE and the meter. In addition, the target harmony element was fed into the context instead of the actual output, thus simulating the interactive influences of harmonic rhythm and meter. 1 Introduction "The poets have a proverb: Metra parant animos (the emotions are animated through verse). They say such quite rightly: for nothing penetrates the heart as much as a well-arranged rhyme scheme [Mat39]". Johann Mattheson's awareness of the cognitive power of underlying metric temporal patterns (be it musical metric feet or rhythmic modes) in music and poetry has been consistently stated and, over the past century, empirically researched. That listening to music involves an initial creation of a metric schema has been well documented. What is not clear, however, is the process in which the listener arrives at a working schema. In this paper we explore and model a possible scenario of metric decision making. As a point of departure we incorporate observations, speculation, *Dan Gang is supported by an Eshkol Fellowship of the Israel Ministry of Science and perceptual studies that suggest: 1. Constructing a metric schema is a task critical to music audition. In Mattheson's words "...the ordering of the feet in poetry and the wellconstructed alternation of meters, even if there were no rhyme scheme, produces something initially so certain and clear in the hearing that the mind enjoys a secret pleasure from the orderliness and accepts the performance so much the easier." 2. Listeners of Western music have preconceived organizational schemas grouping into duple or triple metric units. Listeners count in hierarchies (base 3 or base 4 for most common meters). [Pov81] demonstrated that untrained listeners can accurately distinguish between duple and triple metric units. Furthermore, considerable evidence of preconceived grouping preferences suggest that this is applicable to meter recognition.
Page 00000002 Although generative algorithms (e.g., [LHL82]) and autocorrelative methods (e.g., [DH89]) for meter recognition are successful in their task they do not offer a plausible explanation of how a listener applies schematic based expectation of duple or triple groupings to determine meter. The music theory literature regarding meter (e.g., [LJ83]) similarly fail to account for this basic task. 3. Metric awareness is necessary in building a network of implications and expectations which lies at the heart of the musical experience. London [Lon92] proposes that metric cognition involves a two stage process comprising a recognition phase (establishment of a metrical framework) and a continuation phase (projection of the chosen framework into the future). Thus meter is critical in establishing expectations. London maintains that most computational and experimental studies of meter regard the recognition stage while theoretical studies provide retrospective evidence. Implied here is a failure to provide an adequate study of metric recognition that incorporates prediction and continuation. Our experiments take this challenge as a point of departure. We propose that a listener simultaneously activates two parallel metric schemas each with some degree of independence. When one proves to correlate more consistently with other incoming patterns (dynamic accentuation, harmonic accent, phrase and articulation accents, etc.) the metric schema that fails 'turns off'. Furthermore, our model enables the integration of mutual influences of two interrelated aspects of musical expectations: schematic metric awareness (which influences functional tonal expectations) and learned functional tonal implications that in and of themselves create metric expectations. The merger and integration of these cognitive processes allow for a more refined model of music audition. 2 The network design 2.1 Architecture of the network Figure 1: Simulation of expectancies. From left to right - 4 units represent duple meter and 3 more represent triple meter, the last 12 units are the harmonic expectations represented by 12 PCs. The size of the squares is proportional to the strength of the units' activity. Time proceeds from bottom-up. The right column represents the input and the left column visualizes the net's prediction for the meter and harmony. The progression is - [3/4 I I - vi vi ii - V V V7 - III] In our previous model of fluctuation in DRE (see IOup ____.._____ [GB96] and [BG96]), we adopted a three-layer se- -Otu_ _ __ _ quential net in which 12 state units establish the context of the current chord sequence, and the 12 output layer units represent the prediction of the net for the Figure 2: The progression is - subsequent chord. Both, the state and the output [4/4 I I vi vi -IV IV ii ii -V V V7 V7 units are pitch class (PC) representations of triads and tetrads in the sequence. The output layer is fed -1 I I]
Page 00000003 Figure 3: The progression is - [4/4 I IV V I -vi V7 I IV -ii V V I I I I] back into the state units to influence the next prediction of the net. The value of the state units at time t is the sum of its value at time t- 1 multiplied by decay parameter and value of the output units at time t- 1. By integrating a sub-net with the sequential net we supplemented the model with a simple metrical organizer that supplied a periodic beat stream of four beats per measure of duple harmonic progressions. This model is extended by adding triple meter patterns to the architecture. In so doing we examine how metric expectations can influence the harmonic predictions and how the harmonic progression together with the context of the meter influence the prediction of meter. This architecture differs from the previous model in a number of respects. The representation of meter is extended. We incorporate into the net's state units two pools (or a sub-net) of: 4 units to represent duple meter and 3 more to represent triple meter. These units are connected to the hidden layer together with the pool of the PCs representing the harmonic context. The hidden units are connected to the output layer. The output layer contains three pools of units: a pool for the harmonic expectations represented by the 12 PCs; and the two pools to represent expectations for duple and triple meter. The output of the 7 units of the meter is fed back into the corresponding pools of the state. The output of the prediction of the net for the harmonies' expectation were used to measure DRE and the target was fed into the PC units of the state, to establish the current harmonic context. We thus model the mutual influences of harmony on meter and meter on harmony. We note the enhance ment of this method in quantifying the DRE. The DRE is also influenced by the metric expectations. This is particularly evident in (fig 4) where conflicting metric information greatly affected the DRE. 2.2 The set of learning examples We use a learning set of functional tonal harmonic patterns. The patterns were evenly divided into duple and triple meter progressions. Harmonic rhythm in the learning set ranged from one chord per measure to one chord per beat, although the weighting was on one and two chord changes per measure for both duple and triple patterns. 3 Running the Net 3.1 The Learning Phase For the learning phase the net was given thirty examples containing duple and triple patterns of harmonic progressions. After training, the net was able to reproduce the examples. We have tested the performance of the network with several different learning parameters. For example we found that for this task the net required relatively high value for the decay parameter. 3.2 The Generalization Phase In this phase the net was given four new sequences. The target sequence was compared to the current harmonic and metric prediction of the net. The meter was fed back into the meter's pools of the state units and the target of the current harmonies was fed into the PC units. In analyzing the output we consider the distribution of the units' activation. By calculating how much of the target is present in the harmony pool of the output units, we were able to suggest a quantitative measurement of the DRE. The units of the meter pools in the output reflect duple and triple interpretations and clearly demonstrate conflicting metric and harmonic information. 4 4.1 Data Analysis Figure 1: [3/4 II I- vi vi ii- V V V7 - III] This example represents the output of a standard four measure progression in triple meter. The progression should show a high DRE. The role of the metric subnet is critical in the network's agility in detecting the correct harmonic rhythm by beat five. Of note is
Page 00000004 the openness of the system to change on beat three (resulting from the inconclusive assistance of the metric sub-net). However the downbeat of measure two entrains the network by supposing a metric schema which fully conditions expectation for harmonic progression and change. Thus, in measure two the expectation for a subdominant harmony is progressively strengthened and the expectation for a change to the dominant is highly expected. (The inconclusive expectation for tonic continuance in the final measure is an artifact of 'padding' the example in order to incorporate longer progressions). 4.2 Figure 2: [4/4 I I vi vi -IV IV ii ii -V V V7 V7 -I II I] In this example a harmonic progression in 4/4 with a high DRE is input as a target sequence. In this example the initial willingness for change on beat three (evident in the distribution of strength of PC7 to PC5 and PC9 representing an expectation for shift to the sub dominant) is immediately followed in beat four by an even stronger expectation for change to a subdominant. The lowest DRE in the entire progression occurs in beat five. Here, the downbeat is fully recognized as a point of harmonic shift, with a greater expectation for sub dominant harmony, but with an openness for a dominant downbeat. The arrival of a subdominant in correspondence to the metric downbeat sets a strong expectation for the completion of the progression. 4.3 Figure 3: [4/4 IIV V I -vi V7 I IV -iiV V I -I II I] In this example a distinct conflict between harmonic rhythm and meter results in significant drops in DRE. The hastened harmonic rhythm (a chord already on the second beat, setting up a quarter note harmonic rhythm) is resisted in the output's expectation for continued subdominant harmony in beat 3. The arrival of a tonic on beat four of measure one throws both the metric counter and the harmonic expectations into flux. The drop in DRE is particularly interesting in that the distribution of expectations is not willy nilly but rather reflective of an ambiguity, in which conflicting functional regions (tonic/ dominant) are confused. This conflict persists until the final measure. 5 Discussion Some basic questions regarding the perception of meter in tonal music are raised. Specifically: 1. How does a listener identify the meter, when hearing an unfamiliar work? 2. Is the process of metric cognition one of parallel or sequential testing? That is, do we consider multiple possible meters simultaneously, or do we test one and, failing to achieve a good 'fit', shift to another metric count? 3. What are the implications of these questions on our theory of musical expectations? In our first experiment we extended the initial model by incorporating two parallel and independent counters for three beats and four beats. An experiment currently being considered is to commence with two parallel counters but shut one off when a strong correlation between a high DRE and one of the two pools in the metric sub-net is established. A second experiment under current consideration involves a change of data structure, such that multiple metric possibilities are reflected within a single counter. References [BG96] J. Berger and D. Gang. Modeling musical expectations: A neural network model of dynamic changes of expectation in the audition of functional tonal music. In Proceedings of the Fourth International Conference on Music Perception and Cognition, Montreal, Canada, 1996. [DH89] P. Desain and H. Henkjan. The quantization of musical time: A connectionist approach. Computer Music Jouranal (CMJ), 13(3), 1989. [GB96] D. Gang and J. Berger. Modeling the degree of realized expectation in functional tonal music: A study of perceptual and cognitive modeling using neural networks. In Proceedings of the International Computer Music Conference, Hong Kong, 1996. [LHL82] H. C. Longuet-Higgns and C. S. Lee. The perception of musical rhythms. Perception, 11:115 -128, 1982. [LJ83] F. Lerdahl and R. Jackendoff. A Generative Theory of Tonal Music. Cambridge (MA): MIT Press, 1983. [Lon92] J. London. The cognitive implications of a dynamic theory of meter. In Proceedings of the Fourth International Conference on Music Perception and Cognition, Pennsylvania, 1992. [Mat39] J. Matheson. Der Vollkommene Cappelmeister. Hamburg: Christian Herold, 1739. [Pov81] D. Povel. The internal representation of simple temporal patterns. Journal of Experimental Psychology: Human Perception and Performance, 7:3-18, 1981.