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Page 88 ï~~An Auditory-Motor Model of Beat Induction Neil Todd Department of Music Sheffield University Sheffield, S10 2TN UK Chris Lee Department of Music City University London, EC1V OHB UK Abstract A model of beat induction is presented which attempts to simulate the processes involved in the auditory system which give rise to the percept of a beat. The input to the model is an audio signal and the output is a visual representation of the rhythm in form of a rhythmogram. Rhythmograms are of two kinds (1) timedomain rhythmograms, which resemble Lerdahl and Jackendoff's time-span reductions and (2) frequencydomain rhythmograms, which resemble Lerdahl and Jackendoff's metrical grids. [Research supported by a grant from the MRC.] 1 Overview system which is necessary in order to plan action in advance. The time-domain process can also be thought of as a kind of sensory memory since it is able to account for a number of associated phenomena such as temporal integration and backward masking. 2 Temporal Analysis 2.253 2 1.5 22.214.171.124 Â~, i; o I|r.3 1I A 2:s.5z: 33 time (seonods) 4.S sound model of beat induction. Figure 1. A Figure 1 shows an overview of the model which has the following main components. (1) An auditory periphery simulated with a cochlear model in conjunction with an array of hair-cells; (2) a timedomain process, triggered by onsets, which carries out a temporal segmentation of the activity in the auditory nerve and gives information on the (i) stress or loudness and (ii) the grouping of the neural events; (3) a frequency-domain process, which carries out a periodicity analysis and gives information on the periodicity content of the rhythm; and (4) a sensorymotor filter which selects a tactus from the metrical harmonics. The sensory-motor process is a representation of the dynamic properties of the motor Figure 2. A time-domain rhythmogram. Figure 2 shows the response of the time-domain analysis to a simple rhythm constructed from 50ms tone bursts with inter-onset intervals of 250ms and 500ms. Note that this has captured the principle of grouping by temporal proximity since the tone bursts have been clearly grouped in threes. Further, the last event in a group is represented as more important thus also capturing the well documented phenomenon of interval produced accents. This last feature is vital for determining the phase of the metrical grid. The degree of relative accent is dependent on the relative and absolute intervals between events and thus on the overall tempo. 3 Periodicity Analysis Figure 3 shows the response of the frequency-domain analysis to the same rhythm. The naturalness of placing pitch and rhythm in the same map is evident. Foot-tapping 88 ICMC Proceedings 1994
Page 89 ï~~The tone bursts have two components an octave apart (512Hz and 1024Hz). The main rhythmic harmonics are also spaced by octaves (1,2,&4Hz).,1~... 1001 g10, I am= ber with an action such as foot-tapping, has a natural period of about 600ms (1.7Hz) (Fraisse, 1982). The second system, which we may associate with whole body motion, e.g. body-sway, has a natural period of about 5s (0.2Hz) (Todd, 1994). In the model these sensory-motor components modify the spectrum given by the periodicity analysis so that the harmonic which is nearest to the foot-tapper resonance will be the one which will be favoured for the tactus. 5 Rhythm Space Once the tactus has been identified the metre of a rhythm can be identified from the frequency ratio of the three most important harmonics. The metrical structure can be represented in the form of a compact pattern in rhythm space (X,Y) since the ratio of the metrical harmonics can be simply related by the formula 2AX*3AY (c.f. Longuet-Higgins' three dimensions of harmony). In the example the metre is obviously 2/4 which would form a horizontal bar shape. Figure 5 though, shows the patterns made by the metres 6/8 and 3/4. Metre = 6/8 0.5 1 13 2 2.5 3 3.5 4 4. S time (seconds) Figure 4. A frequency-domain rhythmogram. The important point here is that it is possible to associate each event with numbers of cycles of the metrical harmonics. This number is invariant of the absolute (rhythmic) frequency or tempo since the ratio of harmonics is a fixed property of the rhythm. Thus under those circumstances where an absolute change of tempo is (i) piece-wise continuous and (ii) the rate of change is less than one octave per onset then the ratios of the harmonics will also be invariant. 4 Sensory-Motor Filtering In fact the harmonics which form the metrical grid in the example have already been selected on the basis of sensory-motor feed-back. This is done by adjusting the magnitude of the harmonics according to the response of two band-pass filters which model the motor system (Figure 4). tmy f 9 18 3 12 1 214 3/8 3/4 32 1/16 1/8 1/4 1/2 24 8 16 32 1/24 1/12 1/64 1/3 2/3 4/3 8/3 binary 1/18 11/9 Metre = 3/4 9 18 3/8 3/4 3/2 3 6 12 24 1/16 /8 1/4 1/2 1 2 4 8 16 32 i i I i 1/24 1/12 1/6 I I I1/3 2/3 4/3 8/3 1/18 J 0 Figure 5. Rhythm Space. Figure 4. Sensory-Motor Filters. The sensory-motor filters thus impose a second source of tempo dependency on the perceived rhythm since although the ratio of rhythmic harmonics of a rhythm are a fixed property, the particular harmonic which is selected for the tactus is not. Mathematically the sensory-motor process can be described by a dynamic system which has two degrees of freedom and can be modelled as two weakly coupled mass-spring-damper systems. The first system, which represents the dynamics associated 6 Conclusion The model presented has no "rules" as such and is entirely "bottom-up" in its form of processing. In proposing such a model though, we would not wish to suggest that rhythm perception is entirely so. Clearly, some form of interpretation is required, as is indicated in Figure 1, in order to make sense of the incoming information. However, given that such a large amount of information can be obtained "for free", such as grouping and phenomenal accents, any realistic model of beat induction must be based on the way the auditory system works. ICMC Proceedings 1994 89 Foot-tapping