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
