Smooth Rhythms as Probes of Entrainment Music Perception 10 (1993): 503-508 ABSTRACT If one hypothesizes rhythmic perception as a process employing oscillatory circuits in the brain that entrain to low-frequency periodicities in the neural firings evoked by an acoustic signal, then among the conceptually purest probes of those oscillatory circuits would be acoustic signals with only simple sinusoidal periodicities in the appropriate frequency range (perhaps from 0.3 Hz to 20 Hz). Such signals can be produced by the low-frequency amplitude modulation of an audible carrier wave by one or more sinusoids. The resulting rhythms are smooth in that their amplitude envelopes are smoothly varying with no obvious points of onset or offset. Because preliminary experiments with smooth rhythms have produced some unexpected results, and because smooth rhythms can be precisely controlled and varied (including, for example, the digital filtering of their Fourier components in the frequency domain), they are proposed as versatile stimuli for studies in rhythmic perception.
TEXT Experiments in rhythmic perception employ the stimuli thought to embody rhythm s fundamental elements. For instance, the short-long, 3/4-meter stimuli depicted in Figure 1, typical of innumerable studies, manifest clearly demarcated durations, precisely located points of attack, and easily derived temporal ratios. Yet if rhythmic perception is less a process of measurement and more one of the entrainment of oscillatory circuits in the brain to various periodicities in the neural firings evoked by music or speech (cf. Jones, 1986; Gjerdingen, 1989; Desain, 1992), then those kind of discontinuous stimuli would be among the worst imaginable: their instantaneous onsets and offsets would explosively jangle all such oscillatory circuits. (In this regard, note that researchers generally change idealized vertical onsets and offsets into sloping ramps. To do otherwise is to subject the listener to disturbing clicks. ). An array of oscillators with varying natural frequencies might better be probed by sinusoidal stimuli smooth rhythms. In the hope of advancing a discussion of this alternative paradigm, the following short report surveys results from some informal studies at Stony Brook. There small groups of professional and pre-professional musicians have been asked to comment on, or notate, the rhythms of various sinusoidal stimuli. Modulating an audible tone s amplitude with a single low-frequency sine wave produces, in most cases, the perception of an unending series of equal pulses or even beats. Musically more interesting rhythms result from modulating a tone s amplitude with the product of two low-frequency sine waves. For instance, if two such modulating waves have relative frequencies of f and 2f, then merely shifting the phase between them is sufficient to change the perceived rhythm from a pattern clearly evoking a 2/4 meter to one evoking a 3/4 meter.
Figure 1. A periodic amplitude envelope typical of stimuli used in tests of rhythmic perception. Such a stimulus, though atypical of music or speech, maximizes the researcher s ability to predetermine durations, attack points, and temporal ratios. A prototypical duple meter was perceived when the phase of the faster wave was shifted 45 behind that of the slower wave (see Figure 2 and Appendix). Most musicians transcribed the pattern in 2/4 (or 4/4) meter as a strong quarter (or half) note followed by two weak eighth (or quarter) notes, though about a fourth of the musicians heard it as a strong eighth (or quarter) note followed by three weak eighth (or quarter) notes. Musically, the pattern was strongly reminiscent of the di-di-dum, di-di-dum, di-di-dumrhythm of Rossini s William Tell Overture. If, however, the phase of the faster wave was shifted 90 behind that of the slower wave (see Figure 3 and Appendix), the result was a periodic amplitude
Figure 2. A periodic, smooth amplitude envelope perceived as an alternating series of a long and two shorts in a duple meter. envelope that musicians readily identified as a textbook case of an alternating series of shorts and longs in a 3/4 (or 3/8) meter. The musical effect of Figure 3 is that of a brisk waltz where quarter-note upbeats lead to half-note downbeats. At the perceived tempo (dotted half note = 80 bpm) there is little rhythmical sense of the undulation implied visually in Figure 3. Instead, the tones seem neatly articulated into upbeatdownbeat pairs: di-dum, di-dum, etc. Each tone s perceived moment of initiation or beat in Medieval musical terms, its ictus (pl., ictus) occurs near the halfway point on the ascending slope of each wave. As predicted in Gjerdingen (1988), the ictus thus seems to fall where the amplitude envelope changes from concave up to concave down, that is, where the second derivative of amplitude goes from positive to negative (see Figure 4: points A and B). Moreover, the resulting short-to-iong
Figure 3. A periodic, smooth amplitude envelope perceived as an alternating series of shorts and longs in a triple meter. ratio of 1.00:1.75 essentially matches Gabrielsson s (1988) average results of 1.00: 1.76 for a human performer. Shifting the phase of the faster wave 180 behind that of the slower wave produces the retrograde of Figure 3, in other words, the rhythm of Figure 3 played backwards. (To view the backward amplitude envelope, look at Figure 3 in a mirror.) Most music scholars would in principle deny the possibility that a regular alternation of shorts and longs in 3/4 meter could, when reversed, become other than another regular alternation of shorts and longs in 3/4 meter. Yet the backward version of this smooth rhythm strikes most musicians as a slightly jazzy example of 2/4 time (!) two quarter notes per measure grouped within the bar lines: daah-dip, daahdip, etc.
Figure 4. The smooth rhythm of Figure 3 (solid line) along with the second derivative (central second differences) of its time series of amplitudes (dashed line). In the forward direction, positive-to-negative zero-crossings of the second derivative occur at points A and B; in the reverse direction, at points M and N. These points are good predictors of perceived ictus, that is, moments of perceived onset, attack, or beat. If different oscillatory circuits in the brain respond individually to each lowfrequency periodicity in the amplitude envelope (thus performing a rough Fourier analysis of spectral magnitude), then the same set of oscillators would respond identically to the very different rhythms of Figures 2, 3, and 3-backwards. This undifferentiated response would be due to all three rhythms having been produced by the same three periodicities: the period of the perceived measure (frequency f), the period of half of the measure (2f), and the period of a third of the measure (3f, a sideband result of multiplying frequency f by 2f). For a theory of entrained
oscillators to explain how these very different rhythms would be perceived as different, either the oscillators would need to be responsive to phase relationships or they would need to process not the amplitude envelope as a whole but only extracted features of the envelope. For example, if the envelope of Figure 3 were reduced to just two spikes with the corresponding amplitudes and temporal locations of the predicted ictus (Figure 4, points A and B), then the oscillator representing triple subdivision of the perceived measure would respond considerably more strongly than the oscillator representing a duple subdivision. For Figure 3 played backwards, by contrast, a similar reduction to the predicted ictus (points M and N) would allow the oscillator representing duple subdivision to respond slightly more strongly than the oscillator representing triple subdivision (quintuple and septuple subdivisions would also be evident). These relationships conform to the perceived sense that the rhythm of Figure 3 is strongly triple when played forward, and less strongly duple when played backward. Smooth rhythms provide alternative stimuli for the reexamination of common rhythmic phenomena, and the notion of entrainment as a fundamental process of rhythmic perception provides a provocative conceptual basis for the reexamination of current explanations of those phenomena. The location of ictus in the above stimuli, for instance, is completely explained neither by threshold theories of the perceptual onsets of tones (Vos & Rasch [1981]) nor by theories of the perceptual centers of syllable onsets ( p-centers : Morton, Marcus, & Frankish [1976]; Howell [1988]). Likewise, without some clear theory of ictus such as is given by the secondderivative zero-crossing test, theories of rhythm based on duration will founder on smooth rhythms. I hope other scholars (and especially trained experimentalists) will find these smooth rhythms as intriguing as I do.
APPENDIX The stimuli were produced by modulating the amplitude of a 673 Hz sine-wave tone. The modulating signal for Figure 2 was defined as [(sin(t) + 1] [-cos(2t) + 1], where T = 2π(time in seconds)/0.743 and modulation was 100%. A unit value was added to each function to ensure that computations involved only positive numbers (a common requirement of neural-network modeling the results are different if negative value are introduced). The modulating signal for Figure 3 was defined as [(sin(t) + 1] [-sin(2t) + 1], and for Figure 3 played backwards as [(sin(t) + 1] [sin(2t) + 1]. The wave shown in Figure 3, if examined in detail, shows a tiny rise or hump in the interval between each perceived half note and the following quarter note. Played through loudspeakers, versions of Figure 3 both with and without the suppression of this hump were generally indistinguishable. Played through headphones, listeners sometimes reported a faint eighth note preceding each quarter note.
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