Chapter 5 Rhythm, Timing and the Beat

Keeping time

Singing in time is a skill of sensorimotor synchronisation: producing actions, here sung onsets, in a stable relation to an external beat. Most of what is known about it comes from finger-tapping experiments, which are simple enough to model exactly (Repp 2005; Repp and Su 2013).

Two sources of variability

Evidence

Wing and Kristofferson proposed that each interval between taps is the sum of an interval generated by a central timekeeper and the difference between two motor delays, the delay of the tap that ends the interval and of the one that begins it: $I_n = C_n + D_n – D_{n-1}$ . Because neighbouring intervals share one motor delay with opposite signs, the model predicts a negative correlation between successive intervals, and lets the two sources of variance be separated: the timekeeper variance $\sigma_C^2$ and the motor variance $\sigma_D^2$ (Wing and Kristofferson 1973).

Calculation

A simulation of 4,000 taps at a 500 ms period, with a timekeeper SD of 15 ms and a motor SD of 8 ms, and the model’s estimates recovered from the intervals alone: $\sigma_D^2 = -\text{acov}(1)$ , $\sigma_C^2 = \text{var}(I) – 2\sigma_D^2$ .

import random, statistics

# Wing–Kristofferson model: each inter-tap interval
#     I_n = C_n + D_n − D_(n−1)
# C = central timekeeper interval, D = motor delay. Then
#     var(I) = σC² + 2σD²   and   lag-1 autocovariance = −σD².
random.seed(3)
period, sC, sD = 500.0, 15.0, 8.0                    # ms
D = [random.gauss(50, sD) for _ in range(4001)]
I = [random.gauss(period, sC) + D[n] - D[n - 1] for n in range(1, 4001)]

m = statistics.fmean(I)
var = statistics.pvariance(I)
acov1 = sum((I[k] - m) * (I[k + 1] - m) for k in range(len(I) - 1)) / (len(I) - 1)
sD2 = -acov1
sC2 = var - 2 * sD2
print(f"mean interval {m:.1f} ms, total SD {var ** 0.5:.1f} ms, lag-1 autocorrelation {acov1 / var:+.2f}")
print(f"estimated timekeeper SD {sC2 ** 0.5:4.1f} ms (true {sC:.0f}), motor SD {sD2 ** 0.5:4.1f} ms (true {sD:.0f})")
# mean interval 500.1 ms, total SD 19.2 ms, lag-1 autocorrelation -0.18
# estimated timekeeper SD 15.4 ms (true 15), motor SD  8.1 ms (true 8)

The decomposition recovers both values from the tap times alone. This matters for teaching, because the two sources call for different remedies: timekeeper variability is a matter of the internal sense of the beat; motor variability is a matter of how consistently the body executes each onset.

Correcting towards the beat

When there is a metronome or a band, each asynchrony between the singer’s onset and the beat can be partly corrected on the next one. In the linear phase-correction model, a share $\alpha$ of each asynchrony is corrected (Vorberg and Wing 1996).

Calculation

The variability of asynchrony for different correction shares, with the same timekeeper and motor noise as above.

import random, statistics

# Phase correction when tapping to a metronome (after Vorberg & Wing):
#     A_(n+1) = (1 − α) A_n + (C_n − P) + D_(n+1) − D_n
# A = asynchrony, C = timekeeper interval, P = metronome period, D = motor delay,
# α = the share of each asynchrony corrected on the next tap.
def asynchrony_sd(alpha, sC=15.0, sD=8.0, n=20000, seed=5):
    rnd = random.Random(seed)
    a, d_prev, out = 0.0, rnd.gauss(0, sD), []
    for _ in range(n):
        d = rnd.gauss(0, sD)
        a = (1 - alpha) * a + rnd.gauss(0, sC) + d - d_prev
        d_prev = d
        out.append(a)
    return statistics.pstdev(out[100:])

results = {alpha: asynchrony_sd(alpha) for alpha in (0.05, 0.2, 0.4, 0.6, 0.8, 1.0, 1.3, 1.6)}
for alpha, sd in results.items():
    print(f"α = {alpha:4.2f}: SD of asynchrony {sd:6.1f} ms")
best = min(results, key=results.get)
print(f"smallest variability near α ≈ {best}; too little correction drifts, too much over-corrects")
# α = 0.05: SD of asynchrony   47.2 ms
# α = 0.20: SD of asynchrony   26.2 ms
# α = 0.40: SD of asynchrony   20.7 ms
# α = 0.60: SD of asynchrony   18.9 ms
# α = 0.80: SD of asynchrony   18.5 ms
# α = 1.00: SD of asynchrony   18.8 ms
# α = 1.30: SD of asynchrony   20.8 ms
# α = 1.60: SD of asynchrony   26.0 ms
# smallest variability near α ≈ 0.8; too little correction drifts, too much over-corrects

As with pitch in Chapter 3, there is an optimum. With almost no correction the singer drifts away from the beat; with more correction than the optimum, each correction adds its own noise and the asynchrony swings from early to late. Studies of tapping usually find correction shares below one, consistent with this kind of trade-off (Repp 2005).

The timing networks of the brain

Evidence

Timing of movements and perception of beat involve a network that includes the cerebellum, the basal ganglia and the supplementary and premotor cortex. Patients with cerebellar lesions show increased timekeeper variability in tapping (Ivry and Keele 1989), and functional imaging shows the basal ganglia and supplementary motor area responding more to rhythms with a regular beat than to irregular ones (Grahn and Brett 2007).

Hypothesis · not established

Not established. In beginners, an internal focus on the breath or the larynx makes the execution of each sung onset less consistent without changing the internal sense of the beat; that is, it increases motor variance but not timekeeper variance, compared with an external focus on a metronome placed across the room. The hypothesis predicts that, for sung syllables on a beat, the Wing–Kristofferson motor variance is larger under the internal instruction, while the timekeeper variance is similar under both. It would be refuted if, in a randomised within-singer comparison, the motor variance did not differ between the instructions.

References

Grahn, Jessica A., and Matthew Brett. 2007. “Rhythm and Beat Perception in Motor Areas of the Brain.” Journal of Cognitive Neuroscience 19 (5): 893–906.

Ivry, Richard B., and Steven W. Keele. 1989. “Timing Functions of the Cerebellum.” Journal of Cognitive Neuroscience 1 (2): 136–52.

Repp, Bruno H. 2005. “Sensorimotor Synchronization: A Review of the Tapping Literature.” Psychonomic Bulletin & Review 12 (6): 969–92.

Repp, Bruno H., and Yi-Huang Su. 2013. “Sensorimotor Synchronization: A Review of Recent Research (2006–2012).” Psychonomic Bulletin & Review 20 (3): 403–52.

Vorberg, Dirk, and Alan Wing. 1996. “Modeling Variability and Dependence in Timing.” In Handbook of Perception and Action, Vol. 2: Motor Skills, edited by Herbert Heuer and Steven W. Keele. Academic Press.

Wing, Alan M., and A. B. Kristofferson. 1973. “Response Delays and the Timing of Discrete Motor Responses.” Perception & Psychophysics 14 (1): 5–12.

Fan, Y. (2026). The Biomechanics of Phonation and Attentional Shift: Myoelastic-Aerodynamic Theory, the Singer's Formant, Auditory-Motor Feedback and External Focus of Attention [发声的生物力学与注意力转移] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23719625 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).