Fig 1.
Perceptual bias in entrainment.
Entrainment to an auditory rhythm through synchronized movement reveals a biased response, as in Bayesian inference, towards a listener’s perceptual priors [24, 25]. If a listener does not have enculturated familiarity with musical rhythms related by the ratios 2:2:3, they are likely to interpret a rhythm (approximately) related by these ratios as 1:1:2, and this will be reflected in synchronized movements that are in fact closer to ratios 1:1:2 than 2:2:3.
Fig 2.
An expectation template in PIPPET, as defined in Eq (1), represents the instantaneous rate of events λ(ϕ) at phase ϕ of the underlying process. Expectations for specific events are each represented by a Gaussian centered at the respective phase ϕi, with variance vi representing the precision of the expectation, and scaling factor λi representing the strength of expectations. The constant λ0 accounts for the rate of events unrelated to phase, i.e. background noise.
Fig 3.
Two- and three-interval rhythms.
A) Simple two-interval rhythms of ratios 1:1, 2:1 and 4:3, with a 1000ms period. B) Three-interval rhythm space, where each axis of the ternary plot refers to one of the three intervals in a rhythm. Each point corresponds to a three-interval rhythm of period 2000ms, with each interval constrained to a minimum duration of 300ms. The crosses denote some examples of rhythms related by small-integer ratios (e.g. 1:1:1 and 2:1:1, shown to the left).
Fig 4.
Tracking the phase of a 4:3 rhythm with different timing expectations.
Two pPIPPET models are given patterns of expectations for 1:1 and 2:1 rhythms, but only one with expectations for 4:3 rhythms. The resulting quality of phase tracking—for the first two stimulus repetitions—is shown through adjustments to estimated phase μt on auditory events, alongside changes in uncertainty Vt. Implicit inference of the rhythmic pattern over time is shown through changes in template probability pm. A) European model. Without 4:3 expectations, phase must be adjusted after the first event of each cycle to compensate for the timing shift, causing phase uncertainty to increase until the cycle is complete, when phase is shifted back. B) Malian model. Phase is successfully tracked, with phase uncertainty only growing slightly between events. Note that phase uncertainty always accumulates between events due to expected phase noise (see Methods, σ in Eq 2).
Fig 5.
Production bias when tracking 1:1, 4:3 and 2:1 rhythms.
Performance of pPIPPET in tracking repeating two-interval rhythms, depending on template configuration. A) Distribution of phase corrections following the first interval of the 4:3 stimulus rhythm. Curves are Gaussian kernel density estimates, vertical black dashed line shows an unbiased response, and other vertical dashed lines refer to sample means. B) Phase uncertainty on the time step preceding an auditory event. Error bars show the 95% confidence intervals within each sample. Note that this figure shows qualitative agreement with empirical tapping data shown in [5, Fig 3].
Fig 6.
Categorization maps for three-interval rhythms.
Categorization of three-interval rhythms (pattern duration of 2000ms and minimum interval duration of 300ms) after being presented once, using a pPIPPET filter configured with expectation templates derived from a corpus of German folksongs [46]. Rhythms related by small-integer-ratios (integers less than 3) are marked with crosses and labeled. A) Entropy of the posterior distribution over expectation templates. B) Colors correspond to cyclic permutations of the pattern which has been inferred (i.e. the expectation template which maximizes pm), and transparency the entropy. For comparison, see [1, Fig 10b] and [1, Fig 13a]. This figure, and other ternary plots presented in this work, were made using the python-ternary package [47].
Fig 7.
Simulated iterated reproduction.
Results from the final iteration of all simulated trials, using pPIPPET filters configured with either German folk songs [46] or Turkish makam music [48]. A) Kernel density estimate (KDE) of the underlying data distribution for the German model, using the non-parametric method described in [24], normalized relative to a uniform distribution. B) KDE for the Turkish model. C) Category weights for the two models, obtained by fitting a constrained Gaussian mixture model (GMM), using the modified expectation-maximization procedure described in [25]. Weights for cyclic permutations of each ratio are grouped. Error bars reflect confidence intervals (SD of weights, derived from bootstrapping, N = 250). We draw specific attention to the differences in weights measured for cyclical permutations of the ratios labeled in bold (1:1:1, 1:1:2 and 2:2:3), which relate to differences in the KDE plots at the respective ratios.
Fig 8.
Metrical analysis for German and Turkish corpora.
Relative frequency of onset positions (phase, at the resolution of 16th notes) within a metrical cycle of rhythms belonging to a specific meter (time signature). This analysis from [15, p. 185] is used with permission.