Journal of Mathematical Psychology 111 (2022) 102724 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp A point-process model of tapping along to difficult rhythms David Bulger a,∗ , Andrew J. Milne b , Roger T. Dean b a School of Mathematical & Physical Sciences, Macquarie University, NSW 2109, Australia b The MARCS Institute for Brain, Behaviour and Development, Westmead Innovation Quarter, Western Sydney University, Sydney, Australia article info Article history: Received 11 May 2021 Received in revised form 19 September 2022 Accepted 8 October 2022 Available online xxxx Keywords: Music cognition Rhythm Meter Synchronised tapping Well-formed rhythm Point process Stochastic process Refractory effect abstract Experiments where participants synchronise their taps to rhythmic cues are often used to study human perception and performance of rhythms. This experimental study is novel in two regards: The cyclic rhythms (non-isochronous patterns of cues) presented to participants were more challenging than usual (including many from unfamiliar time signatures), and we have modelled participants’ performance via a conditional point process. Point processes are well suited to describing partly random sequences of events, but have rarely been used previously to model tapping experiments, the only other study we know being Cannon (2021). Our model uses continuous functional parameters to describe participants’ responses to auditory stimuli with much finer temporal resolution than in previous studies. Taking account of both the clock and the dynamic attention theories of sensorimotor synchronisation, we assessed the time course of the propensity to tap within each cycle at a resolution of less than 13 ms, identifying the influence of cues on the tapping propensity and the progress of learning their rhythmic patterns. We also sought to determine the trajectory of the putative refractory period (feedback inhibition of tapping) after each tap, and assessed the distribution of tap- cue asynchronies in a more finely resolved manner than usual. Our models also indicated complex kinetics of the feedback over about 100 ms. © 2022 Elsevier Inc. All rights reserved. 1. Introduction This study uses a non-homogeneous, conditional point process with feedback to model human learning of unusual rhythms. It offers multiple points of difference to most other tapping studies: the complexity of the rhythms, the point-process analysis, and the fine temporal resolution of our models. Many previous studies have used tapping experiments to shed light on perception of metre and rhythm. In spontaneous tapping experiments (Essens & Povel, 1985; Fraisse, 1946, 1956), partici- pants are asked to tap after an imprecise verbal description of a rhythm. Continuation tapping experiments require participants to continue tapping a pulse or rhythm after the cue has ceased. Synchronisation tapping refers to experiments in which par- ticipants try to tap along with an audible ‘‘cue’’. In many of these Aschersleben (2002), Herff, Herff, Milne, Johnson, Shih, and Krusienski (2020), Nozaradan, Peretz, and Mouraux (2012) the cue sequence (that is, the target rhythm) is isochronous. A number of other synchronisation tapping experiments focus on participants’ adaptation to gradually changing rhythms, or perception and performance of the ratio between a long and short ∗ Corresponding author. E-mail addresses: david.bulger@mq.edu.au (D. Bulger), a.milne@westernsydney.edu.au (A.J. Milne), roger.dean@westernsydney.edu.au (R.T. Dean). interonset time (Goldberg, 2015; Polak, Jacoby, Fischinger, Gold- berg, Holzapfel, & London, 2018; Repp, London, & Keller, 2005, 2011, 2012, 2013; Snyder, Hannon, Large, & Christiansen, 2006). Only a few, such as Povel and Essens (1985), Repp et al. (2005) and Snyder et al. (2006), to our knowledge, address challenging rhythms. A variety of statistical methods have been applied to tapping data in the previous literature. Time-series analyses of tapping data are offered in Launay, Dean, and Bailes (2013) and Pressing (1987), with one observation per pulse, typically using a window around each pulse or cue to determine which taps to consider as simultaneous events. One focus has been the distribution (mainly mean and variance) of tap asynchrony (Repp, 2005; Repp & Su, 2013)—each tap sufficiently close to a cue is attributed to that cue, and the timing difference (asynchrony) is taken as an observed variable of interest. Asynchrony is difficult to define in a natural way for taps midway between pulses, or when multiple taps occur near one cue. Furthermore, for complex non-isochronous rhythms, some taps are not meaningfully associated with any actual cue because they result from an incorrect prediction of a cue’s presence. Another focus has been accuracy and bias in participants’ reproductions of the ratio between a long and short beat (Repp et al., 2005, 2012, 2013). In each case, statistical esti- mates or models describe not the raw observations themselves, but higher-level, aggregated variables. https://doi.org/10.1016/j.jmp.2022.102724 0022-2496/© 2022 Elsevier Inc. All rights reserved.