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Edited by: Dezhong Yao, University of Electronic Science and Technology of China, China

Reviewed by: Hugo Merchant, Universidad Nacional Autónoma de México, Mexico; Daya Shankar Gupta, Camden County College, United States

†These authors have contributed equally to this work

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

One curious aspect of human timing is the organization of rhythmic patterns in small integer ratios. Behavioral and neural research has shown that adjacent time intervals in rhythms tend to be perceived and reproduced as approximate fractions of small numbers (e.g., 3/2). Recent work on iterated learning and reproduction further supports this: given a randomly timed drum pattern to reproduce, participants subconsciously transform it toward small integer ratios. The mechanisms accounting for this “attractor” phenomenon are little understood, but might be explained by combining two theoretical frameworks from psychophysics. The scalar expectancy theory describes time interval perception and reproduction in terms of Weber's law: just detectable durational differences equal a constant fraction of the reference duration. The notion of categorical perception emphasizes the tendency to perceive time intervals in categories, i.e., “short” vs. “long.” In this piece, we put forward the hypothesis that the integer-ratio bias in rhythm perception and production might arise from the interaction of the scalar property of timing with the categorical perception of time intervals, and that neurally it can plausibly be related to oscillatory activity. We support our integrative approach with mathematical derivations to formalize assumptions and provide testable predictions. We present equations to calculate durational ratios by: (i) parameterizing the relationship between durational categories, (ii) assuming a scalar timing constant, and (iii) specifying one (of K) category of ratios. Our derivations provide the basis for future computational, behavioral, and neurophysiological work to test our model.

What are

A

Graphical representation of different types of IOI distributions.

Why do rhythms (i.e., patterns of durations) tend to exhibit small integer ratios? Why are humans drawn to rhythms with such a peculiar mathematical property, in both perception and production? Does this property reflect a special quirk of music perception and/or motor sequencing, or could it be explained by domain-general aspects of cognition? Can we explore these alternatives through mathematical formalism? Here, we explore mathematically the possibility that the human bias toward small integer ratios may be explained by a combination of scalar expectancy and categorical perception.

We begin by outlining the relevant classical frameworks for human timing, and go on to summarize the evidence in support of the small-integer ratio bias in rhythm perception. We then present our proposal linking the frameworks to the bias through mathematical formalisms. Specifically, we draw on the scalar property of time interval estimation to formulate a simple model of categorical perception that may result in an integer ratio bias (Figure

Two major theoretical approaches, among several, have been suggested to account for the mechanisms behind human timing (Wing and Kristofferson,

Another relevant approach to timing mechanisms comes from neuroscience and physics. It suggests that neural oscillations entrain (or even “resonate”) with the periodicity of external stimuli at multiple time-scales (Buzsaki,

Recent behavioral research investigated human priors for durations in rhythmic patterns (Ravignani et al.,

Specifically, participants were presented with sequences of IOIs sampled from a uniform distribution

Here we aim to explain the distribution

Our concrete question is: Under which conditions will a distribution

Without any assumptions, distribution

An _{1}, …, _{n−1}) and of ratios _{1}, …, _{n−2}), such that _{i} = _{i+1}/_{i}. Perception of a rhythm _{1}, …, _{n−2}), with a strong tendency to categorize. The vector _{i} = _{i} is attributed to phenomenal category

A general assumption in rhythm research is that perceptual timing can be described as a process combining prior beliefs with sensory input. One way to capture this mathematically is to model time perception as Bayesian inference (Jazayeri and Shadlen, _{i} according to the distribution _{i} = _{i}) ∝ _{i}|_{i} = _{i} = _{i} =

Jacoby and McDermott (

We write the prior as a

Here, the prior assigns to each Gaussian _{k}, which determines its relative prominence as a category; a category mean μ_{k}, which specifies the expected interval ratio underlying this category; and a category variance σ_{k}. The assumption we make is that weights are constant:

Assuming that our indexing of categories under the prior is strictly ordered by the category means, such that _{k} from 200 ms (London,

So far, our assumptions constrain neither category means μ_{k} nor standard deviations σ_{k}. Our final, perhaps most central assumption is that timing exhibits _{k} equals the mean μ_{k} multiplied by a constant, dimensionless factor

Previous empirical reports estimated

All four assumptions are empirically based and independent of each other. Now, _{k} (Figure _{k}. _{k} the cluster _{k+1} the cluster

Combining this idea of a parameterized overlap with scalar properties, each cluster

and their ratio as

Substituting (5) into (4) provides

which can be simplified and rewritten as

Equation (7) requires, to be well-defined, that its right side is positive, namely

Operationally, the category means following from the constraints on

The constraints structure the space of component Gaussians in the prior such that, by specifying μ_{1}, we can compute μ_{k} for all

These quantitative tools enable the formulation of several questions. Given our _{1} to the smallest possible integer ratio?

An alternative approach might be to assume that one ratio is e.g., _{k} in a certain way?

The x-coordinates for the intersection point, expressed as

which simplifies as:

Equation (11) means that the difference of squares between

To make an example with actual numbers, if one substitutes μ_{k} = μ_{1} = 100_{k+1} = μ_{2} = 200

As the right side of Equation (11) is always strictly positive,

Equations (7, 9) support a potential link between scalar timing and integer ratios, as they include the integer ratios _{k} and the scalar constant _{1}, all other μ_{k} are determined by Equation (9): which values of μ_{1} result in

Schematic representation of the perspective introduced by this paper. Black solid-line boxes represent empirically supported assumptions. “Bayesian inference” is outlined in gray to indicate that it is used here as a working assumption and conceptual framework, rather than an empirically supported assumption on cognitive processes (Shi et al., _{i}, a scalar constant _{i}, which is the abbreviation of _{k} (see main text). A deviation from this constancy would result in larger integer ratios, with the deviation accumulating over the categories when iterating equation (8). Empirical work (e.g., Ravignani et al., ^{7/12}≈1.498307 is irrational (Coxeter,

The perspective we offer here creates the basis for expanding not only into theoretical but also empirical work on

We explore quantitative links between scalar timing and the human bias toward small integer ratios. The arguments we provide reduce the explanatory space to a few hypotheses. One possibility is that integer ratios are not a human cognitive primitive, but rather a simple by-product of other cognitive constraints, and their interaction.

Alternatively, the scalar timing framework might not be the most suitable one to explain the integer-ratio phenomenon of human rhythm. If one adopts oscillatory frameworks, integer ratios might simply arise from the oscillatory properties of brain activity, and so can scalar properties and categorical perception. Small integer ratios in particular would just reflect epiphenomena of harmonics of one oscillator or the interaction between two or more oscillators (Collyer et al.,

In any case, scalar timing and oscillatory theories are simplifications, i.e., approximate descriptions derived from confined experimental set-ups. Neurally and behaviorally, the dissociation or compatibility between scalar timing and oscillatory theories is more complex than it may appear in higher level cognitive theories, and only detailed neural models will enable us to define the actual underlying mechanisms.

AR and BT conceived the idea and performed the mathematical derivations. All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are grateful to the editor and the reviewers for their support and helpful comments on earlier versions of this manuscript.

^{2} Marie Curie fellowship 12N5517N awarded to AR). AR and BT were also supported by a visiting fellowship in Language Evolution from the Max Planck Society and ERC grant 283435 ABACUS (awarded to Bart de Boer).