Figure 2.

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., 2013). “Neural oscillations” are dashed because they represent observed neural process whose connection with the other behavioral concepts has not been proven (yet). The quantitative parameters are: category means μ_i, a scalar constant s, and c_i, which is the abbreviation of $c_i^l$ and $c_i^u$, parameterizing the overlap between categories. The proposed way of representing rhythmic structure depends, among other factors, on the constancy of r_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., 2016; Jacoby and McDermott, 2017) has tried to operationalize the connection between the “mathematical perfection” of integer ratios and their empirical counterpart in a number of alternative ways. This perspective does not address how and when a real number is perceived as an integer ratio, leaving this as an empirical question for psychophysics research. In general, large integer ratios, and even irrational-number ratios, can be perceived as small integer ratios if close enough to one. For instance, 2^7/12≈1.498307 is irrational (Coxeter, 1968) but close to 3/2. Virtually all pianos, today, employ this irrational number (1.498307) in their well-tempered tuning, which is “close enough” for human hearing to the integer ratio 3:2. At the same time, the “catchiness” of a rhythm also depends on small deviations from the integer ratios. For instance, delayed occurrences of expected beats even at varying levels of deviation from the underlying rhythms (together with the compensatory temporary speed-ups) are perceived as interesting, while a strictly regular rhythm will quickly appear dull.