In the context of music perception, for instance, the capacity of detecting gradual tempo changes is fundamental.

...this comparison will be made in a context where relative sensitivity to accelerations and decelerations is tested.  This relative sensitivity is argued to be related to the fundamental frequency of some oscillating process (Vos et al., 1997).

Performing gradual tempo changes gracefully as well as accurately is quite demanding, particularly in ensemble playing.

Synchronization becomes a more demanding task when the metronome’s tempo changes systematically. In this case, phase correction alone is not sufficient for keeping synchrony. Rather, the internal timekeeper must be adjusted to the changing period of the metronome. Several models have been proposed for period adjustment. Mates (1994a, 1994b) has suggested that timekeeper adjustments are based on the discrepancy between the current timekeeper interval and the previous metronome interval. This error signal is used to adjust the timekeeper by adding or subtracting a fixed proportion of the interval difference. A simple alternative that we pursue in this article is that, like phase, the timekeeper period is adjusted on the basis of the asynchronies, that is, the temporal difference between perceived events, rather than the difference between time intervals.

In the experiment reported, we observed a novel phenomenon: When synchronizing with a metronome that undergoes accelerando or ritardando, the synchronization errors follow a systematic pattern in the transient phase, which held for all subjects. If the initial and final tempi differ widely, subjects first undershoot the smoothly changing tempo, then catch up and overadjust twice before settling on the goal tempo. The question is whether linear period-adjustment models are compatible with such data patterns. Surprisingly, a new version of the highly successful two- level phase-error correction model (Pressing, 1998; Vorberg & Wing, 1996; Vorberg & Schulze, 2002) fared rather well.  Based on the assumptions that (1) local phase adjustments and global period adjustments are active simultaneously during noticeable tempo changes, that (2) both mechanisms are fed by the same error information, and that (3) adjustments are first-order linear, the model accounts for the qualitative data pattern quite well, although quantitative goodness-of-fit leaves much to be desired. However, even the qualitative fit was satisfactory only when restricted to the transient phase of the data, which implies that there must exist additional control mechanisms that determine when the period adjustment mechanism is started and stopped (e.g., by setting the period correction gain).