# Understanding Xu’s quantitative approximation theory of F0 contours

I am starting to study Vietnamese and have had to revise my assumption that a tone was a specific pitch contour, because it’s clear that a word of a given tone can have quite different pitch contours depending on whether it is being pronounced in isolation or in running speech, and if in running speech, on the context.

Xu’s quantitative approximation theory seems to address this – it is described in terms of a ‘third-order critically damped system’ (Xu et al. 2015) and defined by a correspondingly complex equation. The best way I have found to visualise it is to treat the F0 trace as the path taken by some sort of video game character – call him Sonic – who, at any given time, will have a certain velocity and acceleration. Sonic passes through a series of cells, each of which represents a syllable and has its own pitch target. This target is not the kind you get in archery but more like a lure – it’s a moving target that gets going as soon as Sonic enters the cell and attracts him. The path that Sonic takes then depends on his initial speed and acceleration, the path taken by the lure, and the strength of the attraction. Of these, only the path taken by the lure and the strength of attraction are part of the definition of the target (i.e the underlying tone specification), so the actual pitch contours in running speech will not correspond to the tone specification, but the tone specification will always be recoverable from them if you know the initial speed and acceleration.

Xu hypothesises that the tone specification is always a straight line, and demonstrates that this produces curved contours in running speech (which are reasonably good approximations of the observed contours).

The difficulty I have with this is that I can’t see how it stacks up with the fact that the citation forms of some tones (such as the hỏi tone, in Vietnamese) are curved and/or have turning points. The fact that the actual pitch contours of running speech do not correspond to the tone specification is essentially explained by saying that Sonic is already on some kind of trajectory when the syllable begins* – but in the case of a syllable pronounced in isolation, surely he would start right behind the lure and stay on its tail, making the two trajectories the same. Does this mean that the theory doesn’t work, or if not, what am I missing?

• A more general / earlier statement of this was “The implementation rules are based on possible articulatory constraints on the production of surface F0 contours. Due to these constraints, the implementation of a simple pitch target may result in surface F0 forms that only partially reflect the underlying pitch targets” (Xu and Wang, 2001). The only constraints that seem to have made it into the qTA model described in Xu et al 2015 are the velocity and acceleration passed on from the previous syllable – but in the case of word of one syllable pronounced in isolation, there is no previous syllable.

Xu and Wang 2001: Pitch targets and their realization: Evidence from Mandarin Chinese Speech Communication 33 (2001) 319-337

Xu et al. 2015: Explaining the PENTA model: A reply to Arvaniti & Ladd. Phonology 32(3): 505-535