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Further determination of the upper limit for a is an empirical question. It would rely on languages in which Heuristics 1 is relevant (i.e., Dur(V1)+Dur(R1) > Dur(V2)+Dur(R2)) under the condition Dur(V1) Given the definition of CCONTOUR, we can now construct a Tonal Complexity Scale as in (0). This is a scale of tonal complexity as measured by phonetics. (0) Tonal Complexity Scale: For any two tones T1 and T2, let C1 and C2 be the minimum CCONTOUR values required for the production and perception of T1 and T2 respectively. T1 is of higher Tonal Complexity than T2 iff C1>C2. Therefore, the correlation between CCONTOUR, which is determined by the duration and sonority of the rime, and its ability to carry complex contour tones can be schematized as in (0).
From the discussion of contour tone phonetics, we already know that the following three parameters of a tone influence its position in the Tonal Complexity Scale: the number of pitch targets, the pitch excursion between two targets, and the direction of the slope. In a more rigorous fashion, the influence of these three parameters can be summarized as in (0). (0) For any two tones T1 and T2, suppose T1 has m pitch targets and T2 has n pitch targets; the cumulative falling excursions for T1 and T2 are fF1 and fF2 respectively, and the cumulative rising excursions for T1 and T2 are fR1 and fR2 respectively. T1 has a higher tonal complexity than T2 iff: a. m>n, fF1≥ fF2, and fR1≥fR2; b. m=n, fF1≥fF2, and fR1≥fR2 (‘=’ holds for at most one of the comparisons); c. m=n, fF1+fR1=fF2+fR2, and fR1≥fR2.
Condition (0b) states that if T1 and T2 have the same number of pitch targets, and at least one of T1’s cumulative falling excursion and rising excursion is greater than that of T2’s, and the other one is no smaller than that of T2’s, then T1 is of higher tonal complexity than T2. This is true in virtue of (0b), according to which T1 requires a longer minimum duration in the sonorous portion of the rime than T2. As an example, 535 has a higher tonal complexity than 545, 534, or 435. Condition (0c) states that if T1 and T2 have the same number of pitch targets and the same overall pitch excursion, but the cumulative rising excursion in T1 is greater than that in T2, then T1 is of higher tonal complexity than T2. This is true in virtue of the fact that the percentage of rising excursion in T1 is greater than that in T2, and according to (0c), T1 requires a longer minimum duration in the sonorous portion of the rime than T2. As an example, 435 has a higher tonal complexity than 534, since m=n=3, fF1+fR1=fF2+fR2=3, and fR1=2>fR2=1. These comparisons must be made under the same speaking rate and style of speech, because the pitch excursion of a tone might change under different speaking rates and styles of speech. I assume that the consistent phonological behavior of speakers under different speaking rates and styles is due to their ability to normalize duration and pitch across speaking rates and styles (see Kirchner 1998, Steriade 1999 for similar views). This is discussed in more details in 6.2. Tones are represented phonetically by f0 in Hz throughout this dissertation. This is because that the main perceptual correlate of tone is f0, as I have mentioned, and the relation between the physical and auditory dimensions of f0 (in Hz and Bark respectively) is fairly linear for the sounds of interest in this dissertation (Stevens and Volkman 1940).6
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