  One important aspect of the relationship between spoken and written Chinese
is the ranked syllable-to-character mapping spectrum, which is the ranked list
of syllables by the number of characters that map to the syllable. Previously,
this spectrum is analyzed for more than 400 syllables without distinguishing
the four intonations. In the current study, the spectrum with 1280 toned
syllables is analyzed by logarithmic function, Beta rank function, and
piecewise logarithmic function. Out of the three fitting functions, the
two-piece logarithmic function fits the data the best, both by the smallest sum
of squared errors (SSE) and by the lowest Akaike information criterion (AIC)
value. The Beta rank function is the close second. By sampling from a Poisson
distribution whose parameter value is chosen from the observed data, we
empirically estimate the $p$-value for testing the
two-piece-logarithmic-function being better than the Beta rank function
hypothesis, to be 0.16. For practical purposes, the piecewise logarithmic
function and the Beta rank function can be considered a tie.
