# Syllable structure

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1. Problem and history

If syllables are coded in the canonical form, V, CV, CVC, etc. where the vowel (V) is the syllable nucleus and the consonant (C) is the periphery, we obtain a set of structural types whose number follows a specific two-dimensional distribution. The types can be presented in tabular form

Where the frequency of the type CVVCC is given as n12, i symbolizing the number of consonants before and j that behind the nucleus. There are at most r onsets and s codas in the syllable. Let Pij * be the corresponding probability.

2. Hypothesis

The distribution of canonical syllable types abides by the (modified) bivariate Conway-Maxwell-Poisson distribution.

3. Derivation

Starting from the unified theory we assume that the probabilities of syllable types are joined with the following proportionality relation

(1)$P_{i,j}* \propto P_{i, j-1}*$

$P_{i,j}* \propto P_{i-1, j}*$

i.e. a class is proportional to its left or upper neighbour. In the first step Zörnig and Altmann (1993) used the simple Menzerathian relatioship (see Hierarchic relations) as proprotionality function and obtained

(2)$P_{i,j}* = \frac{b}{j^m}P_{i, j-1}*$ $P_{i,j}* = \frac{a}{i^k}P_{i-1, j}*, \quad m,k \epsilon \mathfrak{R}$

The solution of (2) yields

(3)$P_{i,j}* = \frac{a^1 b^j}{(i!)^k (j!)^m}P_{0,0}*, \quad i= 0,1,...,r, \quad j=0,1,...,s$

where the norming constant is $P_{0,0}^{-1} = \sum_{i=0}^r \sum_{j=0}^s\frac{a^i b^j}{(i!)^k(j!)^m}$ .

Since the particular case examined (Indonesian) has a strong preference for the CVC type ( = n11) the distribution must be modified and renormed. Weighting P11 * by β the other probabilities must be weighted so that βP11 * + α(1 - P11 * ) = 1. As a result we obtain

(4)$P_{i,j}= \begin{cases} \beta P_{i,j}*, & i=j=1 \\ \alpha P_{i,j}*, & i= 0,1,...,r,\quad j=0,1,...,s,\quad i \ne 1 or j \ne 1\end{cases}$

Zörnig and Altmann (1993) show also the estimators won from frequency classes as

(5)$\hat a = n_{10} / n_{00}$

$\hat b = n{01}/n_{00}$

$\hat k \ln(\hat a n_{10}/n_{20})$

$\hat m \ln(\hat b n_{01}/n_{02})$

$\hat \alpha = 1+ \frac{n_{10}\hat b - n_{11}}{N}$

$\hat \beta = \hat \alpha n_{11} /(n_{10}\hat b)$

Example. Structure of Indonesian syllables

In about 15000 Indonesian word forms from different texts Zörnig and Altmann (1993) found 610 syllable types shown in Table 1.

Using (3), (4) and (5) they obtained

$P_{i,j}= \begin{cases} 0.713P*_{i,j}, & i\ne 1 or j\ne 1 \\ 1.291P*_{i,j}, & i=j=1 \end{cases}$

with

$P*_{i,j}=\frac{(o.014)6^{i+j}}{(i!)^{4.585}(j!)^{4.948}}, \quad 0\le i, j<4$.

The estimated parameters are: a = b = 6, k = 4.585, m = 4.948, α = 0.713, β = 1.291, P * 00 = 0.014, N = 610 and the fitting is presented in Table 2. For example NP12 is computed as

$NP_{1,2}= 610\frac{0.713(0.014)6^{1+2}}{1!^{4.585} 2!^{4.948}}=610(0.06985) = 42.61$ .

The result is acceptable without testing.

4. Authors: U. Strauss, G. Altmann

5. References

Lee, Sank-Oak (1986). An explanation of syllable structure change. Korean Language Research 22, 195-213.

Vennemann, T. (1982) (ed.). Zur Silbenstruktur der deutschen Standardsprache. Silben, Segmente, Akzente: 261-305. Tübingen: Narr.

Zörnig, P., Altmann, G. (1993). A model for the distribution of syllable types. Glottometrika 14, 190-196.