Hierarchic relations

1. Problem and history

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration. Up to now the following particular cases have been examined (see Table 1)

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that the greater the whole the smaller its parts (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it Menzerath-Altmann´s law.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chinese characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length (cf. also Menzel 2005). The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with the power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory. It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used. The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14): 1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllable length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds $\leftarrow$)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

2. Hypothesis

The size of the components is a function of the construct size.

3. Derivation

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

$g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}$.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)$\frac{dy}{y-d}= \left(a_0 +\frac{a_1}{x}+ \frac{a_2}{x^2}\right)$.

where d is the minimal value y can attain. In case that there is another independent variable, z, one starts from

(2)$\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad$ and $\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}$

which can be extended to any number of variables. The solution of (1) yields

(3)$y= Cx^{a_1}e^{a_0 x-a_2/x} + d$

the combining (2) the solution is

(4)$y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d$

In different applications the following special cases of (3) have been used (d > 0)

(1a) $y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0$

(1b)$y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0$

(1c)$y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0$

(1d) $y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0$

Solution (4) is still very seldom. Both approaches are special cases of the unified theory ($\leftarrow$).

Examples:

Mean clause length in dependence on sentence length

(in H. Hesse, Der Steppenwolf)

yielding y = 11.5711x^(-0.2285) and R² = 0.97.

4. Authors: U. Strauss, G. Altmann

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