Difference between revisions of "Hierarchic relations"

 
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Up to now the following particular cases have been examined (see Table 1)
 
Up to now the following particular cases have been examined (see Table 1)
  
<div align="center">[[Image:Tabelle1_HR.jpg]]</div>
+
<div align="center">[[Image:Tabelle11_HR.jpg]]</div>
  
 
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''.
 
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 Chines 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).
+
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 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.
+
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.
 
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):
 
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
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3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.
 
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 syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).
+
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.
 
5. Partial reduplication is more frequent in natural languages than full reduplication.
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Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains
 
Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains
  
(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.  
+
(1)<math> \frac{dy}{y-d}= \left(a_0 +\frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.  
  
 
where d is the minimal value y can attain.
 
where d is the minimal value y can attain.
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'''Examples''':
 
'''Examples''':
  
 +
<div align="center">Mean clause length in dependence on sentence length</div>
 +
<div align="center">(in H. Hesse, Der Steppenwolf)</div>
  
'''4. Authors: G. Altmann'''
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[[Image:Tabelle3-neu.JPG]]
 +
yielding y = 11.5711x<sup>(-0.2285)</sup> and R<sup>2</sup> = 0.97.
 +
 
 +
 
 +
'''4. Authors: U. Strauss, G. Altmann'''
  
 
'''5. References'''
 
'''5. References'''
  
'''Abercrombie, D'''. (1967). ''Elements of general phonetics''. Edinburgh: University Press. (p.97)
+
'''Abercrombie, D'''. (1967). ''Elements of general phonetics''. Edinburgh: University Press.  
  
'''Äima, F.''' (1918). Phonetik und Lautlehre des Inarilappischen. ''Mémoires de la Societé Finno-Ougrienne 43, 1-249.'' (p.204)
+
'''Äima, F.''' (1918). Phonetik und Lautlehre des Inarilappischen. ''Mémoires de la Societé Finno-Ougrienne 43, 1-249.''  
  
 
'''Altmann, G.'''  (1980). Prolegomena to Menzerath´s law. ''Glottometrika 2, 1-10''.
 
'''Altmann, G.'''  (1980). Prolegomena to Menzerath´s law. ''Glottometrika 2, 1-10''.
Line 91: Line 97:
 
'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C.''' (2002). ''Einführung in die quantitative Lexikologie.'' Götingen: Peust & Gutschmidt.
 
'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C.''' (2002). ''Einführung in die quantitative Lexikologie.'' Götingen: Peust & Gutschmidt.
  
'''Altmann, G., Beöthy, E., Best, K.-H.''' (1982). Die Bedeutungskomplexität der Wörter und das Menzerathsche Gesetz. ''Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikations-forschung 35, 537-543''
+
'''Altmann, G., Beöthy, E., Best, K.-H.''' (1982). Die Bedeutungskomplexität der Wörter und das Menzerathsche Gesetz. ''Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 35, 537-543''.
  
 
'''Altmann, G., Schwibbe, M'''. (1989). ''Das Menzerathsche Gesetz in informationsverarbeitenden Systemen''. Hildesheim, Olms.
 
'''Altmann, G., Schwibbe, M'''. (1989). ''Das Menzerathsche Gesetz in informationsverarbeitenden Systemen''. Hildesheim, Olms.
  
 
'''Arens, H'''. (1965). ''Verborgene Ordnung''. Düsseldorf: Schwann.
 
'''Arens, H'''. (1965). ''Verborgene Ordnung''. Düsseldorf: Schwann.
 +
 +
'''Asleh, L., & Best, K.-H.''' (2004/05). Zur Überprüfung des Menzerath-Altmann-Gesetzes am Beispiel deutscher (und italienischer) Wörter. ''Göttinger Beiträge zur Sprachwissenschaft'' 10/11, 9-19.
  
 
'''Auer, P., Uhmann, S'''. (1988). Silben- und akzentzählende Sprachen. ''Zeitschrift für Sprachwissenschaft 7, 214-259.''
 
'''Auer, P., Uhmann, S'''. (1988). Silben- und akzentzählende Sprachen. ''Zeitschrift für Sprachwissenschaft 7, 214-259.''
  
'''Bak, P.''' (1996) ''How nature works. The science of self-organized crit''icality. New York: Copernicus-Springer.
+
'''Bak, P.''' (1996) ''How nature works. The science of self-organized criticality''. New York: Copernicus-Springer.
  
'''Bennett, S.G'''. (1935). Vergleichende experimentalphonetische Untersuchung der ungespannten Plosive im Englischen, Deutschen und Französischen. Diss. (Footnote 18)
+
'''Bennett, S.G'''. (1935). Vergleichende experimentalphonetische Untersuchung der ungespannten Plosive im Englischen, Deutschen und Französischen. Diss.  
  
 
'''Bohn, H.''' (1998). ''Quantitative Untersuchungen der modernen chinesischen Sprache und Schrift''. Hamburg: Kovač.
 
'''Bohn, H.''' (1998). ''Quantitative Untersuchungen der modernen chinesischen Sprache und Schrift''. Hamburg: Kovač.
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'''Boroda, M.G., Altmann, G'''. (1991). Menzerath's law in musical texts. ''Musikometrika 3, 1-13.''
 
'''Boroda, M.G., Altmann, G'''. (1991). Menzerath's law in musical texts. ''Musikometrika 3, 1-13.''
  
'''Changizi, M.A'''. (2001). Universal scaling laws for hierarchical complexity in langauegs, organisms, bahaviors and other combinatorial systems. ''J. of Theoretical Biology 211, 277-295.''
+
'''Changizi, M.A'''. (2001). Universal scaling laws for hierarchical complexity in languages, organisms, bahaviors and other combinatorial systems. ''J. of Theoretical Biology 211, 277-295.''
  
'''Collinder, B'''. (1964). Das Wort als phonetische Einheit. In: Collinder, B. (Hrsg.), ''Sprachwissenschaft und Wahrscheinlichkeit: 203-217. Uppsala: Almquist, Wiksells.'' (1939: 66)
+
'''Collinder, B'''. (1964). Das Wort als phonetische Einheit. In: Collinder, B. (ed.), ''Sprachwissenschaft und Wahrscheinlichkeit: 203-217''. Uppsala: Almquist, Wiksells.
  
 
'''Cramer, I.M'''. (2005). The parameters of the Altmann-Menzerath law. ''J. of Quantitative Linguistics 12(1), 41-52.''
 
'''Cramer, I.M'''. (2005). The parameters of the Altmann-Menzerath law. ''J. of Quantitative Linguistics 12(1), 41-52.''
  
'''Cramer, I.M'''. (2005a). Das Menzerathsceh Gesetz. In: Altmann, G., Köhhler, R., Piotrowski, R.G. (eds.), ''Handbook of Quantitative Linguistics: 659-688''. Berlin: de Gruyter.
+
'''Cramer, I.M'''. (2005a). Das Menzerathsche Gesetz. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics - An International Handbook: 659-688''. Berlin: de Gruyter.
  
'''Donner, K.''' (1912). Salmin murteen kvantiteettisuehtista. ''Suomi IV, 9, II''. (p. 33)
+
'''Debowski, L.''' (2007). Menzerath´s law for the smallest grammar. In: Grzybek, P., Köhler, R. (eds.), ''Exact methods in the Study of Language and Tests: 77-85.'' Berlin: de Gruyter.
  
'''Elert, C.C.''' (1965). ''Phonological studies of quantity in Swedish''. Uppsala. (p. 138)
+
'''Donner, K.''' (1912). Salmin murteen kvantiteettisuehtista. ''Suomi IV, 9, II''.  
  
'''Evertz,''' (1929). ''Beiträge zur Phonetik der südfranzösischen Plosive''. Diss. (footnote 21)
+
'''Elert, C.C.''' (1965). ''Phonological studies of quantity in Swedish''. Uppsala.  
  
'''Faure, G., Hirst, D.J., Chafcouloff, M.''' (1980). Rhythm in English: Isochronism, potch, and pereceived stress. In: Linda, R., Schooneveld, C.H.v. (eds.), The ''melody of language: 71-79''. Bartimore: University Park Press.
+
'''Evertz,''' (1929). ''Beiträge zur Phonetik der südfranzösischen Plosive''. Diss.
 +
 
 +
'''Faure, G., Hirst, D.J., Chafcouloff, M.''' (1980). Rhythm in English: Isochronism, pitch, and perceived stress. In: Linda, R., Schooneveld, C.H.v. (eds.), ''The melody of language: 71-79''. Baltimore: University Park Press.
  
 
'''Fenk, A., Fenk-Oczlon, G'''. (1993). Menzerath´s law and the constant flow of linguistic information. In: Köhler, R., Rieger, B.B. (eds.), ''Contributions to quantitative linguistics: 11-31''. Dordrecht: Kluwer.
 
'''Fenk, A., Fenk-Oczlon, G'''. (1993). Menzerath´s law and the constant flow of linguistic information. In: Köhler, R., Rieger, B.B. (eds.), ''Contributions to quantitative linguistics: 11-31''. Dordrecht: Kluwer.
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'''Fenk-Oczlon, G., Fenk, A'''. (1995). Selbstorganisation und natürliche Typologie. ''Sprachtypologie und Universalienforschung 48, 223-238''.
 
'''Fenk-Oczlon, G., Fenk, A'''. (1995). Selbstorganisation und natürliche Typologie. ''Sprachtypologie und Universalienforschung 48, 223-238''.
  
'''Fenk-Oczlon, G., Fenk, A'''. (1999). Cognition, quantitative linguistics, and systemic typology. Linguistic Typology 3, 151-177.
+
'''Fenk-Oczlon, G., Fenk, A'''. (1999). Cognition, quantitative linguistics, and systemic typology. ''Linguistic Typology 3, 151-177.''
  
 
'''Fickermann, I., Markner-Jäger, B, Rothe, U'''. (1984). Wortlänge und Bedeutungskomplexität. ''Glottometrika 6, 115-126.''
 
'''Fickermann, I., Markner-Jäger, B, Rothe, U'''. (1984). Wortlänge und Bedeutungskomplexität. ''Glottometrika 6, 115-126.''
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'''Geršić, S., Altmann, G'''.  (1980). Laut – Silbe – Wort und das Menzerathsche Gesetz. ''Frankfurter Phonetische Beiträge 3, 115-123''.
 
'''Geršić, S., Altmann, G'''.  (1980). Laut – Silbe – Wort und das Menzerathsche Gesetz. ''Frankfurter Phonetische Beiträge 3, 115-123''.
  
'''Grégoire, A'''. (1899). Variation de la durée de la syllable français. ''La Parole 1'' (p. 161).
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'''Grégoire, A'''. (1899). Variation de la durée de la syllable français. ''La Parole 1''.
  
'''Grzybek, P'''. (1998).
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'''Grzybek, P'''. (1999). Randbemerkungen zur Wortlänge im Kroatischen. In: B.Tošović (ed.), ''Die grammatischen Korrelationen: 56-67.'' Graz: Gralis.
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 +
'''Grzybek, P., Stadlober, E.''' (2007). Do we have probelsm with Arnes´ law? A new look at the sentence-word relation. In: Grzybek, P., Köhler, R. (eds.), ''Exact Method in th Study of Language and Text: 203-215.'' Berlin: de Gruyter.
  
 
'''Hammerl, R., Sambor, J'''. (1993a). ''O statystycznych prawach jezykowych''. Warszawa: Polskie Towarzystwo Semiotyczne.
 
'''Hammerl, R., Sambor, J'''. (1993a). ''O statystycznych prawach jezykowych''. Warszawa: Polskie Towarzystwo Semiotyczne.
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'''Hřebíček, L'''. (1992). ''Text in communication: Supra-sentence structure''. Bochum, Brockmeyer.
 
'''Hřebíček, L'''. (1992). ''Text in communication: Supra-sentence structure''. Bochum, Brockmeyer.
  
'''Hřebíček, L'''. (1993b). Text as a construct of aggregations. In: Köhler, R., Rieger, B. (eds.), ''Contributions to quantitative lingusitics''. Dordrecht: Kluwer: 33-39.
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'''Hřebíček, L'''. (1993b). Text as a construct of aggregations. In: Köhler, R., Rieger, B. (eds.), ''Contributions to quantitative lingusitics: 33-39''. Dordrecht: Kluwer.
  
'''Hřebíček, L'''. (1993c). Text as a strategic process. In: Hřebíček, L., Altmann, G. (Eds.), ''Quantitative text analysis'': 136-150. Trier: WVT.
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'''Hřebíček, L'''. (1993c). Text as a strategic process. In: Hřebíček, L., Altmann, G. (Eds.), ''Quantitative text analysis: 136-150''. Trier: WVT.
  
'''Hřebíček, L'''. (1994). Fractals in language. ''J. of Quantitative Linguistics 1'', 82-86.
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'''Hřebíček, L'''. (1994). Fractals in language. ''J. of Quantitative Linguistics 1, 82-86''.
  
'''Hřebíček, L'''. (1994a). The stairway of subsystems. In: ''2nd International Conference on Quantitative Linguistics,'' September 1994, 210-222 Moscow. Moscow: Lomonosov Moscow State University.
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'''Hřebíček, L'''. (1994a). The stairway of subsystems. In: ''2nd International Conference on Quantitative Linguistics, September 1994, 210-222 Moscow''. Moscow: Lomonosov Moscow State University.
  
 
'''Hřebíček, L.'''  (1995). ''Text levels. Language constructs, constituents and Menzerath-Altmann law''. Trier: WVT.
 
'''Hřebíček, L.'''  (1995). ''Text levels. Language constructs, constituents and Menzerath-Altmann law''. Trier: WVT.
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'''Hřebíček, L.''' (1997). ''Lectures on text theory''. Prague: Oriental Institute.
 
'''Hřebíček, L.''' (1997). ''Lectures on text theory''. Prague: Oriental Institute.
  
'''Hřebíček, L.''' (2000). ''Variation in sequen''ces. Prague: Oriental Institute.
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'''Hřebíček, L.''' (2000). ''Variation in sequences''. Prague: Oriental Institute.
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'''Hřebíček, L.''' (2004). The semantic space and Turkish ghazels. ''Archiv orientální 72, 175-191''.
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'''Hřebíček, L.''' (2005). Text laws. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.) (2005), ''Quantitative Linguistics - An International Handbook: 348-361''. Berlin: de Gruyter.
  
'''Hřebíček, L.''' (2004). The semantic space and Turkish ghazels. ''Archiv orientální 72, 175-191''
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'''Hřebíček, L., Altmann, G'''(1993). Prospects of text linguistics. In: Hřebíček, L., Altmann, G. (Eds.), ''Quantitative text analysis: 1-28''. Trier: WVT.
  
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'''Kaumanns, W., Schwibbe, M.H.''' (1983). Strukturmerkmale von Primatensozietäten unter dem Gesichtspunkt der Menzerathschen Regel. In: Altmann, G., Schwibbe, M. (1983): 99-107.
  
 
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'''Krott, A.''' (1996). Some remarks on the relation between word length and morpheme length. ''J. of Quantitative Linguistics 3, 29-37''.
'''
 
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'''Leopold, E.''' (2001). Fractal structures in language. The question of the imbedding space. In Uhlířová, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 163-1176''. Trier: WVT.
  
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Latest revision as of 11:44, 25 June 2007

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)

Tabelle11 HR.jpg

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)

Tabelle3-neu.JPG yielding y = 11.5711x(-0.2285) and R2 = 0.97.


4. Authors: U. Strauss, G. Altmann

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