Analyses of Script: Properties of Characters and Writing Systems (Quantitative Linguistics) (German Edition)

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Analyses of Script

≥

Quantitative Linguistics 63

Editors

Reinhard Köhler Gabriel Altmann Peter Grzybek

Mouton de Gruyter Berlin · New York

Analyses of Script Properties of Characters and Writing Systems

Edited by

Gabriel Altmann Fan Fengxiang

Mouton de Gruyter Berlin · New York

Mouton de Gruyter (formerly Mouton, The Hague) is a Division of Walter de Gruyter GmbH & Co. KG, Berlin.

앝 Printed on acid-free paper which falls within the guidelines 앪 of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication Data Analyses of script : properties of characters and writing systems / edited by Gabriel Altmann and Fan Fengxiang. p. cm. ⫺ (Quantitative linguistics ; 63) Includes bibliographical references and index. ISBN 978-3-11-019641-2 (hardcover : alk. paper) 1. Writing ⫺ Mathematical models. I. Altmann, Gabriel. II. Fengxiang, Fan, 1950⫺ P211.A555 2008 411⫺dc22 2008008072

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

ISBN 978-3-11-019641-2 ISSN 0179-3616 쑔 Copyright 2008 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Cover design: Martin Zech, Bremen. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Printed in Germany.

Preface

For a linguist, script is something that does not belong to language, it is something secondary, left to culture scientists and historians. But for teachers of native languages, orthographers, cryptographers, paleographers, graphologists and especially pupils in elementary and grammar schools, it is an object of primary importance. For all these groups, script is something to be solved, to be used in making inferences about epochs, persons or contents, or to get good grades for. For computer linguists, it is a practical problem of mechanical conversion from written to spoken language or vice versa. Everybody uses script but nobody cares for its inner life in which perhaps there is some kind of self-regulation or control. A group of researchers, not believing in some older unsuccessful endeavours to ﬁnd some essential properties in script, but leaning against the way of thinking in quantitative and synergetic linguistics, started an experiment in conjecturing, quantifying and measuring the properties of script and seeking models of their behavior. This volume presents the results of their research. The results are surprising. Letters or other symbols have complexity, distinctivity, representativity, utility, grapheme size, phonemic load, ornamentality, uncertainty, dimension and perhaps a series of other properties which wait to be established. Some of the properties are associated with one another, some of them compete and there probably is a control cycle which may become basis of a future theory of script. The researchers considered only four script types, namely Latin, Oriya, Japanese and Old Egyptian, which is, of course, not enough to draw general conclusions, but at least a start has been made. Some common problems have been analysed using English, Italian, Swedish, Slovak, Slovenian and German. For Oriya a new weighted distributional calculus has been drafted using letter form and positioning; for the ﬁrst time the strange and unexpected way of simpliﬁcation of hieroglyphs has been expressed quantitatively, leaving open the question of measurement of change from iconism to symbolism; for Japanese the dependence of frequency on polytextuality of kanji has been modelled, and a look has been cast at the capacity dimension of signs. And last but not least, a ﬁrst draft of a future theory of script has been ventured.

vi

For quantitative linguists, a theory is a set of interrelated hypotheses, of which at least one is a law. Though in the present volume laws have not been established – it is a very long way before a statement can be considered a law – a network of hypotheses has been set up and instructions for continuing this work have been offered in the last contribution. Needless to say, further development can fundamentally change the direction of research and the results presented here may become only peripheral consequences of a deeper theory – a usual event in the evolution of science. There are, of course, also practical aspects of this investigation. The representation of phonemes, some load and utility problems etc. can show quantitatively whether an orthographic reform is necessary or whether it is too late to perform one. The resulting numbers must simply be adequately interpreted. The numbers are objective and have nothing to do with the speakers’/writers’ intuition or national feeling. The results can be considered as quantitative descriptions of some writing problems of individual languages but for the involved analysts it is an attempt at laying the foundations of a discipline not existing up to now, opening new vistas and embedding this object in the scope of synergetic linguistics. Finally, we would like to express our sincere gratitude to Peter Grzybek and Reinhard Köhler as the editors of the series Quantitative Linguistics, who have enthusiastically accompanied the whole process of this book coming into existence, from the ﬁrst ideas to the preparation of the layout. In this respect, our thank also goes to Veronika Koch for her competent technical help. Gabriel Altmann (Lüdenscheid, Germany) Fan Fengxiang (Dalian, China)

Contents Preface Gabriel Altmann and Fan Fengxiang

v

I. Introduction Quantitative analysis of writing systems: an introduction Reinhard Köhler

3

II. The phoneme-grapheme relation The phoneme-grapheme relationship in Italian Gerald Bernhard and Gabriel Altmann

13

Graphemic representation of English phonemes Fan Fengxiang and Gabriel Altmann

25

The phoneme-grapheme relationship in Slovene Emmerich Kelih

61

On the distribution of graphemic representations Ján Maˇcutek

75

The phoneme-grapheme relation in Slovak Emília Nemcová and Gabriel Altmann

79

III. Special problems Script ornamentality Karl-Heinz Best and Gabriel Altmann

91

On the decrease of complexity from hieroglyphs to hieratic symbols Ina Hegenbarth-Reichardt and Gabriel Altmann

105

The fractal dimension of script: an experiment Reinhard Köhler

115

On graphemic representation of the Oriya phonemes Panchanan Mohanty and Gabriel Altmann

121

viii Contents

On the relation between types and tokens of Japanese morae Katsuo Tamaoka

141

IV. Towards a theory Towards a theory of script Gabriel Altmann

149

Authors’ Addresses

165

Author Index

167

Subject Index

169

I. Introduction

Quantitative analysis of writing systems: an introduction Reinhard Köhler

1

Introduction

The cultures in the world use various, quite different writing systems (scripts) to ﬁx linguistic material. Linguists distinguish between two principally different categories: logographic (subdivided into pictographic, ideographic and abstract-logographic) and phonographic (subdivided into segmental, syllabic and alphabetic) scripts (cf. Table 1). Sometimes, logographic, syllabic and alphabetic principles occur in a mixture. Table 1: Categories of scripts with examples logographic

pictographic

T

ideographic abstract-logographic phonographic

§

sound-segmental syllabic

mixed systems

alphabetic

ˆş<

various kinds

o;

A recent increasing interest in quantitative descriptions of graphical symbols and scripts can be observed in linguistics. The present contribution aims at giving an overview of measurable properties of signs and sign systems, as well as of functional dependences among symbol properties. 2

Properties

We should distinguish between properties of individual signs and properties of sign systems. Properties such as frequency of occurrence, complexity, pho-

4

Reinhard Köhler

neticity etc. can be attributed to individual signs, whereas scripts can be characterised in terms of inventory size, entropy, efﬁciency, learnability etc. Many properties seem to be determinable with respect to individual signs and to systems as well, such as ambiguity, distinctivity etc. However, in each of these cases we observe, in fact, two different features, i.e. these terms are ambiguous themselves. E.g., ambiguity can be measured with respect to different individual signs, whereas ambiguity with respect to scripts will probably be deﬁned as the mean ambiguity of all the signs belonging to the given script, or in a similar way, thus reﬂecting some kind of a global property. Another distinction should be kept in mind, viz. the distinction between language-dependent and language-independent properties. Many scripts are in use for more than one language. Therefore, some of the properties of signs and even of scripts depend on their function in the given language. A simple example can be seen in the fact that the letter of the Roman alphabet represents a single sound in English, viz. /z/, whereas, in German and Italian, it stands for two sounds: /ts/, in Italian also for /dz/. The letter <s> is unambiguous in Swedish; in German, its pronunciation is /s/ or /z/, depending on its position and context. The Roman alphabet, as used for the German language, has in Germany one letter more than it has in Switzerland, where the <ß> is replaced by <ss> in all cases. Before any attempt to ﬁnd a promising measure of a property, one has to clarify how the corresponding units should be deﬁned. The basic units of alphabetic scripts seem to be clear at a ﬁrst glance. However, different authors use different deﬁnitions. Some authors consider as the basic unit, the grapheme, any letter or combination of letters which represents a sound. The present author prefers the following deﬁnition: Deﬁnition 1 A grapheme is any graphical sign which, on its own, represents in at least one context a portion of linguistic material. Hence, the letter is a grapheme regardless of the fact that it appears also in sequence with for another sound. On the other hand, diacritics such as accents would not be considered as graphemes but as parts of complex graphemes because they do not represent any sound, sound combination, word, or meaning. They are rather distinctive features which serve to differentiate graphemes. Sequences such as will then be considered as syntagmas. However, another point of view may support Altmann’s variant: sequences representing a single sound could also be considered as compound graphemes.

Quantitative analysis of writing systems: an introduction

5

Anyway, the appropriateness of any deﬁnition of a unit or a property depends crucially on the approach and the purpose of the given investigation and on its compatibility with other deﬁnitions within the given approach. Therefore, results of scientiﬁc studies can only be compared if all units and properties are explicitly deﬁned (or, in advanced ﬁelds, are common among all researchers). The same is true of the measures which are employed to determine the properties under study. Let us consider, as an illustrative example, the complexity of individual signs. Bohn (2002) operationalised complexity in terms of the number of strokes a Chinese character consists of (cf. Figure 1a). This measure works perfectly for Chinese; for other scripts, however, the stroke inventory would have to be deﬁned in a different way – if possible at all. Altmann (2004) avoids this difﬁculty. He proposes and uses a measure according to Figure 1b, assigning different scores to dots, straight lines and arches on the one hand and continuous, crisp, and crossing connections on the other hand. Another method is preferred by Peust (2006): he deﬁnes complexity in terms of the maximum number of intersections with a straight line. Figure 1c gives an illustration, which, at the same time, shows the limitations of this approach, since there is no position or angle of a straight line which would correspond to the intuitive complexity of the given symbol.

(a) Bohn

Form: dot straight line arch

1 point 2 points 3 points

Connection: continuous crisp (sharp) crossing

1 point 2 points 3 points

(b) Altmann

(c) Peust

Figure 1: Three different methods to determine ‘complexity’

6

Reinhard Köhler

Another important aspect should also be emphasised: Giving a property a name, such as complexity, does not sufﬁce to ﬁnd an appropriate measure, of course. Before metricising (quantifying) or even measuring, the concept behind the term must be clariﬁed; otherwise any deﬁnition or operationalisation will be unsatisfying or even useless and misleading. Our example, complexity, may serve to illustrate this aspect. A closer look at what can be understood under the term unveils that quite a number of different concepts may be connected with it, depending on the speciﬁc interest of the researcher and of the interrelations one has in mind. Let us consider only two perspectives on complexity, viz. complexity from the point of view of the writer, and complexity from the point of view of the reader. We shall call these two perspectives Production Complexity and Decoding Complexity, respectively. Additionally, we will consider the fact that both kinds of complexity can be measured with respect to different kinds of effort. Again, we shall take into account only two of them: Muscular/Nervous Effort and Cognitive Effort. Combinations of perspectives with efforts yield four different bases for operationalisation: 1. 2. 3. 4.

Production Complexity in terms of Muscular/Nervous Effort Production Complexity in terms of Cognitive Effort Decoding Complexity in terms of Muscular/Nervous Effort Decoding Complexity in terms of Cognitive Effort.

There are certainly more than two perspectives and also more than two forms of effort connected with them. Moreover, it is easy to ﬁnd more than these two aspects, perspective and effort form, which should be taken into account when complexity is concerned.1 And clearly, the speciﬁc selection of (a combination of) aspects determines the way a property can be measured. Let us follow up our example on complexity and discuss the possibilities to ﬁnd operationalisations according to the different aspects. One possibility to measure cognitive effort of sign production is to determine the number of different elements needed for the given sign. Of course, a measuring procedure is to be preferred if it can be applied mechanically, or even better, automatically. This is possible by taking the number of trajectories a sign consists of. Computer fonts in vector representation describe for each symbol the Bézier curves which specify it. Evaluating a font in this way 1. There are other aspects, of course, which are not connected with the ones discussed so far. Peust’s measure for example does not reﬂect any of these aspects but corresponds to a rather abstract, topological concept.

Quantitative analysis of writing systems: an introduction

7

enables automatic processing of the corresponding data. Figure 2 illustrates the procedure.

Figure 2: Number of Bézier curves as speciﬁcations of the trajectories

Although this procedure looks similar to Altmann’s approach there is a signiﬁcant difference. With respect to the production effort, the existence of an intersection does not matter. Drawing two strokes such as “/\” causes the same effort as drawing the two intersecting strokes in the letter “X” in our case. However, if production complexity with respect to muscular and/or nervous effort is to be measured we have to take into account the way the writing instrument is used. Different instruments will require a different number of movements for the same sign. Thus, drawing an “R” with a pencil requires additional movements if the vertical line is drawn from top to bottom. Then the pencil has to be lifted, moved back to the starting position, and lowered again. In any case, an “X” requires an extra movement (cf. Figure 3). Clearly, the situation is different if signs are produced with hammer and chisel, and again different when a typewriter or computer is used. Furthermore, the size of the signs cannot be ignored. Moreover, to have a more realistic picture of production effort, length and angles of the curves should be taken into account, and one should not forget that the effort connected with drawing a concave curve (with the right hand), for example, is less than that of drawing a straight line. Also changes of movement direction cause effort etc. These considerations show that properties as concepts cannot be taken for granted. The more realistic the measure of effort, the more doubts arise as to whether effort is an appropriate operationalisation of complexity at all. If not – what else is complexity? Or, ornamentality? Analogously, decoding (or recognition) effort can be measured in several ways. One of them is the measurement of the time a person needs, ceteris paribus, to recognise a given sign – a rather impractical method. Another one is to analyse the signs with respect to their distinctive elements. However,

8

Reinhard Köhler

Figure 3: Number of Bézier curves as speciﬁcations of the movements of a pen

there are again several aspects and factors which can be taken into account. An additional problem is to single out the features and elements which contribute to the distinctivity of individual signs with respect to all the other ones on the one hand, and those features which contribute to an increased readability in a more global way, such as serifs. Again, the more realistic such a measure the more doubtful it is whether it represents complexity. Nevertheless, there are good reasons to believe that the measures used so far reﬂect properties closely connected to complexity in some way or other, at least to a certain degree, and that they are able to help detect regularities or general patterns which can be used in the process of ﬁnding hypotheses and testing them. Another possible consequence is to abandon ‘complexity’ and to concentrate on the effort measures as such. There is no general solution, of course; one has to solve the problem for any individual property one attempts to investigate. Similar effort must be taken to deﬁne other properties, such as effectivity, efﬁciency, learnability, readability, ornamentality, compositionality, to name just a few. Others can be worked with quite straightforwardly, such as entropy, redundancy, information, polyfunctionality, synfunctionality etc., because they are either sufﬁciently deﬁned, e.g. in information theory, or because there is a strong analogy to properties of signs on other levels of analysis, such as polyfunctionality, which is a direct analog to polysemy.

3

Interrelations and complex models

Starting from concepts (objects and properties), the next step is to look for interrelations, either for interrelations between two or more properties of an object or for interrelations between a property and its frequency of occurrence. Interrelations between two or more properties will be modeled using some kind of function (including dynamic models where time is one of the variables). If the frequency of the observed (or predicted) values of a property

Quantitative analysis of writing systems: an introduction

9

is under study, the corresponding mathematical model will be a frequency distribution. Every new scientiﬁc ﬁeld begins with individual properties, studying their distributions and their dependences or effects on other individual properties. In the beginning years of quantitative linguistics, the study of word frequencies was predominant. Later, other properties of words were detected, such as length and polysemy, and their interrelations with frequency were studied pairwise. The simultaneous investigation of more than two properties of words, i.e. more-dimensional studies, is a recent innovation. In our days, we have the means to model rather complex networks of properties and their interrelations, including theory of dynamic behaviour, thanks to systems theory in general and synergetics in particular. Synergetic models of linguistic phenomena enable us to set up complex models with explanatory power (on the basis of functional explanation, cf. Köhler 1986, 2005). The present volume gives examples of measures, distributions, and functions concerning properties of scripts and signs, and introduces a ﬁrst attempt at a synergetic model of a complex network of script properties.

References Altmann, Gabriel 2004 “Script complexity.” In: Glottometrics, 8; 68–73. Bohn, Hartmut 2002 “Untersuchungen zur chinesischen Sprache und Schrift”. In: Köhler, Reinhard (Ed.), Korpuslinguistische Untersuchungen zur quantitativen und systemtheoretischen Linguistik; 127–177. [http://ubt.opus. hbz-nrw.de/volltexte/2004/279] Köhler, Reinhard, Altmann, Gabriel 1983 “Systemtheorie und Semiotik.” In: Zeitschrift für Semiotik, 5(4); 424– 431. Köhler, Reinhard 1986 Zur linguistischen Synergetik. Struktur und Dynamik der Lexik. Bochum: Brockmeyer. 2005 “Synergetic linguistics”. In: Köhler, Reinhard; Altmann, Gabriel; Piotrowski, Rajmund G. (Eds.), Quantitative Linguistics. An International Handbook. Berlin / New York: Mouton de Gruyter, 760–774. Peust, Carsten 2006 “Script complexity revisited.” In: Glottometrics, 12; 11–15.

II. The phoneme-grapheme relation

The phoneme-grapheme relationship in Italian Gerald Bernhard and Gabriel Altmann

1

Introduction

The graphemic representation of phonemes in a language depends to a considerable extent on its history, on the type of script (letters, syllables, moras, ideograms, mixed scripts), on the strength of foreign inﬂuence, on the time of the introduction of the script, on the number of writing reforms, on the unity of the contiguity of the area where the language is spoken (cf. English vs. Danish), etc. The graphemic representation can accelerate or brake the communication; it is important for the teaching of the native language and the second language. The representation has different properties which become manifest only after having been quantiﬁed. In the present contribution we shall only examine those properties that have been deﬁned by Best and Altmann (2005), namely (1) the orthographic uncertainty of a phoneme, (2) the distribution of grapheme size, (3) the graphemic exploitation of letters and (4) the positional participation of letter in graphemes. Since Italian uses a letter script, a direct comparison of results with German and Swedish is possible. Letters are the symbols of the Latin alphabet, and graphemes are also their combinations and letters with diacritical marks (cf. „e“ and, „è“ is), which can mark also the position of accent (cf. „meta“ vs. „metà“). Italian took over the Latin alphabet and adopted it for its own purposes. The representation of individual phonemes by graphemes is shown in Tables 1a and 1b. Long vowels and long consonants are considered separate phonemes; some consonants obtain the status of allophones because they occur in complementary distribution.

2

The orthographic uncertainty of phonemes

As can be seen in Table 1, individual phonemes are represented by different numbers of graphemes. In the ideal case each grapheme should correspond to

14

Gerald Bernhard and Gabriel Altmann

Table 1a: The phoneme-grapheme relation in Italian: vowels (* = borrowings) Phonemes

Graphemes

Examples

/i/ /i:/ /e/ /e:/ /E/ /E:/ /a/ /a:/ /O/ /O:/ /o/ /o:/ /u/ /u:/ /ao/ /au/ /ai/ /Ei/ /Eu/

<e, he, é, eh> <e> <e, he> <e, è> <ei> <eu>

chicco; hippie/hippy*; così vino pesce; hegeliano*; perché; eh! pelo pesca; herpes* bene; bebè bacca; hanno; ah! baco; città cogliere; ho; yacht [jOte ] toro; oblò torre; holding*; boh! volo, boh! burro; humus*; uh! luna; gioventù ciao, Paolo Sabaudia zaino eidetico Europa

only one phoneme, but since script is either inherited and does not follow the development of language, or was taken over, with the number of phonemes in the target language being originally greater than in the source language, discrepancies arise automatically. The number of graphemes must be enlarged, a process carried out by combining letters or adding new symbols. Different phonological processes result automatically in multiple representations of individual phonemes. Hence these processes give rise to orthographic representation uncertainties, which can, however, be expressed numerically. If the mean uncertainty surpasses a certain threshold, it is a signal for a writing reform. The orthographic uncertainty of a phoneme /x/ can be expressed as U/x/ = log2 nx

(1)

where U/x/ is the uncertainty of the phoneme /x/, log2 is the logarithm with basis 2, and nx is the number of graphemes that can represent the phoneme /x/. In Table 2 one can ﬁnd all Italian phonemes and their uncertainties (Ux ,

The phoneme-grapheme relationship in Italian

15

Table 1b: The phoneme-grapheme relation in Italian: consonants Phonemes

Graphemes

Examples

/;R/ /;R:/ /l/ /l:/ /ń/ [ń, ńń] /m/ /m:/ /n/ /n:/ /˜n/ [ñ, ñ:] /f/ /f:/ /v/ /v:/ /s/ /s:/ /z/ /S/ [S, S:] /ts/ /dz/ /t:s/ /d:z/ /tS/ /t:S/ /dˇz/ /d:ˇz / /w/ /j/

<m> <mm> <s, x> <ss> <s> <sce,i , scia,o,u>

/p/ /p:/ /t/ /t:/ /k/

) , x, k> /k:/ /b/ /b:/ /d/ /d:/ /g/ /g:/ ) >
toro torre male palla gli [ńń]; ﬁglio, maglia, aglio lama mamma nano nanna gnomo; stagno afa bafﬁ lava davvero casa, extra cassa rosa, sbaglio scemo; pesce, sciame, lasciare zio zanzara cozza razzo; mazurca cena; pace; bacio lacci (pl.), laccio gerla, regione; jazz raggio, laggiù quello, guaio chiave, piano; yogurt*; Juventus* juventino papa pappa seta setta cane; chino; questo; extra; karate*; kit* becco; becchi; acqua bibita babbo guado freddo lago; laghi (pl.); hegeliano* leggo (I read); tegghia 16 Gerald Bernhard and Gabriel Altmann with nx denoting the number of representing graphemes, and fx the number of phonemes with uncertainty Ux . Table 2: Orthographic uncertainty of Italian phonemes Phoneme nx /i:/, /e/, /ao/, /au/, /ai/, /Ei/, /Eu/, /;R/, /;R:/, /l/, /l:/, /m/, /m:/, /n/, /n:/, /˜n/, /f/, /f:/, /v/, /v:/, /s:/, /z/, /ts/, /dz/, /t:s/, /w/, /p/, /p:/, /t/, /t:/, /b/, /b:/, /d/, /d:/ /E/, /E:/, /a:/, /O:/, /o:/, /u:/, /ń/, /s/, /S/, /d:z/, /tS/, /t:S/, /d:ˇz/, /g:/ /a/, /O/, /o/, /u/, /dˇz/, /j/, /k:/, /g/ /e/ /i/, /k/ Ux fx 1 0 34 2 3 4 5 1 1.58 2 2.30 14 8 1 2 The mean uncertainty can be computed as the average by means of 1 U¯ = ∑ fx Ux N x∈I (2) where N is the number of all representations. In our case U¯ = [34(0) + 14(1) + 8(1.58) + 1(2) + 2(2.32)]/59 = 0.5641. Comparing this number with the result from Swedish, U¯ = 0.797, and with ¯ 0.965 (cf. Best & Altmann 2005), one could conclude that the German U= Italian orthography is not so vague as the German or Swedish. In order to get a more objective image of these differences we set up an asymptotic test for the difference of two mean uncertainties, i.e. U¯ 1 − U¯ 2 z= V (U¯ 1 ) + V (U¯ 1 ) (3) ¯ is its variance and z is the Here U¯ is the empirical mean uncertainty, V (U) quantile of the normal distribution. The variance of U¯ can be derived using the Taylor expansion as below: 2 N 1 1 1 1 ¯ V (U) = V V (nx ) ∑ log2 nx = N 2 ∑ V (log2 nx ) = N 2 ln2 2 ∑ nx N x=1 E(nx ) The phoneme-grapheme relationship in Italian σ2 , 17 Since nx is the original variable whose expectation is E(nx ) = μ and V (nx ) = we obtain, after substituting in the above formula, ¯ = V (U) σ2 Nμ2 ln2 2 (4) which can be estimated by means of empirical values as ¯ = V (U) s2 0.48N x¯2 (5) In (5) one can easily see that the variance of the uncertainty is a function of only the well known variation coefﬁcient. For Italian we obtain 1 x fx N∑ = [1(34) + 2(14) + 3(8) + 4(1) + 5(2))/59 = 1.694915 1 1 ¯ 2 fx = x2 fx − x¯2 s2 = ∑ (x − x) N N∑ = [12 (34) + 22 (14) + 32 (8) + 42 (1) + 52 (2)]/59 − 1.6949152 = 0.991669 . x¯ = Finally, ¯ Ital = 0.991669/[0.48(59)1.6949152 ] = 0.012189. V (U) ¯ German = In the same way we obtain the variance for German as V (U) ¯ 0.012602 and for Swedish V (U)Swed = 0.022763. If we perform the above test on Italian and German, we obtain 0.9650 − 0.5641 = 2.55 z= √ 0.012189 + 0.012602 and this value is signiﬁcant, i.e. Italian has a signiﬁcantly smaller orthographic uncertainty than German. For the difference between Italian and Swedish we obtain 0.7970 − 0.5641 z= √ = 1.24 0.012189 + 0.022763 which is not signiﬁcant i.e. Swedish and Italian have roughly the same orthographic uncertainty. 18 3 Gerald Bernhard and Gabriel Altmann The distribution of graphemic representations If a language using the Latin alphabet has fewer phonemes than there are letters in Latin, it can represent each phoneme by one letter. In such a case all frequencies are concentrated in point x = 1. We speak then of a deterministic distribution. But if a language has more phonemes than the Latin letter inventory, it must reach for different means in order to build corresponding graphemes. One method is introducing marks placed over or under the letters, as in Slavic languages; another is using some letters to signalize a special quality, such as in German for prolonging the vowel; still another is combining or redoubling some letters, e.g. , , or even in several languages. These new forms can, however, be chosen in such a way that each phoneme can be represented by one unique grapheme. This ideal state is usually considerably disturbed by the interference with morphology or by disregarding the phonological development of language. In this way phonemes acquire multiple representations. From the statistical point of view, a distribution of representation sizes of phonemes arises and it can be captured formally. Since up to now only a small number of languages has been processed in this way, we can start from simple assumptions. At ﬁrst, we assume that the representation size decreases geometrically, i.e. it follows a distribution of the form Px = pqx−1 , x = 1, 2, 3, . . . (6) This is the 1-displaced geometric distribution. For Italian and Swedish this hypothesis would be adequate. However, in German we see (see Table 3) that the distribution does not decrease monotonically but has its mode at x = 2, i.e. more phonemes are represented by means of two graphemes than by one grapheme. Table 3: Distribution of representation size of phonemes in three languages x Italian German 1 2 3 4 5 6 34 14 8 1 2 − 10 18 7 3 0 1 Swedish 16 10 6 1 2 1 The phoneme-grapheme relationship in Italian 19 This circumstance can have different causes which must be analysed individually. In order to keep the original hypothesis, we modify (6) by means of the Gram-Charlier expansion (see Shenton & Skees 1970; Maˇcutek, this volume, pp. 75ff.) and obtain 1 x−1 1+a x− , x = 1, 2, 3, . . . (7) Px = pq p where q = 1 − p, 0 < p ≤ 1, 0 ≤ a ≤ 1/q − 1, q = 1 − p. This distribution is called either Gram-Charlier-geometric or Shenton-Skees-geometric distribution (cf. Wimmer & Altmann 1999). If in (7) p = 1, we obtain the deterministic distribution representing the ideal case, and if a = 0, we obtain the original geometric distribution. The ﬁtting of (7) to the data in these languages can be seen in Table 4. Evidently the ﬁt is in each case very satisfactory, but the hypothesis cannot be corroborated better until more languages have been examined. The ﬁt is shown graphically in Figures 1a–1c. Table 4: Fitting the distribution (7) to data in Table 3 x Italian German Swedish 1 2 3 4 5 6 33.31 14.92 6.37 2.64 1.76 − 9.99 18.00 7.54 2.47 0.73 0.27 15.79 9.99 5.35 2.64 1.24 1.00 p a FG χ2 P 4 0.6488 0.2398 2 1.55 0.46 0.7768 2.3323 1 0.12 0.73 0.6152 0.4588 2 1.36 0.51 Grapheme size A grapheme can consist of one or more Latin letters. Because of unequal size of Latin letter and phoneme inventories of target languages, new letters (e.g. the German <ß>) and several additional marks (tilde, accents, etc.) were 20 Gerald Bernhard and Gabriel Altmann 40 20 35 30 15 25 f(x) NP(x) 20 f(x) NP(x) 10 15 10 5 5 0 1 2 3 4 0 5 1 2 3 (a) Italian 4 5 6 (b) German 20 15 f(x) NP(x) 10 5 0 1 2 3 4 5 6 (c) Swedish Figure 1: Fitting (7) to Italian, German and Swedish data introduced. Thus grapheme inventory can be measured in two ways: (i) as the number of Latin letters without considering additional marks, (ii) as the number of Latin letters plus additional marks. Consequently the German grapheme <ä> can consist of one symbol according to method (i) and of two symbols according to method (ii). For Italian we obtain the results on the basis of Table 1 as shown in Tables 5a and 5b. Table 5a: Size of Italian graphemes: method (i) Size Grapheme 1 2 3 Number 30 36 5 In Table 5b six graphemes with accent passed from size 1 to size 2. The variable “size” has too small a support, which does not allow us to set up a testable model. For the time being it is enough to characterize the graphemics by its average and to compare it with other languages. Using method (i) we obtain the mean size of 1.65 lying between German (1.68) and Swedish (1.61) 21 The phoneme-grapheme relationship in Italian Table 5b: Size of Italian graphemes: method (ii) Size Grapheme 1 2 < ì, é, è, à, ò, ù, hi, he, eh, ha, ah, ho, oh, hu, uh, ao, au, ai, ei, eu, rr, ll, gl, mm, nn, gn, ff, vv, ss, sc, zz, ci, cc, gi, gg, pp, tt, ch, cq, bb, dd, gh> 3 Number 24 42 5 while method (ii) yields the mean size of 1.70 lying also between German (1.78) and Swedish (1.67). 5 The graphemic load of letters Latin letters are used with different frequencies in graphemes of target languages. The exploitation of letters for building graphemes can be designated as graphemic load. One can ask whether the letters present in graphemes have something to do with the phonemic relevance or whether they are merely historical relicts. In German, the letter occurs in 16 graphemes and its function is both segmental (there is a phoneme /h/), purely combinatorial (e.g. in the grapheme <sch>) or suprasegmental, e.g. to prolong the preceding vowel. In Italian occurs in 13 graphemes, but it plays only a secondary role: either it occurs in historically petriﬁed forms or it helps to maintain the phonetic value of the preceding consonant (). In Table 6 the letters are ordered according to their graphemic load. It is not yet possible to set up hypotheses about this distribution because the empirical background is still very restricted and the class occupation very small. For the time being we must content ourselves with the computation of the mean load which results from the numbers in Table 6 as 98/25 = 3.92. For German we get 3.96, for Swedish 3.36. Italian lies between them. 6 Letter usefulness The participation of letters in building graphemes can be weighted. We assume that the role of the letter is the more peripheral the later it appears in the grapheme. The historical and morphological roles of letters is neglected 22 Gerald Bernhard and Gabriel Altmann Table 6: Graphemic load of Italian letters (Participation in grapheme forming) Component in x graphemes 1 2 3 4 5 6 7 8 9 13 Number of letters Letter y, x, j, k r, m, f, v, z, p, t, q, b, d n l, s o u e, g a, c i h 4 10 1 2 1 1 2 2 1 1 in this case. The weighting has a purely positional character. The smaller the weight, the more useful the letter graphemically. Let us consider as an example the letter occurring in the following graphemes (see Table 5a/5b): . Let px gi be the product of the position (px ) of the letter <x> and the number of graphemes gi in which it occurs in this way. Then the positional participation of a letter can be deﬁned as PP<x> = ∑ px gi . (8) gi ∈G If it is weighted in each position by the position itself, then we ﬁnd position 1 eight times and position 2 twice, i.e. PP = 1(8) + 2(2) = 12 . If this operation is performed for each letter, one obtains the results for Italian in Table 7, PP<x> denoting the weight, and fx the number of letters. There is a possible correlation between the relative frequency of individual letters and their graphemic usefulness. For the time being we can merely compute the mean positional weight of letters in the form PW (Language) = 1 fx PP<x> L∑ x (9) where L is the size of the letter inventory. For Italian we obtain PW (Italian) = [1(4) + 3(1) + 4(9) + . . . + 22(1)]/25 = 6.48 . (10) The phoneme-grapheme relationship in Italian 23 Table 7: Positional participation of letters in graphemes PP<x> 1 3 4 6 7 8 9 12 16 18 22 Letter fx y, x, j, k q r, m, f, v, z, p, t, b, d n, s l, o e, a u g c i h 4 1 9 2 2 2 1 1 1 1 1 Comparing with Swedish (5.41) and German (6.12) we see that Italian has a strong letter usefulness (great positional weight). However, it will not be possible to examine historical and morphological dependencies before many languages have been analysed. The same holds for the comparison of individual letters in languages using Latin script and the relationship with the letter/grapheme frequency of occurrence. References Best, Karl-Heinz; Altmann, Gabriel 2005 “Some properties of graphemic systems.” In: Glottometrics, 9; 29–39. Maˇcutek, Ján 2006 “On the distribution of graphemic representations”. This volume, pp. 75– 78. Shenton, Leanne R.; Skees, P. 1970 “Some statistical aspects of amounts and duration of rainfall”. In: Patil, Ganapati P. (Ed.), Random Counts in Scientiﬁc Work. University Park: The Pennsylvania State University, 73–94. Wimmer, Gejza; Altmann, Gabriel 1999 Thesaurus of univariate discrete probability distributions. Essen: Stamm. Graphemic representation of English phonemes Fan Fengxiang and Gabriel Altmann 1 Introduction The grapheme-phoneme analysis of English is radically different from cases analyzed hitherto (German, Swedish, Italian, Slovak). This is caused by (i) the historical origins of English, (ii) its many national and regional varieties and (iii) the borrowing of many foreign words. The grapheme-phoneme mapping has been examined from different aspects (cf. Adams 1990; Berndt, Reggia, Mitchum 1987; Cunningham & Cunningham 1992; Fry 2004; Hanna et al. 1966; Patterson & Morton 1985; Seidenberg et al. 1984), and some probabilities have been computed. Such analysis is relevant not only for linguistics but also for cognition studies and pedagogy. We are interested here only in some measurable properties of the English phoneme-grapheme correspondence, in order to be able to study later on the divergence or convergence of this representation. Our analysis focuses on American English and the results cannot hold for other varieties, i.e., British English, though the methods used here can be applied directly. In order to work with controllable data, we adhere to the phonemic/graphemic analysis based on the American Carnegie Mellon Pronouncing Dictionary hereafter referred to as cmudict1 , which has 129 425 entries with phonological transcriptions. The phonological symbols used in the dictionary are listed below. On typographical grounds we adhere to this way of symbolizing phonemes. The dictionary uses 39 consonant and vowel phonemes. In his Essential Introductory Linguistics, Hudson (2000: 24ff.) uses 38 phonemes without the vowel er, which is used in the cmudict, as well as in the World Book Dictionary (Barnhart & Barnhart 1979). The words analyzed are from the 1 000 000-word Brown Corpus, which has 42 436 word types minus the Arabic numerals. Of these words, the cmudict covers 31 591; the uncovered part mostly consists of personal and place names, and non-word strings. This phonemic/graphemic analysis is the anal1. ftp://ftp.cs.cmu.edu/afs/cs.cmu.edu/data/anonftp/project/fgdata/ dict/ 26 Fan Fengxiang and Gabriel Altmann Table 1: List of phonemes Phoneme Example Transcription /AA/ /AE/ /AH/ /AO/ /AW/ /AY/ /B/ /CH/ /D/ /DH/ /EH/ /ER/ /EY/ /F/ /G/ /HH/ /IH/ /IY/ /JH/ /K/ /L/ /M/ /N/ /NG/ /OW/ /OY/ /P/ /R/ /S/ /SH/ /T/ /TH/ /UH/ /UW/ /V/ /W/ /Y/ /Z/ odd at hut ought cow hide be cheese dee thee Ed hurt ate fee green he it eat gee key lee me knee ping oat toy pee read sea she tea theta hood two vee we yield zee AA D AE T HH AH T AO T K AW HH AY D B IY CH IY Z D IY DH IY EH D HH ER T EY T F IY G R IY N HH IY IH T IY T JH IY K IY L IY M IY N IY P IH NG OW T T OY P IY R IY D S IY SH IY T IY TH EY T AH HH UH D T UW V IY W IY Y IY L D Z IY Graphemic representation of English phonemes 27 ysis of these 31 591 word types using the pronunciation given by the cmudict. The pronunciation of each word in the cmudict is in the following form (0, 1 and 2 represent word stresses): LABORATORY 2 L AE1 B R AH0 T AO2 R IY0. Data The 31 591 word types from the Brown Corpus were automatically separated into graphemes and then paired with their corresponding phonemes with the computer in the following form: i|n|au|g|u||r|a|t|io|n|, /ih n ao g y ah r ey sh ah n/ , i:ih|n:n|au:ao|g:g|u:y ah|r:r|a:ey|t:sh|io:ah|n:n| Computerized analysis is error prone, even with the best commercialized state of the art software. There is no exception in this analysis. Although the result was manually checked, there still may be errors. In addition, there are indeterminable cases. For example, the ﬁrst in the word LABORATORY is not pronounced in American English; should it be paired with the letter and the phoneme /b/ or the letter with the phoneme /r/? The was ﬁnally put together with to become → , or BO:B meaning the grapheme in this word is pronounced as /b/. Another possibility would be to consider as representing nothing, but in that case the analysis would be quite different. The third possibility would be to consider the grapheme as representing the cluster /br/, but in that case the analysis would produce an enormous number of phonemes, clusters and graphemic representations. We chose the ﬁrst alternative, which yielded a reasonable image of this kind of English. On the other hand, we could not avoid the fact that some single graphemes represent a group of phonemes, for example in the word COMPUTER represents the group of phonemes /y uw/, or <x> in BOX represents /k s/. In such cases the given phonemes are (implicit) parts of the graphemic representation and these cases are marked with ∈, e.g. /k/ → ∈ <x>. Another problem was the representation of a phoneme by zero grapheme. For example ABLER has the pronunciation of /ey b ah l er/, in which /ah/ is present phonemically but not graphemically. These cases are interpreted as /ah/ being part of if stays in front of (also ) and 28 Fan Fengxiang and Gabriel Altmann marked as /ah/ → ∈ . There are several cases of this sort as can be seen in Table 9 (see p. 43ff.). The following are the ﬁrst ten cases of the automatic graphemic separation and grapheme-phoneme mapping by the computer. The graphemes are separated with “|”, and “||” means there is an ungraphemically represented phoneme; the word pronunciation is enclosed between “/”; and “:” pairs the phoneme with its corresponding grapheme: a|, /ah/, a:ah| a|b||l|er|, /ey b ah l er/, a:ey|b:b ah|l:l|er: er| a|b||le|, /ey b ah l/, a:ey|b:b ah|le:l| a|b|a|ck|, /ah b ae k/, a:ah|b:b|a:ae|ck:k| a|b|a|n|d|o|n|, /ah b ae n d ah n/, a:ah|b:b|a:ae|n:n|d:d|o:ah|n:n| a|b|a|n|d|o|n|ed|, /ah b ae n d ah n d/, a:ah|b:b|a:ae|n:n|d:d|o:ah|n:n|ed:d| a|b|a|n|d|o|n|i|ng|, /ah b ae n d ah n ih ng/, a:ah|b:b|a:ae|n:n|d:d|o:ah|n:n|i:ih|ng:ng| a|b|a|n|d|o|n|m|e|n|t|, /ah b ae n d ah n m ah n t/, a:ah|b:b|a:ae|n:n|d:d|o:ah|n:n|m:m|e:ah|n:n|t:t| a|b|a|t|e|d|, /ah b ey t ih d/, a:ah|b:b|a:ey|t:t|e:ih|d:d| a|b|d|a|ll|ah|, /ae b d ae l ah/, a:ae|b:b|d:d|a:ae|ll:l|ah:ah| All representations of phonemes by graphemes are shown in Table 9 (see p. 43ff.). Here “/. . . /“ symbolizes a phoneme,“<>” a grapheme, while “∈” means that the given phoneme is part of the grapheme cluster. The condition under which the given phoneme – usually a vowel – is placed (uttered) behind a grapheme is symbolized by a superscript “<>”, e.g. /ah/ * means that in some occasion /ah/ can be pronounced within , e.g. → /ey b ah l/. Though /ah/ is not overtly represented, is considered its representation. There is the possibility of simply ignoring the zero grapheme representation, but we decided for the above alternative. There are 289 different graphemic representations; many of them are used for different phonemes. The graph connecting the phonemes with graphemes is a bipartite graph which, because of its extent, cannot be presented here. 3 Uncertainty The ﬁrst impression of Table 9 (p. 43) is that each phoneme has multiple representations. One would tend to say that there is a very weak connection between the phonemes and graphemes, and that phonemes are represented Graphemic representation of English phonemes 29 by combinations of Latin letters which contain mere phonetic orientations but nothing more. The situation can unreservedly be matched with Accadian writing, in which cuneiform symbols of different directions and sizes are combined, or, still better, with Chinese script containing always a phonetic guide. Hence, English writing resembles and is developing into a kind of linear hieroglyphic or logographic script. The extent of this development can be numerically expressed in different ways. Here we shall show only some very elementary methods. 3.1 Unweighted uncertainty The variation or diversiﬁcation of the way of representing graphically a phoneme can be called in general uncertainty. In the case that the individual representations are not weighted, uncertainty in information theory is deﬁned as the dyadic logarithm of the number of representations, i.e. H0 = log2 K (1) where K is the number of representations. Consider e.g. the phoneme /AA/ having 19 different representations. Its uncertainty can be characterized as H0 (/AA/) = log2 19 = 4.25. There is no maximum of H0 because K is potentially inﬁnite but its minimum is 0. The greater H0 , the more diversiﬁed the phoneme representation. The results for all phonemes are presented in the second and third column of Table 2. It can be seen easily that vowels have more diversiﬁed representations than consonants, though some of them are weakly diversiﬁed (/AE/, /AW/, /OY/). Under other conditions of sampling and interpretation, one would get another picture, but any version is merely an approximation because of the diversity of English. There is no trend, e.g. for normality of distribution of K or H0 because the graphemic representations did not arise by chance but by historical development and adaptation of foreign words. Seeing the phoneme-grapheme relations as a bipartite graph, the number of graphemic representations is, as a matter of fact, the degree of a vertex (phoneme). Thus uncertainty, diversiﬁcation and vertex degree are in this case synonymous. 30 3.2 Fan Fengxiang and Gabriel Altmann Weighted uncertainty Even if a phoneme has a great number of representations (K), not all of them are of the same importance. Their relevance is weighted by their frequency of occurrence. This can be of two sorts: one based on the dictionary and the other based on texts. If one of the representations occurs 1 000 times, it is surely more relevant than one occurring only once. Hence, another measure of uncertainty is the entropy of ﬁrst order taking into account the relative frequencies of individual representations. Usually one uses the Shannon entropy deﬁned as K H1 = − ∑ pi log2 pi (2) i=1 where K is the number of representations and we estimate pi by fi /N, where fi is the absolute frequency, N being the number of occurrences of representations of the given phoneme (N = Σ fi ). The more concentrated the frequencies, the smaller the uncertainty H1 . In order to illustrate the computation, we use the representations of /TH/, where there are N = 741 cases distributed to graphemic variants in proportions: 1, 736, 3, 1. Since formula (2) can be rewritten as H1 = log2 N − 1 K ∑ fi log2 fi N i=1 (2a) we obtain H1 = log2 741 − (1/741)[1 log2 1 + 736 log2 736 + 3 log2 3 + 1 log2 1] = 0.0676 . Though there are four graphemic variants, the uncertainty is very low, because one of them, , occurs in the great majority of cases. Consequently, even if K or H0 are great, H1 yields a more adequate picture of uncertainty/diversity. The results are presented in the fourth column of Table 2. H0 and H1 are characteristics of uncertainty/diversity. The ﬁrst shows the raw diversity, the second the exploitation of this diversity. Theoretically they could be independent but it can be shown that there is a tendential dependence of H1 on H0 . The t and the F-tests show that the curves H1 = aH b0 and H1 = aebx are adequate though the determination coefﬁcient is not high enough. Graphemic representation of English phonemes Table 2: Uncertainties of individual phonemes Phoneme /AA/ /AE/ /AH/ /AO/ /AW/ /AY/ /EH/ /ER/ /EY/ /IH/ /IY/ /OW/ /OY/ /UH/ /UW/ /B/ /P/ /M/ /F/ /V/ /W/ /D/ /T/ /N/ /TH/ /DH/ /S/ /Z/ /R/ /L/ /JH/ /CH/ /SH/ /ZH/ /Y/ /K/ /G/ /NG/ /HH/ K H0 H1 R 19 7 60 19 7 16 18 29 20 22 21 19 6 13 31 5 8 9 12 5 8 5 16 12 4 2 18 16 9 7 7 13 10 8 17 21 8 6 3 4.2479 2.8074 5.9069 4.2479 2.8074 3.9999 4.1699 4.8580 4.3219 4.4594 4.3923 4.2479 2.5850 3.7004 4.9542 2.3219 3.0000 3.1699 3.5850 2.3219 3.0000 2.3219 4.0000 3.5850 2.0000 1.0000 4.1699 4.0000 3.1699 2.8074 2.8074 3.7004 3.3219 3.0000 4.0874 4.3923 3.0000 2.5850 1.5850 1.2742 0.0560 3.1340 1.8927 1.2608 1.0467 0.9189 2.0685 1.3999 1.0632 2.4710 1.0851 1.1616 2.0327 2.7422 0.3049 0.4945 0.5761 1.0432 0.7787 1.1800 0.7994 0.9753 0.4592 0.0676 0.3868 1.5637 1.2667 0.5337 0.9450 1.5289 1.5872 1.8879 1.2482 1.5770 1.9685 0.9152 0.5884 0.0966 0.4956 0.9898 0.1544 0.4166 0.5057 0.6783 0.7185 0.4144 0.5644 0.6196 0.2265 0.6735 0.4963 0.3115 0.2278 0.9158 0.8418 0.8319 0.6553 0.6592 0.5485 0.7292 0.7255 0.8764 0.9866 0.8601 0.5308 0.6300 0.8521 0.6609 0.4649 0.4130 0.3572 0.6181 0.5105 0.4185 0.7339 0.7692 0.9777 N 3989 4884 19983 2411 751 2617 5884 6384 3922 13664 7095 2812 306 535 2091 3790 5999 6379 3370 2711 1738 9082 13740 14163 741 185 12662 6193 10798 11279 3946 1116 2539 184 1060 9163 2380 3370 1510 31 32 3.3 Fan Fengxiang and Gabriel Altmann Concentration Another way of characterizing the diversity is the Herﬁndahl measure of concentration called repeat rate in linguistics. It is deﬁned as the sum of squares of the probability of graphemic representations, K R = ∑ p2i . (3) i=1 The probability is estimated by relative frequency, i.e. pi = fi /N, which gives R= 1 K 2 ∑ fi . N 2 i=1 (3a) Here Kand N are different for each phoneme. This index shows the concentration of graphemic representatives. If all frequencies are concentrated in one grapheme, then R = 1. If all frequencies are equal, i.e. the diversity is maximal, it attains the value R = 1/K. From the geometrical point of view, (3) represents the Euclidean distance in a K-dimensional space, i.e. R is the coordinate of the phoneme. For example with /AE/ there are 7 graphemes but the frequencies are concentrated on , thus R = 0.9898. The smallest concentration (the greatest dispersion) is with /IY/ having R = 0.2265. It is possible to norm R in order to restrict it to the interval <0, 1>, but we leave in its original form. All results are presented in the ﬁfth column of Table 2. 4 Grapheme length distribution In a situation similar to English (i.e. where the number of phonemes is greater than the number of Latin characters), we expect graphemes of different length or modiﬁed graphemes like in Slavic languages. Since there are 26 Latin letters and the number of English phonemes is greater, there must be at least some graphemes consisting of two letters. However, borrowings from other languages automatically amplify the number of graphemes, and since English has other phonemes than the languages of origin, the exploitation of existing graphemes diversiﬁes, i.e. some graphemes are polyphonic (used for different phonemes) and others synophonic (different graphemes used for the same phoneme). In Table 9 we see that the phoneme /AA/ can be represented by 19 Graphemic representation of English phonemes 33 synophonemic graphemes, and the polyphonemic grapheme can represent four phonemes. However, there is no absolute arbitrariness in assigning a grapheme to a phoneme, or lengthening a grapheme by adding more letters to it. If there were no restrictions to length, it would develop randomly according to a Poisson process. Now, the Poisson distribution Px = e−a ax /x! (x = 0, 1, . . .) can be represented by the recurrence formula a Px = Px−1 (4) x and its shape is determined by the parameter a. For a < 1 it is monotonically decreasing, for a = 1 it has two modes, and for a > 1 it is bell-shaped. The greater a, the longer the tail of the distribution. Evidently, a must be greater than 1 because the extent of phonetic changes, borrowing and graphemic conservatism in English produces a great number of graphemes. The ﬁrst step in coping with this proliferation would lead to the exploitation of two-letter graphemes, but not each grapheme can be used to represent each phoneme. Some two-letter graphemes are not allowed. Hence some three-letter graphemes must be applied, etc. However, phonetic reasons are not the only causes restricting the proliferation of grapheme length. It is above all the requirement of economy (or optimality) which cares for balance in all domains of language (cf. Zipf 1935, 1949; Köhler 1986). Thus in formula (4) a more rapid convergence must be built in. Tentatively we replace the proportionality function a/x by the Zipﬁan function a/xb and obtain a Px−1 . xb (5) ax P0 , x = 0, 1, 2, . . . , (x!)b (6) Px = Solving (5) we obtain Px = representing the Conway-Maxwell-Poisson distribution (cf. Wimmer & Altmann 1999) already used in linguistics, being a special case of the Wimmer-Altmann (2005) approach. Since there is no “zero-length” grapheme, we either solve (5) for x = 2, 3, . . . or we displace (6) one step to the right, in order to obtain ax−1 , x = 1, 2, 3, . . . , (7) Px = [(x − 1)!]b T ∞ where T = ∑ j=0 aj , ( j!)b i.e. T is identical with (P0 )−1 . 34 Fan Fengxiang and Gabriel Altmann Table 3: Grapheme length distribution and ﬁtting the Conway-Maxwell-Poisson distribution (7) x fx 1 2 3 4 5 26 159 94 8 2 NPx 25.18 162.86 88.98 11.45 0.54 a = 6.4689, b = 3.5656 χ2 = 0.73, DF = 1, P = 0.39 Applying (7) to our data we obtain the observed and computed values as given in Table 3. The normalizing constant is T = 11.4796. The result is presented graphically in Figure 1. Parameter a can be interpreted as the element of randomness (speaker creativity), conservatism of orthography, borrowing etc., while b means the braking mechanism, the balancing force of economy. 200 150 f(x) NP(x) 100 50 0 1 2 3 4 5 Figure 1: Grapheme length distribution: Conway-Maxwell-Poisson distribution (7) 5 Polyphonemics and synophonemics of graphemes In Table 9 we see that a grapheme can represent several phonemes. Let us call this property graphemic polyphonemics. The great majority of graphemes is, Graphemic representation of English phonemes 35 of course, monophonemic: it can be ascribed only to one phoneme. We distinguish direct ascription of a grapheme to a phoneme from the fact that a phoneme is part of a grapheme. E.g. /k/ is directly represented by <x> (as in “excel”) which differs from representing /k/ as part of ∈<x> (as in “afﬁx”). Thus <x> as a framing grapheme can be ascribed to 6 phonemes and as a direct grapheme to 2 phonemes, i.e. it can be representative in 8 cases. The results of counting can be found in Table 4 (x = number of phonemes represented by a grapheme; fx = number of graphemes with x representations; NPx = computed number of graphemes with x representations). The numbers in the table are to be read as follows: There are 191 graphemes, each of which represents exactly 1 phoneme; there are 43 graphemes, each of which represents exactly 2 phonemes, etc. In order to set up a model, we simply start from the Zipﬁan assumption of setting (relative) frequency proportional to the frequency class using directly the function from (5), namely K , x = 1, 2, 3, . . . , (8) xb where K is the proportionality constant having the function of the normalizing constant, since we use (8) as a probability distribution. The parameter b is, again, a control parameter braking over-strong polyphonemy. Formula (8) is usually called Zipf’s law or zeta distribution. Now, theoretically (8) has an inﬁnite support, which is a nonrealistic situation. For our purposes it will be truncated after x = 10 because no grapheme represents more than 10 phonemes (up to now or in this variant of English). Hence we obtain Px = K , x = 1, 2, . . . , R, (9) xb R being the truncation parameter (here 10). Applying (9) to the data in Table 4 we obtain the result in its third column. The graphic display is in Figure 2. The ﬁt is excellent but it can be made still simpler. In the last row of Table 4 we see that the value of parameter a is approximately 2. Replacing a = 2 in (8) we obtain the so-called Lotka distribution Px = Px = 6 x2 π2 , x = 1, 2, 3, . . . , (10) called also the ergodic distribution of population size (cf. Wimmer & Altmann 1999: 394). However, truncating it at the right side we obtain another 36 Fan Fengxiang and Gabriel Altmann 200 150 f(x) NP(x): (9) NP(x): (11) 100 50 0 1 2 3 4 5 6 7 8 9 10 Figure 2: Polyphonemics of English graphemes: right truncated zeta (9) normalizing constant Px = 1 x2 [π2 /6 − Ψ(R + 1)] , x = 1, 2, 3, . . . , R, (11) where Ψ (.) is the trigamma function (the normalizing constant is simply the sum of 1/x2 in the given deﬁnition domain). Using the right truncated Lotka distribution (11) we obtain the results in the last column of Table 4. The result of ﬁtting is slightly better because we have one degree of freedom more. But more important is the ﬁxed parameter value. Table 4: Graphemic polysemics in English x fx NPx (9) NPx (11) 1 2 3 4 5 6 7 8 9 10 191 43 21 9 9 6 5 1 2 2 187.84 46.37 20.45 11.44 7.29 5.05 3.70 2.82 2.23 1.80 186.48 46.62 20.72 11.65 7.46 5.18 3.81 2.91 2.30 1.86 a = 6.4689, R = 10 X 2 = 3.09, DF = 7, P = 0.88 R = 10 X 2 = 3.12, DF = 8, P = 0.93 Graphemic representation of English phonemes 37 The graphemic synophonemics considers simply the numbers of graphemes representing an individual phoneme. Using Table 9 we get the following numbers in decreasing order: SS*60, 31, 29, 22, 21, 21, 20, 19, 19, 19, 18, 17, 17, 16, 16, 16, 13, 13, 12, 12, 10, 9, 9, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 5, 5, 5, 4, 3, 2.* As can easily be seen, the frequencies of individual representations are rather uniformly distributed; they do not display the same pattern as graphemically “simpler” languages. The only possibility of searching for order is to consider their ranks as the independent variable. In that case, we obtain the data given in the ﬁrst two columns of Table 5. Table 5: Rank-frequency distribution of English graphemic synophones Rank x fx NPx Rank x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 60 31 29 22 21 21 20 19 19 19 18 17 17 16 16 16 13 13 12 12 54.46 36.11 29.80 26.21 23.75 21.89 20.42 19.19 18.14 17.22 16.40 15.66 14.98 14.35 13.77 13.22 12.70 12.21 11.73 11.28 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 fx 10 9 9 8 8 8 8 8 7 7 7 6 6 5 5 5 4 3 2 NPx 10.84 10.41 9.99 9.58 9.18 8.77 8.38 7.98 7.57 7.17 6.75 6.33 5.89 5.42 4.93 4.39 3.79 3.07 2.10 K = 2.1178, M = 0.6708, n = 38 DF = 35, X 2 = 5.43, P ≈ 1.00 For ranking of linguistic units one usually uses the negative hypergeometric distribution or a distribution from the Lerch family containing also the Zipf, Zipf-Mandelbrot, zeta and other distributions (cf. Zörnig & Alt- 38 Fan Fengxiang and Gabriel Altmann 70 60 50 40 f(x) NP(x) 30 20 10 0 5 10 15 20 25 30 35 Figure 3: Synophonemics of English graphemes, negative hypergeometric mann 1995; Köhler & Martináková-Rendeková 1998, Grzybek & Kelih 2003; Grzybek, Kelih, & Altmann 2004; Best 2005a,b,c). Here we adhere to the negative hypergeometric because its ﬁtting turned out to be the best as can be seen in Table 5 and Figure 3, though zeta and Zipf-Mandelbrot both yield very satisfactory results. We use it in 1-displaced form Px = 6 M+x K −M+n−x x−1 n−x+1 , x = 1, 2, . . . , n + 1. K +n−1 n (12) Letter participation The 26 letters of the Latin alphabet used in English are not exploited equally to build graphemes. Letters which in Latin had a vocalic value are used more often than those having consonant value. Again, for individual letters we get exact numbers but the only possibility to capture formally the set of nominal categories (letters) is to rank them according to their participation in graphemes. Since the set of letters is not too large, the best model is again the 1displaced negative hypergeometric distribution (12). For orientation and further research we present the letters not in alphabetic but in ranked order. The result of computing can be seen in Table 6 and Figure 4. The ﬁt is excellent and corroborates once more the adequacy of this model. Graphemic representation of English phonemes 39 Table 6: Letter participation in English graphemes: Fitting the 1-displaced negative hypergeometric distribution (12) Letter Rank x e u o h t a r s i l c g n NPx (12) fx 1 2 3 4 5 6 7 8 9 10 11 12 13 94 54 52 46 40 36 35 34 32 32 25 25 19 Letter 91.64 62.92 51.97 45.30 40.47 36.66 33.48 30.74 28.31 26.11 24.1 22.22 20.46 Rank x p w d m y b f z k x q j v 14 15 16 17 18 19 20 21 22 23 24 25 26 fx NPx (12) 17 14 13 12 11 11 10 9 8 6 5 3 3 18.80 17.21 15.68 14.21 12.78 11.39 10.02 8.66 7.32 5.98 4.61 3.21 1.73 K = 2.5453, M = 0.7095, n = 25, DF = 22, X 2 = 6.95, P = 0.9990 100 90 80 70 60 f(x) NP(x) 50 40 30 20 10 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Figure 4: Letter participation in different graphemes: negative hypergeometric (12) 7 Weighted participation A sightly different aspect is the evaluation of weighted participation of letters. In Section 6 we examined the participation of letters in building graphemes but we did not take into account the polyphonemy of graphemes, which is enormous in English. For example the letter occurs in 36 different graphemes but each of these graphemes can be used to represent different pho- 40 Fan Fengxiang and Gabriel Altmann Table 7: Ranked weighted participation of letter in graphemes Letter Rank x fx NPx (12) e u o h a i t s r l c g y 1 2 3 4 5 6 7 8 9 10 11 12 13 169 110 100 86 85 67 48 46 44 36 35 33 23 169.89 112.03 90.44 77.36 67.95 60.56 54.45 49.21 44.60 40.48 36.75 33.32 30.15 Letter Rank x fx NPx (12) w p n d z m x b f k j q v 14 15 16 17 18 19 20 21 22 23 24 25 26 21 20 20 17 14 13 13 11 11 8 6 5 4 27.20 24.44 21.85 19.41 17.09 14.90 12.82 10.85 8.97 7.20 5.52 3.94 3.62 K = 2.8339, M = 0.6885, n = 26, DF = 22, X 2 = 14.64, P = 0.88 nemes, i.e. it can be polyphonemic. In this section we consider all occurrences of individual letters in graphemes, i.e. we compute the weighted participation of letters. The results are presented in Table 7. Again, the number of cases is too small (26) and, since almost all letters have different participation, no model could be set up. Hence we use, as above, the ranked weighted participation and as expected we obtain again the 1-displaced negative hypergeometric distribution. The result of ﬁtting is graphically displayed in Figure 5. 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 f(x) NP(x) 2 4 6 8 10 12 14 16 18 20 22 24 Figure 5: Ranked weighted participation of letters in graphemes (NHG) 26 Graphemic representation of English phonemes 8 41 Letter utility The last aspect we shall analyze here is the so-called letter utility. In the previous sections we considered the presence of a letter in different graphemes and its presence in all graphemes; here we consider its position in the grapheme. We take into account only different graphemes and ignore their polyphonemics. The position of a letter in a grapheme is a measure of its relevance. The earlier the letter appears in the cluster the more it contributes to its phonetic value. At least this can be assumed to hold in general (although it does not hold in each case). This is especially well expressed in French where the morphology represented by inﬂections is dying out and the ﬁrst letter of a long grapheme is decisive for the phonetic form, e.g. <parlent> → /parl/. Let us illustrate the problem using the graphemes containing , namely < cq, cqu, q, qu, que > . Let nq be the set of graphemes containing and |n | the cardinal number of this set. Let wx be the weight of the letter <x> given by its position in the grapheme. We deﬁne ﬁrst PP<x> = ∑ (13) wx x∈n<X> as the sum of all weights (positions) of <x> in the graphemes of the set nx . For the letter we obtain from the above example nq = 5 and PP = 2 + 2 + 1 + 1 + 1 = 7. For comparative purposes we deﬁne PP<x> = 1 ∑ |n<x> | x∈n <x> wx . (14) In our example we obtain PP = 7/5 = 1.4. The results for all letters are given in Table 8, #G denoting the number of graphemes and MLU mean letter utility. Ordering the letters according to their utility in graphemes we obtain the order: . It would be possible to order the letters according to their absolute weight, too. The mean utility of all letters can be expressed as the ratio of the sum of the second column of Table 8 to the sum of the third column, i.e. 1231/687 = 42 Fan Fengxiang and Gabriel Altmann Table 8: Letter utility in English graphemes Letter a b c d e f g h i j k l m Weight #G MLU Letter 59 17 40 19 201 17 44 100 57 5 10 87 21 38 13 27 15 97 13 27 45 36 4 8 37 14 1.5526 1.3077 1.4818 1.2667 2.0722 1.3077 1.6296 2.2222 1.5833 1.2500 1.2500 2.3514 1.5000 n o p q r s t u v w x y z Weight #G MLU 31 81 26 7 103 75 69 91 4 24 10 19 14 21 54 19 5 45 39 41 47 3 14 5 11 9 1.4762 1.5000 1.3684 1.4000 2.2889 1.9231 1.6829 1.9362 1.3333 1.7143 2.0000 1.7272 1.5555 1.7918, or as the ratio of the sum of the second column to the number of letters, i.e. 1231/26 = 47.35. Comparing this last number with Italian where the ratio is 6.48, (cf. Bernhard & Altmann, this volume, pp. 13ff.), one sees that the diversiﬁcation of graphemics in English is enormous. 9 Conclusions English graphemics is a very complex matter. Some of the measures (indices) introduced here make it evident. They differ drastically from those in other languages. The loss of a unique phonetic value of a letter reduces letters to merely graphical signs obtaining a phonetic value only in a grapheme. The way to hieroglyphism is open. All indices introduced here can be analyzed further statistically. They have their sampling distributions, asymptotic tests can be set up, languages can be classiﬁed according to their graphemics, and there is a possibility to ﬁnd interrelations among all these properties and also between graphemic and nongraphemic properties. The last aim is, of course, to ﬁnd laws of graphemics and join them in a system of laws, i.e. in a theory. At present, any such enterprise would be premature. Graphemic representation of English phonemes Table 9: English phoneme-grapheme correspondences Phoneme Graphemes Frequency Examples /AA/ <e> <ea> <eau> <eah> 1412 3 14 18 1 1 31 6 16 20 2 1 15 1 2427 7 2 4 8 4859 2 1 11 9 1 1 5028 1 7 20 36 10 (a)bo baz(aa)r y(ah) (al)mond arkans(as) baccar(at) astron(au)t (aw)ful s(er)geant wholeh(ea)rtedly bur(eau)cracy exh(au)stively (ho)nors l(i)ngerie abd(o)minal j(oh)n s(ol)der c(ou)gh ackn(ow)ledgement ab(a)ck, zigz(a)gging g(ah)n, p(ah) pl(ai)d beh(al)f, s(al)mon (au)nt, l(au)ghter y(eah) chop(i)n (a)bide, mad(a)m is(aa)c an(ae)sthesia, minuti(ae) abdall(ah), tor(ah) barg(ai)n, vill(ai)ns (au)gusta, nickl(au)s /AE/ /AH/ (continued on next page) 43 44 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme Graphemes Frequency Examples <e> <ea> <eau> <ei> <eo> <eou> <er> ∈ ∈ ∈ ∈ ∈ ∈ ∈ 3619 24 2 7 13 7 2 9 3 8 2379 185 36 1349 66 1 2680 1 5 2 1 17 317 1 2864 1 3 5 53 414 32 5 21 20 43 31 abandonm(e)nt, zab(e)l chang(ea)bl, veng(ea)nce bur(eau)crat, bur(eau)crats for(ei)gn, surf(ei)t bludg(eo)n, surg(eo)ns advantag(eou)s, right(eou)sness paraph(er)nalia, res(er)voir budd(ha), wind(ha)m ve(he)mence, ve(he)mently anni(hi)lation, pro(hi)bition abdom(i)nal, zoolog(i)st acac(ia), venet(ia)n anc(ie)nt, trans(ie)nt abduct(io)n, volit(io)n ambit(iou)s, vivac(iou)s belg(iu)m aband(o)n, zool(o)gy mendelss(oh)n conn(oi)sseur, tort(oi)se linc(ol)n, norf(ol)k m(on)sieur bl(oo)d, ﬂ(oo)d adulter(ou)s, zeal(ou)s mccull(ough) abd(u)ction, y(u)m etiq(ue)tte br(uh)n, (uh) bisc(ui)t, circ(uit)s anal(y)ses, vin(y)l able, ambling babbled, bubbling subtler, subtly article buckled beadles, idling, haydn addle, huddling (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /AO/ Graphemes Frequency Examples ∈ ∈ ∈ ∈ ∈ <m> ∈ ∈ ∈ ∈ <s><m> ∈ <sc> ∈ <ss> ∈ <st> ∈ ∈ ∈ <m> ∈ ∈ <x> ∈ <ea> <eo> 7 13 24 12 10 9 42 18 146 3 1 25 19 44 5 251 2 24 344 1 24 1 257 1 132 3 1 5 7 2 2 1487 54 16 1 72 riﬂe bafﬂe bedraggled ankle mcalister one ample apple activism muscle tussle apostle beetles battle logarithm, algorithm accum(u)lated axle dazzled (a)lbany, y(a)lta ut(ah) b(al)ked, w(al)kways extr(ao)rdinary appl(au)d, v(au)lts v(augha)n (aw), y(aw)ning (awe)some, dr(awe)rs s(ea)n g(eo)rgia, g(eo)rgetown ex(hau)st, inex(hau)stible ex(ho)rtations, ex(ho)rting dig(io)rgio, g(io)rgio abh(o)rrent, zl(o)tys ab(oa)rd, washb(oa)rd d(oo)r, outd(oo)rs f(ore)runner betanc(ou)rt, y(ou)r (continued on next page) 45 46 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme /AW/ /AY/ /EH/ Graphemes Frequency Examples <ei> <ey> <eye> <e> <ea> <ee> <eh> <ei> <eo> <er> <ette> 1 6 32 3 486 3 219 2 1 21 6 2 80 5 10 2293 2 14 22 2 6 6 342 5 494 12 120 4 4955 257 1 2 4 5 2 1 unt(owa)rd l(ao), t(ao)ists aden(au)er, t(au)ssig (hou)r, (hou)rs ab(ou)nd, whereab(ou)ts b(ough), h(ough)s all(ow), y(ow) d(owe)r, h(owe) m(ae)stro al(ai), th(ai)land b(ay)ou, sant(ay)ana (aye), (aye)s alam(ei)n, z(ei)tler ch(ey)enne, m(ey)ers bug(eye)d, (eye)witness ab(i)des, wr(i)ting d(ia)mond, d(ia)monds l(ie), unt(ie) h(igh), th(igh) c(oy)ote, c(oy)otes beg(ui)led, disg(ui)se b(uy), sch(uy)ler acol(y)te, wr(y)ly b(ye), r(ye) actu(a)rial, y(a)rrow (ae)rial, kr(ae)mer ad(ai)r, volt(ai)re pr(ay)er, s(ay)s ab(e)d, z(e)st abr(ea)st, z(ea)lous k(ee)lson g(eh)rig, k(eh)le l(ei)sure, th(ei)rs j(eo)pardizing, l(eo)pards int(er)rogation, int(er)rogator pirou(ette) (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /ER/ /EY/ Graphemes Frequency Examples <ey> <ar> <arr> <ear> <er> <ere> <err> <eur> ∈ 1 3 9 1 11 2 1 1 389 27 4 63 3972 1 30 14 4 2 8 211 3 4 3 2 803 30 50 58 5 485 144 59 6 1 4 2886 r(ey)nolds (hei)r, (hei)rs ch(ie)n, unfr(ie)ndly (oe)dipal b(u)ry, woodb(u)ry marq(ue)tte, velasq(ue)z an(aer)obic lecl(air) afterw(ar)d, wiz(ar)d (arr)anges, re(arr)anged (aur)ora, rest(aur)ateur ath(ear)n, y(ear)nings abl(er), zurch(er) w(ere) ab(err)ation, unint(err)upted amat(eur), restaurat(eur) (her)b, shep(her)ds plag(iar)ism, tert(iar)y croz(ier), sold(ier)s ad(ir)ondack, wh(ir)lwind chesh(ire), staffordsh(ire) st(irr)ed, wh(irr)ing cupb(oar)d, starb(oar)d c(olo)nel, c(olo)nels ab(or)iginal, wh(or)ls c(orr)al, w(orr)ying adj(our)ned, y(our)self ac(re), wi(re)s i(ro)n, i(ro)nside abs(ur)d, z(ur)cher advent(ure), vult(ure) bl(urr)ed, unh(urr)ied b(yr)d, mart(yr)s m(yrrh) ﬁgure (a)bler, z(a)bel (continued on next page) 47 48 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme /IH/ Graphemes Frequency Examples <e> <ea> <ee> <eh> <ei> <eigh> <er> <es> <et> <ete> <ey> ∈ <e> <ea> <ee> <ei> <eou> <ey> <ez> 8 496 2 311 1 66 33 3 2 48 11 1 1 11 1 37 1 2 1 97 1 1 2669 101 55 7 1 1 1 2 6 10415 7 25 4 1 br(ae), vertebr(ae) abig(ai)l, whitet(ai)l g(au)ge, g(au)ged alw(ay)s, yesterd(ay)s m(aye) alfr(e)do, y(e)hudi beefst(ea)k, y(ea)ts b(ee)thoven, soir(ee) l(eh)mann, n(eh)ru alex(ei), w(ei)ghty n(eigh)bor, w(eigh)s bouvi(er) d(es)cartes ball(et), val(et) ricoch(ete)d ab(ey)ance, th(ey) linger(ie) communiq(ue)s, enriq(ue) b(ue)no acre(a)ge, yard(a)ge (ae)gean barg(ai)ning abat(e)d, z(e)ros alv(ea)r, y(ea)rs auction(ee)r, volunt(ee)rs counterf(ei)t, w(ei)rdly cretac(eou)s c(ey)lon rend(ez)vous hemorr(ha)ging, hemorr(ha)ge budd(hi)sm, ex(hi)bits abandon(i)ng, zur(i)ch carr(ia)ges, marr(ia)ge b(ie)rce, s(ie)ve feroc(iou)sly, malic(iou)sly w(o)men (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /IY/ /OW/ Graphemes Frequency Examples <wi> <e> <ea> <ee> <eh> <ei> <eigh> <eo> <ey> <eau> <eaux> <eot> <eou> <ew> 12 1 18 1 238 7 13 1222 611 525 1 53 1 3 193 1 1 1622 281 1 3 1 2 2 1 2551 1 7 4 1 1 1 4 2287 177 30 bacch(u)s, ponti(u)s racq(ue)t b(ui)ld, shipb(ui)lding green(wi)ch ab(y)smal, (y)vette alg(ae), p(ae)an ass(ay), wednesd(ay) abil(e)ne, zo(e) agl(ea)m, z(ea)lously absent(ee), yank(ee)s l(eh)man b(ei)n, w(ei)r l(eigh) p(eo)pled, p(eo)ple abb(ey), yanc(ey) diarr(he)a del(hi) abilit(i)es, compan(i)es p(ie)ce, zomb(ie) cast(ill)o chabl(is), debr(is) pet(it) ph(oe)nix, subp(oe)na mosq(ui)to, mosq(ui)toes marq(uis) abernath(y), zoolog(y) dav(ao) ch(au)ffeur, s(au)ternes b(eau)jolai, tabl(eau) b(eaux) peug(eot) s(eou)l s(ew), s(ew)n abd(o)men, z(o)ology aﬂ(oa)t, wh(oa) b(oe)ing, w(oe)fully (continued on next page) 49 50 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme /OY/ /UH/ /UW/ Graphemes Frequency Examples <eu> <eu> ∈ <eu> ∈ ∈ <eu> <ew> 2 7 2 1 3 16 15 249 4 4 1 181 1 117 2 6 1 9 224 1 24 3 178 1 3 3 1 81 30 130 2 4 79 7 356 2 b(oh)len, c(oh)en c(ol)mer, y(ol)k r(oo)sevelt, r(oo)sevelts aprop(os) dep(ot), p(ot)pourri b(ou)lder, sh(ou)lders alth(ough), th(ough) arr(ow), yell(ow)ish marl(owe), st(owe) bayr(eu)th, r(eu)ther hemorr(hoi)ds adr(oi)t, v(oi)ds iroqu(ois) all(oy), v(oy)age b(uoy)ancy, b(uoy)ant n(eu)ral, n(eu)rotic post(hu)mous b(o)som, w(o)manhood adulth(oo)d, yearb(oo)k w(or)cester bonj(ou)r, y(ou)rselves c(oul)d, w(oul)d acap(u)lco, z(u)rich nieb(uh)r ﬂ(uo)rescent, ﬂ(uo)rine (eu)rasian milieu brav(u)ra br(eu)er, z(eu)s andr(ew), withdr(ew) sil(hou)etted, sil(hou)ette ad(ieu), l(ieu)tenants ad(o), wrongd(o)ing can(oe), sh(oe)string aftern(oo)n, z(oo)ms p(ooh), p(ooh)ed (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /B/ /P/ Graphemes Frequency Examples <wo> ∈ <eau> ∈ <eu> ∈ <ewe> ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ <pa> 73 1 5 2 1 1 818 58 1 46 2 3 4 18 1 7 1 405 26 2 1 3 2 3625 109 52 1 3 1 1 5489 1 114 2 390 1 ac(ou)stic, y(ou)ths den(oue)ment breakthr(ough), thr(ough)put c(oup), c(oup)s rendezv(ous) led(oux) absol(u)tes, y(u)goslavia accr(ue)d, virt(ue)s k(uh)n br(ui)ses, uns(ui)ted b(uo)yed, b(uo)ys t(wo), t(wo)some beautiful eucalyptus ewe interview houston acc(u)mulated arg(ue) h(ugh) (hu)hes deb(ut) vac(uu)m a(b)ack, zom(b)ie a(bb)as, we(bb)er ascri(be), wardro(be) la(bo)ratories cam(pb)ell, cu(pb)oards ha(b)sburg su(bp)oena abru(p)t, zi(p) princi(pa)lly antelo(pe), wi(pe) u(ph)olstered, u(ph)olstery agri(pp)a, zi(pp)er bankru(pt)cy (continued on next page) 51 52 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme Graphemes Frequency Examples /M/ <m> <mb> <me> <mm> <mme> <mn> <mp> <ju> <w> <wh> ∈ ∈ 2 5807 46 220 283 1 16 1 3 2694 26 294 1 1 8 2 27 1 312 2 2 2 3 2123 581 2 4 1 381 1224 116 9 2 1 7688 131 diaphra(gm), paradi(gm) abandon(m)ent, zoo(m)s aplo(mb), whitco(mb) aﬂa(me), wholeso(me) acco(mm)odated, zi(mm)erman fe(mme) autu(mn), sole(mn)ly te(mp)tation gover(nm)ent, gover(nm)entally adol(f), yoursel(f) cha(fe), wildli(fe) a(ff)able, ze(ff)irelli jolli(ffe) uncom(for)tably o(ft)en, so(ft)ens aw(fu)lly, power(fu)lly cou(gh), trou(gh)s (pf)ennig al(ph)a, xeno(ph)obia so(pho)more, so(pho)mores gusta(v), moloto(v) o(f), thereo(f) ste(ph)en, ste(ph)enson abbre(v)iated, y(v)ette aborti(ve), wo(ve) re(vv)ed, sa(vv)y (ju)an, ti(ju)ana biv(ou)ac acq(u)aint, venez(u)elan after(w)ard, wrist(w)atch any(wh)ere, (wh)y (o)ne ch(o)ir b(ue)no aban(d)on, zealan(d) a(dd), yi(dd)ish /F/ /V/ /W/ /D/ (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /T/ /N/ /TH/ /DH/ Graphemes Frequency Examples <de> <ed> <ed> ∈ ∈ <mp> 293 967 3 10 5 4 154 7 436 219 4 11661 693 1 12 500 28 5 1 2 65 2 58 1 13247 5 453 316 10 3 1 1 736 3 1 171 abi(de), worldwi(de) abandon(ed), kill(ed) (t)aoism, (t)aoists de(bt), undou(bt)edly ya(cht), ya(cht)sman conne(ct)icut, indi(ct)ments acquiesce(d), seduce(d) bernhar(dt), schmi(dt) abolish(ed), zipp(ed) aforethou(ght), wrou(ght) (pt)olemaic, recei(pt)s abandonmen(t), zoologis(t) absolu(te), wro(te) descar(tes) apar(th)eid, (th)omson abe(tt)ed, wri(tt)en antoine(tte), yve(tte) na(z)i pi(zz)a we(dne)sday, we(dne)sdays ali(gn), vi(gn)ette colo(gne), champa(gne) ac(kn)owledges, un(kn)own co(mp)troller aba(n)do(n), zo(n)ing gra(nd)children, wi(nd)sor abile(ne), zo(ne) ante(nn)a, wy(nn) a(nne), wy(nne) denoueme(nt), rapprocheme(nt) (pn)eumonia sou(t)hampton aberna(th)y, zeni(th) bli(the)ly, wri(the) ca(tho)lic altoge(th)er, you(th)s (continued on next page) 53 54 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme /S/ /Z/ /R/ Graphemes Frequency Examples <es> <s> <sc> <sce> <se> <ss> <sse> <st> <sw> ∈ <x> ∈ <xe> ∈ ∈ <es> <s> <se> <si> <so> <sp> <ss> <sth> <x> ∈ <x> <er> 14 1209 504 3 7 30 9066 93 5 230 939 4 27 7 29 501 2 5 1 1 409 4877 203 4 1 1 17 1 2 1 6 347 201 30 92 44 ba(the), soo(the) absen(c)es, yan(c)y abeyan(ce), when(ce) glou(ces)ter, wor(ces)ter fawk(es), wilk(es) (ps)alm, (ps)yllium abel(s)on, zoologi(s)t acquie(sc)ence, vi(sc)eral acquie(sce), coale(sce) abu(se), wor(se) abruptne(ss), zei(ss) impa(sse), ru(sse) che(st)nut, wre(st)ling an(sw)er, unan(sw)ered auschwit(z), walt(z) afﬁ(x) a(xe) na(z)i pi(zz)a (cz)ar abiliti(es), zombi(es) abel(s), zoom(s) abu(se)d, who(se) bu(si)ness, bu(si)nessmen unrea(so)ning ra(sp)berry de(ss)ert, sci(ss)ors a(sth)ma clo(thes)horse, plainclo(thes) (ts)ar an(x)ieties, (x)enophobia maga(z)ine, (z)urcher abla(ze), visuali(ze) bli(zz)ard, whi(zz)ing au(x)iliary av(er)ages, vet(er)inary (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /L/ /JH/ /CH/ Graphemes Frequency Examples <wr> <sl> <sle> <j> <s> <sh> 4 9955 379 20 318 1 1 76 1 9032 890 1306 43 4 3 47 50 63 27 770 253 6 453 13 1 627 5 1 2 5 4 321 134 1 1 1 deb(or)a, satisfact(or)y abno(r)mal, zu(r)ich adhe(re), yo(re) go(rh)am, (rh)ythmically abe(rr)ant, ya(rr)ow ca(rre) rappo(rt) a(wr)y, (wr)yly imbro(gl)io ab(l)er, zoo(l)ogy ab(le), ya(le) abda(ll)ah, zefﬁre(ll)i be(lle), wa(lle) i(sl)and, i(sl)es ai(sle), carli(sle) a(d)ulation, une(d)ucated e(dg)y, sto(dg)y acknowle(dge)d, we(dge)d a(dj)acent, una(dj)usted abori(g)inal, zoolo(g)y acrea(ge), yarda(ge) dima(gg)io, exa(gg)erate ab(j)ect, un(j)ustiﬁed an(c)ient, provin(c)ialism ri(cc)i a(ch)eson, wren(ch)ing avalan(che), ni(che) (cz)echoslovakia ri(ght)eous, ri(ght)eousness man(s)ions, ten(s)ions court(sh)ip, wel(sh) absten(t)ion, vul(t)ure backsti(tch), wristwa(tch) ca(ts)up deu(tsch) nie(tzsch)e (continued on next page) 55 56 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme Graphemes Frequency Examples /SH/ <s> <sc> <sch> <sh> <ss> ∈ <x> <s> <sh> ∈ <x> <e> <j> ∈ <eau> ∈ <eu> ∈ <ewe> ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ 157 53 5 47 16 29 906 116 1199 11 16 14 2 143 1 2 3 3 3 104 3 141 4 21 1 1 7 1 736 26 2 1 4 3 2 5711 aca(c)ia, viva(c)ious bro(ch)ures, vi(ch)y bayeri(che), rappro(che)ment apprehen(s)ion, un(s)ure con(sc)ientious, uncon(sc)iously au(sch)witz, (sch)weizer aboli(sh), yiddi(sh) admi(ss)ion, unprofe(ss)ional abduc(t)ion, voli(t)ion an(x)ious ada(g)io, re(g)imes barra(ge), sabota(ge) bon(j)our, di(j)on abra(s)ion, vi(s)ually ca(sh)mere equa(t)ion, equa(t)ions cro(z)ier, sei(z)ure lu(x)urious azal(e)a, heterogen(e)ous aﬁc(i)onado, xav(i)er f(j)ords, (j)ohann back(y)ard, (y)uri beautiful eucalyptus, eurasian ewe milieu interview houston acc(u)m(u)lated, brav(u)ra arg(ue) h(ugh) h(uh)es ﬁgure deb(ut) vac(uu)m abdu(c)tion, zin(c) /ZH/ /Y/ /K/ (continued on next page) Graphemic representation of English phonemes Table 9 (continued from previous page) Phoneme /G/ /NG/ Graphemes Frequency Examples <x> ∈ <x> ∈ <xe> ∈ <x> 58 130 2 252 4 651 2 16 2 2 1277 167 3 1 323 12 15 32 501 2 2031 1 134 25 57 35 2 95 438 2 2923 2 1 4 academi(ca)lly, verti(ca)lly a(cc)ardo, yu(cc)a ba(cch)us, ve(cch)io a(ch)illes, zur(ch)er a(che), heada(che)s aba(ck), yardsti(ck) cho(co)late, de(co)rative a(cq)uaint, ra(cq)uet la(cqu)er, la(cqu)ered len(g)th, len(g)ths a(k)in, yol(k) ali(ke), yo(ke) (kh)aki, (kh)mer tre(kk)ed ade(q)uacy, velas(q)uez con(qu)er, (qu)asimodo anti(que)s, uni(que)ness e(x)cel, overe(x)cited afﬁ(x) a(xe) abi(g)ail, zi(g)zag(g)ing nia(ga)ra a(gg)ies, zigza(gg)ing af(gh)an, u(gh) bala(gu)er, van(gu)ard analo(gue), vo(gue) mor(tg)ages, mor(tg)age coe(x)ist, luxurious adju(n)ct, zi(n)c ha(nd)kerchiefs, ha(nd)kerchief abandoni(ng), zoomi(ng) la(nge), mela(nge) hara(ngu)ing hara(ngue)d, to(ngue)s (continued on next page) 57 58 Fan Fengxiang and Gabriel Altmann Table 9 (continued from previous page) Phoneme Graphemes Frequency Examples /HH/ <wh> <j> 1493 14 3 (h)omage, ab(h)orrent (wh)o, (wh)ole (j)imenez, (j)unta References Adams, Marilyn J. 1990 Beginning to read: thinking and learning about print. Cambridge, MA: MIT Press. Barnhart, Clarence L.; Barnhart, Robert K. (Eds.) 1979 The world book dictionary. Chicago / London: Doubleday. Berndt, Rita S.; Reggia, James A.; Mitchum, Charlotte C. 1987 “Empirically derived probabilities for grapheme-to-phoneme correspondences in English.” In: Behavior Research Methods, Instruments, & Computers, 19; 1–19. Best, Karl-Heinz 2005 “Buchstabenhäuﬁgkeiten im Deutschen und Englischen.” In: Naukovij ˇ visnik Cernivec’kogo universitetu, Germans’ka ﬁlologija, 231; 119– 127. 2005 “Zur Häuﬁgkeit von Buchstaben, Leerzeichen und anderen Schriftzeichen in deutschen Texten.” In: Glottometrics, 11; 9–31. 2005 “Laut- und Phonemhäuﬁgkeiten im Deutschen.” In: Göttinger Beiträge zur Sprachwissenschaft, 10; 21–32. Cunningham, Patricia M.; Cunningham, James W. 1992 “Making words: Enhancing the invented spelling-decoding connection.” In: The Reading Teacher, 46; 106–107. Fry, Edward 2004 “Phonics: a large phoneme-grapheme frequency count revised.” In: Journal of Literacy Research, 36(1); 85–98. Grzybek, Peter; Kelih, Emmerich 2003 “Graphemhäuﬁgkeiten (am Beispiel des Russischen). Teil I: Methodologische Vor-Bemerkungen und Anmerkungen zur Geschichte der Erforschung von Graphemhäuﬁgkeiten im Russischen.” In: Anzeiger für Slavische Philologie, 31; 131–162. Grzybek, Peter; Kelih, Emmerich; Altmann, Gabriel 2004 “Graphemhäuﬁgkeiten (Am Beispiel des Russischen). Teil II: Modelle der Häuﬁgkeitsverteilung.” In: Anzeiger für Slavische Philologie, 32; 25–54. Graphemic representation of English phonemes 59 Hanna, Paul R.; Hanna, Jean S.; Hodges, Richard E.; Rudorf, Erwin H. 1996 Phoneme-grapheme correspondences as cues to spelling improvement. Washington, D.C.: U.S. Department of Health, Education, and Welfare. Hudson, Grover 2000 Essential introductory linguistics. Repr. Malden, Mass.: Blackwell. Köhler, Reinhard; Martináková-Rendeková, Zuzana 1998 “A systems theoretical approach to language and music”. In: Altmann, Gabriel; Koch, Walter A. (Eds.), Systems. New paradigms for the human sciences. Berlin: de Gruyter, 514–546. Patterson, Karalyn E.; Morton, John 1985 “From orthography to phonology: An attempt at an old interpretation”. In: Patterson, Karalyn E.; Marshall, John C.; Coltheart, Max (Eds.), Surface dyslexia: neuropsychological and cognitive studies of phonological reading. London: Erlbaum. Seidenberg, Mark S.; Waters, Gloria S.; Barnes, Marcia A.; Tanenhaus, Michael K. 1984 “When does irregular spelling or pronunciation inﬂuence word recognition?” In: Journal of Verbal Learning & Verbal behavior, 23; 383– 404. Wimmer, Gejza; Altmann, Gabriel 1999 Thesaurus of univariate discrete probability distributions. Essen: Stamm. Zörnig, Peter; Altmann, Gabriel 1995 “Uniﬁed representation of Zipf distributions.” In: Computational Statistics & Data Analysis, 19; 461–473. The phoneme-grapheme relationship in Slovene Emmerich Kelih 1 Introduction In this contribution we discuss some problems concerning the synergetic control cycle of script theory proposed by Altmann (this volume, pp. 149ff.). The following aspects will be treated on the basis of Slovene: (a) the orthographic uncertainty of phonemes and a statistical comparison with other languages, (b) the distribution of grapheme size, (c) the mean grapheme length, (d) the mean graphemic load, and (e) letter utility and its positional weighting. The procedure adheres to the order proposed by Altmann. Let us begin with some qualitative data. The Slovene phoneme system contains 29 phonemes. There are 8 vowels /i, e, E, @, a, o, O, u/. The phonemes /e:/ and /E:/ are long but differ in their degree of openness: /e:/ in [pe:ti] “to sing” is pronounced long and narrow while /E:/ in [pE:ti] “the ﬁfth” is long and open. The phonemes /o:/ and /O:/ differ in their degree of opening, too: long and narrow /o:/ in [mo:ra] “he/she must” and the long open /O:/ in [mO:ra] “nightmare” are examples. Hence the degree of opening is phonemically relevant (cf. Rehder 1998: 231f., Lencek 1982: 160ff. and on Slovene phonemics in general Toporišic 2000: 46ff.).1 While there is a general agreement concerning vowel inventory, the size of the consonant inventory is deﬁned differently in the secondary literature. Lencek (1982: 169) deﬁnes 20 consonants in the inventory: /v, m, n, r, l, j, b, d, g, z, ˇz, p, t, k, s, ˇs, c, ˇc, f, h/ while Rehder (1998: 232) differentiates 22: /p, b, f, v, m, t, d, s, z, n, n’, r, l, l’, ˇs, ˇz, c, ˇc, j, k, g, h/. However, the palatalized /n’/ and /l’/ can be considered as positional variants. The present investigation starts from 21 consonantal phonemes deﬁned according to Toporišic (2000: 85): /p, b, f, v, m, t, d, s, z, n, r, l, ˇs, ˇz, dˇz, c, ˇc, j, k, g, h/. The phoneme /dž/ must be emphasized here because it usually 1. An alternative counting of Slovene vowels would distinguish the opposition stressedunstressed and the quantities short-long respectively. In that case Slovene would have 19 vowels (cf. The Survey in Toporišic 2000: 71). In order to make comparison with other languages possible, we refrain from this maximal system here. We start from the 8 vowels deﬁned above. 62 Emmerich Kelih does not occur in standard presentations of Slovene. Nevertheless, it must be unequivocally identiﬁed as a Slovene phoneme. The respective examples are: [dˇzem] “jam” vs. [ˇcem] “to whom”, [dˇzez] “jazz” vs. [ˇcez] “through” (cf. Toporišic 1978: 73). The Slovene script system contains 25 letters: . The ﬁrst conspicuous characteristic is the fact that phoneme /@/ has no written correspondence. It is represented by <e> or not at all. The second characteristic is the use of diacritics as in <ˇc, š, ž>, which were gradually introduced in Slovene since the forties of the 19th century with the Illyric Movement2 and have become part of the written variant of Slovene up to now. One can ﬁnd them in several other Slavic languages. This elementary presentation of Slovene phoneme and script system is a sufﬁcient basis to begin the investigation. In Table 12 (p. 71ff.) one ﬁnds each of the 29 Slovene phonemes and the respective graphemes. The attribution of a phoneme to a grapheme is based on the Table in Slovenski Pravopis (1990: 143ff.). One starts there, however, from sounds. But the attribution of sounds to letters is irrelevant from the point of view of the investigation. Hence, the inventory of 64 sounds is here “reduced” to 29 phonemes of Slovene in order to study adequately the phoneme-grapheme correspondence. First we state that the 29 phonemes of Slovene are represented by 28 graphemes. We emphasize here that graphemes are the letters or letter combinations that are used to represent the phoneme system of language on the graphemic level. Hence grapheme is deﬁned as a unit constituted from its relation to a phoneme. 2. The inventory of 25 letters has been to a certain extent codiﬁed in the ﬁrst Slovene orthography (Pravopis) in 1899 by Fran Levec (cf. Gložanˇcev 1997: 85f.). This does not exclude the use of some “foreign” letters like <´c, d, ¯ q, w, y, . . . >. The publication of the ﬁrst Slovene orthographic handbook initiated a discussion concerning not only the adequacy of letters for Slovene sound (this dispute has been performed in the so-called ABC-war in the twenties and thirties of the 19th century), but a series of special questions of Slovene orthography (cf. Dobrovoljc 2004: 113ff.). These concern the problem of capitalization, the writing of foreign words, that of personal names etc. The last orthography reform performed in 1990/2001 led to a lively scientiﬁc discussion (cf. Dobrovoljc 2004: 92ff.) and touched in general theses problems. However, this orthography reform did not cause any dramatic changes concerning the relation between phonemes and graphemes. The phoneme-grapheme relationship in Slovene 2 63 Economy of the script system In order to scrutinize the phoneme-grapheme relationship in Slovene, we ﬁrst compute the economy of the script system as the ratio of phoneme number to grapheme number (Best & Altmann 2005: 34). Denoting script economy by SE = #P/#G, for Slovene we obtain SE(Sl) = 29/28 = 1.03. This value can be considered the index of phoneme-grapheme correspondence. For the Slovene script system we actually obtain a very high economy. A comparison with German SE(G) = 0.57 and Swedish SE(Sw) = 0.63 shows the economy of Slovene still more markedly. However, this index is global and does not express the phoneme-grapheme relationship in individual cases. In other words, one must take into account not only the fact that a phoneme can be represented by a single grapheme, but also the fact that graphemes are “polyfunctional” and can represent different phonemes. 3 Orthographic uncertainty of the phonemes Here we examine the question how many graphemes are necessary to express a phoneme. It is reasonable to assume that the more different graphemes are “necessary” to represent a phoneme, the smaller the effectivity of the writing system. An extreme position is taken by English (cf. Fan & Altmann, this volume, pp. 25ff.). Using Best & Altmann’s (2005: 32) deﬁnition the orthographic uncertainty of a phoneme can be given as “. . . the binary logarithm of the cardinal number of the representing grapheme set”. The formula is U/x/ = log2 nx . (1) Here U/x/ is the orthographic uncertainty of the phoneme /x/, log2 is the dyadic logarithm and nx is the number of graphemes representing the phoneme /x/. A survey of representations and uncertainties for individual phonemes is presented in Table 1. Here for each phoneme the number of representing graphemes, the computed orthographic uncertainty and the number of phonemes with the given uncertainty are shown (nx denoting the number of representing graphemes, Ux the uncertainty, and fx the number of phonemes with uncertainty Ux ). It is conspicuous that in Slovene (cf. Table 1) the majority of phonemes (18 out of 29) can be represented by two or more different graphemes. This 64 Emmerich Kelih Table 1: Unweighted orthographic uncertainty of Slovene phonemes Phoneme nx Ux fx a, c, e, e, i, f, h, m, o, O, r b, d, g, k, j, l, n, p, t, z, @, s c, u, dˇz ˇs, v, ˇz 1 2 3 4 0.00 1.00 1.58 2.00 11 12 3 3 is an important hint at the fact that the sole interpretation of the phonemegrapheme correspondence is not enough and must be completed by the unweighted orthographic uncertainty of the phoneme. The mean orthographic uncertainty of the whole system can be computed using data in Table 1. Let N be the number of all phonemes, fx the number of phonemes represented by nx graphemes (or graphemes with uncertainty Ux ), then 1 U¯ = ∑ fx Ux . N x (2) For Slovene we obtain U¯ = [11(0)+12(1)+(3)1.58+3(2)]/29 = 0.7841. Comparing the results with those presented in this volume, namely German (U¯ = 0.965), Swedish (U¯ = 0.797), Italian (U¯ = 0.5641) and Slovak (U¯ = 0.7586), one can state that the Slovene orthographic uncertainty is very similar to Swedish and Slovak. It is placed in the middle of the analysed languages, having the greatest “distance” from German and Italian. In order not to rely on intuitive interpretation of the differences between the values, it is better to perform at least an asymptotic test answering the question about the signiﬁcance of these differences in orthographic uncertainties. The criterion can be set up as U¯ 1 − U¯ 2 z= V (U¯ 1 ) + V (U¯ 1 ) (3) ¯ is its variance. For dewhere U¯ is the mean empirical uncertainty and V (U) tails see Bernhard & Altmann (this volume, pp. 13ff.). In Table 2 the individual values necessary for the computation are presented. Using Slovene as the basis of comparison we obtain the z-values as shown in Table 3. We can state that at the 95% conﬁdence level there is no signiﬁcant difference between Slovene and other languages. We can state that concerning the degree of orthographic vagueness the languages examined show similar behaviour. However, only further research can The phoneme-grapheme relationship in Slovene 65 Table 2: Mean uncertainty and its variance Language U¯ German Swedish Italian Slovak Slovene 0.9650 0.7970 0.5641 0.7586 0.7841 ¯ V (U) 0.012602 0.022763 0.012189 0.012030 0.014057 show whether this fact can be explained by the time of script introduction or other cultural and linguistic factors. In any case, we could show that Slovene, having a small number of phonemes, can be represented by a small number of graphemes, which can partly be used in several functions. Table 3: z-values of the comparison between Slovene and other languages Slovene 4 German 1.13 Swedish 0.07 Italian 1.39 Slovak 0.16 The distribution of graphemic representation In connection with the orthographic uncertainty, we can ask whether the number of graphemes per phoneme can be captured by a theoretical model. Evidently, there is no one-to-one representation, hence the distribution is not deterministic, as can be seen in Table 1. There are phonemes which can be represented maximally by four graphemes. Using Bernhard & Altmann’s (this volume, pp. 13ff.) and Maˇcutek’s (this volume, pp. 75ff.) approach, we try to ﬁt the Shenton-Skees geometric distribution to the data. As can be seen, there is no monotonic decrease; we have to do with a slightly bell-shaped distribution. In Slovene and German the maximum is in x = 2 (cf. Table 4). The empirical data can be successfully captured by the Shenton-Skeesgeometric distribution. Using the chi-square goodness-of-ﬁt test we get P = 0.30 for Slovene, which is a good corroboration of the model. In Table 5 one ﬁnds the computed values for all analysed languages. 66 Emmerich Kelih Table 4: Distribution of representation size of phonemes x Italian German Swedish 1 2 3 4 5 6 34 14 8 1 2 10 18 7 3 0 1 16 10 6 1 2 1 5 Slovak 19 13 10 1 0 1 Slovene 11 12 3 3 Grapheme size The graphemes of a language may consist of one letter or of combinations of several letters. Besides digraphs, trigraphs etc., there is still the possibility of adding diacritical marks to letters in order to enhance the effectivity of a script system. In the Slovene script system there is a special mark above some letters called “haˇcek”: <ˇc,š,ž>. Adhering to the analysis in Best & Altmann (2005), Bernhard & Altmann (this volume, pp. 13ff.) and Nemcová & Altmann (this volume, pp. 79ff.), we shall determine the size of graphemes in two ways. We consider the number of Latin letters 1. in which additional marks are not considered as separate components; 2. in which additional marks are considered as separate components. Table 5: Results of ﬁtting x Italian German Swedish Slovak 1 2 3 4 5 6 33.31 14.92 6.37 2.64 1.76 9.99 18.00 7.54 2.47 0.73 0.27 15.79 9.99 5.35 2.64 1.24 1.00 19.36 13.40 6.58 2.83 1.14 0.69 p a DF X2 P 0.6488 0.2398 2 1.55 0.46 0.7768 2.3323 1 0.12 0.73 0.6152 0.4588 2 1.36 0.51 0.6813 0.7569 2 3.36 0.19 Slovene 12.20 11.33 4.91 2.56 0.7495 1.4216 1 1.07 0.30 The phoneme-grapheme relationship in Slovene 67 In Tables 6a and 6b one ﬁnds the respective grapheme size, the concerned units and their number. Table 6a: Size of Slovene graphemes: Method 1 Size 1 2 Grapheme Number a, b, c, cˇ , d, e, f, g, h, i, j, k, l, m, n, o, p, r, s, š, t, u, v, z, ž dž, lj, nj 25 3 In order to get a compact characterization of the whole system, we compute in the ﬁrst step the mean grapheme length in Slovene. Using Table 6a (without diacritical marks) we obtain 1.11. Table 6b: Size of Slovene graphemes: Method 2 Size 1 2 3 Grapheme Number a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, r, s, t, u, v, z cˇ , š, ž, lj, nj dž 22 5 1 Using method (2), which considers special symbols as grapheme components, we obtain the mean length as 1.25. In order to interpret intuitively the extent of these values we compare them with Slovak, Italian, German and Swedish mean grapheme sizes. The individual values (see Table 7) show in particular that the two Slavic languages have a low mean grapheme size. Table 7: Mean grapheme sizes according to two methods Language German Swedish Italian Slovak Slovene Mean size (Method 1) 1.68 1.61 1.65 1.16 1.11 Mean size (Method 2) 1.78 1.67 1.70 − 1.25 This is not surprising because the use of di- and trigraphs is not a central property of these script systems. The small grapheme size in Slovene can partly be explained historically: in the thirties and forties of the 19th century the ﬁrst orthography leaned against the principle “write as you speak” which 68 Emmerich Kelih means a 1:1 relation of phoneme/sound to grapheme. This principle asserted itself to a great extent. A more thorough interpretation of grapheme size and of the position of Slovene will be possible after more languages have been analysed. Compared with other languages, Slovene has the lowest mean grapheme size, which is an indication of the “simple” construction of the Slovene graphemic system. 6 The graphemic load of letters Having analysed the grapheme size, we examine the graphemic load of individual letters. We consider the presence of a letter in different graphemes. From another point of view we measure the graphemic “polytexty” of Latin letters in Slovene. This property is usually scrutinized with words but it has relevance in graphemics, too. In this analysis letters are taken into account without diacritical marks, i.e. in their Latin form. Hence the inventory consists of 22 letters, see Table 8. In this way comparability with other languages is made possible. Table 8: Participation of Latin letters in Slovene graphemes Component in x graphemes 1 2 3 Latin letter a, b, c, e, f, g, h, i, k, m, o, p, r, s, t, u, v d, l, n j, z Number of letters 17 3 2 Using the numbers in Table 8, we can compute the mean graphemic load of letters, resulting in 1.32, i.e. on the average, a letter occurs in “slightly more” than one grapheme. This is, of course, a very low value explained by the small number of digraphs: . A comparison with other languages is shown in Table 9. 7 Letter utility The participation of a letter in a grapheme can be measured not only directly as presence or absence but also in a weighted way. Best & Altmann (2005: 37) take into account also the position of a letter in the grapheme. It is supposed The phoneme-grapheme relationship in Slovene 69 Table 9: Graphemic load in ﬁve languages Language Graphemic load German Swedish Italian Slovak Slovene 3.96 3.36 3.92 2.23 1.27 that the later a letter occurs in the grapheme, the smaller its relevance. Though this need not be always the case, in the majority of cases it is true. Hence the position of a letter serves as the weight of its utility. We just have to consider all graphemes and for each letter its position. The results for Slovene are shown in Table 10. Table 10: Letter utility Weight 1 2 3 4 5 Number of letters f (x) Letter x a, b, c, e, f, g, h, i, k, m, o, p, r, s, t, u, v d, l, n – z j 17 3 − 1 1 The mean positional weight for Slovene is PW (Sl) = 1.5. Compared with other languages this value is very low. A comparison is shown in Table 11. Table 11: Positional weight in ﬁve languages Language German Swedish Italian Slovak Slovene English Positional weight 6.12 5.41 6.48 2.50 1.45 47.35 Lastly we can compute the distance from the ideal state (cf. Nemcová & Altmann, this volume, pp. 79ff.). The ideal state is achieved if all letters are monographs. For Slovene we get the ideal state as I = 22, and the real 70 Emmerich Kelih state as the vector of frequencies from Table 10 W (Sl) = [17, 3, 2] yielding D(Sl) = 6.16. Consequently, there is a very small distance between the real and the ideal state in Slovene. Slovak (D = 17.78) and Italian (D = 21.6) are more distant. Hence one can state that Slovene has an almost optimal graphemic system. 8 Conclusion In this investigation we ascertain a number of characteristics of the Slovene graphemic system. A central property is a relatively small inventory of phonemes (world wide it is ca. 32) represented also by a small number of graphemes. While the phoneme-grapheme index indicates an “over-economy”; the computation of the orthographic uncertainty shows that Slovene graphemes can represent different phonemes. A statistical comparison of this property with that of other languages has shown that there is no signiﬁcant difference between Slovene and German, Italian, Swedish and Slovak respectively. It could be shown that the Slovene grapheme system has quite optimal characteristics. This concerns grapheme size, letter load, letter utility and positional weighting. Deeper insights can be drawn by further analyses, especially those of Slavic languages. A further step would be the embedding of the characteristics of the Slovene grapheme system in the synergetic control cycle of script theory (cf. Altmann, this volume, pp. 149ff.). The phoneme-grapheme relationship in Slovene 71 Table 12: The phoneme-grapheme relationship in Slovene P = Phoneme, G = Grapheme, = cluster P G /a/ /b/ /c/ /ˇc/ <ˇc> /dˇz/ /d/ /e/ <e> /E:/ <e> /@/ <e> <ø> /f/ /g/ /h/ /i/ /j/ <j> /k/ Examples sam, brat, riba, dna biba, objem, oblati, ob oˇcetu/materi snop daj Cene, stric, Micka peˇc, cˇ lovek, peˇcka bridžka, bridž prinesi, bridž, bridž igra, bridž je stric šepa Džungla odloˇcba, proˇc daj stric Džon dedje, odgnati svatba, svat bo kmet, ribe, pet teta, seja pes, temen rdeˇc, Vltava, Mboja, Nkrumah fant, Škoﬂjica, Štefka glagol, Bogdan kdaj k domu, vsak dan hahljati se, mehko, meh pila, sit, mami jajce celuloid, bo imela krik, kura, pakt bogca, rog, rog odnesi, rog nesi Remarks In front of voiced stops In front of voiceless stops, pause, vowel, in front of the sonorant of the next word In front of voiceless stops In front of voiced stops In front of voiced stops In front of voiced stops Long, open (ê) Semivowel In neighbourhood of consonants (in front of r, l, m, n, v) In front of voiced stops In quoted words; at word beginning, if the previous word ends with a vowel In front of voiceless stops, pause, vowel, in front of the sonorant of the next word (continued on next page) 72 Emmerich Kelih Table 12 (continued from previous page) P G Examples /l/ <m> polje, ohol polj, poljski imam Nina, njena sanj, sanjski nos, gol, bratov noga, voda pipa, pnem, snop Robca, rob strehe, Rob, Rob ima, Rob je /m/ /n/ /o/ /O:/ /p/ /r/ /s/ <s> trava, morda, dar, rdeˇc sila, sneti, stati, trs nizka, niz, niz ima, niz je /ˇs/ <š> <ž> šumiš, smešna, šteti ježka, jež, jež odhaja, jež je /t/ <s> /u/ sˇcasoma iz cˇ asov tat, tknalo, tkati gladka, glad, glad ubija, glad je <e, cl.> kruhu, kruh bukev, barv nesel, nosil, videl, trl /v/ siva, vera siv, sivka v oˇceh Remarks Not in front of a vowel Not in front of a vowel Long, open (ô) In front of voiceless stops, pause, vowel (except of prepositions), in front of the sonorant of the next word (except of prepositions) In front of voiceless stops, pause, vowel, in front of the sonorant of the next word In front of voiceless stops, pause, vowel, in front of the sonorant of the next word In front of voiceless sibilant In front of voiceless sibilant In front of voiceless stops, pause, vowel, in front of the sonorant of the next word In front of stressless semivowels, in consonant clusters With unstressed semivowels, with consonant clusters, labiodental A diphthong, when it alters to [v] (siva) As preposition in front of vowels (continued on next page) The phoneme-grapheme relationship in Slovene 73 Table 12 (continued from previous page) P G /z/ <s> /ˇz/ <ž> <š> <s> Examples Remarks vsak, predvsem Dachaua Nauk bo ušla, bo uran ubiti Afgan, grof gre bral, bralca, kosil, topel, bel trl zet, zrelost, zdaj, cez oceta, mater glasba, glas bobni žaga izvršba iz džungle kos džungle Voiceless bilabial, no diphthong In foreign words In rare Slovene words In word combinations As preﬁx In front of voiced stops In alternation with [l] (brala) At word end In front of voiced stops In front of voiced stops In front of voiced sibilant In front of voiced sibilant References Altmann, Gabriel 2007 “Towards a theory of script”. This volume, pp. 149–164. Bernhard, Gerald; Altmann, Gabriel 2007 “The phoneme-grapheme relation in Italian”. This volume, pp. 13–23. Best, Karl-Heinz; Altmann, Gabriel 2005 “Some properties of graphemic systems.” In: Glottometrics, 9; 29–39. Dobrovoljc, Helena 2004 Pravopisje na Slovenskem. Ljubljana: Založba ZRC. Fan, Fengxiang; Altmann, Gabriel 2007 “Graphemic representation of English phonemes”. This volume, pp. 25– 59. Gložanˇcev, Alenka 1997 “Kratek pregled slovenskego pravopisja od konca devetnajstega stoletja do danes.” In: Jezikoslovni zapiski, 3; 85–104. Lencek, Rado L. 1982 The structure and history of the Slovene language. Columbus, Ohio: Slavia. Maˇcutek, Ján 2007 “On the distribution of graphemic representations”. This volume, pp. 75– 78. 74 Emmerich Kelih Nemcová, Emília; Altmann, Gabriel 2007 “The phoneme-grapheme relation in Slovak”. This volume, pp. 79–87. Pravopis, slovenski 1990 Slovenski Pravopis. 1. Pravila. Ljubljana: Državna Založba Slovenije. Rehder, Peter 1998 “Das Slovenische”. In: Rehder, Peter (Ed.), Einführung in die slavischen Sprachen (Mit einer Einführung in die Balkanphilologie). Darmstadt: Wissenschaftliche Buchgesellschaft, 230–245. Toporišiˇc, Jože 1978 Glasovna in naglasna podoba slovenskega jezika. Maribor: Založba Obzorja. 2000 Slovenska Slovnica. Maribor: Založba Obzorja. On the distribution of graphemic representations Ján Maˇcutek In several articles of the present volume it has been shown that the graphemic representations of phonemes abide by the Shenton-Skees geometric distribution, which is, as a matter of fact, a modiﬁed (1-displaced) geometric distribution. It can be obtained from the geometric distribution by the Gram-Charlier expansion (cf. Shenton & Skees 1970; Berg 1985; Johnson, Kotz & Kemp 1992: 442 ff.). Moreover, a probability distribution can be constructed from a parent distribution P* by modifying P* in the following way, Px = Px∗ (1 + a(x − μ∗)) , (1) where μ* is the ﬁrst moment of P* about zero (mean) and the parameter a fulﬁls certain conditions. In general, a ≤ μ∗1−n , where n is such an integer that μ∗ > n and μ∗ ≤ n + 1. In our case Px∗ = p(1 − p)x−1 , x = 1, 2, 3, . . . (2) with 0 < p < 1. The expectation (mean) of (2) is μ∗ = 1/p, hence we obtain from (1) and (2) that 1 x−1 1 + a(x − ) , x = 1, 2, 3, . . . (3) Px = p(1 − p) p where 0 < p ≤ 1, 0 ≤ a ≤ 1/(1 − p) − 1 (if p = 1 then a = 0) representing the Shenton-Skees geometric distribution (cf. Wimmer & Altmann 1999: 593). If the number of empirical frequency classes is greater than 3, the distribution can be ﬁtted using the available software (which tests the ﬁt by means of the chi-square criterion), but this is not always the case, as can be seen in the articles in this volume. Hence some elementary estimators must be derived. Mohanty & Altmann (this volume) proposed the estimators computed from the frequency classes; we show two other estimators. Since the probability generating function of the Shenton-Skees geometric distribution is (writing q = 1 − p) G(t) = t(p − a) apt , + 1 − qt (1 − qt)2 (4) 76 Ján Maˇcutek from here one easily obtains the moments, namely the mean and the variance as p + a − ap p + aq = , (5) μ1 = p2 p2 q[p2 (1 − a) − a2 q + 2ap] . (6) p4 Let x¯ denote the arithmetic mean and Pˆ1 , Pˆ2 the ﬁrst two observed relative frequencies. We have p + a − ap x¯ = , (7) p2 (8) Pˆ1 = p + a − ap , μ2 = cf. (1) and (5). From (7) and (8) we obtain estimators 1 ± 1 − x¯Pˆ1 , p= x¯ (9) p − Pˆ1 . (10) 1− p In (9) we take only positive numbers for estimators of the parameter p. The advantage of these estimators – if compared with the ones where we restrict ourselves to the ﬁrst two frequencies only (cf. Mohanty & Altmann, this volume) – is that it takes into account all frequencies. Moreover, the estimators using only the ﬁrst two frequencies cannot be computed in the cases where the difference between Pˆ1 and Pˆ2 is too large (see Italian below). However, in theory it is possible that none of these approaches can be applied – if x¯Pˆ1 > 1, the method using the mean will not work. We use the data represented in Table 1 to compare different estimators (cf. Bernhard & Altmann, this volume). a= Table 1: Distributions for graphemic representations for three languages x Italian 1 2 3 4 5 6 34 14 8 1 2 − German 10 18 7 3 0 1 Swedish 16 10 6 1 2 1 77 On the distribution of graphemic representations In the following Table 2 we present estimators obtained by three different methods (the minimal chi-square computed with the software, the ﬁrst two frequencies, the mean and the ﬁrst frequency). Table 2: Parameters estimators χ2 Pˆ1 , Pˆ2 x, ¯ Pˆ1 Italian German Swedish p = 0.649 a = 0.240 − p = 0.777 a = 2.332 p = 0.777 a = 2.336 p = 0.764 a = 2.153 p = 0.615 a = 0.459 p = 0.620 a = 0.463 p = 0.630 a = 0.503 p = 0.681 a = 0.239 The estimator using the ﬁrst two frequencies cannot be used for Italian, as we obtain a negative number under a square root. The maximum likelihood estimators are unfortunately not simple. Setting R L = ∏ p(1 − p) x−1 x=1 fx 1 , 1+a x− p (11) where R is the greatest x value, we obtain 1 x − p fx ∂ ln L = 0, = ∑ 1 ∂a x=1 1 + a x − p R 1 x − p fx ∂ ln L = 0. =∑ 1 ∂a x=1 1 + a x − p (12) R (13) Unfortunately, using these formulas one can compute a and p only iteratively. The moment method estimators involve solving equations of the fourth degree and hence they are also complicated. It is to be noted that if a = 0, the Shenton-Skees geometric distribution yields the original 1-displaced geometric distribution. Acknowledgments. Supported by the research grant VEGA 1/3016/06. 78 Ján Maˇcutek References Berg, Sven 1985 “Generating discrete distributions from modiﬁed Charlier Type B expansions”. In: Lanke, Jan and Lindgren, Georg (Eds.), Contributions to probability and statistics in honour of Gunnar Blom. Lund: University of Lund, 39–48. Bernhard, Gerald; Altmann, Gabriel 2007 “The phoneme-grapheme relation in Italian”. This volume, pp. 13–23. Johnson, Norman L.; Kemp, Adrienne W.; Kotz, Samuel 1992 Univariate discrete distributions. New York: Wiley. Mohanty, Panchanan; Altmann, Gabriel 2007 “On graphemic representation of the Oriya phonemes”. This volume, pp. 121–140. Shenton, Leanne R.; Skees, P. 1970 “Some statistical aspects of amounts and duration of rainfall”. In: Patil, Ganapati P. (Ed.), Random Counts in Scientiﬁc Work. University Park: The Pennsylvania State University, 73–94. Wimmer, Gejza; Altmann, Gabriel 1999 Thesaurus of univariate discrete probability distributions. Essen: Stamm. The phoneme-grapheme relation in Slovak Emília Nemcová and Gabriel Altmann 1 Introduction Slovak is a rather conservative Slavic language spoken by 5 million inhabitants in Middle Europe. The oldest Slavic script was Glagolitic, but in the 13th century, after the fall of the Great Moravian Empire, the Latin script started to be used in Slovakia. Gradually, some letter combinations were added to the Latin alphabet and in the 14th century diacritical marks were introduced, too. In the 15th century, due to Jan Hus, the diacritical way of writing got its way. The Slovak alphabet consists today of the following ofﬁcial letters: and the parallel capitals: ˇ D, D, ˇ DZ, DŽ, E, É, F, G, H, CH, I, Í, J, K, L, ´L, < A, Á, Ä, B, C, C, ˇ ´ S, Š, T, T, ˇ U, Ú, V, W, X, Y, Ý, Z, Ž>. L’, M, N, N, O, Ó, Ô, P, Q, R, R, The orthography of the present-day standard Slovak is based on some rules with a predominant phonemic principle. Based on the phonemic principle, a phoneme, which is able to distinguish word meanings and forms, should be represented by the same grapheme. This rule holds almost consistently within the morpheme. There are some exceptions, namely the phonemes /d’, t’, nˇ, l’/ in front of < e, i, í, ia, ie, iu> are represented by , e.g. deti, telo, leto, nízky, lístie, liat’, lietat’, havraniu, but not in all cases, e.g. in foreign words, in plural masculine adjectives etc. The phonemes /i, i:/ are represented by and <í, ý> respectively. The phoneme /v/ is represented by and in some foreign words by <w>, e.g. <watt> /vat/, or by : /kvibek/. The phoneme /k/ is represented by and in some foreign words by , e.g. /ki:to/, or , e.g. /kvibek/ or in the same word. The phonemes /ia, ie, iu, uo/ are diphthongs (/uo/ is represented by <ô>), but in many foreign words, compounds or derivates, the respective graphemes represent two vowels, e.g. ﬁalka, protialkoholický, diecéza, protiepidemický, diuretikum, protiumelecký, priuˇcit’. 80 Emília Nemcová and Gabriel Altmann The letter combinations are graphemes that respectively represent single phonemes, e.g. medzi, džavot, pichat’, but at the morpheme boundary they respectively become two graphemes representing two phonemes: + , + <ž>, + , e.g. odzemok, odžat’, viachlasný. As can be seen, the juncture between morphemes has a strong inﬂuence on the representation. The orthography abides here by the “morphological principle”, e.g. writing all preﬁxes equally in each case, but the graphemes concerned can represent other phonemes due to assimilation. It affects above all the voicing of consonants. However, other assimilations are possible at the morpheme/syllable boundary, too. For example “predˇcasný” consisting of “pred + cˇ asný” represents /preˇcˇcasni:/, or “sudca” consisting of “sud + ca” represents /succa/. There is only one portmanteau grapheme in Slovak, namely the Latin <x> representing /ks/ or /gz/: <xylofón>, <exemplár>. A purely grammatical, mostly historical principle controls the writing of /i, i:/ against the above mentioned rule, e.g. malý (sg. masc.) and malí (pl. masc). We shall ignore other foreign words which would render the system very complex. In present-day Slovak there are 44 phonemes. There are in all 55 graphemic representations. A letter can participate in at most six graphemes. In Table 1 the phoneme-grapheme representation is shown with examples for each representation. Here, the following signs are used: = {< -i,-í, -e, -ia, -ie, -iu>} ( is the set of given graphemes); *<> = part of (the given phoneme is part of *<>; portmanteau grapheme). The phoneme-grapheme relation in Slovak 81 Table 1: The phoneme-grapheme relationship in Slovak Phoneme Graphemes Examples, comments /a/ /a:/ /¨a/ /e/ /e:/ /i/ /i:/ /o/ /o:/ /u/ /u:/ /ia/ /ie/ /iu/ /uo/ /b/ /c/ < a> < á> < ä> < e> < é> < i, y> < í, ý, i> < o> < ó> < u, v> < ú> < ia> < ie> < iu> < ô> < b, p> < c, dz, d> /ˇc/ < cˇ , dž, d> /ˇZ/ < dž, cˇ > /d/ < d, t> /t/ < t, d, tt> /ď/ < d’,d , t’> /k’/ < t’, d’, t > papier, zajac pás, vták päta, mäso, devät’ pero dcéra, nové pivo, dym pískat’, nový; in foreign word Quito /ki:to/ popol pól puto; at the end of words -ov: chlapov, levov púpava piatok papier, pier cudziu vôl’a, kôˇn žaba, keby; in front of sonorants: kúpme, kopme cap, cedul’a; in front of unvoiced consonants; at the end of words: jedzte, vedzte; jedz, vedz; at a juncture: sudca cˇ eˇcina, cˇ ajník; at the end of words: hádž, sádž; at a juncture: predˇcasný džavot, džungl’a; in front of voiced consonants: lieˇcba, tlˇchuba voda, dom, podnos; in front of voiced consonants: kliatba taký, vata; in front of unvoiced consonants: podcenit’, podchytit’; at the end of words: hlad, plod; in some foreign words: watt d’akovat’, d’alej; in front of : deti, dial’ka; in front of voiced consonants: let’me t’ahat’; in front of unvoiced consonants: lod’ka; at the end of words: lod’, chod’; in front of : ticho, telo (continued on next page) 82 Emília Nemcová and Gabriel Altmann Table 1 (continued from previous page) Phoneme Graphemes Examples, comments /Z/ < dz, c> /v/ < v, f, w, u> /g/ < g, k, *x> /h/ < h, ch> /x/ /j/ /k/ < j> < k, g, *x, c, q, qu> /l/ /l’/ /l:/ /m/ /n/ /ˇn/ < l> < l’, l> < ´l> < m> < n> < nˇ , n > /p/ < p, b> /r/ /r:/ /s/ < r> < ´r> < s, z, *x> /z/ < z, s, *x> /ˇs/ < š, z, ž> /f/ < f,v> dzekat’, hádzat’; in front of voiced consonants: viacboj, viacdielny slovo, vdova; in front of sonorants and voiced consonants: šéﬂekár, šéfdirigent; foreign words: watt; Quebec gagot, agát; in front of voiced consonants: takže, takzvaný, kde; <exemplár> ← /egzempl´ar/ hodina, hrat’; in front of voiced consonants: cechmajster, nechže chata, pochvala; in front of unvoiced consonants: kníhtlaˇciareˇn, l’ahko; at the end of words: prah, pstruh raj, krajský, jest’ kúkol’, koniec; at the end of words: slang, gag; part of portmanteau <x>: foxtrot, saxofón, box; in foreign words: Quebec /kvibek/; Quito /ki:to/ skala, vlak l’avák, zl’aknút’ sa; in front of : lietat’, lízat’ v´lˇca, sk´lznut’ mama, hrom rana, nos baˇna; o nˇ om; in front of : niekde, nikde, nedel’a popol, páperie; in front of unvoiced consonants: robte, hrobka; at the end of words: hrob, snob para, Andrej v´rba, h´rba sneh, súdit’; in front of unvoiced consonants: vzpieraˇc, bezpeˇcný; at the end of words: voz, lez; part of the portmanteau <x>: foxtrot, saxofón, box zima, zaˇciatok; in front of voiced consonants: smena, vˇcasráno; <exemplár> ← /egzempl´ar/ šošovica; rozšírit’; in front of unvoiced consonants: rožtek, rovnobežka; at the end of words: kríž, raž fajka, fúkat’; in front of unvoiced consonants: vták, vtedy (continued on next page) 83 The phoneme-grapheme relation in Slovak Table 1 (continued from previous page) Phoneme /ˇz/ Graphemes Examples, comments < ž, š> žiara; in front of voiced consonants: prášme The analysis will follow the methods proposed by Best & Altmann (2005) and those used by other authors in this volume. 2 Orthographic uncertainty of phonemes Reordering the phonemes according to the number of representing graphemes, we obtain the result presented in Table 2, x denoting the number of representing graphemes, Ux the degree of uncertainty, and fx the number of phonemes with uncertainty Ux . Table 2: Orthographic uncertainty of Slovak phonemes Phoneme x /a, a:, ¨a, e, e:, o, o:, u:, ia, ie, iu, uo, j, l, l:, m, n, r, r:/ /i, u, b, ˇZ, Z, d, h, x, l’, nˇ, p, f, ˇz/ /i:, c, ˇc, t, d’, t’, g, s, z, ˇs/ /v/ /k/ 1 2 3 4 6 Ux 0 1 1.58 2 2.58 fx 19 13 10 1 1 The mean uncertainty computed from Table 2 yields 33.38/44 = 0.7586, which is nearer to Swedish (0.797) than to Italian (0.5641). The variance of ¯ Slovak = 0.01203, mean uncertainty of Slovak, computed directly, yields V (U) ¯ Slovak = 0.014057. and asymptotically1 V (U) 3 The distribution of graphemic representations If the modiﬁcation of the geometric distribution is adequate, the frequencies should be distributed according to the Shenton-Skees-geometric distribution (cf. Maˇcutek, this volume, pp. 75ff.). The ﬁtting is shown in Table 3; the parameters and the goodness-of-ﬁt test are given in the last row of the table. 1. See Bernhard & Altmann, this volume (p. 17), equation (4) 84 Emília Nemcová and Gabriel Altmann Table 3: Fitting the Shenton-Skees-geometric distribution to graphemic representations in Slovak x fx 1 2 3 4 5 6 NPx 19 13 10 1 0 1 19.36 13.40 6.58 2.83 1.14 0.69 p = 0.6813, a = 0.7569 X 2 = 3.36, P = 0.19 The ﬁt is graphically displayed in Figure 1. For the time being it is not possible to ﬁnd a factor in language which is responsible for the special value of the parameters. 20 15 f(x) NP(x) 10 5 0 1 2 3 4 5 6 Figure 1: Fitting the Shenton-Skees-geometric distribution to graphemic representations in Slovak 4 Grapheme size In Slovak, a grapheme can be composed of maximally two letters. This is because Slovak uses several distinguishing symbols (´, ˇ, ’, ¨, ˆ) as substitutes for letter combinations or even in letter combinations (cf. ). There are 43 monographemic symbols and 8 bigraphemic ones if one does not consider The phoneme-grapheme relation in Slovak 85 the diacritic symbols as a separate letter. The mean grapheme size is 1.16, very near to the lower limit, the lowest in European languages analysed up to now (Italian 1.65, German 1.68, Swedish 1.61). 5 Graphemic load of letters Modiﬁcations and combinations of letters result in different graphemic load of letters. Table 4 shows the participation of letters in graphemes. In Slovak, the distribution is more regular than in other European languages but for the time being it is not yet possible to set up a model of this distribution. Table 4: Graphemic load of Slovak letters Component in x graphemes 1 2 3 4 5 6 Latin letter b, f, g, j, k, m, p, v, w, x h, n, q, r, s, y e, o, c, l, t a, d, u, z i Number of letters 10 6 5 4 1 Letter utility Using the procedure proposed by Bernhard and Altmann (this volume, pp. 13ff.) it is easy to compute the utility of letters (positional participation) because there are only eight graphemes composed of two letters; all the rest are single (modiﬁed) letters. Table 5 lists the computed utilities. The mean utility in Slovak is 65/26 = 2.5, differing strongly from Italian (6.64 according to Bernhard & Altmann, this volume, pp. 13ff.) or from English (44.23, Fan & Altmann, this volume, pp. 25ff.), which is extremely distant from the ideal state. One could even measure the distance from the ideal state which is given by the deterministic distribution Px = 1 for x = 1 and 0 otherwise, a case in which all letters are used only in monographemes and do not have any modiﬁcation. One possible way of measuring this distance is to use the Euclidean distance. Let W be the vector of weight frequencies and let M be the minimum (ideal) vector of weight frequencies: M = [I, 0, 0, . . ., 0] 86 Emília Nemcová and Gabriel Altmann Table 5: Utilities (positional weight) of letters in Slovak Weight x Letter 1 2 3 4 5 6 Number of letters fx b, f, g, j, k, m, p, v, w, x n, q, r, s, y o, c, h, l, e, u, d a, i, t z 10 5 4 3 3 1 where I is the inventory of (Latin) letters in the given language. In general, both vectors have K components, where K is the number of weights. Then the distance of the language from the ideal (weightless) state is D= K ∑ (Mx −Wx )2 . (1) x=1 In Slovak, I = 26 and, based on Table 5, WSlovak = [10, 5, 4, 3, 3, 1] . Therefore, DSlovak = [(26 − 10)2 + 52 + 42 + 32 + 32 + 12 ]1/2 = 17.78 . For Italian, using Table 7 in Bernhard & Altmann (this volume, p. 23), we obtain DItalian = 21.6. The greater D, the more it is necessary to perform a reform of orthography, based on purely phonological reasons and disregarding cultural, historical and grammatical backgrounds. References Bernhard, Gerald; Altmann, Gabriel 2007 “The phoneme-grapheme relation in Italian”. This volume, pp. 13–23. Best, Karl-Heinz; Altmann, Gabriel 2005 “Some properties of graphemic systems.” In: Glottometrics, 9; 29–39. Král’, Ábel 1983 Pravidlá slovenskej výslovnosti. Bratislava: Slovenské pedagogické nakladatel’stvo. The phoneme-grapheme relation in Slovak 87 Maˇcutek, Ján 2007 “On the distribution of graphemic representations”. This volume, pp. 75– 78. Pravidlá 1998 Pravidlá slovenského pravopisu. Bratislava: Veda. Mohanty, Panchanan; Altmann, Gabriel 2007 “Some problems of the Oriya graphemics”. This volume, pp. 121–140. III. Special Problems Script ornamentality Karl-Heinz Best and Gabriel Altmann 1 Measurement Ornamentality is a concept used in different domains. In literature and painting there is an ornamental style; one can ﬁnd it also in sculpture, architecture and music. However it is not easy to deﬁne ornamentality exactly. The music of Romanticism is more ornamental than Heavy Metal, but what is it that makes something ornamental? In every domain there are different means that render one thing more ornamental than another. In literature these are words, in music surely some special sequences of notes, other means are used in architecture, etc. In science it is not reasonable to ask what is what, a question that leads to essentialism, but rather to set up operational deﬁnitions which allow us to track down at least some vague correspondences of our own concepts. In nature nothing is ornamental; it is merely an idea projected by us to the outside, our way of viewing. But if we have the concept, we must be able to, or at least try to, ﬁnd its “realisations”. The realisations can be either empirical or some abstract mathematical structures. The concept of ornamentality is not identical with that of beauty which has occupied philosophers’ minds since classical antiquity and still keeps busy specialists in aesthetics. As a matter of fact, they are independent of one another. An ornamental thing need not be beautiful, a beautiful thing need not be ornamental. The concept of ornamentality is not binary, i.e. one cannot say that something is ornamental or not ornamental but in most cases one can say that a thing is more ornamental than another. Thus at least an ordinal scale can be introduced. Now, since ornamentality is an expression of our evaluation of a property, our judgement about the degree of ornamentality is decisive. We suppose that direct measurement of ornamentality is not possible. We can only rely on the judgement of those evaluating the degree of ornamentality. When conducting a test, we are interested in the extent of agreement of test participants regarding the concept of ornamentality. 92 Karl-Heinz Best and Gabriel Altmann As for script, we can consider ornamentality a property which can be created by learning calligraphy and even if the test participants are no experts in calligraphy, they can intuitively grade their judgements. They know what ornamentality “is”, they can pronounce judgement about it but, just like everyone else, they cannot deﬁne it. Since in every script there are different types, e.g. some dozens of them for the Latin script, they all have a different degree of ornamentality. In order to measure it we can proceed in several ways. All such measurements are possible only comparatively, within a given interval. Either we compare the examined script to our own handwriting, a method that yields different results for every participant, or we let participants compare different script types and assign to them degrees of ornamentality on the given scale in the prescribed interval. Another possibility is to ask professional calligraphers to make decisions. More objective methods can be devised, too. One can consider ornamentality as 1. the surplus over the simplest pertinent letter, i.e. as the measure of complexity of a letter minus the measure of complexity of the basic letter; 2. a correlation (function) between test participants’ judgements, but this must be done through all letters of an alphabet; 3. the number of interruptions or sharp changes of direction (more than 45°) in writing a letter – thus writing a “T” there is 1 interruption, a handwritten “F” has two interruptions, but “S”, “C”, “O” and “G” have none. In Russian there are a few interruptions in the written script; 4. a graphical measure analogous to those devised by Birkhoff (1933) and reported and developed further by Bense (1969). Still other methods could be proposed but we are interested in the research participants’ attitudes towards the objectivisaton of a subjective concept. Does there exist a degree of commonality when this objectivisaton is applied to letters? We have chosen the following procedure. Test participants (students of the German department at the University of Göttingen) were given a list of different script types represented by the letter “A” and were asked to grade them according to their concept of “ornamentality” on the scale between 1 (for no ornamentality) and 10 (very high ornamentality). The participants could change their judgements, compare individual letters and assign the degree of ornamentality to the letter. The test sheet is shown in Table 1. Script ornamentality 93 Table 1: Questionnaire concerning script ornamentality Letter Font name Letter Wide Latin Vladimir A Font name Informal Roman A Helvetica Narrow Vivaldi Harrington Viner Hand ITC Harlow Solid Italic Stencil Gigi Snap ITC Freestyle Script Ravie Euclid Math Two Pristina Arial Narrow Parchment Blackadder ITC Papyrus Bradley Hand ITC Old English Text MT Broadway Monotype Corsiva Brushscript Matura MT Script Capitals Century Gothic Magneto Chiller Lucida Fax Colonna Lucida Caligraphy Cooper Black Kunstler Script Coronet Kristen ITC Courier Jokerman Edwardian Script ITC ITC Zapf Chancery Algerian 94 Karl-Heinz Best and Gabriel Altmann We received 51 test sheets altogether. This is, of course, only the ﬁrst of the possible steps in the analysis. Presenting different letters in different scripts would, perhaps, yield other results. In the same way, the comparison of not only Latin scripts would show some other aspects. Hence this examination can be continued in different directions. The results of the grading by participants have been collected in Table 4 (see p. 100ff.) for control and for giving other researchers comparative material and the possibility of different processing. At the ﬁrst glance one sees that the judgements are not unique, some of them having a considerable variation. One can order them in different ways: 1. the mean value of judgements, expressing the degree of ornamentality, 2. the standard deviation of judgements, or 3. the range of judgements, the last two expressing the stability of judgments. The most stable scripts are those with minimal standard deviation (or range) e.g. “Parchment”. In Table 2 we present the ordering according to the mean. As we can see, the judgments are more stable at both ends of the scale. “Parchment” was always given the value of 10, hence it has total stability. If we plot the means against standard deviations in the coordinate system, we obtain the result shown in Figure 1. The ﬁgure shows that the concept of ornamentality has some focal points (here at the extremes, i.e. at 1 and 10) and a zone of uncertainty which is the greater the more distant the judgement is from both extremes. Figure 2, in which the data are displayed as <mean, complexity>, shows the same image. However, if only different calligraphic types or different letters in the same script had been presented, the “Parchment” letter “A” would have displayed different stability. Hence our experiment is preliminary. It merely shows that the “ornamentality” like all other properties is a measurable concept which displays the usual fuzziness. In Figure 1, the last point is <10,0>. The points are not distributed normally. A curve ﬁtted to the data would have the form of an irregular bow. Since we have no theoretical approach – and none is known – the course of the curve can preliminarily be captured by a cubic polynomial: y = 0.52446499x − 0.005364257513x3 , where x is the mean ornamentality and y is the instability of the judgement (standard deviation), yielding D = 0.78 and highly signiﬁcant t- and F-tests. Script ornamentality 95 Table 2: Ordering the scripts according to the mean ornamentality Name Century Gothic Arial Narrow Lucida Fax Papyrus Courier Stencil Bradley Hand ITC Cooper Black Freestyle Script Broadway Chiller Euclid Math Two Snap ITC Colonna MT Wide Latin Kristen ITC Informal Roman Jokerman Viner Hand ITC Ravie Vladimir Script Brush Script MT Lucida Calligraphy ITC Zapf Chancery Coronet Monotype Corsiva Pristina Magneto Vivaldi Matura MT Script Capitals Harlow Solid Italic Algerian Harrington Gigi Kunstler Script Old English Text MT Blackadder ITC Edwardian Script ITC Parchment Mean St.dev. 1.0196 1.0588 1.3725 1.6471 1.8627 2.1273 2.1569 2.4314 2.4902 2.6667 2.6863 2.7843 2.9804 2.9804 3.0784 3.1373 3.3726 3.6078 3.6275 3.7059 3.7451 3.9804 4.0196 4.1373 4.4902 4.5098 4.9804 5.3529 5.5294 5.5686 5.6471 5.6863 5.7843 6.6275 6.6667 7.8824 8.2549 8.5098 10.0000 0.1386 1.4728 0.5928 0.8360 0.7926 1.1885 1.1267 1.3322 1.0914 1.6647 1.3790 1.4728 1.3059 1.5402 1.6785 1.8789 1.4681 1.7044 1.6799 1.8396 1.6668 1.7766 1.7986 1.6571 1.6133 1.7078 1.9553 1.8131 1.9236 1.7960 1.9686 2.1827 2.0705 1.6564 1.6881 1.3669 1.3981 1.2738 0.0000 Grade sum 52 54 70 84 95 109 110 124 127 136 137 142 152 152 157 160 172 184 185 189 191 203 205 211 229 230 254 273 282 284 288 290 295 338 340 402 421 434 510 96 Karl-Heinz Best and Gabriel Altmann 2,5 Standard Deviation 2,0 1,5 1,0 0,5 0,0 0 2 4 6 8 10 Mean Figure 1: <Mean, standard deviation> of ornamentality judgements In any case, this is not a model of the examined dependence, merely a ﬁtting of an empirical curve. The stability of judgments is stronger at the extremes but decreases towards the middle of the domain. The other direction, namely from stability to mean does not bring anything new, no curve can be ﬁtted. It can easily be seen that the above polynomial is the ﬁrst part of the series expansion of a sine function. 2 Ornamentality and complexity In chapter 1 we mentioned the possibility of a correlation of ornamentality with complexity. In general, this relationship is not present (a machine can be very complex but not ornamental), but in script there is a possibility that the surplus to the necessary distinction may be an element of ornamentality. Of course, the correlation may depend also on the chosen complexity measure. Here we shall use Altmann’s proposal (2005) to measure complexity, leaving the Peust-method (2006) and the fractal dimension (Köhler, this volume, pp. 115ff.) untested. The means and the complexities of individual letters are in Table 3. As we can see, complexity increases with increasing ornamentality. Since there are no theoretical approaches as of yet, we preliminarily Script ornamentality 97 capture the dependence by an empirical curve y= c , d − aebx yielding in our case y= 4.7569 0.30749 − 0.041605e0.18975x with the values given in the last column of Table 3. In Figure 2, the complexity points and the ﬁtted curve can be observed. 180 160 140 Complexity 120 100 80 60 40 20 0 0 2 4 6 8 10 12 Mean Ornamentality Figure 2: Mean ornamentality (x) and complexity (y) At least a part of ornamentality can be accounted for by complexity, but intuitively we know that in the case of script it is complexity that arises because of ornamentality. Thus we can state that these two properties are interrelated, but it is not reasonable to consider one of them as primary. Complexity can arise also by the requirement of distinctivity, e.g. in Chinese script, where it would be impossible to devise thousands of well distinguished and at the same time non-complex signs (cf. Bohn 1998, Yu 2001). Hence all properties of script seem to build a control cycle similar to Köhler’s system. From case to case one of the properties dominates and reduces the importance of some other ones, e.g. in calligraphic scripts like “Parchment” ornamentality 98 Karl-Heinz Best and Gabriel Altmann Table 3: Ornamentality means and complexities of individual letter types Type Century Gothic Arial Narrow Lucida Fax Papyrus Courier Stencil Bradley Hand ITC Cooper Black Freestyle Script Broadway Chiller Euclid Math Two Snap ITC Colonna MT Wide Latin Kristen ITC Informal Roman Jokerman Viner Hand ITC Ravie Vladimir Script Brush Script MT Lucida Calligraphy ITC Zapf Chancery Coronet Monotype Corsiva Pristina Magneto Vivaldi Matura MT Script Capitals Harlow Solid Italic Algerian Harrington Gigi Kunstler Script Old English Text MT Blackadder ITC Edwardian Script ITC Parchment Ornamentality mean 1.0196 1.0588 1.3725 1.6471 1.8627 2.1273 2.1569 2.4314 2.4902 2.6667 2.6863 2.7843 2.9804 2.9804 3.0784 3.1373 3.3726 3.6078 3.6275 3.7059 3.7451 3.9804 4.0196 4.1373 4.4902 4.5098 4.9804 5.3529 5.5294 5.5686 5.6471 5.6863 5.7843 6.6275 6.6667 7.8824 8.2549 8.5098 10.0000 Complexity 12 12 21 13 22 13 19 21 17 13 15 34 23 20 23 17 24 28 19 25 18 16 22 20 17 20 22 18 29 18 25 81 23 30 33 34 42 40 159 Script ornamentality 99 dominates pulling along complexity but reducing distinctivity (reading ease, decoding economy). It is to be noted that the properties of letters need not be always homogeneous. Analysing letters other than “A”, we could obtain other ranking, but the general results would probably hold true. 3 Conclusion Ornamentality, distinctivity and complexity of a script are interrelated properties. The present article merely gives hint for further examination, not ﬁnal results. Further investigations must analyse other scripts and script types, propose better and more voluminous questionnaires, compare different scripts and test participant reactions, combine the study with psychological methods, taking into account economy, perception and learning ease, etc. And last but not least, theoretical considerations must be done in order to get models of the above mentioned phenomena. Here we wanted merely to show the existence of the problem. 100 Karl-Heinz Best and Gabriel Altmann Table 4: The evaluation of ornamentality by individual test participants Type Judgements Wide Latin 6, 4, 6, 1, 2, 4, 2, 4, 2, 1, 2, 1, 2, 4, 2, 4, 5, 2, 2, 2, 3, 5, 2, 2, 3, 5, 3, 1, 2, 1, 3, 2, 5, 3, 2, 2, 2, 2, 5, 5, 3, 5, 7, 8, 2, 1, 5, 4, 1, 3, 2 5, 8, 4, 6, 5, 6, 8, 3, 2, 2, 3, 7, 3, 3, 7, 5, 3, 3, 4, 4, 2, 7, 3, 3, 4, 5, 3, 3, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 6, 2, 4, 3, 3, 3, 4, 3, 6, 4, 2, 2, 4 10, 8, 8, 5, 6, 9, 10, 5, 3, 4, 5, 6, 6, 5, 7, 8, 9, 4, 7, 5, 4, 5, 4, 7, 5, 6, 5, 4, 5, 3, 3, 3, 8, 4, 4, 3, 3, 4, 6, 3, 7, 6, 5, 9, 7, 3, 5, 6, 6, 4, 5 6, 6, 5, 5, 5, 5, 10, 2, 2, 2, 4, 4, 7, 4, 6, 4, 3, 2, 4, 3, 2, 6, 3, 5, 3, 4, 3, 2, 4, 3, 2, 3, 3, 2, 2, 2, 2, 3, 5, 2, 6, 3, 2, 3, 4, 2, 4, 5, 2, 2, 2 6, 3, 3, 1, 2, 5, 3, 3, 1, 1, 2, 3, 1, 2, 2, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 5, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 2, 4, 2, 1, 5, 2, 1, 1, 3, 2, 2, 1 7, 3, 6, 2, 2, 4, 4, 4, 2, 1, 2, 3, 4, 3, 2, 5, 2, 3, 3, 4, 3, 3, 3, 4, 2, 5, 3, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 4, 3, 5, 4, 2, 6, 3, 2, 1, 2, 3, 3, 1 7, 3, 7, 2, 3, 8, 7, 4, 3, 1, 2, 5, 8, 4, 5, 5, 3, 3, 4, 6, 4, 4, 4, 5, 2, 6, 3, 1, 2, 3, 2, 2, 4, 3, 4, 2, 2, 4, 4, 2, 6, 6, 1, 6, 3, 2, 1, 2, 3, 4, 2 9, 5, 6, 6, 4, 5, 7, 5, 3, 4, 3, 7, 9, 5, 6, 7, 9, 4, 6, 5, 4, 7, 7, 4, 2, 6, 7, 3, 5, 4, 3, 3, 9, 6, 3, 3, 3, 4, 6, 2, 6, 4, 5, 8, 4, 3, 2, 7, 4, 3, 2 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10 1, 4, 2, 1, 2, 2, 2, 2, 2, 1, 5, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 4, 2, 1, 1, 2, 2, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1 Vladimir Vivaldi Viner Hand ITC Stencil Snap ITC Ravie Pristina Parchment Papyrus (continued on next page) Script ornamentality 101 Table 4 (continued from previous page) Type Judgements Old English text MT 9, 7, 9, 7, 7, 9, 9, 6, 7, 7, 7, 9, 9, 6, 8, 10, 10, 9, 9, 8, 6, 8, 9, 7, 8, 8, 8, 9, 8, 9, 8, 10, 9, 5, 6, 8, 8, 8, 4, 8, 8, 8, 9, 8, 7, 4, 9, 7, 8, 8, 10 1, 4, 5, 6, 6, 5, 4, 5, 3, 5, 6, 7, 8, 4, 3, 5, 4, 4, 6, 7, 5, 7, 7, 2, 6, 3, 4, 3, 4, 4, 4, 5, 8, 7, 3, 4, 3, 3, 6, 2, 3, 6, 4, 8, 4, 3, 3, 3, 3, 3, 2 9, 4, 7, 6, 6, 7, 7, 5, 5, 5, 5, 4, 9, 4, 7, 8, 6, 4, 7, 6, 5, 7, 8, 6, 6, 7, 4, 4, 4, 5, 4, 6, 10, 8, 3, 3, 5, 3, 7, 3, 6, 6, 7, 9, 3, 4, 4, 5, 3, 4, 4 9, 2, 6, 5, 5, 7, 8, 5, 5, 5, 5, 4, 9, 4, 6, 8, 5, 4, 7, 6, 4, 6, 8, 6, 6, 7, 3, 4, 4, 5, 4, 6, 10, 8, 3, 3, 5, 3, 7, 3, 5, 6, 4, 8, 3, 4, 5, 6, 3, 4, 5 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1 2, 2, 5, 6, 4, 5, 7, 6, 3, 3, 5, 6, 3, 3, 6, 5, 4, 4, 6, 5, 3, 7, 7, 2, 2, 4, 4, 4, 3, 2, 2, 4, 8, 6, 3, 4, 4, 3, 3, 2, 3, 3, 2, 8, 2, 3, 3, 8, 2, 2, 2 5, 4, 7, 7, 7, 6, 9, 7, 2, 5, 6, 8, 8, 5, 8, 8, 9, 6, 7, 7, 5, 8, 9, 5, 5, 7, 7, 9, 8, 6, 3, 9, 9, 9, 6, 5, 7, 7, 6, 3, 7, 5, 8, 9, 8, 5, 6, 8, 7, 6, 7 10, 1, 4, 2, 6, 3, 3, 3, 2, 1, 5, 3, 7, 2, 4, 3, 2, 2, 3, 6, 3, 4, 5, 2, 2, 4, 3, 2, 1, 2, 2, 2, 7, 6, 2, 3, 3, 2, 4, 1, 3, 3, 1, 7, 1, 2, 3, 2, 3, 2, 1 4, 2, 8, 3, 7, 6, 4, 5, 2, 1, 4, 3, 5, 3, 4, 3, 3, 3, 2, 6, 4, 5, 5, 4, 2, 6, 3, 1, 2, 2, 3, 2, 7, 6, 3, 3, 3, 3, 5, 1, 3, 4, 2, 7, 1, 3, 3, 4, 4, 4, 1 2, 4, 5, 6, 7, 5, 5, 5, 3, 3, 5, 7, 4, 3, 4, 5, 4, 4, 6, 5, 4, 7, 7, 2, 2, 3, 4, 3, 3, 3, 3, 3, 9, 6, 3, 4, 4, 3, 5, 2, 3, 4, 5, 8, 3, 2, 2, 3, 4, 3, 2 5, 3, 4, 3, 4, 4, 3, 4, 3, 2, 5, 6, 7, 3, 7, 3, 2, 3, 5, 5, 3, 6, 6, 3, 2, 6, 3, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 2, 5, 1, 3, 3, 3, 3, 3, 3, 1, 2, 2, 4, 1 10, 4, 7, 7, 5, 9, 5, 5, 4, 3, 5, 8, 9, 6, 8, 7, 9, 5, 7, 7, 5, 9, 7, 8, 5, 7, 6, 4, 4, 4, 3, 6, 10, 5, 4, 3, 5, 3, 5, 2, 8, 4, 9, 8, 5, 5, 4, 4, 6, 5, 2 Monotype Corsiva Matura MT Script Capitals Magneto Lucida Fax Lucida Calligraphy Kunstler Script Kristen ITC Jokerman ITC Zapf Chancery Informal Roman Harrington (continued on next page) 102 Karl-Heinz Best and Gabriel Altmann Table 4 (continued from previous page) Type Judgements Harlow Solid Italic 10, 2, 7, 6, 6, 7, 9, 5, 5, 3, 7, 4, 8, 5, 5, 8, 5, 4, 8, 6, 6, 7, 8, 6, 5, 7, 3, 4, 4, 5, 4, 5, 10, 8, 4, 3, 5, 3, 6, 3, 6, 7, 2, 9, 8, 5, 6, 3, 6, 6, 4 6, 5, 5, 7, 8, 4, 8, 7, 5, 8, 6, 8, 10, 5, 8, 9, 10, 5, 8, 7, 6, 8, 8, 4, 6, 6, 7, 8, 8, 6, 7, 8, 9, 8, 4, 5, 6, 6, 7, 3, 8, 6, 8, 7, 8, 4, 7, 3, 5, 7, 6 2, 1, 3, 2, 3, 3, 2, 3, 2, 2, 5, 5, 3, 3, 3, 2, 2, 2, 3, 4, 2, 5, 5, 4, 2, 3, 2, 1, 1, 2, 2, 2, 3, 4, 2, 2, 2, 2, 4, 1, 3, 3, 1, 2, 2, 3, 2, 1, 2, 1, 1 5, 1, 3, 1, 3, 6, 3, 3, 1, 1, 2, 3, 2, 4, 5, 3, 2, 3, 2, 2, 3, 5, 4, 3, 2, 4, 3, 1, 4, 1, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 2, 8, 5, 2, 2, 1, 2, 5, 4, 1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 7, 6, 8, 10, 10, 8, 10, 8, 9, 9, 6, 9, 10, 7, 9, 10, 10, 9, 9, 7, 8, 9, 9, 8, 8, 8, 8, 9, 9, 8, 8, 9, 10, 9, 5, 7, 8, 7, 9, 4, 8, 8, 10, 9, 9, 6, 8, 5, 8, 9, 10 1, 1, 3, 3, 6, 2, 3, 3, 2, 1, 5, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 5, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 4, 1, 2, 4, 2, 2, 1, 2, 2, 1, 2, 1, 1 7, 2, 4, 1, 5, 7, 2, 2, 1, 1, 1, 2, 2, 4, 2, 3, 1, 2, 1, 5, 2, 8, 2, 3, 2, 5, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 4, 2, 5, 1, 1, 4, 4, 2, 3, 2 6, 2, 5, 3, 5, 4, 7, 4, 3, 3, 3, 7, 5, 4, 8, 5, 9, 3, 5, 4, 4, 6, 7, 3, 2, 3, 3, 2, 4, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 5, 6, 8, 3, 2, 6, 2, 3, 3, 2 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 6, 3, 6, 2, 7, 3, 2, 3, 2, 2, 3, 4, 2, 4, 2, 4, 2, 2, 4, 3, 3, 4, 4, 3, 1, 3, 2, 1, 4, 2, 2, 1, 5, 2, 2, 2, 2, 2, 5, 1, 3, 3, 1, 2, 1, 2, 1, 2, 2, 2, 1 5, 2, 5, 2, 6, 6, 4, 4, 1, 1, 2, 3, 1, 5, 3, 3, 2, 3, 2, 3, 4, 6, 3, 3, 3, 4, 3, 1, 2, 1, 2, 1, 3, 2, 4, 2, 4, 2, 4, 2, 3, 3, 7, 5, 1, 1, 0, 3, 5, 3, 1 Gigi Freestyle script Euclid Math Two Arial Narrow Blackadder ITC Bradley Hand ITC Broadway Brushscript Century Gothic Chiller Colonna (continued on next page) Script ornamentality Table 4 (continued from previous page) Type Judgements Cooper Black 2, 1, 4, 1, 7, 2, 2, 3, 1, 1, 3, 2, 1, 3, 2, 3, 2, 2, 2, 4, 3, 6, 3, 2, 2, 3, 3, 1, 1, 2, 2, 1, 6, 1, 2, 2, 2, 3, 3, 2, 2, 3, 2, 5, 1, 2, 0, 3, 3, 3, 1 5, 3, 6, 5, 6, 6, 8, 7, 2, 3, 3, 7, 4, 4, 7, 6, 4, 3, 6, 6, 4, 8, 7, 4, 4, 3, 4, 3, 4, 3, 3, 3, 3, 5, 6, 3, 3, 4, 4, 1, 4, 4, 7, 4, 5, 4, 5, 6, 5, 2, 3 1, 1, 4, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 3, 1, 2, 2, 1, 5, 1, 1, 0, 2, 2, 3, 2 9, 6, 9, 10, 10, 7, 9, 8, 7, 9, 8, 9, 10, 5, 8, 10, 10, 8, 9, 9, 8, 10, 9, 7, 9, 8, 8, 10, 9, 9, 9, 9, 10, 9, 7, 6, 9, 8, 9, 4, 8, 8, 10, 9, 9, 8, 9, 9, 9, 9, 9 8, 2, 6, 7, 9, 6, 5, 6, 4, 2, 6, 5, 9, 4, 7, 9, 5, 5, 6, 6, 6, 8, 4, 6, 9, 5, 6, 2, 6, 4, 3, 4, 7, 10, 2, 4, 8, 3, 7, 2, 7, 3, 9, 8, 5, 2, 6, 8, 7, 8, 4 Coronet Courier Edwardian Script ITC Algerian 103 104 Karl-Heinz Best and Gabriel Altmann References Altmann, Gabriel 2004 “Script complexity.” In: Glottometrics, 8; 68–73. Bense, Max 1969 Einführung in die informationstheoretische Ästhetik. Grundlegung und Anwendung in der Texttheorie. Reinbek: Rowohlt. Birkhoff, George D. 1933 Aesthetic measure. Cambridge: Harvard University Press. Bohn, Hartmut 1998 Quantitative Untersuchungen der modernen chinesischen Sprache und Schrift. Hambung: Kovaˇc. Köhler, Reinhard 2006 “The fractal dimension of script: an experiment”. This volume, pp. 115– 119. Peust, Carsten 2006 “Script complexity revisited.” In: Glottometrics, 12; 11–15. Yu, Xiaoli 2001 “Zur Komplexität chinesischer Schriftzeichen.” In: Göttinger Beiträge zur Sprachwissenschaft, 5; 121–129. On the decrease of complexity from hieroglyphs to hieratic symbols Ina Hegenbarth-Reichardt and Gabriel Altmann 1 Introduction The development of script is not necessarily a step towards simplicity. The form of the script depends on the writing instruments, the material to write on, and on the purpose of the document. Different calligraphic scripts are more complex than their original models and even present-day computer fonts differ considerably in complexity. However, in general we expect that a “natural” transition of a script from its iconic shape to a symbolic form is accompanied by some kind of simpliﬁcation, as was the case in the development of the hieratic script from the Egyptian hieroglyphs (Altenmüller 2005: 76). But scholars always stress the serious problems when the hieratic script is transferred in “egyptological” hieroglyphic writing (Junge 1996) for their studies. Because of that, in 1929 Gardiner felt the necessity to set up rules for those “transcriptions”, especially for the Late Egyptian texts. The earliest hieroglyphs were found in the necropolis of Abydos and can be dated to the predynastic period of Naqada III (ca. 3300-3050 BCE). The hieratic script was formed very early out of these hieroglyphic signs as a cursive script for the Ancient Egyptian administration; possibly both scripts were developed at the same time (Satzinger 1977: 187). Furthermore, they both inﬂuenced each other throughout their existence in Egyptian history: while the hieroglyphs got more signs, which were transferred into the hieratic, the hieratic script itself created its own signs, which were adopted by the hieroglyphic script. In contrast to the “sacred” character of the hieroglyphs as a script mainly for monuments and stelae, the hieratic writing was used for (more or less) profane texts written on “movable” materials. This is why Egyptian scribes started their studies by learning hieratic and most of them, who often worked in the Egyptian administration, could only read and write this cursive script during their lifetime. Only a few, very specialized and well skilled scribes learned to read and write the hieroglyphic script. Furthermore, most hiero- 106 Ina Hegenbarth-Reichardt and Gabriel Altmann glyphs were carved in stone by artists; sometimes their more cursive variants were written with black and red ink on papyrus (standard texts for the monuments, but also the “Book of the Dead”) or on stone(walls), for example the walls of the sarcophagus-chambers in the early royal tombs in the Valley of the Kings. While representative texts were written in hieroglyphs on the monuments (in order to be there for eternity), hieratic texts were written on papyrus, leather, wood and ostraca of stone or pottery. All the texts were written in black (and red) ink and with a small reed brush. Therefore, the scribe had to chew on the pointed end of a small piece of reed (Schlott 1989). Considering these facts, one needs to know that cursive scripts and especially the hieratic one show many variants, not only because of local differences or the various handwritings, but mainly because the script was used for such a long time during Egyptian history. There were many changes and even reforms (Altenmüller 2005: 77) regarding both the language and the script, which are also mirrored in the hieratic texts. Figure 1 shows the many variants of one hieroglyph in hieratic script. Figure 1: The development of the sign mn from the Old Kingdom to the beginning of the New Kingdom In this ﬁgure the development of the hieroglyph (senet-game) with the phonetic value mn (Gardiner list Y5) is shown from the end of the Old Kingdom to the beginning of the New Kingdom. Apparently, there are many variants, not only through the ages, but also in the scripts of the same period of time. In order to show the extent of simpliﬁcation, the complexity of both the hieroglyphs and the corresponding hieratic signs must be measured. This can be accomplished in many different ways: one can count the number of strokes (as it has been done in the case of Chinese or Accadian scripts), one can use some measures developed in linguistics (cf. Altmann 2004, Peust 2006), or those used in fractal geometry (like capacity dimension, information dimension, correlation dimension, probability dimension etc.), or some special complexity measures whose number grows from year to year. The choice is not easy but we prefer a method which can be used without programming or software in order to make it available for those Egyptologists who do not use a computer daily. On the decrease of complexity 107 The second problem is the variability of the hieratic script. It is true that handwriting differs from one person to next and the variability is inﬁnite (even for one person), but usually the complexity of identical symbols does not differ signiﬁcantly. However, with the hieratic script which itself developed and diversiﬁed geographically, the variation is enormous. Hence we necessarily must concentrate on one variant only and we choose here the old hieratic manuscript of Abusir ascribed to the ﬁfth dynasty. This papyrus appeared in 1893 in Cairo, where it was offered for sale by antiquity-dealers. Most of its sheets were brought to Europe and only a few (small and damaged) sheets stayed in Egypt and are now in the National Museum of Cairo. The papyrus originally belonged to the pyramid complex, more precisely to the mortuary temple of Neferirkare (third king of the ﬁfth dynasty) at Abusir. It primarily contains bills of the sacriﬁces and of the distributions of food in that temple at the time of the king Asosi (the last but one king of the ﬁfth dynasty). The manuscript does not contain all subsequent hieratic forms of hieroglyphs but since our source (Möller 1908/1927/1965) divides the symbols in 26 groups we randomly take one example from each group if the Abusir variant is present. The method can be used for scrutinizing the change of complexity in all periods. In the case of hieroglyphs, we have usually taken the contours into consideration, as opposed to the strokes in the case of hieratic symbols. Some distorted places in hieroglyphs have been slightly modiﬁed but these small “improvements” do not encroach on the generality of the results. The hieratic symbols were not considered as joined areas but as joined strokes resulting from the way of writing. 2 The method It is sufﬁcient initially to use a simple method which can be performed with pencil and paper in order to make the ﬁrst step in the formal corroboration of the hypothesis that graphical iconism leads to simpliﬁcation as it passes to symbolism. Later on, when software is available, the complete development of script can be studied from this point of view and more sophisticated measures from fractal geometry should be used. Here we shall adhere to the simple evaluation system proposed by Altmann (2004) in which different graphical ﬁgures and their connections are scaled as shown in Table 1. 108 Ina Hegenbarth-Reichardt and Gabriel Altmann Table 1: Evaluation system dot: straight line: arch: continuous joint: crisp joint: crossing: 1 point 2 points 3 points 1 point 2 points 3 points For the sake of illustration, a square has four straight lines (2 · 4) and four crisp joints (2 · 4) yielding a complexity of 16 points; a circle has two arches (2 · 3) and two continuous joints (2 · 1) yielding 8 points (an arch cannot exceed 180°; a continuous joint is a smooth continuation of a curve; a crisp joint is a point at which two curves meet forming a cusp). A dot can be dimensionless; it can be ﬁlled or it can be of any form, but it must be very small in comparison to other ﬁgures. The results of computation of complexity of 20 randomly chosen signs from different domains of Egyptian life are presented in Table 2 (p. 109). As can easily be seen, the hieroglyphic signs are in all cases more complex than their hieratic counterparts. The difference is somewhat drastic but other methods could perhaps show that the difference in complexity is in some aspects not so abysmal. But even the inner variation of hieratic signs is very great. In any case, this evaluation is preliminary. As can easily be seen, the hieroglyphic signs are in all cases more complex than their hieratic counterparts. The difference is somewhat drastic but other methods could perhaps show that the difference in complexity is in some aspects not so abysmal. But even the inner variation of hieratic signs is very great. In any case, this evaluation is preliminary. Any ordering of signs according to their complexity would be premature. Undoubtedly, we see that the hieroglyphic script is very complex. However, the hieratic script seems to get the dimension of a usual alphabetical script (though it is not alphabetical at all, because, like the hieroglyphs, it consists of ideograms, phonograms and determinatives). Though our sample is preliminary and small, we can compute the mean complexity from Table 3 yielding Mhieratic = 17.50, with the empirical variance VMean,hieratic = 106.35/20 = 5.3 and Mhieroglyphs = 53.40, VMean,hieroglyphs = 700.74/20 = 35.04. These values were compared asymptotically. Since the difference in the means is enormous, we get z = 5.65 for a two-sided test showing that the decrease is highly signiﬁcant. On the decrease of complexity 109 Table 2: Complexities of hieroglyphs and the corresponding hieratic symbols No. in Möller Hieroglyph Hieratic (Abusir) hieroglyph Complexity hieratic symbols 15 109 22 61 32 18 79 67 12 129 65 23 148 75 55 192 101 19 233 36 17 250 34 13 263 40 12 269 33 10 315 56 29 350 54 22 374 45 22 421 74 12 444 37 8 465 40 12 495 13 8 522 28 15 537 101 8 552 28 13 110 Ina Hegenbarth-Reichardt and Gabriel Altmann Comparing hieratic with the two scripts analyzed by Altmann (2004), we obtain MArial = 9.65 and VMean,Arial = 15.303/26 = 0.5886, yielding z = 3.23, which is signiﬁcant. However, for the mean of the Courier font with MCourier = 17.81 and VMean,Courier = 1.361, the test for the difference of means yields z = 0.120, i.e. hieratic and Courier have almost identical complexity. The present article brings hints rather than ﬁrm results. Not only that the inventory of symbols used for the computation must be signiﬁcantly enlarged in the future, but the methods of computing complexity must be variegated. Even whole Egyptian manuscripts could be compared using the current methods of image resolution. However, our aims are linguistic and historical. 3 The simpliﬁcation curve The fact that the hieratic script has a mean complexity comparable to a simple Latin script results automatically in the next hypothesis, namely that the simpliﬁcation of hieroglyphs is not linear but follows another trend. In other words, it is not necessarily true that the more complex a hieroglyph, the more complex its hieratic simpliﬁcation. In a simpliﬁcation process there are two circumstances that must be taken into account: 1. the complexity of hieroglyphs is not inﬁnite though they may be quite complex (compare the Maya script); 2. if a simpliﬁcation arrives at a (fuzzy) boundary at which the given resulting simpliﬁed hieratic symbol would be more complex than a simple hieroglyphic symbol, the linear relationship between hieroglyph complexity and hieratic simpliﬁcation must change drastically. The slope of the representing curve ceases to be constant, i.e. the acceleration cannot simply be disregarded, and the curve must be modiﬁed. This can be done in different ways. The ﬁrst simple way is to take into consideration the second derivation (acceleration) of the curve resulting in the differential equation d2y + ay = 0 (1) dx2 where a is a positive parameter. The solution yields y = c1 sin (bx) + c2 cos (bx) (2) On the decrease of complexity 111 √ where b = a. In order to test this hypothesis, we order the data from Table 2 according to the complexity of the hieroglyphs (x) and take averages in the given intervals. The result is presented in Table 3. Table 3: The extent of simpliﬁcation depending on hieroglyph complexity Interval Mean hieroglyph complexity Observed hieratic complexity Computed hieratic complexity 10-19 20-29 30-39 40-49 50-59 60-69 70-79 100-109 14.50 24.50 34.50 44.50 54.50 64.50 74.50 104.50 8.00 14.00 13.20 15.30 25.50 17.50 33.50 16.33 7.38 12.09 16.25 19.66 22.17 23.67 24.07 18.84 For our data it can be shown by means of a software (N LREG) that the particular solution y = a·sin (bx) is sufﬁcient, yielding y = 24.0815 sin (1.2299x) where both the t-tests for the parameters and the F-test for the regression are signiﬁcant. The variable x in the formula has been transformed in x = x · π/2(104.5 − 14.5), or x = x · π/2(90), because of different phases and different period. The graphical representation is displayed in Figure 2. Simplification (hieratic complexity) 40 30 20 10 0 0 20 40 60 80 100 120 Complexity (mean hieroglyph complexity) Figure 2: The dependence of the extent of simpliﬁcation on complexity 112 Ina Hegenbarth-Reichardt and Gabriel Altmann The above physical or geometrical way of derivation can be substituted by a least-effort (cf. Zipf 1949) model known from quantitative linguistics (cf. Job & Altmann 1985). The same argumentation leads to the conclusion that the relative rate of change of the simpliﬁcation curve is proportional to the difference between the relative rate of change of hieroglyph complexity and that of simpliﬁcation endeavor, because a special upper boundary of complexity (after simpliﬁcation) must not be surpassed. We can write this argument formally as dy a b = − dx, (3) y x K −x yielding the solution y = cxa (K − x)b , 0 < x < 105 . (4) Since the greatest x-value is 104.5, we set K = 105 and using optimization, we obtain the curve y = 0.1823x1.0036 (105 − x)0.20004 , which can be “rounded” to y = 0.1823x(105 − x)0.2 . The ﬁt is slightly better than using the sinus curve. The result is displayed in Table 4 and graphically in Figure 3. Table 4: Fitting (1) to hieratic simpliﬁcation Interval Mean hieroglyph complexity Observed hieratic complexity Computed hieratic complexity 10-19 20-29 30-39 40-49 50-59 60-69 70-79 100-109 14.50 24.50 34.50 44.50 54.50 64.50 74.50 104.50 8.00 14.00 13.20 15.30 25.50 17.50 33.50 16.33 6.57 10.87 14.93 18.69 22.10 25.04 27.34 16.87 The second approach predicts zero complexity of the hieratic sign if the hieroglyph attained a complexity greater that 105. This prediction is, however, not realistic. There are more complex hieroglyphs. In that case, one simply shifts the upper limit K appropriately to the right. The hypothesis is, of course, weakly corroborated. More data and tests are necessary in other languages in order to be able to generalize it. In its general form, the hypothesis means that in script development, simpliﬁcation On the decrease of complexity 113 Simplification (hieratic complexity) 40 30 20 10 0 0 20 40 60 80 100 120 Complexity (mean hieroglyph complexity) Figure 3: The least effort model of simpliﬁcation goes parallel to complexity but only to a certain modal point. Thereafter, the simpliﬁcation is the more drastic the greater the complexity. Otherwise, the gain for the economy would not be sufﬁcient. References Altenmüller, Hartwig 2005 Einführung in die Hieroglyphenschrift. Hamburg: Buske. Altmann, Gabriel 2004 “Script complexity.” In: Glottometrics, 8; 68–73. Gardiner, Alan H. 1927 Egyptian grammar. Being an introduction to the study of hieroglyphs. 3rd ed. London: Oxford University Press. Job, Ulrike; Altmann, Gabriel 1985 “Ein Modell für anstrengungsbedingte Lautveränderung.” In: Folia Linguistica Historica, 6(2); 401–407. Junge, Friedrich 1996 Einführung in die Grammatik des Neuägyptischen. Wiesbaden: Harrassowitz. Möller, Georg 1927 Hieratische Paläographie. Die aegyptische Buchschrift in ihrer Entwicklung von der fünften Dynastie bis zur römischen Kaiserzeit. Osnabrück: Zeller. 114 Ina Hegenbarth-Reichardt and Gabriel Altmann Peust, Carsten 2006 “Script complexity revisited.” In: Glottometrics, 12; 11–15. Satzinger, Helmut 1977 “Hieratisch”. In: Helck, Wolfgang and Westendorf, Wolfhart (Eds.), Lexikon der Ägyptologie. Wiesbaden: Harrassowitz, 1187–1189. Schlott, Adelheid 1989 Schrift und Schreiber im Alten Ägypten. München: Beck’s Archäologische Reihe. Zipf, George Kingsley 1949 Human behavior and the principle of least effort. 2nd ed. New York: Hafner. The fractal dimension of script: an experiment Reinhard Köhler 1 Introduction To our knowledge, Hˇrebíˇcek (cf., e.g., 1995) was the ﬁrst to propose the measurement of fractal structures in language. The present paper makes an attempt at a corresponding measurement of the fractal dimension of written symbols. The dimension of regular geometric objects is well deﬁned and its calculation is straightforward. The Hausdorff dimension, for example, is obtained as the (limit of the) quotient of the logarithm of the initiator and the logarithm of the generator. A simple example is the fractal dimension of the Koch curve. The corresponding Hausdorff dimension is DH = lim r→0 log 4 log N = = 1.26 . . . log (1/r) log 3 (1) The measurement of the dimension of an object which is irregular (either because it is of a stochastic nature or because of its empirical origin) is more complicated. Several methods have been developed to estimate the dimension of such objects: Lyapunov exponents, which we, however, exclude from our considerations because they are too hard to calculate; the compass dimension, which is applicable only to time series; and three other measures called 1. correlation dimension, 2. Hull dimension, and 3. capacity dimension. Hull and capacity dimension measures can be calculated in two variants each: using block city or Euclidean geometry. Four dimension measures were tested on data describing letters from the Roman alphabet in different styles (fonts): correlation dimension using block city and Euclidean geometry, capacity dimension, and an accelerated version of the capacity dimension. The most stable and promising results were obtained with the capacity dimension. 2 Data The data for the experiment were collected and prepared in the following way. First, selected letters from the Roman alphabet (I, L, E, H, Q) and a Cyrillic 116 Reinhard Köhler letter were individually inserted into empty graphics ﬁles using a graphics editor. The letters were inserted in text mode, one letter per ﬁle, each letter in 72pt size. Style variants were stored separately. All letters were prepared at least in variants without serifs (Arial) and with serifs (Times). Some letters (I, H, Q) were prepared also in various ornamental styles. Then, the ﬁles were stored in 300 dpi BMP format, resulting in objects of 167 × 164 pixels up to 363 × 388 pixels, depending mainly on the width of the letters. A program was written to extract from these graphic ﬁles the coordinates of the non-white dots which formed the letters. The results of these conversions were stored and used as objects of the dimensional analysis. 3 Method In a ﬁrst study, the available methods were applied to the data. Table 1 shows the results of the calculations. As we can see, the different measures yield surprisingly different results, and only one of the methods seems to be appropriate, viz. the capacity dimension (Peitgen & Saupe 1988: 60f.; Hunt & Sullivan 1986). This measure is calculated iteratively, determining in each step the number of rectangles which contain at least one dot belonging to the object. The size of the rectangles is reduced stepwise by a contraction factor, which is a parameter of the procedure. Clearly, the number of rectangles which are checked during each iteration increases with the number of iterations and with the decreasing size of the rectangles. The capacity dimension is calculated as Ni−1 ln Ni di = , (2) ln δ where N is the number of rectangles containing at least one dot of the object, δ the contraction factor, and i the running number of iterations. For the fractal analysis, a previously written program AAFD was used, which implements the four above-mentioned methods and its two variants. All methods show a rather instable behaviour during the iteration process: There is no smooth asymptotic approach toward a ﬁnal value. Instead, the values vary, jump up and down from step to step until they ﬁnally disappear in a sudden fall. Therefore, in general, cumulative values or regression lines are used instead of the original dimension values, which avoid the extremes but follow, of course, the ﬁnal breakdown. Because of this, even the smoother The fractal dimension of script: an experiment 117 Table 1: Selected letters and stylistic variants with their fractal dimensions according to different measures and methods Correlation dimension Block City Iterations: Letter | I I L E E H O Ö W Capacity dimension (accelerated) Correlation dimension Euclid Capacity dimension 20 10 15 20 5 10 15 20 0.92 1.58 1.38 1.25 1.34 1.45 1.38 1.49 1.45 1.28 1.39 1.38 1.43 1.33 1.27 1.12 1.29 1.34 1.45 1.44 1.37 1.30 1.47 0.89 1.46 1.48 1.48 1.47 1.13 0.60 0.73 0.79 0.83 0.83 0.80 0.73 0.86 0.48 0.86 0.87 0.88 0.90 0.87 1.61 1.56 1.27 1.39 1.50 1.41 1.49 1.49 1.30 1.35 1.42 1.69 1.37 0.89 2.03 1.78 1.68 1.38 2.03 1.96 1.99 1.81 1.78 1.75 1.81 1.82 1.73 0.91 1.87 1.58 1.43 1.53 1.63 1.57 1.74 1.61 1.55 1.61 1.76 1.61 1.67 0.91 1.75 1.64 1.51 1.70 1.72 1.51 1.41 1.71 1.41 1.68 1.67 1.67 1.57 0.91 1.75 1.64 1.52 1.70 1.72 1.51 1.41 1.71 1.41 1.70 1.68 1.69 1.59 regression or cumulative values have to be observed individually, and in each case, the appropriate number of iterations has to be determined by the user during the process. The calculation can be automatised, therefore, only partially. Figure 1 shows screenshots of the program AAFD (result presentations). Figure 1a shows the graphic display for a typical case: the line which represents the dimension measure is the one which displays the most extreme upward and downward movements; the others represent regression and cumulation. The x-axis represents the scale, the y-axis the dimension calculated at the given step. Figure 1b gives the corresponding values as presented by the program. 118 Reinhard Köhler (a) Graphical representation of an iterative calculation (iterations from right to left) (b) Table of iteratively calculated dimension and smoothing values Figure 1: Screenshots of the program AAFD (result presentation dialogs) The data were also processed by another software package (NLyzer); in particular, the demo version available from the Technische Universität Darmstadt. The mathematical model implemented by this program is slightly different from ours, behaving roughly inversely. 4 Results and Preliminary Evaluation The various applicable dimension measures and the corresponding methods yield incompatible values and trends as results. Only one of them, viz. the capacity dimension, seems to be useful for the characterisation of symbols. The results are represented in Figure 2. The values obtained for the selected letters using the capacity dimension (cf. Figure 2a) correspond roughly to an intuitively assigned rank order of complexity. The same is true – in an inverse form – of the values obtained from the NLyzer (cf. Figure 2b). Only one of the letters, an ornamental version of an H, does not match this general pattern. However, the fractal dimension is certainly not a complexity measure. Whereas it is possible to consider a plane as more complex than a line, and a line as more complex than a point, it would be difﬁcult to ﬁnd good reasons to support the idea that the Cantor set with dimension 0.63 . . . is less complex than a line with dimension 1, or the Koch curve (dimension 1.26 . . .) less complex than a plane (dimension 2) etc. This is not an argument against the fractal dimension as an interesting characteristic of signs but rather a word of caution concerning possible in- The fractal dimension of script: an experiment 119 (a) Graph of the capacity dimension values (b) Graph of the dimension values calculated (program AAFD) by the NLyzer Figure 2: Fractal dimensions of selected characters as calculated by two different algorithms terpretations or even identiﬁcations. More experimentation and thinking will provide more insight. References Hˇrebíˇcek, Ludˇek 1995 Text Levels. Language Constructs, Constituents, and the MenzerathAltmann Law. Trier. Hunt, Fern Y.; Sullivan, Francis 1986 “Efﬁcient algorithms for computing fractal dimensions”. In: MeyerKress, Gottfried (Ed.), Dimension and entropy in chaotic systems. Berlin: Springer, 83–93. Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar 1988 The science of fractal images. New York / Berlin: Springer. On graphemic representation of the Oriya phonemes Panchanan Mohanty and Gabriel Altmann 1 Introduction Up to now only languages with the Latin script have been analysed as to the graphemic representation of their phonemes (cf. the contributions in this volume). The aims and the methods are frequently very different (cf. Venezky 1970; Katz & Frost 1992; Klima 1972; Van den Bosch & Daelemans 1993; Daelemans & Van den Bosch 1994; Van den Bosch et al. 1994), and many times the direction of description is opposite, i.e. from grapheme to phoneme, in order to enable automatic reading. The literature is sheerly enormous. The development of script in languages using Latin letters was relatively simple. If the number of phonemes in any of these languages was greater than that of the letters in the Latin alphabet or if there were sounds foreign to Latin, then either some letters were combined (e.g. dz, sch) or they were used with diacritics (e.g. ä, é, š, c´ ) to represent the additional phonemes. But one-to-one correspondence between the phonemes and the graphemes in these languages was a rarity. Some letters represented more than one phoneme, and sometimes one phoneme was represented by several letters. In this way, the weight of the participation of letters in graphemic representation changed. At the same time phonemes got a kind of graphemic uncertainty/variability/diversiﬁcation and a very regular distribution of representations. Such problems, perhaps, may not be found in scripts that did not arise from the Latin alphabet, but new ones do arise. This is the case e.g. in Oriya (and other Indian scripts), whose script is derived from Brahmi, having an inherent /@/ in all consonant graphemes. This /@/ is dropped in different conditions, e.g. in the case of non-ﬁnal consonants in a cluster, in a consonant that occurs word-ﬁnally, or when a consonant is followed by another explicit vowel. The situation is quite different in a language like Japanese that has syllabic or moraic scripts along with some special ways of expressing consonant reduplication, length, etc. In this paper we intend to discuss the Oriya phonemes and their graphemic representations, i.e. we start from phonemes and seek their graphemic expression (and not vice versa). Table 1 displays all of these, as well as examples for each representation. We are not interested in some possible practical aspects 122 Panchanan Mohanty and Gabriel Altmann of this relationship but in the possibility of expressing its properties numerically. We conjecture that there are some regularities in the development of script which can be better captured if everything is expressed numerically. We assume the existence of laws in this development but any generalization in this direction would be premature. Besides being a true representation of its phonemes, the Oriya script has some properties that are not found in the Latin scripts. Languages using either the Latin or the Greek script (with modiﬁcations) show a linear arrangement, i.e. the letters are placed in a linear array. On the other hand, languages like Korean use syllabic script and write the letters in a square. Thus, this script is syllabic-linear, but as to letters it is planar. It clearly uses two dimensions. The Japanese Kana-scripts are moraic-linear, but they do not have single letters (except ). Of course, each letter, syllable, or mora as a symbol is written in two dimensions. In Korean the sequence of letters need not be written linearly. There are two possibilities: to the right and below. In Oriya the situation is sightly more complicated. The letter-script is evidently planar and all possibilities are given. Of course, the positions of subsequent letters are prescribed and the sequence of phonemes is linear allowing the (Harary-Paper) distributional calculus of phonemes to be performed with usual means but the distributional calculus of graphemes would need a quite different evaluation procedure, as will be shown below. There are the following possibilities concerning the second grapheme in its relation to the ﬁrst: – no signalization – four different positions of the second grapheme – two different positions of the ﬁrst grapheme – doubling of the second grapheme – assimilation and creation of new graphemes – positional alternatives combined with modiﬁcations of shape and size. All these combinations are given in Tables 1 and 3. On graphemic representation of the Oriya phonemes 123 Table 1: Graphemic representation of the Oriya phonemes # = Serial no., P = Phoneme, G = Grapheme(s) P # G Examples A A> B D F H A’> /@da/ ‘ginger’ Vowels 1 2 3 4 5 6 @ a I Ú e o A>fç /ajI/ ‘today’ Bs> /Ita/ ‘brick’ ˙ DB /ÚI/ ‘white ant’ FL /ek@/ ‘one’ Hs /ot@/ ‘camel’ ˙ þœ /m@n@/ ‘mind’ À >f> /raja/ ‘king’ L ¯ ç /k@bI/ ‘poet’ L ëL ëÀ /kÚkÚr@/ ‘dog’ {Ó /se/ ‘s/he’ {þ>s> /mota/ ‘fat’ ˙ Consonants L 7 8 k kh 9 10 11 12 13 g gh c ch j 14 15 16 17 18 19 20 jh t ˙ th ˙ d ˙h d ˙ n ˙ t ä S W ` d f ¾ l s v x | ~ † 21 22 th d $ ’ ä L œ> /k@na/ ‘cloth’ Qs> /kh @ta/ ‘sour’ ˙ äêÀ /kh Ir@/ ‘milk’ Sd /g@ch@/ ‘tree’ WÀ /gh@r@/ ‘house/home’ ` Ì> /c@sa/ ‘farmer’ d¯ ç /ch @bl/ ‘picture’ fÀ /j@r@/ ‘fever’ ¾æœ /jan@/ ‘vehicle’ lúÀ ~> /jh @r@n˙a/ ‘stream’ sÓÀ /t@s@r@/ ‘silk’ v L /th˙@k@/ ‘cheat’ x À /d˙ @r@/ ‘fear’ ˙ h {| D /d eÚ/ ‘wave’ ¯ ~ /b@n˙ @/ ‘forest’ ˙ † Á /t@l@/ ‘below’ ˙ D‡s /Útk@t @/ ‘pungent’ $Á ç /th @lI/ ˙‘bag’ ˙ ’ À /d@r@/ ‘cost’ (continued on next page) † Ô À /t@sk@r@/ ‘thief’ ÕÁ œ /skh @l@n@/ ‘deviation’ ˙ ‘thin’ ÓíäR /sUkh m@/ þë’SÀ /mÚdg@r@/ ‘cudgel’ ¨Ê>†ú /p@scat/ ‘afterwards’ þfk> /m@jja/ ‘marrow’ Aƒ> /@nda/ ‘egg’ ˙˙ {þ„> /men dh a/ ‘sheep’ ˙ ¯ ‚ öœ æ /b@rn˙@na/ ‘description’ À N /r@kt@/˙ ‘blood’ L >¡ /kanth@/ ‘wall’ 124 Panchanan Mohanty and Gabriel Altmann Table 1 (continued from previous page) # P G Examples 23 24 25 26 dh n p ph ™ œ ¨ ü ™Á > /dh @la/ ‘white’ œ À /n@r@/˙ ‘man’ ¨’> /p@da/ ‘ﬁeld’ ü ~> /ph @na/ ‘hood’ 27 28 29 30 b bh m y 31 r 32 33 34 35 l ˙ l w s 36 h œ çþ§ /nImn@/ ‘low’ ß ˙ (of a snake) ¯ Á /b@l@/ ‘strength/force’ ˙ μ ß /bh @y@/ ‘fear’ þ† /m@t@/ ‘opinion’ /j@y@/ ‘victory’ fß Qþ´ /kh @mb@/ ‘pillar’ ’ » /d@mbh @/ ‘strong’ fœ k /j@nm@/ ‘birth’ Óμ¿ /s@bh y@/ ‘member/ À À Ó /r@s@/ ‘juice’ S¯ ö /g@rb@/ ‘pride’ H´>À ~s /war@nt/ ‘warrant’ ’´>À Ü >† fÜ § /j@hn@/ ‘moon’ ¯ μ þ Á à H´ Ó ÉÀ Ì„ Ü civilised’ μ ’ ÷ /bh @dr@/ ‘polite/decent’ fÁ /j@l˙@/ ‘water’ ¯ ç¨È¯ /bIpl˙@b@/ ‘revolution’ þà Èâ /m@ll@/ à † > /l@ta/ ‘creeper’ ‘wrestler’ ˙˙ Ó>œ /san@/ ‘small’ ÉÀ /s@r@/ ‘arrow’ Ì„ /s@ndh @/ ‘bull’ ˙˙ /hat@/ ‘hand’ /dwar@/ ‘door’ On graphemic representation of the Oriya phonemes 2 125 Uncertainty/Diversiﬁcation I As can be seen, phonemes have an independent and one or more dependent graphical forms. The ofﬁcial orthography exactly states how to write; hence the graphical representation is fraught rather with variation/diversiﬁcation than with uncertainty. In many cases, the phonemes themselves are fraught with uncertainty (e.g. in French, see Hug 1997) which is a common phenomenon in dialectology, especially in a language like English. The writing of an Oriya phoneme depends on the circumstances, speciﬁcally – on the neighbourhood. In any case, we have to work with a typical case of codiﬁed diversiﬁcation. In handwriting, diversiﬁcation is automatic since everybody has a different handwriting, but this is not so in print. Usually, this variation/uncertainty/diversiﬁcation (U) is characterized as the logarithm (with basis 2) of the number of variants x, i.e. Ux = log2 x. (1) In Table 2 one ﬁnds the phonemes, the number of representing graphemes, the uncertainty (we shall adhere to this concept) and the number of phonemes with x variants (cf. Best & Altmann 2005). Retroﬂex phonemes are written with capitals, x denotes the number of representing graphemes, Ux the degree of uncertainty, and fx the number of graphemes with uncertainty Ux . Table 2: Graphical uncertainty of Oriya phonemes Phoneme gh , jh , Th , ph , kh , /χ, g, p, h/ /a, U, e, o, c, ch , k, j, th , T, d, dh , N, n, b, bh , m, y, L, l, w/ / I, t, d, Dh , r, s/ ˙ ˙ x Ux 1 2 3 0 1 1.58 fx 9 21 6 The mean uncertainty computed from Table 2 yields U¯ I = [(0)9 + 1(21) + 1.58(6)]/36 = 0.8467. This can be compared with the known results for certain other languages: Italian = 0.564, Swedish = 0.797, Oriya = 0.847, German = 0.965. The difference between the languages can be tested either using the mean of x (which would be 1.92 here), or using the mean of Ux . The variance of Ux has been shown in Bernhard & Altmann (this volume, pp. 13ff.). We simply state that the orthographic uncertainty in Oriya lies between Swedish and German, hence Oriya does not represent an exception but rather corroborates the conjecture that some regularity is in the background. 126 Panchanan Mohanty and Gabriel Altmann 3 Uncertainty II The situation changes drastically if we consider not only the shape but also the placement of a grapheme in its relation to the preceding/following grapheme and its size. Here we must propose a scaling procedure giving different weights to different placements and sizes. One of the possible ways will be detailed below. Before that, let us capture the distribution of Oriya graphemes as displayed in Table 3. Here the symbols have the following meaning: – Z A B H F S b * = = = = = = = = = P = O = O1 = O2 = S1 = S2 = no bigram Zero realization of the vowel after the consonant letter Above the ﬁrst consonant Below the ﬁrst consonant beHind or to the rigHt of the ﬁrst consonant beFore or to the leFt of the ﬁrst consonant on both Sides of the ﬁrst consonant second grapheme used in small size with no change of shape the grapheme representing the ﬁrst consonant is used below the second one A special conjunct letter is used The grapheme representing the ﬁrst consonant is used above the second one The grapheme representing the ﬁrst consonant is used above the second one and a special grapheme is used for the ﬁrst one The grapheme representing the ﬁrst consonant is used above the second one and the second grapheme is doubled palatal sibilant letter retroﬂex sibilant letter On the basis of these facts, we can consider the Oriya script as written in different dimensions. The ﬁrst dimension is the usual direction of writing from left to right, the second dimension is above or below. We identify the counter direction (i.e. from right to left) as the third dimension, which cannot be displayed explicitly in planar writing. The fourth and the ﬁfth attributedimensions are the size and the shape. That is, the grapheme is characterized in the form of a vector G = [right, above or below, left, size, shape]. On graphemic representation of the Oriya phonemes 127 For measurement purposes we can introduce I binary decisions, i.e. occurrence (1) or not (0) in the given dimension, or II a scaling procedure combined with binary decisions, i.e. the dimensions obtain special values in correlation with learning and recognition difﬁculty. For example: G = [1, 2, 3, 4, 5], (2) i.e. a change in shape gets the value 5; a change in size, the value 4; and a change in direction, the value 3, etc. III Another possibility is to normalise each dimension, place it in interval [0, 1] and ascribe the graphemes some values according to the extent of modiﬁcation. In the ﬁrst three dimensions only the extreme values 0 and 1 are possible but the size and the shape modiﬁcations can be scaled. This scaling can be performed by anybody, not necessarily by an Oriyaknowing person. Now, whatever the values assigned to the individual dimensions of a grapheme may be, we can consider each value the grapheme coordinate if the grapheme is present in that dimension. We can thus determine the grapheme diversiﬁcation as a function of its coordinates. There are several possibilities how to characterize the diversiﬁcation of a grapheme with a single number. The most usual is the grapheme’s Euclidean distance from the origin – in this case in the ﬁve-dimensional space. The general approach would be to set up a vector for each grapheme x, G(x) = [d1 , d2 , d3 , d4 , d5 ] (3) and compute its distance from the origin as D(x) = 5 ∑ I j d 2j 1/2 , (4) j=1 where I j is the index of the dimension, I j = 1 if the grapheme is present in the dimension and I j = 0 otherwise. Since this must be done for n combinations with other graphemes representing individual phonemes, we can use at last the mean distance that represents the mean distortion of a grapheme in combinations with n other graphemes. The magnitude n can be different for individual graphemes and it depends on the number of graphemes with which the given one can be combined. 128 Panchanan Mohanty and Gabriel Altmann Let us illustrate procedures (I) and (II) with examples. Using method (I), we ascribe 1 to the presence and 0 to the absence of the grapheme in the given dimension. For example, if a grapheme x in second position is written below the preceding one (B in Table 3) without any other change, we could write (A) D(x) = (0 + 12 + 0 + 0 + 0)1/2 = 1 which is the minimal Euclidean distance from the origin in method (I), in which d j = 0.10. If x occurs on both sides of a grapheme, i.e. right and left, we get √ (B) D(x) = (1 + 0 + 1 + 0 + 0)1/2 = 2 = 1.41. If it occurs above the other grapheme and is modiﬁed in size, the result is (C) D(x) = (0 + 1 + 0 + 1 + 0) = 1.41. Using method (II), which evaluates the distance more intuitively, the more a grapheme deviates form its linear placement, shape and size, the greater the weight of the given dimension should be. Thus the ﬁrst proposal would be the following weighting: 0 1 2 – – – 3 4 5 – – – no representation of a grapheme, D(x) = 0 placed linearly behind (H) placed above (A) or below the ﬁrst (B); the ﬁrst is placed above (O) or below (*) the second placed before the ﬁrst (F) change of shape (O1 , O2 , S1 , S2 , P) change in size (b). If part of a subsequent grapheme is written on both sides of the “preceding” one it is posited in dimension 1 (H) and in dimension 3 (F) yielding (without shape and size changes): D(x) = (1 + 0 + 32 + 0 + 0)1/2 = 3.16. If the grapheme is implicit (Z), it gets the placement value 0. This is the case of /χ/ which gets a sign only if it builds an independent syllable, i.e. not occurring behind a consonant. The same is ascribed to a grapheme if it does not occur at all behind another consonant (–). In practice, it means that using this method we simply multiply the elements of the vector G in (2) by 1 in the case of presence and by 0 in the case of absence of a grapheme. Thus, from the above examples we would obtain (A’) D(x) = (0 + 22 + 0 + 0 + 0)1/2 = 2, (B’) D(x) = (1 + 0 + 32 + 0 + 0)1/2 = 3.16, (C’) D(x) = (0 + 22 + 0 + 42 + 0)1/2 = 4.47. @ a I U e o k kh g gh c ch j jh T Th D Dh N t th d dh n p ph b bh m y r L l w s h a H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H @ H H H Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z H H H H H H A,P A,B A A A,P A,P A,P A A A A,P A,P A A,P A,B A,P A,B A,P A A A,P A,P A A A,P A,P A,P A A A,P I H H H H H H B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B H H H F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F U e H S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S H H H H H H B * O B - H H H H H H O b - H H H H H H b O - C ch j jh T Th N t th d dh n H H H H H H P O b - H H H H H H P B O - m y r L l w s h H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H - B H B B B B b - B H B - - - - - B H B B B B - - H B - - - - - H - - - - - - - - - - - - - H B - - B - - - - - - - - - H B - - - - - H - - - - - - - B - - - - - H - - - - - - H - - - - - - *,B H B - - B * - H - - - - - P B H B - - B - - H B - - B - - B H - - - - - * - H B B B - b - - B - - - - - H B - B - - - H B - - - - B b H B B - - - * - - - - - - - O O - - - O - O O - B - - - - - - - B - - - - - - - - - - - - - - B H B B B B - - H B B B B - - p ph b bh H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H - B - B b - - - b - - b B - - B - - - - - - - - - - - - - - - - - - - - - - - - B B - - - P b - *,B * - - - - - P P - - - - - B B P P B - B - B - - - - - P B - - - - - - B P - - - - O2 O1 O O1 O1 O O - - - b - - - b - - - - B B B B b - - - - D Dh H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H - B - - - P P - - - B b - - P - - b - - B b B - - - - - - B - B b - - - O O1 O1 O2 O O - O - - - - b,S1 - - b,S2 b,S2 - - o k k h g gh Table 3: Qualitative distribution of Oriya graphemes On graphemic representation of the Oriya phonemes 129 130 Panchanan Mohanty and Gabriel Altmann This method seems to better discriminate the individual modalities. We shall adhere to this method until “better” ones are proposed. Now, this computation must be performed for a grapheme in each of ng possible combinations. At the end, the mean of all results will be computed, i.e. the mean distance is n 1 g ¯ (5) D(g) = ∑ D(xi ) ng i=1 where ng is the number of positions in which the distance has been measured. In order to express the uncertainty or learning/perceiving difﬁculty, we take logarithms of (5) and obtain uncertainty II of a grapheme as ¯ UgII = log2 D(g). (6) For the sake of illustration consider the phoneme /k/ and its representations given in Table 3. Let us look at both the column and the row of /k/. In the columns we should search for symbols A, B, H, F, S, b, P, O2 , S1 , S2 concerning the grapheme in second position (column), and in the rows for symbols ∗ , P, O, O1 , O2 , S1 , S2 concerning it in the ﬁrst position (row). The vowels have alternatives only in the second position. Z and – simply mean no representation. For /k/ in the ﬁrst position (row) there is no relevant symbol. In the column, we ﬁnd one B and 3 b (O and * are relevant only for the pertinent rows). Thus n/k/ = 1 + 3 = 4. Now, for the B, which signalizes the position in the second dimension (below) without change of shape or size, we get D(k) = (0 + 22 + 0 + 0 + 0)1/2 = 2. For each b, which signalizes a position in the second dimension and a change of size, we get D(k) = √ 2 2 1/2 (0 + 2 + 0 + 4 + 0) = 20 = 4.47. Hence the mean diversity of the /k/representation is 1 ¯ D(/k/) = (2 + 4.47 + 4.47 + 4.47) = 3.85 4 and the second kind of uncertainty yields II U/k/ = log2 3.85 = 1.94 . (7) We deﬁne log2 0 = 0. In the “plain” computation the uncertainty I of /k/ was 1. The new uncertainty is just a variable whose sampling distribution can easily be derived. The computation of the uncertainty II can be performed On graphemic representation of the Oriya phonemes For columns: For rows: A B H F S * P O = = = = = = = = 2 2 1 3 3.16 2 5 2 b P O2 S1 S2 O1 O2 S1 S2 = = = = = = = = = 131 4.47 5 5 5 5 5.28 2 5 5 mechanically because the values of distances can be ascribed to the individual symbols according to whether they occur in the column, in the row, or in both. We have As can be seen, some of the symbols are valid both for rows and columns, but do not necessarily have the same value (e.g. O). Using these evaluations, we can compute for each phoneme its graphemic uncertainty II. It is given in Table 4. Summing the numbers in Table 4 and dividing by n = 36 (number of phonemes) we obtain U¯ II = 53.86/36 = 1.4961. The pure form uncertainty I above yielded U¯ I = 0.847. 4 The distribution of graphemic representations In all the languages studied so far (i.e. German, Swedish, Italian) it has been stated that the frequency of representations follows the Gram-Charlier expansion of the geometric distribution yielding the so-called Shenton-Skeesgeometric distribution Px = pq x−1 1 1+a x− , x = 1, 2, 3, . . . p (8) Unfortunately, in the Oriya case we have merely three classes (x = 1, 2, 3) and the distribution has two parameters, thus no degrees of freedom would remain for a chi-square test. The software does not function in such cases though the ﬁt must be perfect under these conditions. Nevertheless, we are interested in the values of the parameters a and p for comparative purposes. 132 Panchanan Mohanty and Gabriel Altmann Table 4: Graphemic uncertainties II of Oriya phonemes (deviation from linear regularity) Phoneme Graphemic uncertainty II Phoneme /@/ /a/ /I/ /U/ /e/ /o/ /k/ /kh/ /g/ /g h/ /c/ /c h/ /j/ /j h/ /T/ /Th/ /D/ /Dh/ 0 0 1.00 1.00 1.58 1.66 1.94 2.16 2.16 0.00 2.28 2.32 1.93 2.16 1.97 2.22 1.69 1.00 /N/ /t/ /th/ /d/ /dh/ /n/ /p/ /ph/ /b/ /bh/ /m/ /y/ /r/ /L/ /l/ /w/ /s/ /h/ Graphemic uncertainty II 1.84 1.20 1.56 2.32 2.04 1.32 2.20 2.24 2.27 1.81 1.32 0 1.25 1.00 1.00 1.00 2.42 0 We can estimate them easily from the ﬁrst two frequency classes as follows: pˆ − P1 . pˆ = P1 + P12 − P1 + P2 , aˆ = 1 − pˆ (9) Since P1 = 9/36 = 0.25 and P2 = 21/36 = 0.5833, we obtain from (9) p = 0.8791 and a = 5.2035, yielding, of course, automatically a perfect ﬁt. The last value (x = 3) is of course NP3 = 1 − ∑2x=1 NPx = ∑∞ x=3 NPx which can easily be checked. A graphical display is not necessary because the observed and the computed values are identical. A comparison with other languages shows that Oriya conspicuously differs from some European Latin-script languages. We have in Italian p = 0.6488, a = 0.2398, in German p = 0.7768, a = 2.3323, and in Swedish p = 0.6152, a = 0.4588. On graphemic representation of the Oriya phonemes 133 Table 5: Fitting the Shenton-Skees-geometric distribution to Oriya graphemic representations x fx NPx 1 2 3 9 21 6 9 21 6 p = 0.8791, a = 5.2035, C = χ2 /N = 0.00 5 Weighted graphemic distributional calculus Up till now only the distribution of phonemes (bigrams) has been examined, but n-grams have been analysed only for letters. The literature concerning the ﬁrst aspect is scarce, though for the second aspect it is rich, because letters are accessible also to mathematicians, whereas phonemes are only accessible to linguists. In linguistics, distributionalism lost its relevance due to the impact of generative linguistics, but came back after its decline in many forms due to the possibility of automatic processing of texts. It appears in cryptography, automatic language recognition, study of collocations, study of denotational text structures, information theory, psychology, authorship attribution, etc. It has many aspects and many possible applications. In this paper, we want to apply some aspects of the Harary-Paper distributional calculus (cf. Harary & Paper 1957, Altmann & Lehfeldt 1980) to Oriya graphemic representations, however, using their weighted variants. The weighting is performed as shown above (cf. Section 3) namely using uncertainty II. The occurrence of a grapheme y behind grapheme x is not any more a Bernoulli variable with two outcomes (1 for occurrence and 0 for nonoccurrence) yielding a comfortable computation using the binomial distribution, but rather a continuous variable whose distribution is not yet known. In any case it is at least possible to deﬁne some distributional properties of the Oriya graphemic representations of phonemes and to compute them directly from Table 3. Based on the numerical evaluation of the symbols in Table 3, we can rewrite it as shown in Table 6. In this table two values are simply added if there is a double impact, e.g. A,P = 2 + 5 = 7. If the evaluation in the column differs from that in the row, the values are averaged, e.g. an O2 yields 5 in the column and 2 in the row, thus O = (2 + 5)/2 = 3.5. If a grapheme does 134 Panchanan Mohanty and Gabriel Altmann not occur behind another, it is not evaluated (−), if it is implicit, it gets the value 0. The present calculus is rather a measurement of the distributional graphemic impact. Of course there are many possibilities to set up such a calculus, and we simply want to present a starting idea which can be further developed for several Indian languages. We restrict ourselves to the actual state in one language but it would be interesting to study the development of the Oriya script and of all Indian languages using this or an akin method. Let us now state the following deﬁnitions: 1. G = {g| g ∈ S}: the set of graphemes of the given script S 2. #G = the cardinal number of the set G (number of graphemes in the inventory) 3. R = the relation of immediate successorship, e.g. gRh means that stays immediately behind 4. Ag = {h|h ∈ G, hg ∈ R} = the set of predecessors of grapheme 5. gB = {h|h ∈ G, gh ∈ R} = the set of successors of grapheme 6. Av g ={hv |h ∈ G, hg ∈ R} = the set of values of predecessors of grapheme (as given in Table 6) 7. gBv = {{hv |h ∈ G, gh ∈ R} = the set of values of successors of grapheme (as given in Table 6). In order to illustrate these sets, consider the row of the grapheme in Table 6. We get Bv , which may be called the set of ’s graphemic impact on successors, and Av , which may be called ’s graphemic impact on predecessors), with: < k > Bv = {0, 1, 7, 2, 3, 3.16, 2, –, –, –, –, –, –, –, 2, –, –, –, –, 2, –, –, –, 2, –, –, –, –, 2, 1, 2, 2, 2, 2, 4.47}, v A < k > = {1, 1, 1, 1, 1, 1, 2, –, –, –, –, –, –, –, –, –, –, –, –, 2, –, –, –, –, –, –, –, –, –, –, 2, –, –, –, 2, –}. The impact cannot be measured if the grapheme does not occur in the immediate neighbourhood (symbolized as “-” in the respective sets), but it is evaluated as 0 if it is implicit – a special feature of many Indian scripts. As a matter of fact, the symbol “-” need not be written, but its inclusion helps to identify the given grapheme. Further we deﬁne 8. #Av g = the number of elements in the set Av g 9. #gBv = the number of elements in the set gBv ; e.g. we have #Bv = 17, and #Av = 10. As a matter of fact, these numbers are the num- @ a I U e o k kh g gh c ch j jh T Th D Dh N t th d dh n p ph b bh m y r L l w s h 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 7 2 2 7 7 7 2 2 2 7 7 2 7 7 7 7 7 2 2 7 7 2 2 7 7 7 2 2 7 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 @ a I U e 1 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 3.16 o kh g gh C ch j jh T Th D Dh N t th d dh n p s h 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - 2 1 2 2 2 2 4.5 - 2 1 2 - - - - - 2 1 2 2 2 2 - - - 1 2 - - - - - - 1 - - - - - - - - - - - - - - - 1 2 - - 2 - - - - - - - - - - - 1 2 - - - - - - 1 - - - - - - - - 2 - - - - - - 1 - - - - - - - 1 - - - - - - 4 1 2 - - 2 2 - - 1 - - - - - 5 2 1 2 - - 2 - - - 1 2 - - 2 - - 2 1 - - - - - 2 - - 1 - - 1 - 5 - - - 2 - - - - - - 1 2 - 2 - - - - 1 2 - - - - 2 4.5 1 2 2 - - - 2 - - - - - - - - 2 2 - - - 2 - 2 2 - 2 - - - - - - - 2 - - - - - - - - - - - - - - - 2 1 2 2 2 2 - - - 1 2 2 2 2 - - ph b b h m y r L l w 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 - 2 - 2 - - - 4.5 - - - - - 4.5 - 4.5 2 - - - 2 - - 5 5 - - - - - - - - 2 4.47 - - - - - - - - 5 - - - - - - - - - 4.47 - - - - - - - - 2 4.47 2 2 2 - - - 2 - 5 4.5 - 4 2 - - - - - - - 4.47 - 5 5 - - - - - - - - 2 2 5 5 2 - - 2 - 2 - 2 - - 2 - - - - - 4.47 - 5 2 - - 5 - - - - - - 2 5 5 2 - - - - 2 2 2 2 5.28 5.28 3.50 2 2 2 - 3.5 5.3 2 5.3 5.3 2 2 2 2 - - - 4.5 - - - - 4.5 - - - - - 2 4.47 - 9.47 - 9.47 9.47 - 2 2 2 - 2 4.5 4.5 - - - - - k Table 6: Weighted distribution of Oriya graphemes On graphemic representation of the Oriya phonemes 135 136 Panchanan Mohanty and Gabriel Altmann bers of interactions of the given grapheme (used in the simple HararyPaper calculus). 10. ∑ hv = the sum of the values of graphemes h that are successors of h∈gB grapheme g 11. ∑ hv = the sum of the values of graphemes h that are predecessors of h∈Ag grapheme g. For example, the sum of the values of the successors of is given as ∑ hv = 0 + 1 + 7 + 2 + 2 + 3.16 + 2 + 2 + 2 + 2 + 2 + 1+ h∈B + 2 + 2 + 2 + 2 + 4.47 = 38.63. The sum of the values of predecessors of in the above example is 14. Finally we deﬁne some measures which characterize the impact of graphemes on their immediate neighbours: 12. the Mean successor modiﬁcation: MSM(g) = ∑ hv /#gBv h∈gB 13. the Mean predecessor modiﬁcation: MPM(g) = ∑ hv /#Av g. h∈Ag We illustrate these measures using the above example. For MSM(< k >) we obtain MSM(< k >) = 38.63/17 = 2.27 and for MPM(< k >) = 14/10 = 1.40. The result of evaluation for individual graphemes is shown in Table 7 (p. 137). In languages with Latin script this phenomenon is extremely rare and could not be studied systematically. It is to be noted that inﬂuences of this kind are well known from phonetics and phonemics, e.g. the Sanskrit sandhi, the French liaison, all types of assimilation and coarticulation in various languages. They have been observed and described, but have never been treated quantitatively. The overall measure of graphemic modiﬁcations is given as the sum of all successor (or predecessor) values divided by the number of realized interactions. In practice, the sum of all values in Table 6 divided by their number (each occurrence of zero is counted, but “−” is not). Formally one can deﬁne this measure as ∑ ∑ hv GM(L) = g∈G h∈gB ∑ #gB g∈G ∑ ∑ hv = g∈G h∈Ag ∑ #Ag g∈G . (10) On graphemic representation of the Oriya phonemes 137 Table 7: Mean impact of individual graphemes in Oriya Grapheme #gBv ∑ hv MSM #Avg ∑ hv MPM @ a I U e o k kh g gh c ch j jh T Th D Dh N t th d dh n p ph b bh m y r L l w s h 32 34 36 35 32 32 17 10 15 9 9 6 11 6 9 7 8 7 12 16 7 14 9 14 12 8 13 8 16 6 30 8 8 6 23 11 32.00 34.00 36.00 35.00 32.00 32.00 39.63 25.63 33.10 16.16 27.16 16.16 27.63 11.16 19.16 12.16 22.63 17.16 24.63 44.63 17.16 42.63 21.16 37.16 24.16 15.16 37.63 19.16 38.63 11.16 83.56 22.63 22.63 11.16 73.98 25.16 1.00 1.00 1.00 1.00 1.00 1.00 2.33 2.56 2.21 1.80 3.02 2.69 2.51 1.86 2.13 1.74 2.83 2.45 2.05 2.79 2.45 3.04 2.35 2.65 2.01 1.90 2.89 2.40 2.41 1.86 2.79 2.83 2.83 1.86 3.22 2.29 33 32 36 36 33 31 10 8 8 7 9 8 9 8 13 8 9 7 11 12 10 10 11 15 12 9 9 9 17 27 22 11 13 14 10 9 3.00 32.00 156.00 66.00 93.00 95.80 14.00 12.47 12.47 8.00 25.78 16.28 15.98 12.47 30.47 19.94 14.47 8.00 22.44 24.28 16.47 26.28 27.75 25.00 28.41 17.47 15.00 15.00 32.47 27.00 38.00 16.00 19.00 22.00 19.47 12.00 0.09 1.00 4.33 1.83 2.82 3.09 1.40 1.56 1.56 1.14 2.86 2.04 1.77 1.56 2.34 2.49 1.61 1.14 2.04 2.02 1.65 2.63 2.52 1.73 2.37 1.94 1.67 1.67 1.91 1.00 1.73 1.45 1.46 1.57 1.95 1.33 138 Panchanan Mohanty and Gabriel Altmann For the Oriya script we obtain (using Table 7) GM(Oriya) = 1041.13/536 = 1.94 . In the non-weighted version of the Harary-Paper distributional calculus this number must always be less than 1. 6 Symmetry of the graphemic impact In Table 7 one can see that MSM and MPM of the given grapheme are seldom equal. In order to test whether the successor and the predecessor modiﬁcations are equal or signiﬁcantly different, one can apply a number of different non-parametric tests. Then, to test the homogeneity of marginal sums, one could apply approximately the Stuart test. Since we are interested in individual differences of MSM(g) and MPM(g), we can compare them using the normal test for the difference of two means. We consider both kinds of impact as independent (the successor and the predecessor impacts are stated independently) and ignore the covariances. In that case we obtain the asymptotic criterion MSM(g) − MPM(g) z= , (11) Var(MSM(g)) +Var(MPM(g)) where Var(.) is the variance of the given mean. The variance of the successors can be computed from the rows of Table 6 and the variance of the predecessors, from the columns of Table 6. First, we obtain Var(gsucc ) = and Var(g pred ) = 1 ∑ (hv − MSM(g))2 #gBv − 1 h∈gB 1 #Av g − 1 ∑ (hv − MPM(g))2 (12) (13) h∈Ag and ﬁnally Var(MSM(g)) = Var(gsucc ) #gBv (14) Var(MPM(g)) = Var(g pred ) #Av g (15) and On graphemic representation of the Oriya phonemes 139 respectively. The formulas look a little strange but (14) and (15) are simply the mean square deviations of the rows and columns of Table 6 respectively. For example, the MSM(< k >) can be found in Table 7 as 2.33 (rounded), hence Var(< k >succ ) = [(0 − 2.33)2 + 2(1 − 2.33)2 + (7 − 2.33)2 + 9(2 − 2.33)2 + + (3 − 2.33)2 + (3.16 − 2.33)2 + (4.47 − 2.33)2 ]/(17 − 1) = 37.4731/16 = 2.3421 from which Var (MSM (< k >)) = 2.3421/17 = 0.1378. For MPM(< k >) = 1.4 we get Var(MPM(< k >)) = 0.026667 thus 2.33 − 1.4 z = √ = 2.29 0.1378 + 0.0267 which is signiﬁcant at the 0.05 level. This means the impact of on its successors is signiﬁcantly higher than the impact of the predecessors on it. We restrict ourselves to the presentation of the method without performing tests for each grapheme. The relevance of this method could be evaluated both in the historical perspective, i.e. by comparing the graphemic impacts in the development of the Oriya script in different periods or by comparing different Indian scripts. References Altmann, Gabriel; Lehfeldt, Werner 1980 Einführung in die quantitative Phonologie. Bochum: Brockmeyer. Bernhard, Gerald; Altmann, Gabriel 2006 “The phoneme-grapheme relation in Italian”. This volume, pp. 13–23. Best, Karl-Heinz; Altmann, Gabriel 2005 “Some properties of graphemic systems.” In: Glottometrics, 9; 29–39. Bosch, Antal van den; Daelemans, Walter 1993 “Data-oriented methods for grapheme-to-phoneme conversion”. In: Proceedings of the 6th Conference of the EACL. 45–53. Bosch, Antal van den; Content, Alain; Daelemans, Walter; Gelder, Beatrice de 1994 “Measuring the complexity of writing systems.” In: Journal of Quantitative Linguistics, 1; 178–188. 140 Panchanan Mohanty and Gabriel Altmann Daelemans, Walter; Bosch, Antal van den 1994 “A language-independent, data-oriented architecture for grapheme-tophoneme conversion”. In: Proceedings of ESCA-IEEE Speech Synthesis Conference ’94. Harary, Frank; Paper, Herbert H. 1957 “Toward a general calculus of phonemic distribution.” In: Language, 33; 143–169. Hug 1997 “Disturbing factors in a linguistic usage test.” In: Journal of Quantitative Linguistics, 4(1-3); 110–121., Marc Katz, Leonard; Frost, Ram 1992 “The reading process is different for different orthographies: the orthographic depth hypothesis”. In: Haskins Laboratories Status Report on Speech Research 1992. New Haven, Conn.: Haskins Laboratories, 147–160. Klima, Edward 1972 “How alphabets might reﬂect language”. In: Kavanagh, James F. and Venezky, Richard L. (Eds.), Orthography, Reading and Dyslexia. Baltimore: University Park Press, 137–153. Venezky, Richard L. 1970 The structure of English orthography. The Hague: Mouton. On the relation between types and tokens of Japanese morae Katsuo Tamaoka 1 Type and token frequencies of Japanese mora The fundamental set of Japanese sounds, which are described in phonological scripts of two types of kana symbols: hiragana or katakana, consists of 101 basic kana including the special sound of the nasal /N/ (written in hiragana as S and in katakana as S). Each Japanese kana symbol fundamentally corresponds to a speciﬁc moraic sound. For example, a kana symbol in hiragana or in katakana represents a moraic sound /ka/ consisting of consonant /k/ and vowel /a/. Using the word printed-frequency database created by Amano and Kondo (2000, and 2003 for CD-ROM version)1 , Tamaoka & Makioka (2004) studied the frequency (token frequency) and the context of Japanese morae, consisting of words in which they occur (type frequency). Since type frequency represents here the property of cotextuality, i.e. the number of different contexts of a mora, and token frequency represents the frequency, the present study proposes the hypothesis that there is some relationship between these two properties known from synergetic linguistics (cf. Köhler 1986, 2002). Köhler tried to express this hypothesis by means of a straight line and obtained a signiﬁcant correlation coefﬁcient signaling the existence of a linear relationship. However, the ﬁtted straight line is merely a ﬁrst exploratory approximation because the lower frequency values of the computed straight line are negative, which is empirically impossible. The 1. A very large lexical corpus of 341 771 morphemes was established from newspapers containiannot attain inﬁnite sizes and their changes tend to slow down depending on the already attained value. Thus we postulate a very simple relationng 287 792 797 morphemes, all of which were taken from the Asahi Newspaper printed from 1985 to 1998. At present, this is the largest and the most up-to-date lexical database created from calculating the word frequency of occurrence in Japanese written texts. There are some coding errors in this database. For instance, Tokyo, written in two kanji as !, is phonologically stored as /higasikyoR/ (/R/ refers to a long vowel) instead of /tokyoR/. In addition, a long vowel /R/ is not clearly distinguished in the database. Thus, I did not include the sounds of the long vowel /R/ and the geminate sound /Q/ in this study. 142 Katsuo Tamaoka present study therefore tries to ﬁnd a more adequate expression of this relationship. The present study starts from the general theory of language laws (Wimmer & Altmann, 2005) stating that the majority of relationships between linguistic variables can be expressed by means of the relation between their relative rates of change, i.e. between dy/y and dx/x. Here y is frequency (tokens), x is cotextuality (types). This assumption is reasonable because linguistic variables c dy b = c+ dx (1) y x where b is the proportionality coefﬁcient and c is interpreted either as a kind of “noise” or disturbance, or it represents the ceteris paribus condition, i.e. a constant state of all other possible factors. The solution of this equation is y = axb ecx (2) where a = eC with integration constant C. This relationship follows directly from the respective part of the synergetic control cycle joining polytexty with frequency. Since in synergetic linguistics one considers the variables usually in their logarithmically transformed form, we speak about L-polytextuality and L-frequency. In the cycle presented below (Köhler, private communication, cf. Köhler 1986; 2002) one can see that polytextuality affects frequency proportionally, with an additive constant coefﬁcient of usage requirement L-Appl (“communicative relevancy” of the given meaning); at the same time it affects the frequency by means of an exponential operator with a constant C, which is assumed to represent the mean sign complexity (or length). The cycle is a new variant in Köhler’s theory, allowing a non-logarithmic, “plain” effect of one variable on another, using the exponential operator. Thus, the formula becomes ln(F) = R ln(Appl) + B ln(PT ) −C exp(ln(PT )), i.e. ln(F) = R ln(Appl) + B ln(PT ) −C(PT ) from which it follows that F = Appl R PT B e−C(PT ) . On the relation between types and tokens of Japanese morae 143 L-Appl R L-Polytextuality L-Frequency B exp –C Figure 1: The relationship between polytextuality and frequency in general Since ApplR can, in the framework of a synchronic study, be considered a constant, for example, A, while PT = x, and F = y, we can obtain y = Axb e−cx which is identical to the above solution of the differential equation. Iterative computing of this curve yielded y = 26.57366832x1.31502554 exp(−0.0000125937521x) . (3) The determination coefﬁcient is R2 = 0.93, indicating that relationship (2) can be preliminarily accepted. The observed and the computed values are shown in Table 1 . The computation has been performed in logarithmic transformation, thus R2 holds for the logarithmic values in the given interval. Though in many cases the deviation is considerable, the variability of token frequencies is so enormous that it renders R2 very high and the F-test very signiﬁcant. 2 Conclusion The problem of type and token frequencies can in this case be reduced to that of cotextuality and frequency, a relationship that has been very lightly studied in quantitative linguistics. In any case, the result conﬁrms Köhler’s 144 Katsuo Tamaoka 4.4e+007 3.52e+007 2.64e+007 1.76e+007 8.8e+006 0 0 18000 36000 54000 72000 90000 Figure 2: Relation between types and tokens of Japanese morae assumption. In the majority of linguistic cases, frequency is considered as an independent variable but in the case of Japanese morae it is exactly the other way round. This can be explained by the fact that frequency, as most of the other linguistic quantities in the synergetic control cycle, has direct as well as indirect effects on other variables and is, at the same time, a dependent variable, e.g. depending on polytextuality. References Amano, Shigeaki; Kondo, Tagahisa 2000 Nihongo-no goi tokusei. Dai-7-kan. [Lexical properties of Japanese. Volume 7.] Tokyo: Sanseido. , 2003 Nihongo-no goi tokusei. Dai-2-ki CD-ROM-ban. [Lexical properties of Japanese. The second volume of CD-ROM version.] Tokyo: Sanseido. Ferrer i Cancho, Ramon; Solé, Richard V.; Köhler, Reinhard 2004 “Patterns in syntactic dependency networks.” In: Physical Review, E 69, 051915. Köhler, Reinhard 2002 Korpuslinguistische Untersuchungen zur quantitativen und systemtheoretischen Linguistik. [http://ubt.opus.hbz-nrw.de/volltexte/ 2004/] On the relation between types and tokens of Japanese morae 2005 145 “Synergetic linguistics”. In: Köhler, Reinhard; Altmann, Gabriel; Piotrowski, Rajmund G. (Eds.), Quantitative Linguistics. An International Handbook. Berlin: de Gruyter, 760–774. Tamaoka, Katsuo; Makioka, Shogo 2004 “Frequency of occurrence for units of phonemes, morae, and syllables appearing in a lexical corpus of a Japanese newspaper.” In: Behavior Research Methods, Instruments & Computers, 36(3); 531–547. Wimmer, Gejza; Altmann, Gabriel 2005 “Uniﬁed derivation of some linguistic laws”. In: Köhler, Reinhard; Altmann, Gabriel; Piotrowski, Rajmund G. (Eds.), Quantitative Linguistics. An International Handbook. Berlin: de Gruyter, 791–807. 146 Katsuo Tamaoka Table 1: Types (cotextuality) and tokens (frequency) of Japanese morae in a corpus #Types #Tokens 9 45 69 69 82 83 113 117 147 170 175 183 201 207 400 504 551 700 738 1122 1165 1166 1216 1343 1814 1898 2034 2084 2183 2442 2633 2937 3024 3125 3208 3480 3623 3769 4139 4929 5175 5317 5464 5569 5736 5965 6249 6260 6422 7088 8336 8670 95 511 3304 3087 125380 16016 14646 15343 29907 53284 10406 50769 15494 59424 33782 162751 166750 577939 446886 214108 182551 188263 199251 192847 929206 995900 411356 937044 507561 1541224 1254559 984460 1646454 936779 2138138 445557 1482387 1591698 2721853 2041928 2858438 1104998 4484240 1177824 1384993 7744822 2009223 2771906 2328945 2549499 4064902 1644763 (3) 477.81 3964.81 6953.57 6953.57 8724.03 8864.09 13294.87 13916.47 18781.27 22731.02 23612.77 25039.87 28321.35 29436.08 69829.51 94506.13 106199.38 145211.16 155590.11 268616.69 282083.06 282397.60 298240.76 339317.30 500865.36 531023.82 580621.35 599085.76 635993.97 734627.19 809153.65 930614.90 965975.25 1007342.14 1041580.88 1155274.00 1215911.60 1278397.76 1439209.44 1792936.88 1905609.90 1971142.40 2039337.01 2088262.07 2166436.68 2274317.01 2409134.24 2414376.53 2491787.48 2813332.09 3427819.93 3594413.22 #Types #Token (3) 8688 8694 8857 8951 9007 9174 9263 9566 10011 10038 10491 11507 11707 12158 12180 12822 12919 13622 13922 13925 14103 14413 14435 14873 15529 16836 17622 17737 18207 18898 19482 20442 21459 22195 22624 22763 22853 23598 23656 25431 27131 27384 27551 28364 31183 38359 40585 43982 67643 81463 84624 3740881 2577386 4303939 1210586 2206487 4118162 2758994 5777889 2749654 14757618 1620066 3554252 3067681 5306817 6791877 11037767 12594714 3432278 3777201 2927552 4767153 19326674 7858931 6823149 6550231 8127590 9953427 11996376 10498607 6149909 5808199 6947107 3972466 14053377 7932086 9271734 9612206 3607169 28899891 6116529 17102180 16442465 12154255 6289027 9926212 17211261 16518016 20530065 43985426 35719268 39052254 3603412.66 3606409.18 3688009.24 3735140.99 3763249.39 3847166.66 3891956.70 4044762.26 4269991.40 4283689.66 4513869.51 5032502.51 5134888.29 5366028.81 5377309.08 5706765.14 5756559.53 6117584.83 6271623.81 6273169.18 6364539.70 6523625.18 6534915.86 6759522.22 7095418.24 7762131.57 8160955.32 8219151.77 8456526.23 8804141.22 9096493.22 9574022.17 10075369.91 10435061.19 10643439.60 10710752.90 10754271.59 11112889.60 11140687.70 11981880.94 12769738.39 12885432.49 12961581.65 13329688.48 14571893.89 17480465.76 18306022.84 19494944.36 25488953.21 27348656.72 27630516.29 IV. Towards a theory Towards a theory of script Gabriel Altmann 1 Introduction A letter, whether written, printed, painted or chiselled, has a number of properties which are in some mutual relations. Since up to now only a small number of scripts has been analysed quantitatively, there is only a handful of hypotheses that can be formulated; however, they may turn out to be false if tested. Only testing can help corroborate a theory. As customary in quantitative linguistics, a theory is considered a system of hypotheses out of which at least some are laws. Though for the time being, only the frequency distributions and the resulting entropy can be considered laws, we shall try to construct a net of dependencies which can be easily analysed theoretically. The analyses can be postponed until more empirical data from testing are at our disposal. To this end we present an itinerary for research on individual scripts and refer to the literature in which the methods have been shown. Here we restrict ourselves to alphabetic scripts and hope that similar hypotheses can be set up for syllabic or pictographic scripts, too. We begin with formulating some deﬁnitions which are based merely on agreement and can be reformulated mutatis mutandis according to the script analysed. Deﬁnition 2 A letter is a single sign adopted from Latin, Greek or other alphabetic scripts not necessarily attributable to a single phoneme. E.g. the Latin <x> can be attributed to two phonemes, the Italian to no phoneme. Deﬁnition 3 A grapheme is a letter, or a combination of letters, or a letter with additional diacritical marks (such as those in Slavic languages, French, Spanish, German etc.) used as a whole in a language and attributable to a phoneme. Hence a letter is a grapheme only if it can be directly attributed to a phoneme (or more phonemes). For example, the Italian is a letter, not a grapheme, because it has no phonetic value. It is used on historical grounds and as a means controlling the pronunciation of other letters or graphemes. 150 Gabriel Altmann Deﬁnition 4 A portmanteau grapheme and a framing grapheme contain an implicit phoneme but do not represent it explicitly. For example, in Oriya (or other Indian languages) “consonantal” letters contain an intrinsic /@/, which must be suppressed by special rules, hence consonantal letters are portmanteau graphemes; the same holds for the Latin <x> in many languages; in American English a vowel is uttered in “able” between and , thus * is a framing grapheme representing the given vowel. Up till now letters have been revealed to have the following properties. There is a (1) ﬁxed inventory size of letters and graphemes in every alphabet used to represent the phoneme inventory, (2) , (3) frequency, (4) ornamentality, (5) distinctivity, (6) variability, (7) phonemic load, (8) grapheme size, (9) graphemic load, (10) graphemic utility or letter usefulness. All these properties have been at least touched upon in the literature, either in older works which did not care for quantiﬁcation, in this volume, or in the journal Glottometrics. The frequency of letters itself has been known and studied for 150 years and belongs to the oldest disciplines of quantitative linguistics. The ﬁeld of letter properties can be shown graphically in Figure 1. Inventory Complexity Ornamentality Frequency Distinctivity Letter Variability Grapheme size Phonemic load Graphemic load Letter utility Figure 1: Properties of letters It is to be expected that in the future more properties will be revealed, especially after considering scripts other than the alphabetic ones. Figure 1 shows the present scope of research possibilities, but it does not contain any hypotheses. All the letter properties can be deﬁned operationally in different ways. Now, following a meta-hypothesis based on synergetic linguistics we assume that each of these properties is linked to at least one other property. That is, the properties constitute a system whose structure is made up of their mutual relations, even if we still do not see its exact form and cannot corrob- Towards a theory of script 151 orate the links in a satisfactory way. Nevertheless, we can begin to discuss the possibilities and show some vistas for future research. 2 The properties 2.1 Inventory size Inventory size is the most stable factor of script, especially for alphabetic or syllabic ones, which are very restricted. It is ﬁnite even for non-alphabetic scripts. It can be easily ascertained and for a given time period or a given writer (in hand-written documents) it is a ﬁxed quantity even if there are variants. The inventory can change not only if the script gets borrowed (cf. the Latin script, its reduction in Hawaiian and its ampliﬁcation in Slavic languages) but also if it develops or diversiﬁes (cf. the Brahmi script and the modern Indian scripts; or the different versions of the hieratic scripts in Egypt). The inventory is necessarily associated with some other properties. One must differentiate three kinds of inventories, namely the L-inventory (letter inventory), the G-inventory (grapheme inventory) and the P-inventory (phoneme inventory), all of which can have different impacts on other properties. 2.2 Complexity Complexity, as is well known from the science of complexity, is itself a complex concept with a large number of operational deﬁnitions. As to the script, up to now three methods have been proposed to measure the complexity of signs: 1. The stroke counting in Chinese script, where the strokes can be either simple or complex, but the counting follows the way of handwriting. 2. The intersection method proposed by Peust (2006) counting complexity as the maximal number of lines of a letter that can be crossed by a straight line (of any direction). 3. The composition method proposed by Altmann (2004) scaling the form and the kind of connections of lines. 152 Gabriel Altmann For example, the Latin (Arial) letter would have a complexity 3 by method (1) and (2), but 10 by method (3). Other different methods can be devised and proposed, e.g. the different approximations to the Hausdorff dimension – cf. Schroeder (1991), Falconer (1993) Tricot (1993), Youssif & Purdy (2002), Cha, Tappert & Srihari (2003), Feixas et al. (1999), and many others; see also Köhler (this volume, pp. 115ff.). Complexity concerns the optical image of a sign. The measures used by OCR must be performed with a computer program, in linguistics one needs simple “pencil and paper” procedures. 2.3 Frequency Of all the letter properties the simplest and the easiest to capture is letter frequency, which can be done mechanically, though grapheme or phoneme counting is a rather complex task for a computer. Ranked phoneme/letter/grapheme frequency follows a very stable probability distribution (usually the negative hypergeometric), though there is a number of different approaches (cf. http://lql.uni-trier.de). The ﬁrst extensive statistics were obtained by Förstemann (1846, 1852) and since then there has been an incessant increase of publications on the topic with different orientations (language recognition, cryptology, Morse alphabet, printing devices, genre and style differentiation, euphony, informatics, etc.). Letter frequency is not identical with phoneme frequency, grapheme frequency or “letters + punctuation marks frequency” or “graphemes + punctuation marks frequency”, hence we shall distinguish P-frequency, L-frequency, G-frequency, LM-frequency and GMfrequency (cf. Best 2005). Frequency itself can have several different global properties (moments, entropies, etc.) 2.4 Ornamentality Ornamentality is a property which is intuitively or psychologically evident, but its deﬁnition has not been presented up to now. It can, nevertheless, be estimated; every graphical sign has a mean ornamentality and its variation (cf. Best & Altmann, this volume, pp. 91ff.). It is not identical with complexity, though there is surely a correlation between them. Ornamentality is an optical property and concerns individual signs (letters, not graphemes). In Towards a theory of script 153 the arts (literature, sculpture, music, architecture) it can be established and understood in different ways. 2.5 Distinctivity Distinctivity. Each letter must be distinguishable from all other ones. Consequently, this property can be measured only in relation to the other elements of the inventory. It can be formally deﬁned as a function of all differences between the given and the other letters. Not only individual letters can be characterized in this way but the whole script system (cf. Anti´c & Altmann 2005). Again, the measurement can be performed in many different ways using curve characterisation, Bézier curves, fractal properties, geometric transformations, OCR techniques, etc. 2.6 Variability In the given context, variability does not mean the different font types such as Arial, Courier etc.; but rather the variability arising a) due to the writing modality, e.g. writing with a pen or a brush, b) due to the adaptation of the script to other language (cf. the forms of the Greek alphabet in Slavic languages), c) due to geographic or school diversity, and even d) due to the development of the script forms. The best example is the diversiﬁcation of letter forms in the South East Asian area which contains several variabilities (b, c, d). The variability of the Egyptian hieratic script written on papyrus and the simpliﬁcation of hieroglyphs which was regionally diversiﬁed are of the types (a), (c) and (d) (cf. Möller 1927; Hegenbarth-Reichardt & Altmann, this volume, pp. 105ff.). Variability can be measured as a function of differences between letters originating form the same letter. Variability is again an optical property in this context. 2.7 Phonemic load A letter can be used to represent one or more phonemes, especially if it occurs in different graphemes. This phenomenon is well known in English and is the 154 Gabriel Altmann object of a well developed discipline. The measurement can be performed in different ways and has different aspects. One of them has been shown in (Fan & Altmann 2006). Again, one can distinguish L-phonemic load and Gphonemic load. 2.8 Grapheme size This is a simple grapheme property, with size referring to the number of letters. Usually it appears in languages having a greater phoneme inventory than the inventory of Latin letters or having an antiquated or morphologically oriented orthography. One can count also the diacritical marks as grapheme modiﬁcations; it depends on the kind of quantiﬁcation and the aim of research (cf. Bernhard & Altmann, this volume, pp. 13ff.). The operational deﬁnitions for iconographic scripts are very complex but important for the development of a theory. 2.9 Graphemic load of letters (L-graphemic load) This is a letter property expressing the participation of letters in building graphemes. If there are graphemes consisting of more than 1 letter, then some of these letters must have a large graphemic load. For example, in Italian, the letter occurs in 14 graphemes either on historical grounds or as a kind of diacritical mark controlling the pronunciation of some graphemes, hence its graphemic load is greater than that of the other letters. 2.10 Graphemic usefulness of a letter (L-utility) This is a kind of weighting for the previous properties. The weight can be ascribed in different ways, e.g. depending on the position in the grapheme. For example the letter in the grapheme can obtain the weight 3, or one can count the position in reverse order. Probably there are still other properties which may be established in the future. In what follows we shall try to link these properties by hypotheses. The testing of the hypothesis is, unfortunately, not yet possible because the number of scripts and languages analysed is not sufﬁcient. Towards a theory of script 3 155 Some hypotheses Let us begin with the L-inventory, which is the basic property of any alphabetic script. If the inventory is small, the letters can be shaped in such a way that they yield the maximal mean distinctivity. This tendency is best shown in handwriting: if two letters tend to be confused, an additional modiﬁcator is added. For example the original Arabic number <7> tends to be confused with <1>, hence many Europeans write seven with a dash through its lower part. For fear of confusing with in handwriting, one writes <ˇu> or < u¯ > in order to acquire greater distinctivity. In general, one can suppose that Hypothesis 1 The greater the L-inventory, the smaller the mean distinctivity of letters. The more letters there are in the inventory, the more frequently some of their parts (which may be called distinctive features in analogy to phonemics) must be repeated in other letters. This fact reduces mean distinctivity and increases mean similarity. From phonemics and graphemics it is well known that the more letters there are in the inventory, the ﬂatter the rank-frequency distribution of letters. This fact has been shown in countless number of cases (cf. e.g. Altmann & Lehfeldt 1980; Martindale et al. 1996; Grzybek & Kelih 2003, 2005; Grzybek, Kelih & Altmann 2004). Flatter hyperbolic curves have different global properties than steep curves. For example, the moments, the entropy, the repeat rate etc. are different for large and small inventories. A relatively well corroborated hypothesis says that Hypothesis 2 The greater the size of the (P-, L-, G-)inventory, the greater the entropy and the smaller the repeat rate of the distribution. It has been already shown that for entropies and repeat rates there are expressed dependencies in which inventory size is the independent variable (cf. Lehfeldt & Altmann 1980; Zörnig & Altmann 1983, 1984). Analogous dependencies on inventory size can be expected for any sampling functions. Whatever the manner of measurement of complexity , as soon as a language begins to create new graphemes not by letter combinations (like e.g. English) but by letter modiﬁcation or addition of new letters, the mean complexity of symbols must increase. Hence it is reasonable to conjecture that 156 Gabriel Altmann Hypothesis 3 The greater the inventory of phonemes, letters, graphemes or even ideograms, the greater the mean complexity of letters or graphemes or ideograms. If we consider the graphemes of English as optical units, the hypothesis would hold, too, but usually one measures only the complexity of letters or of ideograms. The L-inventory may have an inﬂuence upon the graphemic load of letters. As a matter of fact, it is the P-inventory in its relation to L-inventory that causes different graphemic loads. If the P-inventory is great and the Linventory is small, it is necessary to create new graphemes in order to secure adequate representation of phonemes. Either the letters are combined or subsidiary modiﬁcations are introduced. For example in Slovak (cf. Nemcová & Altmann, this volume, pp. 79ff.) is a component of 5 graphemes , the last three being diphthongs. Hence it is reasonable to suppose that Hypothesis 4 The greater the P-inventory and the smaller the L-inventory, the greater the mean graphemic load of letters. Since graphemic load is directly associated with orthographic uncertainty, we can suppose that Hypothesis 5 The greater the mean G-load of letters, the greater the orthographic uncertainty of phonemes. Hence, indirectly, orthographic uncertainty of phonemes is a function of the relation between the P-inventory and the L-inventory. The measurement of orthographic uncertainty can be found in this volume (see the contributions by Bernhard & Altmann, Kelih, Nemcová & Altmann, Fan & Altmann, and Mohanty & Altmann). Further, a great P-inventory and a small L-inventory automatically increase the grapheme size, because new graphemes are made up of letter combinations and modiﬁcations. Hence Hypothesis 6 The relationship between P-inventory and L-inventory inﬂuences the mean size of graphemes. Towards a theory of script 157 This relationship must take into account at least three variables, a dependent one and two independent ones. Thus hypothesis 5 and 6 can evidently be joined because ﬁrst a grapheme size increases and then its letters get greater graphemic load. Hence more directly Hypothesis 7 The greater the size of graphemes, the greater the graphemic load of letters. Since letter utility is a weighted expression of the graphemic load, it automatically holds that Hypothesis 8 The greater the size of the graphemes, the greater the L-utility or alternatively Hypothesis 9 The greater the inventory of graphemes, the greater the L-utility. Again, these two hypotheses go back to the ratio of P-inventory, and Linventory, which is the cause of G-inventory. Consider now the frequency of letters. We must distinguish between their rank-frequency distribution which is a global property just as the different characteristics of the distribution (mean, variance etc.) and their individual frequency. As mentioned above, the rank-frequency distribution depends on the inventory size; here we consider individual frequencies. As to alphabetical scripts there is no necessity to suppose that more frequent letters are less complex. The script can be borrowed but the frequencies in the source language need not. If a language develops and diversiﬁes, the phonemic and orthographic roles of its letters may change drastically. The relationship does not hold even for syllabic scripts. However, in the development of an alphabetic script out of a pictographic one, the tendency can assert itself. In pictographic scripts this tendency must be present: if no religious or other grounds hinder simpliﬁcation, then either a frequent sign will be simpliﬁed or out of two synonymous signs the simpler will be chosen. Though we do not want to treat pictographic scripts, we merely mention that it is very probable that out of two synonymous pictograms the simpler is more frequent. Here it is, however, simplicity/complexity that controls frequency, not vice versa. 158 Gabriel Altmann Frequency has much to do with the phonemic load, the graphemic load and the graphemic utility of letters. Here the following hypotheses can be set up: Hypothesis 10 The greater the P-load of a letter, the more frequently it occurs. Hypothesis 11 The greater the G-load of a letter, the more frequently it occurs. If a letter is used in representations of several phonemes, or if it occurs in many graphemes as e.g. pronunciation catalyzer (e.g. Italian ), then its frequency automatically increases. This increase is surely not linear, otherwise it would strongly contribute to the increase of redundancy. The third hypothesis is Hypothesis 12 The greater the graphemic utility of a letter, the more frequently it occurs. This is partially a consequence of the previous hypothesis. In handwriting, frequency is a direct “cause” of change in distinctivity. letters that occur very frequently have the tendency to be shortened, simpliﬁed, written indistinctly, i.e. Hypothesis 13 The more frequent a letter in handwriting, the more it loses its distinctivity. Here distinctivity is not meant in the deciphering (reading) process but in its optical appearance. The letter loses some of its distinctive features, becomes more similar to other letters, and its mean distinctivity decreases. Its deciphering is due to redundancy contained in the rest of the word in which it occurs. On the other hand, complexity contributes to distinctivity. The more strokes there are in the letter and the more complex their mutual positioning, the better it can be distinguished from other letters. Hence, one can suppose that Hypothesis 14 The greater the complexity of a letter, the more distinctive it is. However, this hypothesis must be used cum grano salis. Distinctivity is not meant as ease of reading but as the difference from other letters. It is Towards a theory of script 159 known that calligraphic scripts which are very complex are not easy to read. Distinctivity is a graphically measurable property, a function of the difference to all other letters. Ornamentality is an artistic supplement to the letter having no relation to the majority of the properties. It arises on aesthetic grounds, and in exaggerated form it can cause a loss of deciphering (reading) distinctivity. There is only one hypothesis possible: Hypothesis 15 The more ornamental a letter, the greater its complexity. Here, too, the deciphering (reading) distinctivity should not be mixed up with distinctivity in relation to other letters. Joining the last two hypotheses we obtain: Hypothesis 16 The more ornamental a letter, the greater its distinctivity (in relation to other letters) 4 The control cycle Let us ﬁrst present all the hypotheses in graphical form indicating the dependence with arrows without trying to derive the form of the dependence. They are presented in Figure 2. Whatever the mathematical form of these hypotheses, we can try to set up a theory based on the following expectations. 1. None of these relationships is linear. In some cases we know it already, e.g. inventory vs. entropy, but for other hypotheses we expect some (non-linear) proportionality relations. In other cases we conclude per analogiam leaning against synergetic linguistics. 2. The system of all these properties is not balanced. Self-regulation works well in spoken language, but orthography is always behind the times. If orthographic reforms are not made continuously, at a certain point in the development of language it is not possible to carry out the reforms without evoking a cultural catastrophe. Hence, the above system (Fig. 2) is not a self-regulating system. In spoken language the majority of changes is spontaneous; in orthography unauthorized changes are punished. 160 Gabriel Altmann Entropy L-Inventory size – Mean L-distinctivity + Grapheme size – + + Orthographic uncertainty + + L-utility + + + – + G-Inventory Graphemic load of letters Complexity L-distinctivity P-load of letter – + + Ornamentality + P-Inventory + + L-Frequency – Figure 2: Some relations among letter properties 3. The individual properties considered as vertices of a graph have different in-degrees and out-degrees. Some of them do not seem to be affected by other properties, but this statement is preliminary. The more properties are discovered the denser will be the net of their mutual relations. 4. The above network does not evolve spontaneously; new edges are added in dependence on the advance in this domain. It is a theoretical, not a natural system. 5 Proposal for the analysis of alphabetic scripts In order to facilitate research on script theories for different languages, we append a list of steps that can be done in a systematic way. Not all hypotheses can be tested because for some tests one needs many languages. But if the database increases, it will be possible to search for laws of script formation. The analysis is appropriate for alphabetic scripts, but it also holds for scripts with inherent (same) vowel, e.g. Indian scripts. It is not adequate for iconographic scripts, which must be treated differently. The individual steps are as follows: Towards a theory of script 161 1. Set up the inventory of phonemes. Do not adhere to historically or morphologically based orthographies; realize that an allophone exists only if there is no independent phoneme, e.g. in German there is a phoneme /N/ (“Wange”) written as ; here in “Bank” does not represent an allophone of /n/ but the phoneme /N/. In the German words “Grad” and “Grat” the last phonemes are identical. The number of phonemes in the inventory is |P|. One should rely on the work of phonemicists if their work is not morphological or historical. One should decide in advance whether phonemes occurring only in foreign words are part of the inventory, or set up two different inventories. Write phonemes in slant lines //. 2. For each phoneme, write all letters and graphemes which can represent it. This is the most difﬁcult task. As shown in step 1, assimilations usually “change” one “phoneme” in another. Look at the completed analysis of some languages in this volume or elsewhere. Write letters or graphemes in brackets <>. 3. For each phoneme, write the number of representing graphemes. For phoneme x it is Gx . Set up the inventory of graphemes G (each grapheme occurs in it only once, even if it represents several phonemes). 4. Count all Gx to get the number of graphemes in a language, |G|, i.e. ascertain the cardinal number of G. 5. Take several text samples and count the frequency of phonemes or letters or graphemes or all. To this purpose one can also use a corpus. 6. Compute the relative frequencies of phonemes/letters/graphemes and use them to compute the entropies of the frequency distributions. Use the Shannon entropy but try to compute also other ones if you like (there are about 40 different deﬁnitions of entropy). 7. Compute the phonemic load of letters, PLx . 8. Compute the graphemic load of letters, GLx . 9. Compute the orthographic uncertainty of each phoneme, Ux . 10. Compute the mean grapheme size from a list of graphemes set up in step 2 (each grapheme counted only once). 11. Compute letter utility (useful only in some languages). 12. Compute the complexity of all letters. Use the connection method and the fractal dimension or develop new methods. 13. Compute the L-distinctivity of letters and the mean distinctivity of the script. Propose new methods if necessary. 14. If you have students ready to ﬁll in questionnaires, compute the orna- 162 15. 16. 17. 18. 19. 20. 21. 22. 23. Gabriel Altmann mentality of different letter types (a ﬁrst step has been made for Latin script types in MS WORD, cf. Best & Altmann in this volume). Fit the negative hypergeometric distribution to the (absolute) frequency counts. Then try to ﬁt simpler distributions called Zipﬁan. Take the known entropy curves for phoneme frequencies and localize your language/script on this curve. Prepare a table containing the P-load and the frequency of a letter. Test whether there is a correlation between them and try to ﬁnd the shape of the dependence (P-load is the independent variable). Display the points graphically < P-load, Frequency>. Prepare a table containing the G-load and the frequency of a letter and proceed as in step 17, < G-load, Frequency>. Prepare a table containing the graphemic utility of a letter and its frequency. Proceed as in step 17, < G-utility, frequency>. Prepare a table containing the complexity and the distinctivity of a letter. Proceed as in step 17, . If you have computed ornamentality, prepare a table containing ornamentality and complexity of letters. Proceed as in step 17, . If you have computed ornamentality, prepare a table containing ornamentality and distinctivity of letters. Proceed as in step 17, . Present distinctivity as a function of ornamentality and complexity. If new properties of script are established, the number of hypotheses will automatically increase. Steps 1 to 5 are purely “qualitative”. All decisions made later on depend on this analysis. Thus these steps must be performed with much care. References Altmann, Gabriel; Lehfeldt, Werner 1980 Einführung in die quantitative Phonologie. Bochum: Brockmeyer. Altmann, Gabriel 2004 “Script complexity.” In: Glottometrics, 8; 68–73. Anti´c, Gordana; Altmann, Gabriel 2005 “On letter distinctivity.” In: Glottometrics, 9; 46–53. Towards a theory of script 163 Bernhard, Gerald; Altmann, Gabriel 2007 “The grapheme-phoneme relationship in Italian”. This volume, pp. 13– 23. Best, Karl-Heinz; Altmann, Gabriel 2007 “Script ornamentality”. This volume, pp. 91–104. 2004 “Some properties of graphemic systems.” In: Glottometrics, 9; 29–39. Best, Karl-Heinz 2005 “Zur Häuﬁgkeit von Buchstaben, Lesezeichen und anderen Schriftzeichen in deutschen Texten.” In: Glottometrics, 11; 9–31. Cha, Sang-Hyuk; Tappert, Charles C.; Srihari, Sargur N. 2003 “Optimizing binary feature vector similarity measure using genetic algorithm and handwritten character recognition”. In: Proceedings of the Seventh International Conference on Document Analysis and Recognition (ICDAR 2003). Los Alamitos, CA: IEEE Computer Society. Falconer, Kenneth J. 1993 Fraktale Geometrie. Heidelberg: Spectrum. Fan, Fengxiang; Altmann, Gabriel 2006 “Graphemic representation of English phonemes”. This volume, pp. 25– 59. Feixas, Miguel; del Acebo, Esteve; Bekaert, Philippe; Sbert, Mateu 1999 “An information theory framework for the analysis of scene complexity.” In: Eurographics, 18(3); 95–106. Förstemann, Ernst 1846 “Ueber die numerischen Lautverhältnisse im Deutschen.” In: Germania, 7; 83–90. 1852 “Numerische Lautverhältnisse im Griechischen, Lateinischen und Deutschen.” In: Zeitschrift für vergleichende Sprachforschung, 1; 163–179. Grzybek, Peter; Kelih, Emmerich 2003 “Graphemhäuﬁgkeiten (am Beispiel des Russischen). Teil I: Methodologische Vor-Bemerkungen und Anmerkungen zur Geschichte der Erforschung von Graphemhäuﬁgkeiten im Russischen.” In: Anzeiger für Slavische Philologie, 23; 131–162. Grzybek, Peter; Kelih, Emmerich; Altmann, Gabriel 2004 “Graphemhäuﬁgkeiten (am Beispiel des Russischen). Teil II: Modelle der Häuﬁgkeitsverteilung.” In: Anzeiger für Slavische Philologie, 32; 25–54. Grzybek, Peter; Kelih, Emmerich 2005 “Graphemhäuﬁgkeiten im Ukrainischen. Teil I: Ohne Apostroph (´)”. In: Altmann, Gabriel; Levickij, Viktor; Perebejnis, Valentina (Eds.), Problemi kvantitativnoï lingvistiki – Problems of Quantitative Linguisˇ tics 2005. Cernivci: Ruta, 159–179. 164 Gabriel Altmann Hegenbarth-Reichhardt, Ina; Altmann, Gabriel 2007 “On the decrease of complexity from hieroglyphs to hieratic symbols”. This volume, pp. 105–114. Kelih, Emmerich 2007 “The phoneme-grapheme relationship in Slovene”. This volume, pp. 61– 74. Martindale, Colin et al. 1996 “Comparison of equations describing the frequency distribution of graphemes and phonemes.” In: Journal of Quantitative Linguistics, 3; 106–112. Möller, Georg 1927 Hieratische Paläographie. Die aegyptische Buchschrift in ihrer Entwicklung von der fünften Dynastie bis zur römischen Kaiserzeit. Osnabrück: Zeller. Peust, Carsten 2006 “Script complexity revisited.” In: Glottometrics, 12; 11–15. Schroeder, Manfred 1991 Fractals, chaos, power laws. Minutes from an inﬁnite paradise. New York: Freeman. Strauss, Udo; Altmann, Gabriel; Best, Karl-Heinz 2006 “Phoneme frequency.” [http://www.uni-trier.de/uni/fb2/ldv/ lql_wiki/index.php/Phoneme_frequency] Tricot, Claude 1993 Curves and fractal dimesnions. New York: Springer. Youssif, Roshdy S.; Purdy, Carla N. 2002 Fuzzy similarity measures for signal pattern classiﬁcation. [Citeseer. ist.psu.edu/578632.html] Zörnig, Peter; Altmann, Gabriel 1983 “The repeat rate of phoneme frequencies and the Zipf-Mandelbrot law.” In: Glottometrika, 5; 205–211. 1984 “The entropy of phoneme frequencies and the Zipf-Mandelbrot law.” In: Glottometrika, 6; 41–47. Authors’ Addresses Bernhard, Gerald Universität Bochum Romanisches Seminar, GB 8/131 D–44780 Bochum, Universitätsstraße 150, Germany email: [email protected] Best, Karl-Heinz Universität Göttingen Seminar für deutsche Philologie D–37073 Göttingen, Käte-Hamburger-Weg 3, Germany email: [email protected] Fan, Fengxiang Dalian Maritime University School of Foreign Languages C–116026 Dalian, China email: [email protected] Hegenbarth-Reichardt, Ina Asien-Afrika-Institut: Aethiopistik D–20146 Hamburg, Edmund-Siemers-Allee 1, Germany email: [email protected] Kelih, Emmerich Universität Graz Institut für Slawistik A–8010 Graz, Merangasse 70, Austria email: [email protected] Köhler, Reinhard Universität Trier Linguistische Datenverarbeitung D–54296 Trier, Universitätsring 15, Germany email: [email protected] 166 Maˇcutek, Ján Comenius University Department of Applied Mathematics and Statistics SK–84248 Bratislava, Mlynská dolina, Slovakia email: [email protected] Mohanty, Panchanan University of Hyderabad Centre for Applied Linguistics and Translation Studies Gachibowli, I–500046 Hyderabad, India email: [email protected] Nemcová, Emília Universität Trnava Lehrstuhl für Slovakische Sprache und Literatur SK–91701 Trnava, Námestie Jozefa Herdu 2, Slovakia email: [email protected] Tamaoka, Katsuo Hiroshima University International Student Center 1-1, 1-chome Higashihiroshima, Japan 739–8524 email: [email protected] Author Index A Acebo, E.D. . . . . . . . . . . . . . . . 163 Adams, M.J. . . . . . . . . . . . . . 25, 58 Altenmüller, H. . . . 105, 106, 113 Altmann, G. . . . . vi, 4, 5, 7, 9, 13, 16, 19, 23, 33, 35, 38, 42, 58, 59, 61, 63–66, 68–70, 73–76, 78, 83, 85–87, 96, 104, 106, 107, 110, 112, 113, 125, 133, 139, 142, 145, 151–156, 162–164 Amano, N. . . . . . . . . . . . . 141, 144 Anti´c, G. . . . . . . . . . . . . . . 153, 162 D Daelemans, W. . . . . 121, 139, 140 Dobrovoljc, H. . . . . . . . . . . . 62, 73 B Barnes, M.A. . . . . . . . . . . . . . . . . 59 Bekaert, P. . . . . . . . . . . . . . . . . . 163 Bense, M. . . . . . . . . . . . . . . 92, 104 Berg, S. . . . . . . . . . . . . . . . . . 75, 78 Berndt, R.S. . . . . . . . . . . . . . 25, 58 Bernhard, G. . . . . . 42, 64–66, 73, 76, 83, 85, 86, 125, 139, 154, 156, 163 Best, K.-H. . . . . . . . 13, 16, 23, 38, 58, 63, 66, 68, 73, 83, 86, 125, 139, 152, 162–164 Birkhoff, G.D. . . . . . . . . . . 92, 104 Bohn, H. . . . . . . . . . . .5, 9, 97, 104 G Gardiner, A.H. . . . . 105, 106, 113 Gelder, B. de . . . . . . . . . . 121, 139 Gložanˇcev, A. . . . . . . . . . . . . . . . 73 Grzybek, P. . . . . . 38, 58, 155, 163 C Cha, S.-H. . . . . . . . . . . . . . 152, 163 Content, A. . . . . . . . . . . . . . . . . 139 Cunningham, J.W. . . . . . . . . 25, 58 Cunningham, P.M. . . . . . . . 25, 58 F Förstemann, E. . . . . . . . . 152, 163 Falconer, K.J. . . . . . . . . . . 152, 163 Fan, F. . . vi, 63, 73, 85, 154, 156, 163 Feixas, M. . . . . . . . . . . . . . 152, 163 Ferrer i Cancho, R. . . . . . . . . . 144 Frost, R. . . . . . . . . . . . . . . 121, 140 Fry, E. . . . . . . . . . . . . . . . . . . 25, 58 H Hˇrebíˇcek, L. . . . . . . . . . . . 115, 119 Hanna, J.S. . . . . . . . . . . . . . . 25, 59 Hanna, P.R. . . . . . . . . . . . . . . 25, 59 Harary, F. . . . . . . . . . . . . . 133, 140 Hegenbarth-Reichhardt, I. . . . 164 Hodges, R.E. . . . . . . . . . . . . 25, 59 Hug, M. . . . . . . . . . . . . . . . 125, 140 Hunt, F. . . . . . . . . . . . . . . . 116, 119 J Job, U. . . . . . . . . . . . . . . . . 112, 113 Johnson, N.L. . . . . . . . . . . . . 75, 78 Junge, F. . . . . . . . . . . . . . . 105, 113 168 AUTHOR INDEX K Köhler, R. . . 9, 33, 38, 59, 96, 97, 104, 141–145 Katz, L. . . . . . . . . . . . . . . . 121, 140 Kelih, E. . . 38, 58, 155, 156, 163, 164 Kemp, A.W. . . . . . . . . . . . . . 75, 78 Klima, E. . . . . . . . . . . . . . 121, 140 Kondo, T. . . . . . . . . . . . . . 141, 144 Kotz, S. . . . . . . . . . . . . . . . . . 75, 78 L Lehfeldt, W. . . 133, 139, 155, 162 Lencek, R.L. . . . . . . . . . . . . . 61, 73 Levec, F. . . . . . . . . . . . . . . . . . . . . 62 M Möller, G. . . . . 109, 113, 153, 164 Maˇcutek, J. . . . . 23, 65, 73, 83, 87 Makioka, S. . . . . . . . . . . . 141, 145 Martináková-Rendeková, Z. . . 59 Martindale, C. . . . . . . . . . 155, 164 Meyer-Kress, C. . . . . . . . . . . . . 119 Mitchum, C.C. . . . . . . . . . . . 25, 58 Mohanty, P. . . 75, 76, 78, 87, 156 Morton, J. . . . . . . . . . . . . . . . 25, 59 N Nemcová, E. . . . . . 66, 69, 74, 156 P Paper, H.H. . . . . . . . . . . . . 133, 140 Patil, G.P. . . . . . . . . . . . . . . . . . . . 78 Patterson, K.E. . . . . . . . . . . . 25, 59 Peitgen, H.-O. . . . . . . . . . 116, 119 Peust, C. . . . 5, 6, 9, 96, 104, 106, 114, 151, 164 Purdy, C.N. . . . . . . . . . . . 152, 164 R Reggia, J.A. . . . . . . . . . . . . . 25, 58 Rehder, P. . . . . . . . . . . . . . . . 61, 74 Rudorf, E.H. . . . . . . . . . . . . . 25, 59 S Satzinger, H. . . . . . . . . . . 105, 114 Saupe, D. . . . . . . . . . . . . . 116, 119 Sbert, M. . . . . . . . . . . . . . . . . . . 163 Schlott, A. . . . . . . . . . . . . 106, 114 Schroeder, M. . . . . . . . . . 152, 164 Seidenberg, M.S. . . . . . . . . . 25, 59 Shenton, L.R. . . . . . 19, 23, 75, 78 Skees, P. . . . . . . . . . . 19, 23, 75, 78 Solé, R.V. . . . . . . . . . . . . . . . . . . 144 Srihari, S.N. . . . . . . . . . . . 152, 163 Sullivan, F. . . . . . . . . . . . . 116, 119 T Tamaoka, K. . . . . . . . . . . . 141, 145 Tanenhaus, M.K. . . . . . . . . . . . . 59 Tappert, C.C. . . . . . . . . . . 152, 163 Toporišic . . . . . . . . . . . . . 61, 62, 74 Tricot, C. . . . . . . . . . . . . . 152, 164 V Van den Bosch, A. . 121, 139, 140 Venezky, R.L. . . . . . . . . . 121, 140 W Waters, G.S. . . . . . . . . . . . . . . . . 59 Wimmer, G. . . 19, 23, 33, 35, 59, 75, 78, 142, 145 Y Youssif, R.S. . . . . . . . . . . 152, 164 Z Zipf, G.K. . . . . . . . . . 33, 112, 114 Zörnig, P. . . . . . . . . . . . 37, 59, 164 Subject Index A abstract-logographic script . . . . . 3 alphabetic script . . 3, 4, 108, 149, 151, 155, 157, 160 B Bézier curve . . . . . . . . . . . . . . . 6, 7 C calculus, distributional . . . v, 122, 133, 138 capacity dimension . . v, 115, 116, 118 Chinese . . . . . . 5, 29, 97, 106, 151 cognitive effort . . . . . . . . . . . . . . . 6 complexity v, 3, 5–8, 96–99, 105– 113, 118, 142, 150–152, 156–159, 161 decoding c., 6, 99 measure of c., 6, 8, 92, 96, 106, 118, 151, 152, 155, 156 measure of complexity, 6, 8, 92, 96, 106, 118, 151, 152, 155, 156 production c., 6, 7 concentration . . . . . . . . . . . . . . . 32 Conway-Maxwell-Poisson distribution . . . . . . . . . . . . . . . 33, 34 correlation dimension . . 106, 115 cotextuality . . . . . . . 141–143, 146 curve, Bézier . . . . . . . . . . . . . . . 6, 7 D decoding complexity . . . . . . .6, 99 degree, vertex . . . . . . . . . . . . . . . 29 dimension . v, 106, 108, 115–119, 122, 126–128, 130 capacity d., v, 115, 116, 118 correlation d., 106, 115 fractal d., 96, 115, 117–119, 161 Hausdorff d., 115, 152 Hull d., 115 distinctivity . . v, 4, 8, 97, 99, 153, 155, 158, 159, 161, 162 distribution, frequency 9, 37, 149, 155, 157, 161 distributional calculus . . . . v, 122, 133, 138 distributions Conway-Maxwell-Poisson d., 33, 34 geometric d., 18, 19, 75, 77, 83, 131 Lotka d., 35, 36 negative hypergeometric d., 37– 40, 152, 162 Poisson d., 33 Shenton-Skees-geometric d., 19, 65, 75, 77, 83, 84, 131, 133 zeta/Zipf d., 35, 37, 162 Zipf-Mandelbrot d., 37 diversiﬁcation . . 29, 42, 121, 125, 127, 153 E economy . 33, 34, 63, 70, 99, 113 effectivity . . . . . . . . . . . . . . . 63, 66 efﬁciency . . . . . . . . . . . . . . . . . . 4, 8 170 SUBJECT INDEX effort . . . . . . . . . . . . . 6–8, 112, 113 cognitive, 6 muscular, 6, 7 English . . v, 4, 13, 25, 27, 29, 32, 33, 35–39, 42, 43, 63, 69, 85, 93, 95, 98, 101, 125, 150, 153, 155, 156 entropy . . 4, 8, 30, 119, 149, 155, 159, 161, 162 explanation, functional . . . . . . . . 9 exploitation, graphemic 13, 21, 33 exponent, Lyapunov . . . . . . . . 115 F fractal dimension . . . . . . . 96, 115, 117–119, 161 French . . . . . . . . 41, 125, 136, 149 frequency v, 3, 8, 9, 22, 30, 32, 35, 43–58, 75, 77, 131, 132, 141–144, 146, 150, 152, 157, 158, 161, 162 grapheme, 21, 23, 32, 152 letter, 23, 150, 152, 157, 162 frequency distribution 9, 37, 149, 155, 157, 161 function, Zipﬁan . . . . . . . . . 33, 35 functional explanation . . . . . . . . . 9 G geometric distribution 18, 19, 75, 77, 83, 131 German . . . . . . . . . v, 4, 13, 16–21, 23, 25, 63–67, 69, 70, 76, 77, 85, 92, 125, 131, 132, 149, 161 grapheme . . . . . . . . . . . . . 4, 13–16, 18–23, 25, 27–29, 32–58, 62–71, 79–85, 121–123, 125–130, 133–139, 149– 158, 161 g. frequency, 21, 23, 32, 152 g. length, 32–34, 61, 67 g. size, v, 13, 19–21, 61, 66– 68, 70, 84, 85, 126, 128, 150, 154, 156, 157, 161 graphemic g. exploitation, 13, 21, 33 g. impact, 75, 76, 134, 136– 139 g. load of letters, 21, 22, 61, 68, 69, 85, 154, 156–158, 161 g. representation, 13, 18, 27– 29, 32, 65, 80, 83, 84, 121, 123, 131, 133 H Hausdorff dimension . . . 115, 152 hieratic script . . . . . 105–114, 151 hieratic symbols . . . . . . . 105–114 hieroglyphs . . . . . v, 105–112, 153 Hull dimension . . . . . . . . . . . . . 115 I ideographic script . . . . . . . . . 3, 13 impact, graphemic . . 75, 76, 134, 136–139 Indian script . . . . . . . . . . . . . . . 151 information . . . . . . 8, 29, 106, 133 inventory size 4, 22, 61, 150, 151, 155, 157 SUBJECT INDEX Italian . . . . . . . . . . v, 4, 13–23, 25, 42, 64–67, 69, 70, 76–78, 83, 85, 86, 125, 131, 132, 149, 154, 158 J Japanese . . v, 121, 122, 141, 144, 146 K Korean . . . . . . . . . . . . . . . . . . . . 122 L languages Chinese, 5, 29, 97, 106, 151 English, v, 4, 13, 25, 27, 29, 32, 33, 35–39, 42, 43, 63, 69, 85, 93, 95, 98, 101, 125, 150, 153, 155, 156 French, 41, 125, 136, 149 German, v, 4, 13, 16–21, 23, 25, 63–67, 69, 70, 76, 77, 85, 92, 125, 131, 132, 149, 161 Italian, v, 4, 13–23, 25, 42, 64– 67, 69, 70, 76–78, 83, 85, 86, 125, 131, 132, 149, 154, 158 Japanese, v, 121, 122, 141, 144, 146 Korean, 122 Latin, 38, 121 Oriya, v, 78, 121–123, 125– 127, 129, 131–135, 137– 139, 150 Slavic, 18, 32, 62, 67, 70, 79, 149, 151, 153 Slovak, v, 25, 64–67, 69, 70, 79–81, 83–86, 156 171 Slovene, 61–71, 73 Spanish, 149 Swedish, v, 4, 13, 16–21, 23, 25, 63–67, 69, 70, 76, 77, 83, 85, 125, 131, 132 Latin . . . . . . . . . . . . . . . . . . . 38, 121 law Zipf’s, 35 learnability . . . . . . . . . . . . . . . . 4, 8 length, grapheme . . 32–34, 61, 67 letter . . . . . . . . . . . . . . . . . . . . . . v, 4, 7, 13, 14, 18–23, 27, 29, 32, 33, 38–42, 62, 66, 68, 69, 79, 80, 84–86, 92–94, 96, 98, 99, 115–118, 121, 122, 126, 133, 149–161 frequency, 23, 150, 152, 157, 162 participation, 13, 21–23, 38– 40, 68, 80, 85, 154 usefulness/utility, 21–23, 41, 42, 61, 68–70, 85, 86, 150, 154, 157, 158 weighted participation, 21, 39, 40, 121 linguistics . . 3, 9, 25, 32, 33, 133, 143, 149, 150, 152 synergetic l., v, vi, 141, 142, 150, 159 load of letters, graphemic . 21, 22, 61, 68, 69, 85, 154, 156– 158, 161 logographic script . . . . . . . . . 3, 29 Lotka distribution . . . . . . . . 35, 36 Lyapunov exponent . . . . . . . . . 115 M mixed script . . . . . . . . . . . . . . 3, 13 172 SUBJECT INDEX muscular effort . . . . . . . . . . . . . 6, 7 N negative hypergeometric distribution . . . . . . . .37–40, 152, 162 network . . . . . . . . . . . . . . vi, 9, 160 O Oriya . . . . . . v, 78, 121–123, 125– 127, 129, 131–135, 137– 139, 150 ornamentality . . . v, 7, 8, 91–100, 150, 152, 159 orthographic uncertainty . 13, 14, 16, 17, 61, 63–65, 70, 83, 125, 156 P participation . . . . . . . . . . . . . 22, 39 letter p., 13, 21–23, 38–40, 68, 80, 85, 154 positional p. of letters, 13, 22, 23, 85 weighted letter p., 21, 39, 40, 121 phonographic script . . . . . . . . . . . 3 pictographic script . . . 3, 149, 157 Poisson distribution . . . . . . . . . . 33 polyfunctionality . . . . . . . . . . 8, 63 polyphonemics . . . . . . . 34, 36, 41 polysemy . . . . . . . . . . . . . . . . . . 8, 9 polytextuality . . . . . . . v, 142–144 positional participation of letters 13, 22, 23, 85 production complexity . . . . . . 6, 7 R readability . . . . . . . . . . . . . . . . . . . 8 redundancy . . . . . . . . . . . . . . 8, 158 repeat rate . . . . . . . . . . . . . . 32, 155 representation, graphemic 13, 18, 27–29, 32, 65, 80, 83, 84, 121, 123, 131, 133 S script . . . . . . . . . . v, 3–5, 9, 13, 14, 23, 29, 61–63, 65–67, 70, 79, 92–97, 99, 105–108, 110, 112, 121, 122, 126, 132, 134, 136, 138, 139, 141, 149–151, 153, 154, 157, 159, 160, 162 hieratic s., 105–114 abstract-logographic s., 3 alphabetic s., 3, 4, 108, 149, 151, 155, 157, 160 Chinese s., 151 hieratic s., 151 ideographic s., 3, 13 Indian s., 151 logographic s., 3, 29 mixed s., 3, 13 phonographic s., 3 pictographic s., 3, 149, 157 syllabic s., 3, 13, 121, 122, 149, 151, 157 self-regulation . . . . . . . . . . . v, 159 Shenton-Skees-geometric distribution . . 19, 65, 75, 77, 83, 84, 131, 133 simpliﬁcation . . . . . . . v, 105–107, 110–113, 153, 157 size 7, 18–21, 29, 35, 66, 67, 116, 122, 126–128, 130, 141 SUBJECT INDEX grapheme s., v, 13, 19–21, 61, 66–68, 70, 84, 85, 126, 128, 150, 154, 156, 157, 161 inventory s., 4, 22, 61, 150, 151, 155, 157 Slavic languages . . 18, 32, 62, 67, 70, 79, 149, 151, 153 Slovak . . . . . v, 25, 64–67, 69, 70, 79–81, 83–86, 156 Slovene . . . . . . . . . . . . . . 61–71, 73 Spanish . . . . . . . . . . . . . . . . . . . . 149 Swedish . . v, 4, 13, 16–21, 23, 25, 63–67, 69, 70, 76, 77, 83, 85, 125, 131, 132 syllabic script . . . 3, 13, 121, 122, 149, 151, 157 symbols hieratic s., 105–114 symbols, hieratic . . 107, 109, 110 symmetry . . . . . . . . . . . . . . . . . . 138 synergetic linguistics . . v, vi, 141, 142, 150, 159 synfunctionality . . . . . . . . . . . . . . 8 synophonemics . . . . . . . 34, 37, 38 T type-token . . . . . . . . 141–144, 146 U uncertainty . v, 14, 16, 17, 28–30, 63–65, 83, 94, 121, 125, 126, 130–133, 161 orthographic u., 13, 14, 16, 17, 61, 63–65, 70, 83, 125, 156 unweighted u., 29, 64 weighted u., 30 unweighted uncertainty . . . 29, 64 173 usefulness, letter . . . . . . . . . 21–23, 41, 42, 61, 68–70, 85, 86, 150, 154, 157, 158 utility, letter . . . 21–23, 41, 42, 61, 68–70, 85, 86, 150, 154, 157, 158 V variability 107, 121, 143, 150, 153 vertex degree . . . . . . . . . . . . . . . . 29 W weighted letter participation . . 21, 39, 40, 121 weighted uncertainty . . . . . . . . . 30 Z zeta/Zipf distribution . 35, 37, 162 Zipf’s law . . . . . . . . . . . . . . . . . . 35 Zipf-Mandelbrot distribution . . 37 Zipﬁan function . . . . . . . . . . 33, 35

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