The Chronology of the Qur’ān: A Stylometric Research Program

Arabica 58 (2011) 210-299 brill.nl/arab he Chronology of the Qur ān: A Stylometric Research Program Behnam Sadeghi1 Abstract I verify a chronology in which seven groups of passages represent consecutive phases. A proposed chronology is verified if independent markers of style vary over its phases in a smooth fashion. Four markers of style follow smooth trajectories over the seven phases: he first is average verse length. he second encompasses the 28 most common morphemes in the Qur ān. he percentages of these morphemes in a text constitute its stylistic profile. he thus-defined stylistic profile is shown to vary in a smooth fashion over “time”, i.e. over the proposed chronological sequence of phases. hird, a similar thing holds for a profile based on the frequencies of 114 other common morphemes. Fourth, similar results are obtained for a list of 3693 relatively uncommon morphemes. In addition to establishing a relative chronology in seven phases, this essay demonstrates the stylistic unity of many large passages. It also shows that the Qur ān has one author. Keywords Qur ān, sūras, chronology, stylometry, Islamic origins, Sīra, Prophet Muḥammad, Mehdi Bazargan 1. Introduction And a Koran We have divided, for thee to recite it to mankind at intervals, and We have sent it down successively.2 1 his work was supported through the research fund of Michael Cook at Princeton University in 2005. he essay was presented in Nov. 2006 in the American Academy of Religion Conference in Washington, DC. It was submitted to Arabica in 2009, but was updated before publication with references to recent literature. I thank Michael Cook for his written comments on this essay, Shuly Wintner for generous assistance with the “tagged Qur ān” he developed jointly with the late Rafael Talmon, Andrei Radulescu for invaluable assistance with computers, Abdolali Bazargan and Mohammad Hossein Bani Asadi for gifting several volumes of Mehdi Bazargan’s Sayr, and Asad Ahmed and Patricia Crone for useful discussions. 2 Kor 17, 106. See also Kor 25, 32. Arthur Arberry, he Koran Interpreted, London, Oxford University Press, 1983. © Koninklijke Brill NV, Leiden, 2011 DOI: 10.1163/157005810X529692 B. Sadeghi / Arabica 58 (2011) 210-299 211 he Goal Using stylometry, I answer three related questions: First, how many authors does the Qur ān have? Second, is the basic textual unit a small fragment normally no more than a few sentences long, or do many relatively large passages form stylistically coherent units? hird, what is the relative order in which the passages of the Qur ān were disseminated? his last problem, viz. relative chronology, has long been a topic of scholarly controversy, and it is what provides the impetus for this essay. Knowing the relative chronology of the Qur ān is important if one hopes to interpret it properly and use it to understand the formation of Islam. he answers to the first two questions emerge as corollaries of the analysis conducted for chronology. he analysis here covers the entire Qur ān. he last time a publication appeared with a similar scope was in 1976, when Mehdi Bazargan (Mahdī Bāzargān) published the first volume of his landmark Sayr-i taḥ awwul-i Qur ān.3 Bazargan’s work has inspired mine and provided the starting point 3 he publication history of the three parts of Bazargan’s book requires clarification. he first volume, comprising about 200 pages, was published in HS [Solar Hejri=modern Iranian calendar] 1355/1976-7, many years after it was written (Mahdī Bāzargān, Sayr-i taḥ awwul-i Qur ān, vol. I, Tehran, Qalam, HS 1355/1976-7). his volume contained the main results and a quantitative study of the distribution of themes in the Qur ān. It also contained a summary in the French language. he second volume, offering a sūra-by-sūra discussion of the block divisions, was published in HS 1360/1981 (Mahdī Bāzargān, Sayr-i taḥ awwul-i Qur ān, vol. II, Tehran, Širkat-i Sahāmī-i Intišār, HS 1360/1981). In HS 1362/1983-4 a “complementary volume” (Mutammim) was published with no overlap with vols I and II, offering a qualitative discussion of the evolution of ideas and language in the Qur ān. In HS 1377/1998-9, a revised edition of the first volume was published, expanded to include a 200-page addition comprising the complementary Mutammim published earlier and incorporating corrections to the results in the first edition (Mahdī Bāzargān, Sayr-i taḥ awwul-i Qur ān, vol. I, Tehran, Širkat-i Sahāmī-i Intišār, HS 1377/1998-9, 428 p.). In this edition, the citations of page numbers in the French part were not updated. Finally, in HS 1386/2007, a new edition of the book was published in which a group of researchers used computers to make corrections (Mahdī Bāzargān, Sayr-i taḥ awwul-i Qur ān, vols I and II in one volume, Tehran, Širkat-i Sahāmī-i Intišār, HS 1386/2007, 617 p.). his edition includes the revised versions of the old volumes I and II in one volume, but does not include the contents of the abovementioned complementary Mutammim volume. Unfortunately, volume I in this latest edition is missing some materials included in the previous editions. his includes, for example, the final section, corresponding to p. 157-210 in the 1976-7 edition and p. 181-234 in the 1998-9 edition, which is devoted to plotting the distribution of different themes against time. In my citations of vol. I, three page numbers are given separated with slashes, referring respectively to the editions of 1355/1976-7, 1377/1998-9, and 1386/2007. In my citations of vol. II, two page numbers are given separated with slashes, referring respectively to the editions of 1360/1981 and 1386/2007. For the Mutammim volume, I use the abbreviation Sayr Mutammim, and the page numbers refer to the 1377/1998-9 edition. 212 B. Sadeghi / Arabica 58 (2011) 210-299 for my analysis. Its focus on style and its quantitative cast are methodological features that carry over into my work. However, four features distinguish my contribution from the works of Bazargan and other researchers. First, this essay answers the above three questions (viz. the number of authors, the basic textual unit, and chronology) without any recourse to the statements of early Muslims about the history of the Qur ān. It uses neither the reports of individuals about Islamic origins nor the broad historical framework taken for granted in the huge literature that these reports comprise collectively. he reliability of such statements has been the subject of an academic debate. Focusing on the style of the Qur ān, this article bypasses the sayings of early authorities altogether and therefore is immune to doubts regarding their authenticity and reliability. In fact, the results here constitute an independent test of the broad outline of Islamic beginnings given in the literary sources.4 he second contribution involves the distinction between two things: (1) the criterion used for generating a sequence of groups of passages that one conjectures as representing the chronological order, and (2) the criterion one uses to corroborate or verify that sequence. Bazargan and other scholars used style to generate such sequences, but largely failed to use style to corroborate them. heir sequences could claim corroboration from considerations of meaning and external literary evidence—the kind of sources I disregard for the purpose of this article, but they were for the most part uncorroborated by style.5 By contrast, this essay provides a purely style-based method of verifying My analysis of Bazargan’s chronology is based on the original 1976-7 edition and does not take into account the corrections that were introduced later. his is because when I finished my analysis in 2005, I did not yet have access to the other volumes. My study concerns broad and robust patterns, so the kinds of small deviations introduced in later editions are immaterial. 4 his is not to say that I am neutral in the debate. I recognize the difficulty of evaluating the historical reports bearing on chronology. For example, the reports on the occasions of revelation (asbāb al-nuzūl ) are often contradictory and speculative. But I do not grant the plausibility of the revisionist claims that (1) the study of the literary sources cannot shed light on Islamic origins and that (2) the broad outline of Islamic history found in those sources is unreliable. See Behnam Sadeghi, “he Codex of a Companion of the Prophet and the Qur ān of the Prophet”, Arabica, 57/4 (2010), p. 343-436. he point is that the method in the present essay circumvents the problems and debates associated with the literary sources. 5 hese statements apply to pre-modern scholars, Nöldeke, and Bazargan. Some premodern authorities distinguished two phases using stylistic elements, including word choice. See Rāmyār, Tārīḫ-i Qur ān, p. 604-9; Muḥammad b. Abd Allāh al-Zarkašī, al-Burhān fī ulūm al-Qur ān, ed. Abū l-Faḍl al-Dimyāṭī, Cairo, Dār al-ḥadīt ̠h, 1427/2006, p. 132-4. For example, some said that the words kallā and yā ayyuhā l-nās occur in Meccan sūras, while yā ayyuhā llad̠īna āmanū occurs in Medinan sūras. For the Weil-Nöldeke chronology, verse length was a key criterion. See Gustav Weil, “An Introduction to the Quran. III” (translated by Frank Sanders, et al.), he Biblical World, 5/5 B. Sadeghi / Arabica 58 (2011) 210-299 213 proposed sequences, and it applies this method to corroborate a seven-phase chronology. he approach is rooted in what I call the “Criterion of Concurrent Smoothness”. he criterion does not generate chronological sequences; rather, it judges sequences that have been derived by other means. It does not matter how a proposed sequence was obtained: as long as it satisfies the criterion, it is confirmed as genuinely chronological. hird and related, previous style-based proposed sequences were generated based on the assumption that the style of the Qur ān changed in one direction without reversals. his is the way certain pre-modern authorities thought about the classical binary Meccan-Medinan division. It is also the approach taken in the four-phase Weil-Nöldeke chronology and the more detailed chronology of Bazargan. Bazargan assumed that verse length tended to increase over time, without reversing course, and he used this principle to rearrange the passages. By contrast, seeking to approach the problem with minimal premises, I make no a priori assumptions about how the style of the Qur ān developed over time. I do not presuppose that style must have progressed irreversibly, nor that it must have changed gradually. I do not even assume that style must have evolved in a manner satisfying my criterion for evaluating chronological sequences, namely the Criterion of Concurrent Smoothness. hus, if a sequence does not satisfy this criterion, that does not mean that it is not the true chronological sequence. But if it does satisfy it, then it is. Fourth, in the literature there are several quantitative investigations of some aspects of the Qur ān.6 hese contributions are valuable, but relatively limited (May 1895), p. 343-59; idem, “An Introduction to the Quran. IV”, he Biblical World, 5/6 ( June 1895), p. 438-47; heodor Nöldeke, Geschichte des Qorāns, 2nd ed., ed. Friedrich Schwally, et al., Hildesheim, Olms, 1961, p. 58-234. Nöldeke’s Geschichte has been translated by George Tamer into Arabic as Tārīḫ al-Qur ān, Beirut, Konrad-Adenauer-Stiftung, 2004. In a recent essay, Nicolai Sinai takes the first phase (early Meccan period) of the Weil-Nöldeke chronology (corresponding approximately to the first three groups in Bazargan’s chronology) and breaks it up into four sub-phases: I, II, IIIa, and IIIb. he last sub-phase (IIIb) is distinguished from the previous three by longer verses. In this respect, Sinai’s work mirrors that of Bazargan. However, Sinai also considers sūra length as a criterion. he four sub-phases are arranged in order of increasing sūra length. For the first three sub-phases it is not possible to claim a convergence of the criteria of verse length and sūra length, since their verse-length profiles are similar. However, if one combines the first three sub-phases (I, II, IIIa) into one subphase, the outcome is two sub-phases that exhibit a convergence of the two stylistic criteria of verse length and sūra length. Sinai also considers the number of sub-sections in a sūra, a variable that is probably not independent from sūra length. See Nicolai Sinai, “he Qur’an as Process”, in he Qur ān in Context: Historical and Literary Investigations into the Qur ānic Milieu, ed. Angelika Neuwirth, et al., Brill, Leiden, 2010, p. 407-40. 6 Mehdi Bazargan has done the most substantial work and he is the only one to use quantitative methods to synthesize a chronological sequence. However, he does not use the techniques of multivariate statistics. His book is cited in footnote 3. 214 B. Sadeghi / Arabica 58 (2011) 210-299 in either scope or methodology. On the one hand, studies conducted by scholars like Bazargan who know the historical problems are circumscribed in their use of quantitative techniques. On the other hand, researchers with the best statistical and computer skills lack a historical and literary background, which hampers them in formulating questions that advance Qur ānic Studies. he present article helps close the gap. It is a systematic analysis of the entire Qur ān that uses some methods that are commonly used in the field of stylometry but which have not been applied to the study of the Qur ān before (e.g. principal component analysis). he essay also uses some methods that are not common in stylometry, such as the cutting-edge technique of weight optimization. It offers a statistical treatment of many stylistic features: common words and morphemes, function words, uncommon words, word length, verse length, and hapax legomena. It needs to be stressed, however, that I have taken Naglaa habet uses multivariate techniques, namely hierarchical clustering. She compares the vocabulary of twenty-four long sūras, excluding function words. his reveals two major clusters of sūras which she identifies with the Meccan-Medinan division. See Naglaa habet, “Understanding the thematic structure of the Qur’an: an exploratory multivariate approach”, Proceedings of the ACL Student Research Workshop, 2005, p. 7-12. Hermann Moisl follows up on the work of habet. He discusses the limitations that sūra length places on meaningful clustering. He clusters 47 sūras on the basis of the frequencies with which they use nine lexical items (llāh, lā, rabb, qāla, kāna, yawm, nās, yawma id̠in, and šarr). He does not discuss chronology beyond the Meccan-Medinan division. See Hermann Moisl, “Sura Length and Lexical Probability Estimation in Cluster Analysis of the Qur’an”, ACM Transactions on Asian Language Information Processing (TALIP), 8/4 (Dec. 2009), article 19. Nora Schmid has written an essay that has two parts. he first part shows that in Nöldeke’s fourphase chronology the sūras in each phase tend to have longer verses than those in the preceding phase. (Bazargan had made a similar observation.) his is to be expected since verse length was one of the criteria used to define the phases in that chronology. More significantly, she shows that in the Meccan period average verse length is somewhat correlated with sūra length. hus, verse length as a criterion seems, at least in the Meccan period, somewhat corroborated by sūra length. See also the essay of Nicolai Sinai cited in the previous footnote. Note that the idea that in the Meccan period the length of a sūra is related to its chronological rank is implicit in I timād al-Salt ̣ana’s chronological sequence (see below, footnote 24), which lists Meccan sūras in the reverse order of their place in the Ut ̠mānic Qur ān, hence roughly in order of increasing sūra length. he second part of Schmid’s essay shows that in the Middle Meccan sūras, a short verse tends to be adjacent to a short verse, and a long verse tends to be adjacent to a long verse, thus refuting the hypothesis that individual verses were put next to each other in a random fashion. Whether this entails coherence at the sūra level requires more discussion. See Nora Schmid, “Quantitative Text Analysis and the Qur ān”, he Qur ān in Context: Historical and Literary Investigations into the Qur ānic Milieu, ed. Angelika Neuwirth, et al., Brill, Leiden, 2010, p. 441-60. Hans Bauer considered the extent to which the arrangements of the sūras in the Ut ̠mānic codex and the codices of Ibn Mas ūd and Ubayy b. Ka b depend on the lengths of the sūras. See Hans Bauer, “Über die Anordnung der Suren und über die geheimnisvollen Buchstaben im Qoran”, Zeitschrift der Deutschen Morgenländischen Gesellschaft, 75 (1921), p. 1-20. B. Sadeghi / Arabica 58 (2011) 210-299 215 pains to make the treatment accessible to historians who have not studied statistics and hardly even remember any math from high school. Beyond what has been accomplished, the prospect of what can be done in future is exciting. he corroboration of a seven-phase chronology may be gratifying, as it turns a sequence that is hypothetical into one that is verified, and as it nearly doubles the number of the phases in the Weil-Nöldeke chronology. However, there is reason to believe that the methods used here, especially the Criterion of Concurrent Smoothness and weight optimization, will enable future stylometric studies to verify more precise chronologies, increasing the number of phases beyond seven. Before us now lies the vista of a new research program in the chronology of the Qur ān. Motivating the Approach in his Essay I call the seven-phase chronology corroborated here the “Modified Bazargan” chronology. Figure 1 shows how this sequence stands in relation to some others. he Weil-Nöldeke chronology divided the Meccan sūras into three phases, yielding a total of four phases. Bazargan offered a more detailed chronology that is portrayed here as twenty-two phases. he Modified Bazargan sequence is obtained by combining some consecutive phases of Bazargan. For example, the passages in the last three phases of Bazargan are combined into a single phase in the Modified Bazargan chronology. his results in a reduction from twenty-two to seven phases as shown. It is necessary to begin with a discussion of Bazargan’s work. Bazargan divides the 114 sūras of the Qur ān into 194 blocks, preserving some sūras intact as single blocks while dividing others into two or more blocks. He then rearranges these blocks approximately in order of increasing average verse length. his order, he proposes, is the chronological order. His working assumption is that over time the style of the Qur ān, as represented by verse length, changed gradually—indeed not only gradually but also monotonically, i.e. irreversibly in one direction. He stresses that his proposed chronology should not be taken as rigid because it is statistical in nature and because statistical methods sustain firm conclusions about averages of aggregates rather than individual items. Bazargan corroborates his proposed chronology in a number of ways. He points to broad agreement with Blachère’s chronology, which is almost the same as Nöldeke’s redaction of Weil’s phases.7 In addition, he examines fifteen instances where historical information suggests a date for a passage. In his 7 Bāzargān, Sayr, vol. I, p. 25 / 43 / 50; and p. 100-13 / 122-35 / 128-45. 216 B. Sadeghi / Arabica 58 (2011) 210-299 Mecca Medina Traditional Weil, et al. 1 2 3 4 Bazargan 1 2 3 4 Modified Bazargan 5 6 7 Figure 1. he Modified Bazargan chronology is obtained by combining some consecutive phases of the Bazargan sequence to yield seven phases. For example, phase 7 consists of all the passages in the last three phases of Bazargan (20-22). Corroboration is not claimed for Bazargan’s first phase, which contains only 415 words. chronology, thirteen of these passages line up in the expected sequence.8 I performed a similar test using nine Medinan passages discussed by Neal Robinson, including three passages not found in Bazargan’s list.9 Bazargan’s chronology puts seven of them in the expected order. As another test, Bazargan traces the development of themes over time. He notes, for example, that the verses on wine line up in the expected way, in order of increasing severity.10 Furthermore, his chronology divides the Qur ān into two halves whose thematic profiles fit the Meccan and Medinan phases of the Prophet’s career. For example, with only a few exceptions, the theme of war crops up in the second (Medinan) half of Bazargan’s chronology.11 Ibid., p. 127-36 / 149-55 / 160-9. Neal Robinson, Discovering the Qur ān: A Contemporary Approach to a Veiled Text, 2nd ed., Washington, D.C., George Washington University Press, 2003, p. 37-44. 10 Bāzargān, Sayr Mutammim, p. 239-42. He also discusses ribā, sexual norms, war and ğihād, the Hypocrites, Abraham, man’s creation, spiritual purification and zakāt, and Moses and the Children of Israel; see Sayr Mutammim, p. 243-409. 11 he exceptional war-related verses in the Meccan part of Bazargan’s chronology are as follows: Kor 2, 244 (Block 113), Kor 22, 58 (Block 124), and Kor 73, 20 (Block 132). hese are surely Medinan. See Bāzargān, Sayr, vol. I, p. 183 / 207 / missing; and Sayr Mutammim, p. 258-9. I have not included references to ğihād among the exceptions, since they need not be 8 9 B. Sadeghi / Arabica 58 (2011) 210-299 217 he agreement with Blachère’s chronology is to be expected to the extent that the latter uses verse length as a criterion. But that chronology relies also on other criteria and sources, such as early traditions and considerations of meaning. his suggests that there is more to Bazargan’s reasoning than circular justification. Furthermore, the backing provided by agreement with the historical information is very impressive and completely devoid of circularity. While Bazargan orders the blocks using a stylistic criterion, in order to support the validity of his ordering, he reaches beyond style, adducing considerations of meaning and historical evidence. Ultimately, semantic and historical evidence should play an indispensable role in the evaluation of any proposed chronology. However, it is useful to ask how far one can go in corroborating a chronology using only style. he advantage of initially limiting oneself to style is that subsequently it may enable independent corroboration in a more meaningful fashion. hat is, to the extent that style-based indications agree with other forms of evidence, one can speak of joint corroboration by genuinely independent strands of evidence. Yet if from the outset one mixes different kinds of analysis, then in the end it may be less straightforward to tell independent corroboration apart from the duplication of the same evidence in different guises. here is, therefore, merit in initially preventing semantic and historical considerations from shaping one’s analysis of chronology. he Approach in his Essay he present task, then, is to see to what extent one can corroborate Bazargan’s chronology through purely stylistic considerations. he first step is to ask how much corroboration his chronology enjoys at present if its semantic and historical justifications were stripped away. he answer would have to be: relatively little. Although it may be questioned, his postulate that the style of the Qur ān changed gradually is not implausible. But he does not stop at that, as he also assumes that it evolved only in one direction. But why should average verse length have increased monotonically? Why, for example, could it not have increased for some time, slowly leveled off, and then decreased gradually for a while more, before leveling off again or resuming an upward trend? Indeed, one can take the Qur ān, or any other corpus of texts, and rearrange it in many different ways that all make a particular marker of style, such as sentence length, change in a relatively continuous manner. Yet, most such orderings will not reflect chronology. understood in the military sense. For chronological graphs of other themes (eschatology, past prophets, the People of the Book, etc.), see Sayr, vol. I, p. 176-96 / 201-20 / missing. 218 B. Sadeghi / Arabica 58 (2011) 210-299 Clearly, then, a single marker of style cannot strongly corroborate any particular chronology. But what if one took into account additional markers of style: the relative frequencies of the most common words, the distribution of word lengths, and so on? In general, given a corpus of texts, it would appear impossible to rearrange it to make many independent markers of style vary gradually, as opposed to in a jagged fashion. his is because as soon as one rearranges a text to make one marker of style vary smoothly, that will affect the behaviors of all the other markers, interfering with their smoothness. Finding an arrangement that makes all variations smooth would be quite extraordinary. Such a pattern could not be due to chance, thus requiring an explanation. Chronological development would usually be the only plausible explanation. hat this would be a natural explanation must be obvious enough: if style did indeed change gradually, continuous change could very well characterize more than one marker of style (though it would not have to), and the chronological sequence would thus make multiple markers of style vary smoothly. hat this is usually the only plausible explanation reflects our inability to adequately explain the pattern in other ways.12 Concurrent smoothness, when observed, cannot be a coincidence; thus, potential critics of my approach have the burden of explaining it without resort to chronology. Methodology: he Criterion of Concurrent Smoothness In sum, the principle underlying my study is that if different, independent markers of style vary in a relatively continuous fashion over a particular ordering, then that sequence reflects the chronological order. he point is that while it is easy to find many orderings of a corpus over which one particular marker of style varies smoothly, it is highly unlikely that an ordering will yield smooth variation simultaneously for different, independent markers of style (“concurrent smoothness”). his is so because if one reorders the corpus to make one 12 here is a caveat and a potential objection. he caveat: here is a way to artificially construct a sequence that makes different markers smooth without the sequence reflecting chronology. Mixing two different, discrete styles in gradually changing proportions can accomplish that. he Conclusion explains this possibility and discusses whether it characterizes Bazargan’s sequence. he potential objection: his essay shows that average verse length correlates with morpheme frequencies. It may be objected that this correlation reflects facts of linguistics: certain grammatical structures can be used more easily in shorter sentences. In response, I note two things. First, the person who makes this objection has the burden of demonstrating the claim. Second, it would not be implausible for such a phenomenon to affect a few morphemes; but the results of this essay do not change if several morphemes are deleted from the lists of morphemes used. he essay uses three lists including respectively 28, 114, and 3693 morphemes; and only those conclusions are accepted that are confirmed by all three lists or at least two of them. B. Sadeghi / Arabica 58 (2011) 210-299 219 marker of style vary smoothly, usually this reordering will disturb the smoothness of other markers. In the unlikely event that one does find a reordering that achieves concurrent smoothness, usually the only plausible explanation is that it reflects the gradual variation of different markers of style over time.13 I thus examine whether markers of style other than verse length also vary in a smooth fashion over Bazargan’s sequence. If they do, and to the extent that they do, that confirms this ordering as a reflection of chronology. Note that my approach does not take it as a premise that as the Qur ān was revealed, its style changed gradually. If in reality there were major breaks in the style of the Qur ān over time, then that would simply mean that one would not be able to find an ordering that achieves concurrent smoothness. Where I cannot find such a sequence, I simply conclude nothing.14 I do not begin with a working assumption of gradual change, even though such a hypothesis would not be implausible. he reader may have noted a certain asymmetry in my approach: concurrent smoothness confirms a chronology, but the converse does not hold. If an arrangement does not achieve concurrent smoothness, while that fact is certainly suggestive, I would be reluctant to conclude that the order is not chronological, preferring instead to suspend judgment. Gerard Ledger takes a similarly asymmetric approach to the authenticity of disputed Platonic texts. If a disputed text is stylistically identical or close to undisputed works, that is evidence of authenticity; but if a disputed text is stylistically atypical, “that in itself does not constitute proof that it was written by another, unless it can be shown that the author in question never departed from his standard established style”.15 Ledger’s cautiously asymmetric approach arises from his unwillingness to make gratuitous assumptions about Plato’s manner of writing. Similarly, the asymmetry in my approach is meant to minimize preconceptions about how the Qur ān’s style changed over time. To take lack of smoothness as a sign that a sequence is not chronological would be tantamount 13 Of course, if one sequence achieves such smoothness, others that are broadly similar to it will do so as well. So, the claim of uniqueness of sequences yielding concurrent smoothness must be understood as valid only within the range of broad similarity. 14 here is one type of break, however, that if it did occur in reality, would lead to error in my approach. I have in mind a discontinuous reversion to the style of a specific moment in the past, with respect to all markers of style (vocabulary choice, verse length, etc.). I discuss this possibility below in Section 9 (“he Conclusion”). 15 Gerard Ledger, Re-counting Plato: A Computer Analysis of Plato’s Style, Oxford, Clarendon Press, 1989, p. 168-9. However, one may corroborate that a disputed work is ascribed incorrectly to an author if its style is shown to be identical to that of a known alternative author who is known as a plausible candidate for its authorship; but, even here, considerable caution is required. 220 B. Sadeghi / Arabica 58 (2011) 210-299 to positing gratuitously that the Qur ān’s style must have varied smoothly over time. In sum, concurrent smoothness confirms a proposed sequence as chronological, but lack of concurrent smoothness does not discredit a proposed sequence. his is so because there is no reason a priori for assuming that a chronological sequence will yield concurrent smoothness. his does not mean, however, that proposed sequences are unfalsifiable. A proposed sequence is disconfirmed if a different sequence with which it is incompatible is confirmed by concurrent smoothness. his asymmetry may occasion discomfort among those familiar with Karl Popper’s philosophy. Popper believed that a theory may be corroborated only to the extent that it passes a test that could have potentially falsified it. hus, if a type of evidence cannot potentially falsify a theory, then it cannot corroborate it either.16 It would appear, then, that if the absence of concurrent smoothness does not refute a proposed sequence, then its presence cannot confirm a sequence either. In response, I note that Popper’s observation applies strictly to theories of the form “All A are B”, e.g. “All swans are white”. he situation is often different. Take the following mundane example: if two broken, jagged shards of window glass fit each other perfectly, that is strong evidence for their being from the same broken window. But if they do not fit, that is not strong evidence against their being from the same window. Stylometry Demystified To quantify style, I use techniques from the field of stylometry. Bazargan’s book itself is a fine example of a stylometric study, although he knew little about the discipline and did not use multivariate techniques. Stylometry has conceptual similarities to the traditional methods of stylistic analysis.17 he primary difference lies in its use of quantitative methods, which can provide an additional measure of objectivity and help detect statistically significant patterns that might otherwise be difficult or impossible to discern. he use of computers nowadays has become the second distinguishing characteristic of stylometry, as they have made quantitative approaches immensely more practical. Whereas Bazargan did his counting by hand, thanks to computers I do not have to. Karl Popper, Realism and the Aim of Science, London, Routledge, 1983, p. 235. For a methodological comparison of stylometric and traditional techniques, see Section 8 below (“Multivariate Analysis (List C): he Generalized Smoking Gun Technique”). 16 17 B. Sadeghi / Arabica 58 (2011) 210-299 221 here is a wide range of stylometric approaches, arising in part from the diversity of features used as markers of style. Such features may include punctuation marks, consonants and vowels, character strings, syntactical features, parts of speech, rhythm, hapax legomena (once-occurring words), sentence length, word length, and so on. Probably, the most popular approach is to use the relative frequencies of common function words such as “the”, “a”, “an”, “or”, “upon”, and “it”. My study, too, considers lists of common morphemes. he most common application of stylometry is establishing the authorship of disputed texts. To do so, typically one compares the style of a disputed text with the styles of known authors. Less commonly, scholars have used stylometry to study issues of chronology.18 he problem in stylometric literature 18 See e.g. B. Brainerd, “he Chronology of Shakespeare’s Plays: A Statistical Study”, Computers and the Humanities, 14 (1980), p. 221-30; Leonard Branwood, he Chronology of Plato’s Dialogues, Cambridge, Cambridge University Press, 1990; Fazli Can and Jon Patton, “Change of Writing Style with Time”, Computers and the Humanities, 38/1 (2004), p. 61-82; A. Devine and L. Stephens, “A New Aspect of the Evolution of the Trimeter in Euripedes”, Transactions of the American Philological Association, 111 (1981), p. 45-64; W.E.Y. Elliott and R.J. Valenza, “Can the Oxford Candidacy be Saved? A Response to W. Ron Hess: ‘Shakespeare’s dates: the Effect on Stylistic Analysis’”, he Oxfordian, 3 (2000), p. 71-97; J. Fitch, “Sense-Pauses and Relative Dating in Seneca, Sophocles, and Shakespeare”, American Journal of Philology, 102 (1981), p. 289-307; Richard Forsyth, “Stylochronometry with Substrings, or: A Poet Young and Old”, Literary and Linguistic Computing, 14/4 (1999), p. 467-78; Bernard Frischer, Shifting Paradigms: New Approaches to Horace’s Ars Poetica, Atlanta, Georgia, Scholars Press, 1991; MacD. Jackson, “Pause Patterns in Shakespeare’s Verse: Canon and Chronology”, Literary and Lingustic Computing, 17/1 (2002), p. 37-46; Patrick Juola, “Becoming Jack London”, Proceedings of Qualico-2003, Athens, Georgia, 2003; Anthony Kenny, he Computation of Style, Oxford, Pergamon Press, 1982; Gerard Ledger, Re-counting Plato: A Computer Analysis of Plato’s Style, Oxford, Clarendon Press, 1989; S. Michaelson and A.Q. Morton, “hings Ain’t What hey Used to Be: A Study of Chronological Change in a Greek Writer”, in he Computer in Literary and Linguistic Studies, Proceedings of the hird International Symposium, eds Alan Jones and R.F. Churchhouse, Cardiff, University of Wales Press, 1976, p. 79-84; Charles Muller, Étude de statistique lexicale: Le vocabulaire de théâtre de Pierre Corneille, Paris, Larousse, 1967; Debra Nails, Agora, Academy, and the Conduct of Philosophy, Dordrecht, Kluwer, 1995; Behnam Sadeghi, “he Authenticity of Two 2nd/8th-Century Ḥ anafī Legal Texts: he Kitāb al-Āt̠ār and al-Muwaṭtạ ’ of Muḥammad b. al-Ḥ asan al-Shaybānī”, Islamic Law and Society, 17/3 (Nov. 2010), p. 291-319; J.A. Smith and C. Kelly, “Stylistic Constancy and Change across Corpora: Using Measures of Lexical Richness to Date Works”, Computers and the Humanities, 36/4 (2002), p. 411-30; Constantina Stamou, Dating Victorians: An Experimental Approach to Stylochronometry, Saarbrücken, Verlag Dr. Müller, 2009; D.K. Simonton, “Popularity, Content, and Context in 37 Shakespeare Plays”, Poetics, 15 (1986), p. 493-510; idem, “Imagery, Style, and Content in 37 Shakespeare Plays”, Empirical Studies of the Arts, 15 (1997), p. 15-20; Tomoji Tabata, “Investigating Stylistic Variation in Dickens through Correspondence Analysis of Word-Class Distribution”, in English Corpus Linguistics in Japan, ed. Toshio Saito, Junsaku Nakamura and Shunji Yamazaki, Amsterdam, Rodopi, 2002, p. 165-82; J.T. Temple, “A Multivariate Synthesis of Published Platonic Stylometric Data”, Literary and Linguistic Computing, 11/2 (1996), p. 67-75; David Wishart and 222 B. Sadeghi / Arabica 58 (2011) 210-299 most similar to the Qur ānic case is perhaps that of the chronology of Plato’s dialogues.19 According to Gerard Ledger, a common shortcoming of such studies of Plato is that they assume that a marker of style must have changed over time in one direction.20 his is just the kind of questionable premise that my approach, as described above, is designed to circumvent. he basic idea in stylometry is to use a statistic, such as the frequency of occurrence of a certain feature, as a marker of style. A limited sample of writing is used to gain an estimate of an abstract, hypothetical quantity, namely the frequency with which an author would use a certain feature if he wrote an infinite amount of text. For example, within the 694 characters in this paragraph, the letter “i” occurs forty-nine times: a relative frequency count of 49/694 = 7.1 %. Moreover, to get better results, instead of just the letter “i”, one may consider many features simultaneously, such as all letters of the alphabet. Doing so would characterize my style by a vector, i.e. list of relative frequency counts. When one represents texts with vectors instead of single numbers, the data are called multivariate. Observed relative frequency counts may be thought of as approximations to an abstraction that may be called an author’s “true style”. For example, the observed 7.1 % frequency count in my last paragraph may be thought of as an approximation to my “true style” of, say, 8 to 9 %. he more precise the “true style” defined by the chosen signifier(s) of style, the more effective the method is likely to be. hus, some markers of style will be better discriminators than others. Furthermore, larger sample sizes are helpful. To get a better estimate of my usage of i’s, one should use a larger sample than the mere 141 words in the last paragraph. he greater the sample size is, the smaller the sampling error. he sampling error is a product of the uncertainty introduced by the limited size of the sample.21 he most immediate and fundamental task of stylometry is to determine how similar two or more texts are in terms of style. Similarity is always relative. One may say, “A is stylistically similar to B”; but one really means, “text A is more similar to text B than to text C”. Furthermore, similarity depends entirely Stephen Leach, “A Multivariate Analysis of Platonic Prose Rhythm”, Computer Studies in the Humanities and Verbal Behavior, 3 (1970), p. 90-9. 19 Ledger, Re-counting Plato; Nails, Agora, p. 97-114; Branwood, Chronology; and especially Wishart and Leach, “A Multivariate Analysis of Platonic Prose Rhythm”. For the full citations, see the previous footnote. 20 Ledger, Re-counting Plato, p. 175; cf. Nails, Agora, p. 113-4. 21 For an accessible introduction to the basic concepts of (univariate) statistics, see Kenny, he Computation of Style, cited above. I also benefited from working through Larry Christensen and Charles Stoup, Introduction to Statistics for the Social and Behavioral Sciences, Belmont, California, Brooks/Cole, 1986. Following this essay does not require any background in statistics. B. Sadeghi / Arabica 58 (2011) 210-299 223 on the marker of style chosen. Suppose the marker of style is the prevalence of the letter i. By this marker, if 6 % of the characters in text A are i’s, 7 % of characters in B are i’s, and 18 % of characters in C are i’s, then one may say that A and B are more similar to each other than either is to C. But if one changed the criterion, say to the frequency of the letter t, then one might obtain different results. A keystone of my study, and a very common technique in stylometry, is the graphical representation of similarity. Stylistic dissimilarity may be represented by the spatial metaphor of distance. “Stylistically different” becomes “far”, and “stylistically similar” becomes “near”. Texts are represented by points on a page. he closer two points are on the page, the more similar their stylistic profiles are. Obtaining distances between texts is straightforward. he following example illustrates the method, and grasping it is essential for understanding this essay. Suppose that the chosen marker of style is the frequencies of five common morphemes in the Qur ān: wa (“and”), u (nominative case ending), an (accusative, indefinite case ending), fa (punctuation), and llāh (“God”). Suppose one wishes to compare the styles of three chapters of the Qur ān: sūra 13 “hunder”, sūra 23 “Believers”, and sūra 36 “Yā-Sīn”. In Figure 2, each sūra is represented by a row of five numbers representing the relative frequencies of the five morphemes. he heights of the columns represent the morpheme frequencies: the higher a morpheme frequency is, the taller the column representing it. For example, the tallest column in Figure 2 indicates that 7.2 % of the morphemes in sūra 13 consist of the nominative case ending u. Each row of numbers forms the stylistic profile of a sūra. he back row, for example, represents hunder. Simply glancing at these rows in order to compare the sūra profiles, it is evident that the sūra hunder stands apart from the others, and that the sūras Believers and Yā-Sīn are stylistically closer to each other than either is to hunder. he question is how to translate these relationships into distances. To obtain the distance between hunder and Believers, one takes into account each of the morphemes. Let us begin with the leftmost morpheme, wa. In hunder, 6.6 % of the morphemes consist of wa. For Believers, the number is 5.3 %. he difference in the heights of the wa columns of the two sūras is 6.6 – 5.3 = 1.3. For the u morpheme, the difference is 7.2 – 4.5 = 2.7. For an, it is 2.4 – 1.6 = 0.8. For fa, it is 2.7 – 1.1 = 1.6. Finally, the difference between the percentages of llāh in these two sūras is 1.9 – 0.7 = 1.2. An overall measure of the difference between hunder and Believers is the sum of these individual differences, i.e. 1.3 + 2.7 + 0.8 + 1.6 + 1.2 = 7.6. his number is the distance between hunder and Believers. In general, the distance between any pair of texts is calculated in a similar way: for each morpheme, one takes the 224 B. Sadeghi / Arabica 58 (2011) 210-299 8 7.2 6.6 7 6 5.7 5 5.3 4.5 4 4.2 1.6 3 2.7 2.4 1.1 1.9 2 2.1 1 2.3 0.7 ievers” h sin” llaa fa an under” Sura 23 “Bel 0.2 Su ra 36 “Ya- u wa 0 Sura 13 “h Figure 2. he relative frequency counts of five common morphemes (wa, u, an, fa, or llāh) in three sūras, i.e. their percentages in each sūra. he distance between two sūras is found by subtracting their morpheme frequencies and adding up the differences. difference between its percentages in the two texts, and one adds up the differences for all the different morphemes. It was just shown that the distance between hunder and Believers is 7.6. It can be shown, in a similar fashion, that the distance between hunder and Yā-Sīn is 7.3, and that the distance between Believers and Yā-Sīn is 1.8. hese relationships are represented graphically in the following figure. As expected, hunder stands far apart from the other two sūras. Yā-Sīn Believers hunder B. Sadeghi / Arabica 58 (2011) 210-299 225 his diagram is a concise epitome of a great deal of information. he present essay examines up to twenty-two texts at a time based on lists of dozens or even thousands of morphemes; yet the method used to analyze those cases is the same as that in the above example. he graphical method is a convenient and intuitive way of dealing with data of such complexity. he only hitch is that moving from a list of distances to a diagram is not straightforward, as in general it is not possible to reproduce the distances exactly in a two-dimensional picture. Approximations are therefore made using standard methods such as “principal component analysis” to produce diagrams that capture the main trends in the data while filtering out some of the noise. he above diagram can also be used to illustrate the idea of smoothness. Suppose one wishes to arrange the three sūras in a chronological or reverse-chronological sequence. he three choices are listed here in order of decreasing smoothness (1) Believers å Yā-Sīn å hunder (2) Yā-Sīn å Believers å hunder (3) Believers å hunder å Yā-Sīn he smoothest sequence is the one in which style progresses most gradually, which is to say that texts that are near in time are stylistically similar. Translated into the language of distances, smoothness means that texts that are close to each other in the chronological sequence are located near each other in terms of stylistic distance. If the texts be thought of as cities and the progression from one text to another as a traveling salesman’s itinerary, then the smoothest trajectory is the one that minimizes the total distance traveled by the salesman if he were to stop at every city exactly once. he smoothest path is thus the shortest. In this case, the first sequence above is evidently the smoothest one, the second sequence is slightly less smooth, while the third one is decidedly not smooth. If the first sequence yielded stylistic smoothness for several independent markers of style, that would amount to concurrent smoothness.22 hat takes care of the main ideas behind the method. Now the reader should be able to interpret most of the diagrams. However, the potential remains for a family of misconceptions. he best way to address them is to deal with an example of the kind of objections such misunderstandings generate: It may be asked, for example, why the different uses of a morpheme are not distinguished. he morpheme wa sometimes means “and” and sometimes means 22 A mathematically precise definition of smoothness is given in the Appendix. 226 B. Sadeghi / Arabica 58 (2011) 210-299 “by” (as in an oath). Surely, the objection goes, it does not make sense to treat the two as the same. his criticism concerns the choices made about what to count. One can readily come up with a long list of other objections in this vein, since every choice involves an element of arbitrariness that may occasion the question, “why not do it in this other way?” he objection is based on three misunderstandings. he first mistaken assumption is that the plausibility of these various initial decisions is what validates the outcome. Actually, it is the reverse. he validation comes at the end when one confirms that there is concurrent smoothness. As an analogy imagine somebody who picks two jagged shards of glass from a heap of broken windows, shows that they fit perfectly, and claims that they are from the same window. What validates his claim is the improbable fit, not how he went about finding the matching fragments. In fact, had he picked them in a random fashion, his argument would be no less valid. he second mistaken assumption is that we need to understand how the results follow from the initial choices made. he idea again is best illustrated by an example. Suppose we enter a stadium and see that most people are wearing red. We know immediately that there has to be a reason for it other than chance. We may thus put forward an explanation, e.g. that the crowd is showing support for a soccer team, or that this is a political rally organized by communists. It would not undermine such an explanation to point out that Maryam who is in the stadium is in fact wearing red for an entirely different reason. What supports the explanation is the statistical significance of the overall pattern. Individual cases need not fit the pattern. By the same token, it is not necessary for us to understand the role played by individual morphemes such as wa. A person making this objection might be surprised to know that some important stylometric studies rely on the frequencies of the letters of the alphabet. hus, the “s” in “sin” is treated as the equal of the “s” in “soup”, and the word “funeral” can be replaced with “real fun” without consequence. he third mistake is to assume that if the initial decisions (such as the choice of morphemes) are poor, one will end up with a wrong result. In reality, what the initial decisions determine is not the accuracy of the outcome, but its precision. Bad choices will lead to fewer phases or none satisfying concurrent smoothness. hus, either the conclusions will be less precise or one will arrive at no conclusion. But a less precise result is not necessarily wrong. A twophase chronology is not “less correct” than a three-phase one. It is like using a magnifying glass instead of a microscope: it may capture much less detail, but what it reveals is not wrong. B. Sadeghi / Arabica 58 (2011) 210-299 227 he Organization of the Essay his outline may help readers navigate the rest of this essay by highlighting the most important points. Section 2 (“Bazargan’s Chronology”) defines twentytwo selections or “groups” of Qur ānic passages, numbered 1 through 22, which according to Bazargan’s chronology were disseminated by the Prophet in the order enumerated. he ultimate goal is to see how various markers of style behave over this sequence and whether they exhibit a smooth trajectory. Section 3 (“Univariate Assessments of Smoothness”) examines several univariate markers of style for smoothness over Bazargan’s sequence of groups, e.g. average word length and the frequencies of hapax legomena. he reader must examine Figure 4, which shows the variation of average verse length over the sequence of the twenty-two groups defined in Section 2. Noteworthy, too, is Figure 8, which shows a strikingly smooth initial trajectory for the percentages of hapax legomena. he central task of this article is to see how multivariate markers of style vary over the twenty-two phases. Before that task is accomplished, however, there are the two intervening Sections 4 and 5. Historians might find some of the details in these sections difficult, but the larger ideas should be clear. Section 4 offers a “Non-technical Introduction to Multivariate Methods”. he discussion above, under “Stylometry Demystified”, is even less technical, and unlike Section 4 it is necessary that the reader come to understand it thoroughly. Section 5 (“Morphemes: Weighting and Weight Optimization”) hones the multivariate techniques. It tests whether stylistic profiles based on morpheme frequencies succeed in showing the stylistic similarity of texts that we may expect in advance to belong together. he texts that are chosen for this purpose are halves of large passages. Morpheme frequencies indeed assign the halves to each other, which shows not only the reliability of the method, but also the stylistic coherence and unity of large passages. he section also explains that one may choose to not accord all morphemes equal weight when forming stylistic profiles of passages and calculating the distances between them. It introduces weight optimization, a method for finding the best weights automatically. Being a little involved, Section 5 may be skimmed by casual readers, except that Table 5 and Table 6 require attention, as do the concluding words of the section. he three sections that follow constitute the heart of this essay: Sections 6, 7, and 8 use three different, independent multivariate markers of style based on morpheme frequencies. Each section examines whether the stylistic profile based on one particular list of morphemes varies in a smooth manner over 228 B. Sadeghi / Arabica 58 (2011) 210-299 the sequence of twenty-two groups. Section 6 does this for the top twentyeight most frequent morphemes in the Qur ān. Section 7 considers 114 other common morphemes. Section 8 uses a list of 3693 relatively uncommon morphemes. he results from the three independent multivariate markers ought to be compared in order to assess the degree of concurrent smoothness. Section 9 (“he Conclusion”) accomplishes that. It finds that if some of Bazargan’s groups are combined in accordance with Figure 1 above, then all the three different markers of style will exhibit smooth trajectories. Specifically, the following sequence of seven clusters yields concurrent smoothness and hence represents the true chronological order: {Group 2}, {Group 3}, {Group 4}, {Group 5}, {Groups 6-11}, {Groups 12-19}, {Groups 20-22}. he claim is not that the passages in one cluster all came after those in the preceding clusters, but that only on average they did so. In addition, the chronology of the passages within a cluster is indeterminate. For example, I have not confirmed or refuted that passages in Group 8 on average came after those in Group 7, since the two groups belong to the same cluster. he upshot is that the first half of Bazargan’s chronology is broadly confirmed. Its second half, consisting of Groups 12-22, which happen to correspond to Medina in the traditional reckoning, remains largely unconfirmed, although it is at least clear that it comes after the first half. 2. Bazargan’s Chronology Mehdi Bazargan (d. HS 1373/1995) was one of the pillars of Islamist and democratic thought in Iran.23 His interest in chronology arose during Qur ān 23 Mehdi Bazargan was born in HS 1286/1907 in Tehran. He was a professor of thermodynamics and Dean of the Engineering Faculty in Tehran University, where, incidentally, he was the first to establish group prayers on campus. In HS 1329/1950-1, appointed by Muḥammad Muṣaddiq as the first Iranian chief executive of the National Iranian Oil Company, he helped end British control with minimal disruption to the industry. As head of the Water Organization in HS 1332/1953-4, he brought running water to Tehran for the first time. He spent 1963 to 1967-8 (HS 1341 to 1346) in prison for criticizing the Shah. Bazargan became the first prime minister of the Islamic Republic in HS 1357/1979, but resigned after nine months, went into opposition and served a term in the Parliament. Bazargan spent the last years of his life working in private industry. He also continued his religious scholarship and political activism until his death on January 20, 1995 (HS 10/30/1373). he Qur ān lay at the center of his spiritual, political, and intellectual life. It not only provided the framework and language of much of Bazargan’s thought, but also formed the subject of a number of his important works, including a work of exegesis (tafsīr) that he arranged according to the chronological order of the Qur ān. he above biographical information is based mostly on Mahdī Bāzargān, Ḫ ātirāt-i Bāzargān: B. Sadeghi / Arabica 58 (2011) 210-299 229 study sessions held with his colleagues, mostly from his Freedom Movement, in prison in Burāzğān in the Ramaḍān of HS 1344/1965-6, and it culminated in the completion of the book in Qaṣr Prison. He began his study of verse length upon encountering a chronological list offered in a Qur ān printed by I timād al-Salṭana (d. 1330/1912). his list gives a year-by-year break down of the sūras.24 Bazargan used this list to construct a graph of the mean length of verses versus time, obtaining a rather jagged graph with an overall increasing trend. Having also made graphs of themes vs. time, he noted that some of the breaks in the trends in the two types of graph went together. He then rearranged the sūras to make mean verse length increase monotonically. his removed the sharpest breaks from the plots of the themes and improved agreement with the known provenance of the sūras as Meccan or Medinan. In other words, he had encountered the correlation between style, content, and historical information. Encouraged, he further refined his chronology by breaking up the sūras into blocks and reordering these blocks instead of whole sūras.25 Eventually, some secondary sources were sent to him in prison, including a copy of Blachère’s Introduction which allowed him to make comparisons Šaṣt sāl ḫidmat wa muqāwamat, Tehran, Mu assasa-i Ḫ adamāt-i Farhangi-i Rasā, HS 1375/1996. I also used the biography on http://www.bazargan.com, a site maintained by Abdolali Bazargan. Citations to Bazargan’s work on chronology are given above in footnote 3. 24 According to Bazargan, the historian and litterateur Muḥammad Ḥ asan I timād al-Salṭana does not give a source for his chronology. I think the chronology was probably his own. Comparing his chronological sequence with the thirteen lists of Muslim and European origin provided by Mehdi Abedi, one finds that it is unique (Michael Fischer and Mehdi Abedi, Debating Muslims: Cultural Dialogues in Postmodernity and Tradition, Madison, University of Wisconsin Press, 1990, p. 445-7). Bazargan points out that the subject index (kašf al-maṭālib) that I timād al-Salṭana provided was, in the words of the latter, composed on the basis of an index created by an European scholar, “one of the ulamā of the Maġrib”. After quoting this, Bazargan wonders whether the same European scholar, or maybe I timād al-Salaṭana himself, was not the source of the chronology (Bāzargān, Sayr, vol. I, p. 21-3 / 39-41 / missing). In fact, the European author of the subject index was not the source of the chronology. Muḥammad Nūrī and Maḥmūd Rāmyār both identify the source of the subject index as Jules La Beaume, Le Koran analysé d’après la traduction de M. Kazimirski et les observations de plusieurs autres savants orientalistes, Paris, Maisonneuve & Cie (“Bibliothèque Orientale”, 4), 1878 (see Muḥammad Nūrī, “Tafṣīl āyāt al-Qur ān al-ḥ akīm”, Faṣlnāma-i kitābhā-i Islāmī, no. 7, available online at http://www.i-b-q. com/far/07/article/09.htm; Maḥmūd Rāmyār, Tārīḫ-i Qur ān, 2nd ed., Tehran, Amīr Kabīr, HS 1362/1983, p. 667). La Beaume, however, does not offer any chronological list. I timād al-Salṭana’s list is as follows (note that its Meccan part is simply the known Meccan sūras listed in the reverse of the official order): Mecca: 1, 114-99, 96-67, 56-50, 46-34, 32-25, 23, 21-10, 7-6; Medina: 97, 64, 63, 62, 98, 61, 57, 47, 58, 59, 60, 49, 22, 4, 24, 65, 66, 33, 48, 8, 9, 2, 3, 5. 25 See Bāzargān, Sayr, vol. I, p. 16-25 / 34-43 / 41-52. 230 B. Sadeghi / Arabica 58 (2011) 210-299 with Blachère’s version of Nöldeke’s chronology.26,27 Bazargan’s book has been received well.28 As arguably the most impressive work on the subject, it deserves the praise it has received; but a reevaluation is long overdue. As mentioned above, Bazargan recognizes that a sūra may contain material from different periods. He thus divides sūras into blocks and rearranges these blocks rather than the sūras. In this process, Bazargan takes into account the distribution of verse lengths in different blocks by considering their “characteristic curves”. Figure 3 depicts the characteristic curves for six different sūras 26 Régis Blachère, Introduction au Coran, 2e éd. partiellement refondue, Paris, Besson & Chatemerle, 1959. 27 My own interest in this research was sparked, first, by Bazargan’s work, and, second, by noting in my study of al-Šaybānī’s Muwaṭtạ that word frequency could correlate with time. See Sadeghi, “he Authenticity of Two 2nd/8th-Century Ḥ anafī Legal Texts”, cited above in footnote 18. Subsequently, an inquiry about word frequency in stylistics led me to stylometry and its techniques. 28 See e.g. Rāmyār, Tārīḫ-i Qur ān, p. 665. Among Islamicists impressed with the Sayr, the most famous would be Āyatullāh Murtaḍā Mut ̣ahharī, who called the book “highly valuable” (Mutạ hharī, Muqaddama-ī bar ğahānbīnī-i Islāmī, Qumm, Ṣabā, n.d., p. 194; I owe this reference to Hossein Modarressi). Incidentally, it was on Muṭahharī’s recommendation that Āyatullāh Khomeini appointed Bazargan prime minister. Among religious public intellectuals, the most famous admirer of the Sayr was the influential and charismatic Alī Šarī atī, although he liked the book perhaps for the wrong reason. An excerpt from his overly sanguine letter to Abdolali Bazargan ( Abd al- Alī) dating from 1968-9 appears in the preface of the expanded edition of the Sayr. For some criticisms of the Sayr, see Alī Riḍā Ṣadr al-Dīnī, “Naẓarī bih Sayr-i taḥ awwul-i Qur ān”, Kayhān-i Farhangī, 3/12 (Isfand HS 1365/1986), p. 14-8. he skeptical Ṣadr al-Dīnī writes, “his book has gained a foothold among the ulamā and scholars, and its deductions have been met with acceptance. As of late, some of its contents are even being printed in the appendices of some Qur āns”. For another criticism, see the introduction written by Aḥmad Mahdawī Dāmġānī for Muḥammad Riḍā Ğalālī Nā īnī, Tārīḫ-i ğam -i Qur ān-i karīm (with an introduction by Aḥmad Mahdawī Dāmġānī), Tehran, Našr-i Nuqra, HS 1365/1986, p. xv. Western scholars who specialize in the Qur ān do not discuss Bazargan’s work, not even mentioning it in a footnote. his is so despite the fact that Bazargan wrote a summary of his work in French and presented his work in Germany. It is not uncommon for Western scholars to overlook Islamic secondary scholarship on Islamic religion. Aside from the issue of language, this is partly due to the notion that a person motivated by religion is unlikely to reason properly on historical matters. his assumption renders the works of religious Muslim scholars ineligible as valid secondary sources, relegating them to the status of primary sources. hat is, the assumption is that works of Muslim scholarship may shed light on their authors and their milieus, but not on the historical questions with which they grapple. Referring to Nöldeke’s work on the chronology of the Qur ān, Fredrick Denny writes, “his type of scholarly-critical operation is not accepted by most Muslims, because it means treating the holy text just like any other text: generated by historical circumstances and understandable by means of historical-critical methodology” (Fredrick Denny, An Introduction to Islam, New York, MacMillan, 1985, p. 157). Denny is describing what he expects to be true of most Muslim scholars. Whatever the cause of Western scholars’ occasional disengagement from the secondary scholarship produced by Muslims, the result is that at times they have missed key insights with regard to the history of the Qur ān and, even more so, the Ḥ adīt̠. 231 B. Sadeghi / Arabica 58 (2011) 210-299 chosen randomly from among the sūras that Bazargan does not break up into smaller blocks. he curves show the percentages of the verses with different verse lengths, where length is measured in the number of words.29 For example, in sūra 77 almost 40 % of the verses have exactly three words. he different plots have a skewed bell shape, with a rapid rise followed by a slower decline. For each curve, Bazargan computes three parameters: mean verse length (MVL), mode, and height. Mean, or average, verse length is simply the total number of words divided by the total number of verses. he mode is the most frequent verse length. It is thus the verse length at which the curve achieves its peak. For example, in sūra 77, mode = 3. Finally, the height is simply the value of the peak, namely the percentage of verses that have the mode as their length.30 Bazargan observes that the mean, mode, and inverse of height tend to increase together in the Qur ān, and often in like proportions. his gives him the idea to represent each block by the average of these three parameters in order to temper the aberrations of individual parameters—except that before taking the average, he makes the parameters comparable in size by dividing each by an appropriately chosen constant. hus for each block he calculates a characteristic number given by 10 × 1 3 ( mean 10 + mode 8 + 15 height ) and reorders the blocks in order of increasing characteristic number to obtain his chronology. Table 1 shows the resulting chronology. his table lists and numbers the blocks in the chronological order—e.g. Block 2 came after Block 1. he notation “(2) 74: 1-7” means that “Block 2 is defined as verses 1-7 in sūra 74”. Table 2 organizes the same data according to sūra number. In this table, “(2) 164: 40-152” means “sūra 2 contains Block 164, consisting of verses 40-152”. he question remains as to how Bazargan determines the block divisions. Bazargan leaves fifty-nine sūras intact and divides the rest into smaller blocks. His doing so is consistent with the pre-modern and modern scholarly insight that sūras may contain materials from different periods. In dividing the 29 Bazargan does not count lā, lam, law, hal, or yā as distinct words, but he counts wa-lā, fa-lā, a-lam, a-fa-lā, law lā, yā ayyuhā, and a-fa as one word apiece (Bāzargān, Sayr, vol. I, p. 17-20 / 35-8 / 47). his may not be a standard way of defining words, but as Bazargan points out, for our purposes all that matters is that one be consistent in the way one counts. My own way of counting follows the usual definition of what a word is. his is reflected in all the statistics and plots I provide. 30 Incidentally, I have found that results obtained by using the mode and height are too sensitive to the exact number of words, meaning that a slight change in the number of words in a verse could make a large difference in the mode or height. In future studies mode and height should be replaced with more stable markers that still capture what they are intended to get at. 232 B. Sadeghi / Arabica 58 (2011) 210-299 Sura 77 Sura 50 20 % of verses % of verses 40 0 0 5 verse length 0 10 0 Sura 21 Sura 11 % of verses % of verses 20 10 20 0 0 10 20 verse length 5 0 0 30 Sura 12 20 verse length 40 Sura 13 15 % of verses 8 % of verses 10 verse length 0 0 10 20 30 verse length 40 50 0 0 20 40 verse length 60 Figure 3. Characteristic curves for sūras 77, 50, 21, 11, 12, 13, respectively Blocks 25, 87, 94, 118, 130, 169, listed in the chronological order according to Bazargan’s scheme. he graphs show the percentage of verses with a given verse length, where verse length is measured in number of words. Note that as verse length increases from sūra to sūra, the peak tends to decline. sūras, Bazargan is guided by considerations of thematic unity, rhyme patterns, historical information, as well as verse length distribution, which he investigates by graphing characteristic curves for all sūras and blocks. He observes, for example, that in many cases dividing a sūra as he does helps resolve incongruous characteristic curves into more typical-looking ones. In cases where he divides a sūra into more than two blocks, if two or more blocks in the sūra display similar verse length profiles, then because they belong to the same period in Bazargan’s scheme, he combines them into a single block, even if they do not form a contiguous passage. he second volume of his book is devoted to discussing the block divisions for every sūra. Table 2 presents Bazargan’s division of sūras into blocks. B. Sadeghi / Arabica 58 (2011) 210-299 233 Table 1. Bazargan’s Chronology. he blocks are numbered in the chronological order. A block is defined according to this format: (block number) sūra: verses, exclude verses. For example, “(1) 96: 1-5” means that Block 1 consists of verses 1-5 in sūra 96. Verses marked “exclude” are excluded from calculations. (1) 96: 1-5. (2) 74: 1-7. (3) 103: 1-2. (4) 51: 1-6. (5) 102: 1-2. (6) 52: 1-8. (7) 112: 1-4. (8) 88: 1-5, 8-16. (9) 86: 11-17. (10) 82: 1-5. (11) 91: 1-10. (12) 108: 1-3. (13) 87: 1-7. (14) 85: 1-7, 12-22. (15) 81: 1-29. (16) 94: 1-8. (17) 93: 1-11. (18) 114: 1-6. (19) 79: 1-26. (20) 74: 8-10. (21) 92: 1-21. (22) 107: 1-7. (23) 70: 5-18. (24) 91: 11-15. (25) 77: 1-50, exclude 19, 24, 28, 34, 37, 40, 45, 47, 49. (26) 78: 1-36. (27) 74: 11-30, 32-56. (28) 106: 1-4. (29) 53: 1-22, 24-25. (30) 89: 1-14, 27-30. (31) 84: 1-25. (32) 80: 1-42. (33) 104: 1-9. (34) 109: 1-6, exclude 5. (35) 96: 6-19. (36) 88: 6-7, 17-26. (37) 75: 7-13, 20-40. (38) 95: 1-8, exclude 6. (39) 75: 1-6, 14-19. (40) 56: 1-96. (41) 55: 1-7, 10-27, 46-77, exclude 16, 18, 21, 23, 25, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77. (42) 87: 8-19. (43) 1: 1-7. (44) 100: 1-11. (45) 69: 38-52. (46) 79: 27-46. (47) 111: 1-5. (48) 113: 1-5. (49) 90: 1-20. (50) 102: 3-8. (51) 105: 1-5. (52) 68: 1-16. (53) 89: 15-26. (54) 99: 1-8. (55) 86: 1-10, exclude 7. (56) 53: 33-62. (57) 101: 1-11. (58) 37: 1-182. (59) 82: 6-19. (60) 69: 1-3, 13-37. (61) 70: 19-35. (62) 83: 1-36. (63) 44: 43-59. (64) 23: 1-11. (65) 26: 52-227, exclude 127, 145, 164, 180, 67, 103, 121, 174, 190, 68, 104, 122, 140, 159, 175, 191, 110, 126, 131, 144, 150, 163, 179. (66) 38: 67-88. (67) 15: 1-5, 49-99. (68) 69: 4-12. (69) 97: 1-5. (70) 51: 7-60. (71) 54: 1-55, exclude 22, 32, 40, 21, 30. (72) 68: 17-52. (73) 44: 1-42. exclude 1. (74) 70: 1-4, 36-44. (75) 52: 9-20, 22-28. (76) 43: 66-80. (77) 71: 1-28. (78) 55: 28-45, 8-9, 78, exclude 28, 30, 32, 34, 36, 38, 40, 42, 45, 77-78, 8, 9. (79) 73: 1-19. (80) 20: 1-52, exclude 1. (81) 19: 75-98. (82) 52: 21, 29-49. (83) 15: 6-48. (84) 26: 1-51, exclude 1. (85) 76: 1-31. (86) 38: 1-25, 30-66. (87) 50: 1-45. (88) 36: 1-83, exclude 1. (89) 103: 3. (90) 23: 12-118. (91) 41: 1-8, exclude 1. (92) 43: 1-65, 81-89, exclude 1. (93) 33: 1-3, 7-8, 41-48, 63-68. (94) 21: 1-112. (95) 72: 1-28. (96) 78: 37-40. (97) 85: 8-11. (98) 19: 1-33, 41-74, exclude 1. (99) 98: 1-8. (100) 31: 1-11, exclude 1. (101) 30: 1-27, exclude 1. (102) 25: 1-77. (103) 20: 53-135. (104) 67: 1-30. (105) 14: 42-52. (106) 19: 34-40. (107) 16: 1-32, 41-64, 98-105, 120-128. (108) 18: 1-8, 60-110. (109) 32: 1-30, exclude 1. (110) 74: 31. (111) 17: 9-52, 61-65, 71-81, 101-111. (112) 40: 1-6, 51-60, exclude 1. (113) 2: 1-20, 153-157, 159-163, 204-209, 244-245, exclude 1. (114) 27: 1-93. (115) 39: 29-37, 53-66. (116) 45: 1-37, exclude 1. (117) 64: 1-18. (118) 11: 1-123. (119) 41: 9-36. (120) 30: 28-60. (121) 17: 1-8, 82-100. (122) 7: 59-155, 177-206. (123) 24: 46-57. (124) 22: 18-29, 42-69. (125) 6: 1-30, 74-82, 105-117. (126) 29: 1-69, exclude 1. (127) 34: 10-54. (128) 10: 71-109. (129) 38: 26-29. (130) 12: 1-111. (131) 28: 1-46, 85-88, 47-75, exclude 1. (132) 73: 20. (133) 40: 7-50, 61-85. (134) 53: 26-32, 23. (135) 18: 29-59. (136) 31: 12-34, exclude 15, 27. (137) 14: 1-5, 7-30, 32-41. (138) 42: 1-53, exclude 1. (139) 2: 30-39, 190-195. (140) 35: 4-7, 9-11, 13-17, 19-45. (141) 39: 1-28, 38-52. (142) 47: 1-38. (143) 8: 1-75. (144) 61: 1-14. (145) 41: 37-54. (146) 17: 53-60, 66-70. (147) 46: 1-14, 27-28, exclude 1. (148) 16: 33-40, 65-89, 106-119. (149) 5: 7-11, 20-26, 33-40. (150) 62: 1-11, exclude 3. (151) 3: 32-180. (152) 63: 234 B. Sadeghi / Arabica 58 (2011) 210-299 Table 1 (cont.) 1-11. (153) 22: 1-17, 30-41, 70-78. (154) 3: 1-31, 181-200, exclude 1. (155) 7: 1-58, 156-176, exclude 1. (156) 59: 1-24, exclude 7, 15, 21-24. (157) 39: 67-75. (158) 34: 1-9. (159) 9: 38-70. (160) 10: 1-70. (161) 57: 1-29. (162) 16: 90-97. (163) 24: 1-34. (164) 2: 40-152. (165) 33: 4-6, 9-40, 49-52, 56-62, 69-73. (166) 4: 44-57, 131-175. (167) 6: 31-73, 83-104, 118-134, 154-165. (168) 13: 1-43. (169) 9: 71-129. (170) 48: 1-29. (171) 65: 8-12. (172) 5: 51-86. (173) 49: 1-18. (174) 28: 76-84. (175) 4: 1-43, 58-126. (176) 18: 9-28. (177) 9: 1-37. (178) 46: 15-26, 29-35. (179) 110: 1-3. (180) 14: 6, 31. (181) 58: 1-22. (182) 5: 27-32, 87-120. (183) 2: 21-29, 158, 165-189, 196-203, 210-242, 254, 261-283. (184) 60: 1-13. (185) 35: 1-3, 8, 12, 18. (186) 66: 1-12. (187) 6: 135-153. (188) 65: 1-7. (189) 4: 127-130, 176. (190) 24: 35-45, 58-64. (191) 5: 12-19, 41-50. (192) 2: 164, 243, 246-253, 255-260, 284-286. (193) 33: 53-55. (194) 5: 1-6. Table 2. he division of the sūras into blocks. he information is sorted by sūra number. he format is (sūra) block: verses, exclude verses. For example, “(1) 43: 1-7” means that sūra 1 contains Block 43, which covers verses 1-7. Verses marked “exclude” are excluded from calculations. (1) 43: 1-7. (2) 113: 1-20, 153-157, 159-163, 204-209, 244-245, exclude 1. (2) 183: 21-29, 158, 165-189, 196-203, 210-242, 254, 261-283. (2) 139: 30-39, 190-195. (2) 164: 40-152. (2) 192: 164, 243, 246-253, 255-260, 284-286. (3) 154: 1-31, 181-200, exclude 1. (3) 151: 32-180. (4) 175: 1-43, 58-126. (4) 166: 44-57, 131-175. (4) 189: 127-130, 176. (5) 194: 1-6. (5) 149: 7-11, 20-26, 33-40. (5) 191: 12-19, 41-50. (5) 182: 27-32, 87-120. (5) 172: 51-86. (6) 125: 1-30, 74-82, 105-117. (6) 187: 135-153. (6) 167: 31-73, 83-104, 118-134, 154-165. (7) 155: 1-58, 156-176, exclude 1. (7) 122: 59-155, 177-206. (8) 143: 1-75. (9) 177: 1-37. (9) 159: 38-70. (9) 169: 71-129. (10) 160: 1-70. (10) 128: 71-109. (11) 118: 1-123. (12) 130: 1-111. (13) 168: 1-43. (14) 137: 1-5, 7-30, 32-41. (14) 180: 6, 31. (14) 105: 42-52. (15) 67: 1-5, 49-99. (15) 83: 6-48. (16) 107: 1-32, 41-64, 98-105, 120-128. (16) 148: 33-40, 65-89, 106-119. (16) 162: 90-97. (17) 121: 1-8, 82-100. (17) 111: 9-52, 61-65, 71-81, 101-111. (17) 146: 53-60, 66-70. (18) 108: 1-8, 60-110. (18) 176: 9-28. (18) 135: 29-59. (19) 98: 1-33, 41-74, exclude 1. (19) 106: 34-40. (19) 81: 75-98. (20) 80: 1-52, exclude 1. (20) 103: 53-135. (21) 94: 1-112. (22) 153: 1-17, 30-41, 70-78. (22) 124: 18-29, 42-69. (23) 64: 1-11. (23) 90: 12-118. (24) 163: 1-34. (24) 190: 35-45, 58-64. (24) 123: 46-57. (25) 102: 1-77. (26) 84: 1-51, exclude 1. (26) 65: 52-227, exclude 127, 145, 164, 180, 67, 103, 121, 174, 190, 68, 104, 122, 140, 159, 175, 191, 110, 126, 131, 144, 150, 163, 179. (27) 114: 1-93. (28) 131: 1-46, 85-88, 47-75, exclude 1. (28) 174: 76-84. (29) 126: 1-69, exclude 1. (30) 101: 1-27. exclude 1. (30) 120: 28-60. (31) 100: 1-11, exclude 1. (31) 136: 12-34, exclude 15, 27. (32) 109: 1-30, exclude 1. (33) 93: 1-3, 7-8, 41-48, 63-68. (33) 165: 4-6, 9-40, 49-52, 56-62, 69-73. (33) 193: 53-55. (34) 158: 1-9. (34) 127: 10-54. (35) 185: 1-3, 8, 12, 18. (35) 140: 4-7, 9-11, 13-17, 19-45. (36) 88: 1-83, exclude 1. (37) 58: 1-182. (38) 86: 1-25, 30-66. (38) 129: 26-29. (38) 66: 67-88. (39) 141: 1-28, 38-52. (39) 115: 29-37, 53-66. (39) 157: B. Sadeghi / Arabica 58 (2011) 210-299 235 Table 2 (cont.) 67-75. (40) 112: 1-6, 51-60, exclude 1. (40) 133: 7-50, 61-85. (41) 91: 1-8, exclude 1. (41) 119: 9-36. (41) 145: 37-54. (42) 138: 1-53, exclude 1. (43) 92: 1-65, 81-89, exclude 1. (43) 76: 66-80. (44) 73: 1-42, exclude 1. (44) 63: 43-59. (45) 116: 1-37, exclude 1. (46) 147: 1-14, 27-28, exclude 1. (46) 178: 15-26, 29-35. (47) 142: 1-38. (48) 170: 1-29. (49) 173: 1-18. (50) 87: 1-45. (51) 4: 1-6. (51) 70: 7-60. (52) 6: 1-8. (52) 75: 9-20, 22-28. (52) 82: 21, 29-49. (53) 29: 1-22, 24-25. (53) 134: 23, 26-32. (53) 56: 33-62. (54) 71: 1-55, exclude 22, 32, 40, 21, 30. (55) 41: 1-7, 10-27, 46-77, exclude 16, 18, 21, 23, 25, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77. (55) 78: 28-45, 8-9, 78, exclude 28, 30, 32, 34, 36, 38, 40, 42, 45, 78, 8, 9. (56) 40: 1-96. (57) 161: 1-29. (58) 181: 1-22. (59) 156: 1-24, exclude 7, 15, 21-24. (60) 184: 1-13. (61) 144: 1-14. (62) 150: 1-11, exclude 3. (63) 152: 1-11. (64) 117: 1-18. (65) 188: 1-7. (65) 171: 8-12. (66) 186: 1-12. (67) 104: 1-30. (68) 52: 1-16. (68) 72: 17-52. (69) 60: 1-3, 13-37. (69) 68: 4-12. (69) 45: 38-52. (70) 74: 1-4, 36-44. (70) 23: 5-18. (70) 61: 19-35. (71) 77: 1-28. (72) 95: 1-28. (73) 79: 1-19. (73) 132: 20. (74) 2: 1-7. (74) 20: 8-10. (74) 27: 11-30, 32-56. (74) 110: 31. (75) 39: 1-6, 14-19. (75) 37: 7-13, 20-40. (76) 85: 1-31. (77) 25: 1-50, exclude 19, 24, 28, 34, 37, 40, 45, 47, 49. (78) 26: 1-36. (78) 96: 37-40. (79) 19: 1-26. (79) 46: 27-46. (80) 32: 1-42. (81) 15: 1-29. (82) 10: 1-5. (82) 59: 6-19. (83) 62: 1-36. (84) 31: 1-25. (85) 14: 1-7, 12-22. (85) 97: 8-11. (86) 55: 1-10, exclude 7. (86) 9: 11-17. (87) 13: 1-7. (87) 42: 8-19. (88) 8: 1-5, 8-16. (88) 36: 6-7, 17-26. (89) 30: 1-14, 27-30. (89) 53: 15-26. (90) 49: 1-20. (91) 11: 1-10. (91) 24: 1115. (92) 21: 1-21. (93) 17: 1-11. (94) 16: 1-8. (95) 38: 1-8, exclude 6. (96) 1: 1-5. (96) 35: 6-19. (97) 69: 1-5. (98) 99: 1-8. (99) 54: 1-8. (100) 44: 1-11. (101) 57: 1-11. (102) 5: 1-2. (102) 50: 3-8. (103) 89: 3. (103) 3: 1-2. (104) 33: 1-9. (105) 51: 1-5. (106) 28: 1-4. (107) 22: 1-7. (108) 12: 1-3. (109) 34: 1-6, exclude 5. (110) 179: 1-3. (111) 47: 1-5. (112) 7: 1-4. (113) 48: 1-5. (114) 18: 1-6. In rearranging the blocks, Bazargan follows his ordering method strictly, but makes an exception for the first five verses (twenty words) of sūra 96, which he makes the first block.31 In fact, their proper location is somewhat later, between blocks 36 and 37, although one may also join these verses with the rest of the sūra, namely with what is now Block 35. In my own investigations, I have strictly preserved Bazargan’s ordering with this one exception. I will now describe my decisions about what to count. hese decisions are mostly about minor things that do not affect the overall results, but it’s useful to specify them to make it possible for other researchers to reproduce my results. I have excluded certain verses from calculations, as indicated in Table 1 and Table 2. hese verses fall into three categories. First, Bazargan counts only once verses that are repeated to serve as a refrain. For example, the sentence fa-bi-ayyi ālā i rabbikumā tukad̠d̠ibān occurs thirty-one times in sūra 55, 31 Bazargan deferred to some historical reports that identify the beginning verses of sūra 96 as the first revelation. 236 B. Sadeghi / Arabica 58 (2011) 210-299 but Bazargan excludes the repetitions, observing that doing so yields a more typical-looking characteristic curve. I too have excluded the refrains, counting only their first occurrences.32 Second, Bazargan excludes some verses from his calculations due to thematic and stylistic considerations without assigning these excluded verses to any particular block.33 He also excludes twelve words from Kor 60, 4 (Block 184). In my own calculations, I adopt Bazargan’s exclusions except, merely for ease of computation, in this last case. One may argue with some of Bazargan’s choices, but due to the small number of such cases, such disputation would be immaterial to my results. hird, I have removed the mysterious “detached letters” from computation. Where these letters take up entire verses of their own, I have excluded the verses.34 Where the mysterious letters form only part of a verse, I have left aside the letters while including the remainder of the verse.35 Such details will not change my results, concerned as they are with broad patterns. here are various systems of dividing up the text into verses. hey differ among themselves typically by one, two, or a few verses per sūra.36 Bazargan’s numbering of verses follows a particular Qur ān published in Iran in HS 1328/1949 to which I do not have access.37 Examining his citations, its numbering appears to agree with that of the Flügel edition through sūra 79, after which it switches to another system. I have gone through his citations and converted them into the numbering system of the Egyptian standard edition, i.e. the Kūfan system, which is what all my tables and references reflect. A few of the smaller blocks are highly sensitive to the uncertainties in verse division. For example, Block 110 (sūra 74: 31) is three verses in Bazargan’s reckoning and only one in mine, and Block 179 (sūra 110: 1-3) is one verse in his reckoning and three in mine. hat translates into a three-fold difference in mean verse length. hese passages, however, are small, and their placement 32 Here are the repeated verses I have excluded: in sūra 26: verses 127, 145, 164, 180, 67, 103, 121, 174, 190, 68, 104, 122, 140, 159, 175, 191, 110, 126, 131, 144, 150, 163, 179; in sūra 54: verses 22, 32, 40, 21, 30; in sūra 55: verses 16, 18, 21, 23, 25, 28, 30, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77; in sūra 77: verses 19, 24, 28, 34, 37, 40, 45, 47, 49; in sūra 109: verse 5. 33 Here is a list: in sūra 31: verses 15, 27 (Block 136); in sūra 55: verses 8-9, 78 (Block 78); in sūra 59: verses 7, 15, 21-24 (Block 156); in sūra 62: verse 3 (Block 150); in sūra 86: verse 7 (Block 9); in sūra 95: verse 6 (Block 38). 34 hese cases occur in sūras 2, 3, 7, 19, 20, 26, 28, 29, 30, 31, 32, 36, 40, 41, 42, 43, 44, 45, 46. 35 hese cases include sūras 10, 11, 12, 13, 14, 15, 27, 38, 50, 68. 36 Anton Spitaler, Die Verszählung des Koran nach islamischer Überlieferung, Munich, Verlag der Bayerischen Akademie der Wissenschaften, 1935. 37 Bazargan cites it as “Qur ān in the handwriting of Mr. Ḫ ušnavīs, Tehran, Tīr 1328, Kitābfurūšī-i Islāmiyya” (Bāzargān, Sayr, vol. I, p. 14 / 32 / 39). B. Sadeghi / Arabica 58 (2011) 210-299 237 makes little difference in the results of this essay, which are broad-based, concerned as they are with average properties of large groups of text. Conversely, the methods used in this article are ill-suited to placing blocks of such small size in the chronological sequence. he fact that semantic and thematic features, in addition to stylistic ones, play an important role in the process of segmentation of sūras into blocks may elicit the objection that non-stylistic criteria are applied in a task that was supposed to be free of them.38 One may address this concern by noting that the ultimate determinant of the chronology is the reordering procedure, which is purely style-based. his reordering procedure can help mitigate mistakes made in the process of segmentation. Should Bazargan mistakenly have divided passages that really belong together, the reordering procedure, if it is sound, should assign them to about the same period. However, this does not mean that mistaken divisions will be without a cost. Once divided, the text becomes smaller, and its analysis more vulnerable to sampling error, leading to loss of precision in reordering.39 he number of phases displaying concurrent smoothness may be reduced, leading one to confirm less of the chronological sequence than one might do otherwise. Loss of precision, however, is not loss of accuracy. “Stanford University is on earth” may be less precise than “Stanford University is in California”, but it is no less accurate. In sum, reliance on meaning at the stage of segmentation does not fundamentally prejudice the final chronology nor makes it less accurate, even though it may entail loss of information and make it less precise. I have not discussed whether the principles behind Bazargan’s proposed chronological sequence are sound. he reason why is that the Criterion of Concurrent Smoothness shall be the judge of the sequence of passages that result from them. How Bazargan arrived at his chronology is immaterial as long as it exhibits concurrent smoothness. In fact, if one proposed a chronology based on a purely random procedure, and if this chronology happened to yield greater concurrent smoothness, then it would be a superior proposal. If a proposed sequence achieves concurrent smoothness, others that are broadly similar to it may do so as well. here may be various proposed chronologies with large numbers of small differences which achieve a similar degree of concurrent smoothness, and these will be considered as equally corroborated. his is because the corroboration offered by statistical methods involves average characteristics of long texts or large aggregates of short texts, not small units of text in isolation. his is one reason why readers must resist the temptation to take either Bazargan’s chronological list or my own recasting of it in a more precise way than they are intended. Where I identify one group of texts 38 39 See “Motivating the Approach in this Essay”, above in Section 1. For sampling error, see “Stylometry Demystified”, above in Section 1. 238 B. Sadeghi / Arabica 58 (2011) 210-299 as having come after another, this claim holds only in an average sense; it does not mean that every text in one group came after every text in the other group. Increasing precision is a long-term goal, and this essay represents only the beginning of the journey. he definition of “groups” It is convenient (and in the case of small blocks necessary) to combine the blocks into larger groups. his has the advantage of simplifying graphs and presentations of results. he manner of aggregating the blocks into groups is arbitrary as long as consecutive blocks, which have similar verse-length profiles, are grouped together. I find it convenient to combine the 194 blocks into twenty-two groups corresponding in a rather approximate fashion to the twenty-three years of revelation as defined by Bazargan, which is not to say that I am committed to his reckoning of years. Even he noted that the true date of a passage may be two or three years off its assigned date. I will speak loosely of these as “Bazargan’s groups”. For each group, Table 3 lists the blocks it contains and the number of words in it. It should be noted that the first group is rather small, which entails a larger sampling error, warranting caution about whether it can be characterized adequately by the chosen markers of style. I will come back to this point in the Conclusion. As each group includes a set of adjacent blocks, verse length tends to increase from one group to the next. I now proceed to examine how different markers of style behave over these groups. Table 3. he Groups Defined. he block numbers are those defined in Table 1. Each group contains passages that belong to the same period according to Bazargan. Our task is to see how style varies over these consecutive groups. Group Blocks Words Group Blocks Words Group Blocks Words 1 2 3 4 5 6 7 8 2-16 17-34 1, 35-65 66-82 83-91 92-101 102-110 111-116 415 1256 3916 3214 3624 3676 4430 3479 9 10 11 12 13 14 15 16 117-121 122-126 127-131 132-138 139-143 144-150 151-154 155-160 3494 4463 4431 3660 3423 2338 4389 3920 17 18 19 20 21 22 3504 3860 3959 4018 4019 3766 161-164 165-167 168-174 175-178 179-183 184-194 239 B. Sadeghi / Arabica 58 (2011) 210-299 3. Univariate Assessments of Smoothness Univariate markers of style are those that involve just one variable, thus representing the twenty-two groups by one number apiece. Variables considered here include mean verse length, mean word length, the standard deviation of word length, and the frequency of hapax legomena. A key question is whether these markers display smooth behavior over Bazargan’s twenty-two groups. hat is, do groups that are near each other in Bazargan’s sequence tend to have similar stylistic profiles? he extent to which different, independent markers of style vary in a smooth fashion over a sequence is a measure of concurrent smoothness, hence of the degree of confirmation of the sequence as the true chronological one. Univariate Marker of Style: Mean Verse Length (MVL) Mean Verse Length 35 30 25 20 15 10 5 0 1 3 5 7 9 11 13 15 17 19 21 Group Figure 4. Mean Verse Length (MVL) vs. Group number. he height of the third column, for example, gives the size of MVL for the third group. It can be seen that in Bazargan’s sequence, average verse length varies relatively smoothly over time. 240 B. Sadeghi / Arabica 58 (2011) 210-299 Figure 4 shows the mean (i.e. average) verse length (MVL) of each group, where the length of a verse is the number of words in it. Each column represents a group, and its height the size of the MVL of that group. MVL tends to increase over Bazargan’s sequence. his is to be expected, as his sequence was devised precisely to achieve this effect: he posited that verse length increased gradually over time. One cannot yet claim concurrent smoothness, since I have not yet checked whether markers of style other than verse length also vary smoothly. But one may make a number of observations. First, note the great range in the variation of verse length. Between the group with the shortest verses and that with the longest, the difference in mean length is greater than a factor of nine! his huge difference hints that verse length might be an effective discriminator of style. Second, the variation in verse length is not discrete in nature: there is a continuum between the extremes of MVL. his is not a trivial point, as one would have had no reason to particularly expect this in advance. his suggests that the hypothesis of gradual change is worth examining. hird, if it is true that verse length is an effective indicator of relative time, one should expect lesser discriminatory efficacy over Groups 9-19, which exhibit significantly less variation. Since these groups have relatively similar verse lengths, it becomes easier to imagine them occurring in a different order than shown. For example, if one were to switch Groups 11 and 12, that would make a smaller dent in smoothness than if one switched Groups 4 and 5. his undermines confidence in the accuracy of the second half of Bazargan’s proposed sequence. All of the above hunches will be confirmed more rigorously in later sections using multivariate markers of style. One way to examine the efficacy of MVL as a criterion is to see what it does to passages that we already suspect belong to the same period. his is because MVL is effective only to the extent that passages from the same period have similar mean verse lengths. he simplest way to check this is to take a relatively coherent passage, bisect it, and see if the two halves have similar MVLs. To that end, I consider all sūras which Bazargan leaves intact and which have 570 or more words. here are twelve such sūras, and they are shown in Table 4. I have divided each text in Table 4 into two nearly equal parts, which I loosely call “halves”. his table and Figure 5 give the MVL for each half-text. he texts are listed in order of increasing block number, hence approximately increasing MVL. he distance between two halves is defined as the difference between their MVLs. In the topmost graph, the length of each line segment represents the distance between two halves. It is evident that, on the whole, the two halves of the texts have similar MVLs. his similarity is greatest, however, for the “earlier” ones, which have shorter verses.40 40 his remains true if one measures dissimilarity as the percentage of the difference relative to the MVL, i.e. as the distance between the two halves divided by the MVL. 241 B. Sadeghi / Arabica 58 (2011) 210-299 Table 4. Twelve intact sūras, arranged in order of increasing block number. Each sūra is divided into two halves with almost the same number of words. he table gives the MVL for each half, the distance between each half (defined as the absolute value of the difference of their MVLs), and clustering quality. Text number Sūra (block) 1 2 3 4 5 6 7 8 9 10 11 12 37 (058) 36 (088) 21 (094) 25 (102) 27 (114) 11 (118) 29 (126) 12 (130) 42 (138) 08 (143) 57 (161) 13 (168) Number of words MVL in 1st half MVL in 2nd half Distance betwn. the two halves Clustering Quality (m=3) 865 729 1174 896 1158 1945 977 1793 861 1243 574 853 4.97 8.71 10.7 12.2 12.9 17.0 14.6 16.2 19.0 15.3 18.3 21.1 4.56 9.08 10.2 11.1 12.0 14.8 14.1 16.1 14.8 18.1 21.7 18.7 0.41 0.36 0.52 1.12 0.89 2.21 0.50 0.06 4.15 2.77 3.44 2.36 16.7 9.72 5.42 2.09 2.63 0.63 4.02 24.7 0.36 0.51 0.83 1.19 MVL 20 15 10 Clustering Quality 5 2 4 2 4 6 8 10 12 8 10 12 10 5 0 6 (Halved) Texts Figure 5. Pair distances and Clustering Quality. Top: MVLs for each pair of halves (vertical axis) vs. text number (horizontal axis). he halves of the first eight texts have nearer MVLs than do the halves of the last four texts. Bottom: Clustering Quality (vertical axis) vs. text number (horizontal axis). Clustering Quality for a given text indicates how easily one may recognize its two halves as belonging together. he larger the Clustering Quality is, the easier that task. here is a decreasing trend, thus clustering is more successful for the earlier blocks. 242 B. Sadeghi / Arabica 58 (2011) 210-299 In the topmost graph in Figure 5, I use our prior knowledge of which halves belong together to examine how close the halves are. Now imagine the reverse of this process. Suppose you were given the MVL values of all twenty-four halves, but were not told which half goes with which, and were asked to guess the correct pairings. How successful would you be? You would form the correct clusters in five cases (texts 1, 2, 3, 7, and 8), but probably fail in the other seven cases. Given a text, two factors contribute to one’s success or failure in assigning its halves to the same cluster: how close they are to each other, and how far apart they are from the halves of other texts. he farther apart from other pairs, and the closer together they are, the greater one’s ability to correctly join the halves. hus one may define Clustering Quality for a given pair of halves as: the mean of distances of the given pair from, say, the three nearest pairs (m =3) divided by the distance between the two halves of the given pair. (Here, the distance between any two pairs of halves is defined as the distance between their respective midpoints.)41 his quantity signifies the ease with which one can successfully assign the two halves of a given pair together. Figure 5 (bottom graph) gives a plot of Clustering Quality for each pair of halves, while Table 4 lists its values. he test confirms the validity of MVL as a marker of style that can provide information about chronology. For, as seen above, texts that we know already date from the same time tend to have similar MVLs. Such correspondence is very striking for passages with shorter verses. he test, however, also suggests that the discriminatory effectiveness of MVL may decline with increasing MVL. If this is true—and more tests are needed—then given that Bazargan’s ordering is based entirely on verse length, confidence in its later parts is less justified. he possibly better performance of MVL with passages with shorter verses is perhaps due to two facts. First, as already noted, MVL appears to vary more rapidly over the “earlier” groups. Second, in earlier groups verse length tends to be more consistent. hat is, within each group, verses tend to have lengths that are close the MVL. By contrast, verses in “later” passages display larger deviations from the mean verse length. Figure 6 shows graphs of standard deviation, which is a measure of the average amount of deviation of the verse lengths from the mean verse length, MVL. 41 For example, suppose the first two halves have 1 and 2 as their average verse lengths respectively, and the next two halves have 3 and 4 as that value. hen the midpoint is 1.5 for the first 243 B. Sadeghi / Arabica 58 (2011) 210-299 group 21 19 15 17 13 9 11 7 5 0 1 21 19 15 17 13 9 11 0.2 0 7 0.4 5 5 0.6 10 1 15 3 Standard Deviation / MVL 0.8 3 StD Standard Deviation 20 group Figure 6. Left : Standard deviation of verse length indicates how much, on average, verse length differs from the MVL. Passages with longer verses display a greater amount of spread around the mean verse length. Right : Dividing the standard deviation of verse length by MVL gives the percentage by which verse length deviates from MVL. he deviation is about 40% in the earlier groups and 50% in the later ones. Univariate Markers of Style: Word Length Mean and Standard Deviation By the length of a word, I mean the number of consonants and vowels it contains.42 Figure 7 shows the mean and standard deviation of word length for each group. he mean word length is simply the average word length, which is obtained by summing the word lengths and dividing by the number of words. he standard deviation is a measure of how much, on average, word length differs from the mean word length. It indicates whether word length tends to stay at about the same value. A small standard deviation means that word lengths tend to stay close to the mean, while a large value indicates greater diversity in word length. (It is obtained by subtracting the length of each word from the mean, squaring this quantity, obtaining the average of these squares for all the words, and then taking the square root. In other words, it is the square root of the mean of the squares.) As shown in the figure, mean length shows very little variation compared to standard deviation. Mean word length ranges from about 6.3 to 6.7, a variation of only 6 %. On the other hand, standard deviation ranges from 0.54 to pair, and 3.5 for the second. Subtracting these, one obtains 2 as the distance between the first pair of halves and the second pair of halves. 42 I count vowels that are not written. I do not count elided alifs. I count the lām in the definite particle regardless of whether what follows it is a sun letter or moon letter. I count consonants with šadda twice. As usual, the exact way one defines a thing is unimportant. What matters is that one is consistent in applying whatever definition one has picked. 244 B. Sadeghi / Arabica 58 (2011) 210-299 Mean Word Length 6.8 6.6 6.4 6.2 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Group Standard Deviation of Word Length 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Group Figure 7. Mean word length (left) shows only slight variation, making the shape of the curve more vulnerable to sampling error. By contrast, there is a significant variation in the standard deviation (right), making the differences more meaningful and hinting at genuine variation in style. 1.37, a variation of 154 %. his indicates that the shape of the graph of standard deviation is less vulnerable to the random fluctuations of sampling error, as these oscillations are drowned out by the larger trends. hus the pattern captures a genuine variation in style, not just random fluctuations arising from sampling error. he graph of the mean is perhaps slightly smoother than if the blocks were arranged at random. here is only a very vague tendency for groups of similar mean lengths to be grouped together. One cannot conclude much. A variable varies “smoothly” if groups that are located near each other in the sequence have similar values. he more this holds, the smoother a graph. In a smooth variation, if one takes the differences in values of consecutive groups and adds them up, this sum will be smaller than what one would obtain on average from a random arrangement of the groups. Visually, this translates into a graph with fewer jags or smaller jags. And the more there are local or global trends, the greater the amount of smoothness locally or globally. he graph of standard deviation displays fairly smooth behavior in its left side. After that, there is no general smoothness except at the very end, although one may discern the outline of an overall trend. But this lack of smoothness in the middle part might be a sign of the fact that the groups have about the same height, so sampling error may be the cause of the jaggedness. In addition, note that although the internal chronology of Groups 6-22 remains in doubt, at least it is clear that as a whole they belong to the right side of Groups 1-5; if they were moved to the left of Group 1, that would create a severe discontinuity. his graph hints at the possible validity of the first quarter of Bazargan’s chronology, for it means that two independent markers of style vary in a smooth fashion, the markers being (1) mean verse length and (2) the standard B. Sadeghi / Arabica 58 (2011) 210-299 245 deviation of word length. he graph also suggests that Groups 21 and 22 belong together. Univariate Marker of Style: Hapax Legomena I use the term “hapax legomena” a bit loosely. It properly refers to words that occur only once in a corpus. Here, I use it to refer to morphemes that occur just once in the entire Qur ān. A morpheme can be a word or part of a word. So, for example, if the term ǧamīl occurs here with a definite article (l-ǧamīl-u) and there with an indefinite accusative case ending ( ǧamīl-an), then I count these as two occurrences of the same morpheme, not as two different morphemes. I use the transliteration of the Qur ān developed by Rafael Talmon and Shuly Wintner, which uses hyphens to divide words into morphemes.43 Do they divide words in the right way? Actually, there is no one correct way of dividing words. he chief requirement is that the division be done in a consistent manner, and the Wintner-Talmon transliteration meets this requirement. here are about 4,000 hapax legomena in the Qur ān, accounting for slightly over half of the total number of distinct morphemes. hese, however, are not distributed in the groups in an even fashion. To get a sense of their distribution, I have, for each group, added up the total number of hapax legomena and divided this number by the total number of words, yielding the percentage of words that have a hapax legomenon. he resulting percentages, graphed in Figure 8, provide a measure of vocabulary richness. he graph is striking. he amount of variation represents a six-fold difference over its range. One gets a sense of rapid stylistic change in the left side towards a steady state, which is reached by Group 5. hus, in the left side of the graph one observes not only smoothness, but also a trend. As for the 43 See Rafi Talmon and Shuly Wintner, “Morphological Tagging of the Qur ān”, in Proceedings of the Workshop on Finite-State Methods in Natural Language Processing, an EACL’03 Workshop, Budapest, Hungary, April 2003; and Judith Dror, Dudu Shaharabani, Rafi Talmon, and Shuly Wintner, “Morphological analysis of the Qur ān”, Literary and Linguistic Computing, 19/4 (2004), p. 431-52. For the on-line tagged Qur ān developed by Rafi Talmon and Shuly Wintner (and their students), see http://cl.haifa.ac.il/projects/quran/. One may use the program to search for morphemes as in the following example. To search for the morpheme “wa”, type the following command in the spot for SQL expressions in the upper right part of the program’s window, and then click on “Analyze”: “SELECT DISTINCT tbl0.location , tbl0.Word , tbl0. full_analyse FROM qortbl2 AS tbl0 WHERE ( tbl0.Word LIKE “%-wa” OR tbl0.Word LIKE “%-wa-%” OR tbl0.Word LIKE “wa-%” OR tbl0.Word LIKE “wa”) ORDER BY location”. To search for other morphemes, replace all occurences of “wa” in this expression with the morpheme of interest. 246 B. Sadeghi / Arabica 58 (2011) 210-299 Hapax legomena 25% 20% 15% 10% 5% 0% 1 3 5 7 9 11 13 Groups 15 17 19 21 Figure 8. Relative frequencies of hapax legomena, i.e. the fraction of words in each group that have a hapax legomenon, obtained by dividing the total number of hapax legomena in each group by the number of words in the group. remainder of the groups, one can conclude nothing about their correct sequence other than observing that they belong together on the right side. If they were moved to the left of Group 1, a great discontinuity would appear. Univariate Markers of Style: Summary of Results We have seen graphs of four independent indicators of style: mean verse length, mean word length, standard deviation of word length, and hapax legomena. In these graphs, except perhaps for mean word length, one observes fairly smooth behavior over the first five or so groups. his speaks in favor of the proposition that these groups are arranged in the true chronological sequence (or reverse-chronological sequence). With even greater justification, one can say that the remaining groups collectively belong to the right side of this sequence, since if one moved them to the left side of Group 1, then that would create a drastic discontinuity in three of the graphs. However, so far one cannot say much about the ordering of Groups 6-22 relative to one another. Multivariate methods of analysis, which are more effective, will shed more light on that part of the sequence. B. Sadeghi / Arabica 58 (2011) 210-299 247 4. A Non-Technical Introduction to Multivariate Methods he most powerful stylometric methods employ many variables simultaneously. For example, in a later section I represent each of Bazargan’s groups by a row of twenty-eight numbers consisting of the relative frequency counts of twenty-eight morphemes. he goal will be as before: to assess the similarity of nearby groups in Bazargan’s sequence in order to check for smoothness. But while such a task is easy for univariate markers of style such as word length, it is more difficult to see how to compare texts that are represented by twentyeight numbers apiece. One may do so using the techniques of multivariate analysis. In the present section, I will illustrate two multivariate techniques, PCA and MDS, by using the frequency counts of only three morphemes to represent each group. his allows pictorial representation of the groups as points in a three-dimensional space, making it easier to grasp the concepts. When applied to this three-dimensional dataset, the PCA and MDS techniques generate a two-dimensional (flat-surface) diagram that represents the dissimilarity of two texts by the distance between them. hus, two texts that are stylistically similar in that they have similar morpheme-frequency profiles are placed near each other. he resulting diagram can be used to determine whether the progression over the sequence of Bazargan’s groups is smooth or jagged. Figure 9 shows the relative frequency counts of three morphemes per group, namely llāh (marked as “llaah”) and the indefinite case endings an and in. he relative frequency count of llāh in Group 2, for example, is the percentage of morphemes in Group 2 that are llāh. It is obtained by taking the total number of occurrences of llāh in Group 2 and dividing it by the total number of occurrences of morphemes in that group. Each group is thus represented with a vector, i.e. a row of numbers: in this case three numbers. To assess the smoothness of the transitions, one needs to discern which groups are similar to which, since smoothness means that consecutive groups have similar profiles. You can see that Groups 1 and 3 have very similar profiles, but an aberrant Group 2 interposes between them. Other similar pairs include, for example, Groups 4 and 5, 6 and 7, and 21 and 22. he similarity or dissimilarity of two groups can be expressed through the notion of the distance between them, which can be defined in terms of the difference between their respective frequency count vectors. Suppose you wish to assess the similarity of Groups 1 and 3. To quantify their dissimilarity, one first calculates the difference in height of the llāh-columns of the two groups— by subtracting the height of the shorter column of llāh from the taller one. One then does the same for the pair of an-columns and the pair of in-columns. 248 B. Sadeghi / Arabica 58 (2011) 210-299 frequency count 10% 8% llaah an in 6% 4% 2% 0% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 group Figure 9. Group profiles in terms of the relative frequency counts of three morphemes. One then adds up these differences to get a sense of the overall difference. his calculation gives what is called the city-block distance between the two groups, which is the notion of distance used in this essay.44 Obviously, similar texts have a small city-block distance. A crucial point is that just as, in the last paragraph, I calculated distances between pairs of groups represented in three dimensions, one may also do so in higher dimensions. hat is, if one represents each group by the frequency counts of twenty-eight morphemes, one may calculate the city-block distances between any two groups just as before. he difference is that we can readily visualize data in a two- or three-dimensional space, whereas higher dimensions require special techniques to enable visualization. Considering that each group is represented by three frequency counts, one may visualize the groups as points in a three-dimensional space, where each axis represents the frequency count of one morpheme. Figure 10 provides visualizations of the same set of points from two different angles. Labeling each group would make the images too cluttered, so I have opted for connecting consecutive groups with straight lines. he topmost point is Group 1. he city-block distance between two points is the length of the shortest path between them if one were allowed to move only in lines parallel to the axes.45 Observe that Groups 1 and 3 are close to each other as expected, but in the path from 1 to 3, the aberrant Group 2 has caused a jag along the direction 44 Alternatively, instead of simply adding up the differences in height, one may first square these differences, then add them up, and then take the square root of the sum. his gives what is known as the Euclidean distance or the l2 distance. his is an equally popular measure of distance, though not the one used in this essay. he results in this article are conservative and robust enough to be unaffected by the choice of the distance measure. 45 On the other hand, the Euclidean distance between two groups (as defined in the last footnote) would be simply the length of the straight line connecting them. 249 B. Sadeghi / Arabica 58 (2011) 210-299 Group 1 Group 1 7% 7% in in 2% 2% 0 llaah an 9% 10 % 2% 10 % an 00 llaah 7% Figure 10. Bazargan’s groups represented as points in three-dimensional space. he two images depict the same points, but from different angles. Group 1 is the topmost point. Consecutive groups are connected with straight lines. of the an axis. By contrast, the path from Group 7 to Group 9 is smooth, with Group 8 in a nice, intermediate position. As mentioned before, data in a twenty-eight-dimensional space cannot be visualized in the way three-dimensional data can. However, there are multivariate techniques for reducing the dimensionality of data to two or three, at the cost of some error, so as to enable visualization. One such method is MDS, or multidimensional scaling. Given a number of points in a, say, twenty-eight-dimensional space, with each group being represented by a vector of twenty-eight morpheme frequencies, MDS arranges the same number of points in a lower dimensional space, e.g. in two or three dimensions, with the property that the distances46 between the points in the lower dimensional space approximate the original inter-point distances.47 To illustrate MDS with my example of the three morphemes, Figure 11 provides a two dimensional representation, as produced by MDS, of the three-dimensional data. PCA (Principal Component Analysis) is another method of dimension reduction. Suppose we have a three-dimensional dataset, with each group represented by three morpheme frequencies. Suppose we would like to obtain a 46 hat is, Euclidean distances, which give the length of the straight line connecting two points; see the last two footnotes. 47 Some MDS algorithms, rather than approximating the inter-point distances, try to preserve the order of the distances, so that greater inter-point distances show up as greater distances in the lower-dimensional representation, without trying to preserve the exact proportions of the original inter-point distances. Incidentally, the original distances may be city-block, Euclidean, or indeed expressed in any other measure of dissimilarity. 250 B. Sadeghi / Arabica 58 (2011) 210-299 Group 1 Figure 11. MDS (Multidimensional Scaling). A two-dimensional representation of the twenty-two groups obtained by MDS. he distances between the points in the two-dimensional graph approximate the city-block distances between the points in the original three-dimensional space, depicted above in Figure 10. Group 1 is the point on the upper-left side. Consecutive groups are connected with straight lines. he axes are inconsequential, the only relevant feature being the inter-point distances. two-dimensional representation of the groups. One way to do so is to take a snapshot with a camera. However, snapshots taken from different angles cover differing amounts of variance in the data. For example, the snapshot in Figure 12 (Left) is much less illuminating in this respect than those shown in Figure 10. he aim is finding a perspective for the snapshot that maximizes the captured variation. his is equivalent to finding a flat surface, say, a sheet of paper, that cuts through the data in an optimal fashion. he two perpendicular edges of such an optimally-placed sheet are the first two principal components. Figure 12 (Right) shows the projection of the twenty-two groups on the plane (flat sheet of paper) spanned by the first two principal components. In general, the first principal component is defined as the direction along which the data shows the greatest variation. he second principal axis is 251 B. Sadeghi / Arabica 58 (2011) 210-299 0 9 2 PCA 2 Group 1 S e c o n d 7 7 C o m p o n e n t First Principal Component Figure 12. Left : Groups observed from an angle that obscures much of the variation in the data. hese points are the same as those in Figure 10. Right : PCA is tantamount to viewing the dataset shown on the left side from an optimal perspective, i.e. one that maximizes the observed variation in the data. he horizontal axis represents the first principal component (i.e. the direction of the greatest variation), and the vertical axis the second principal component (i.e. the direction of the second greatest amount of variation). hese two principal components together account for 91 % of the variance. Variance along the third principal component, that jutting out of the page, is merely 9 % of the total. Group 1 is the point at the upper-left corner. Consecutive groups are connected with straight lines. chosen from among axes perpendicular (“orthogonal”) to the first principal component. Of all such axes, the one is chosen along which the data show the greatest variance. he third principal component (if any) is chosen as perpendicular to the first two, again with the criterion of maximal variance. One may continue in this manner until one has as many principal components as the number of axes (or dimensions) in the original dataset. However, if one desires a lower dimensional representation, one retains fewer principal components than that. In the example shown in Figure 12 (Right), I have retained two principal components. he third principal component, deliberately omitted, is perpendicular to the two shown, thus jutting out of the page towards us. herefore, the true position of each point is a bit directly above or below the point on the sheet, and this information is lost. However, the variation along this omitted component accounts for only 9 % of the total variance in the data, the first two components accounting for 91 % of it. In sum, I have successfully reduced 252 B. Sadeghi / Arabica 58 (2011) 210-299 the number of dimensions, though at the cost of failing to capture 9 % of the variance. Such a “loss” is not a bad thing if it eliminates noise in the data that tends to obscure larger trends. 5. Morphemes: Weighting and Weight-Optimization Basic Concepts: Features, Feature Vectors, Textual Distances he rest of this essay is devoted to analyzing relative frequencies of morphemes. Henceforth I sometimes use the word “feature” to refer to a morpheme. A morpheme, it will be remembered, is a word or part of a word, delimited by hyphens or spaces in the Wintner-Talmon transliteration of the Qur ān. For example, the transliterated word l-raḥ mān-i consists of three morphemes. On average, there are slightly over two morphemes per word. In this essay I work with three separate lists of morphemes. I now introduce the first two. Feature List A, given in Table 5, consists of the top twenty-eight most common morphemes in the Qur ān. It can be seen that these features are ones that can be used independently of the subject matter at hand; that is, they are in principle non-contextual. hey also tend to be function features (pronouns, case-endings, etc.), as opposed to content features. Feature List B, provided in Table 6, consists of 114 morphemes that are function features or otherwise are relatively non-contextual, and which are not included among the top twenty-eight most frequent features. hese are among the most frequent 536 features in the Qur ān. I eliminated many elements such as qāla, arḍ, yawm, qālū, and ad̠āb that are firmly tied to specific themes. I did so in a manner that eliminated the possibility of cherry-picking.48 he idea is to reduce the influence of subject matter in favor of style. In addition, it is important to note that these features occur in the Qur ān far less frequently than those in Feature List A. Tests indicate that a medium-frequency feature tends to convey less information about style than a high-frequency one. To compensate for this effect, it is important to take a larger number of features, hence the larger size of List B. he fact that I eliminated some morphemes may raise the suspicion that I picked features that guaranteed the desired result. I eliminated the possibility of such cherry-picking by first removing morphemes on the grounds explained above and then proceeding with the analysis without going back to modify my choices of morphemes. hus, neither at the moment of elimination nor later did I come to see if or how the results changed. At any rate, the results are too robust too be alterable by the removal or addition of several morphemes. One could randomly add or subtract many morphemes without changing the overall picture. 48 253 B. Sadeghi / Arabica 58 (2011) 210-299 Table 5. Feature List A, consisting of the twenty-eight most frequent morphemes in the Qur ān. he features are listed here in order of descending total frequency, ranging from 9567 occurrences in the Qur ān for wa to 1080 for llad̠īna. he third and fourth columns give the relative frequencies of each feature respectively in Groups 2 and 3. For example, wa makes up 6.12 % of the morphemes in Group 2. feature grp 2 grp 3 1 2 3 4 5 6 7 8 9 10 wa i l u a an min ūna fa hum 6.12 5.42 6.78 4.41 3.33 4.30 0.78 1.36 3.14 0.89 5.23 5.04 6.84 3.93 3.87 1.37 1.48 3.01 2.40 1.54 feature grp 2 grp 3 11 12 13 14 15 16 17 18 19 20 llāh mā in bi un la kum hu lā fī 0.27 1.70 2.21 1.12 1.55 1.24 0.78 2.01 1.01 1.01 0.37 1.58 2.76 1.37 1.85 1.72 0.67 1.57 1.09 1.02 21 22 23 24 25 26 27 28 feature grp 2 grp 3 0.68 1.24 1.01 0.97 1.24 1.05 0.31 0.23 2.14 1.04 0.59 0.60 0.91 1.20 0.73 0.33 īna ka hi li hā inna nā allad̠īna Table 6. Feature List B, consisting of 114 either relatively non-contextual morphemes or function morphemes, selected from among the top 536 most frequent morphemes in the Qur ān but excluding those in Feature List A. he features are listed here in order of descending total frequency, ranging from 1050 occurrences for in to 20 for laday. An initial hamza is indicated with an apostrophe. in, him, ū, rabb, man, alay, alā, illā, an, ī, a, huwa, ilā, d̠ālika, id̠ā, kāna, an, qad, kull, nī, yā, lam, anna, id̠, t̠umma, qul, ilay, llad̠ī, aw, šay , law, kānū, bayn, qabl, hād̠ā, ayy, ūlā ika, ba d, kuntum, ind, anna, mim, lammā, ba ḍ, ma a, amr, gȧ yr, dūn, ḥ attā, hunna, n, am, antum, bal, la alla, sa, ayni, lan, ā, awna, āni, hal, naḥ nu, kayfa, takūn, anta, ni, ya, d̠, mit̠l, aḥ ad, laysa, yakūn, lākin, ay, aw, anā, llatī, hinna, hiya, iy, lākinna, na, ammā, d̠āli, al, ‘am, fal, hād̠ihi, unna, hā ulā i, tarā, ūl, sawfa, tilka, kunta, kānat, kallā, yakun, d̠ā, ḥ ayt̠, alā, d̠āt, im, rijāl, annā, ki, wal, iyyā, ma, balā, takun, kam, laday. Given a feature list, any passage or any group of passages may be represented by a list of numbers consisting of the relative frequency counts of each feature in that text. Such a list of numbers is called a feature relative frequency count vector, or feature vector for short. Table 5, for example, provides the feature vectors for Groups 2 and 3 relative to Feature List A. he concept of a relative 254 B. Sadeghi / Arabica 58 (2011) 210-299 frequency count may be illustrated with an example. Take the following verse, in which the morphemes are separated by hyphens: wa-qur ān-an faraqnā-hu li-taqra -a-hu alā l-nās-i alā mukt̠-in wa-nazzalnā-hu tanzīl-an (Kor 17, 106). In this verse there are twenty-one occurrences of morphemes. he feature wa makes two appearances; so, its relative frequency count is two divided by twenty-one, or 9.5 %. his number is the relative frequency count of wa in this short passage. he task in the following sections is to use feature vectors to examine the distances between groups, where I use the city-block distance as a measure of the inter-group similarities. his section addresses two issues. First, it considers whether the different features must contribute equally to the calculation of inter-group distances, or whether they should be “weighted” differently. In my discussion of weighting, I will explain the concepts of normalization, standardization, and feature weight optimization. his last involves a new methodological contribution to stylometry. Second, this section tests the utility of Lists A and B by dividing some sūras into halves and checking if the halves display similar frequency count profiles. Feature Weights: Normalization and Standardization As one goes down the list of features in Table 5, the frequencies fall sharply. his rapid decline raises a question. Remember from the last section how one calculates the distance between two groups, say, Groups 2 and 3: first, for each feature, one takes the difference in its frequency count. For example, for the first item, wa, we get 6.12-5.23 = 0.89. hen one adds up all these differences (for all the features), the sum representing the distance between Groups 2 and 3. Now, as one goes down the list, and as the magnitudes of individual frequency counts decrease, so do the differences. herefore, items lower in the list contribute significantly less to the sum that defines the distance than do items higher on the list. With feature lists much larger than just twenty-eight items, this discrepancy becomes even more pronounced, with features at the bottom of the list barely contributing anything. In order to make different features count about the same, one may choose to magnify some of them to make them all have comparable magnitudes before adding them up to get the overall distance. For example, it makes sense to multiply smaller-magnitude features by larger numbers. In general, when one multiplies a feature frequency by a number to adjust its contribution, this number is called a feature weight. he list of weights for all the features is called the weight vector. One common way of weighting is normalization. It means dividing each feature vector by its average magnitude. In the case at B. Sadeghi / Arabica 58 (2011) 210-299 255 hand, it means dividing a morpheme frequency by the mean value of the frequencies of that morpheme over the twenty-two groups. In this scheme, since smaller-magnitude features are divided by smaller numbers, and larger-magnitude features by larger ones, the feature frequencies become comparable. Another method is standardization. It involves dividing the values of a feature by the standard deviation of the feature (over all the groups).49 Both methods equalize the features. he difference is that normalization happens to give greater weight to features that vary significantly over the groups. Incidentally, in both procedures, it is customary to subtract the feature mean from feature values before multiplying by the weight. Weight Adjustment he ultimate goal is to see how similar or different Bazargan’s groups are stylistically. he dissimilarity of two groups is defined as the sum of the differences of morpheme frequencies in those two groups. But to simply add up these differences is to give all morphemes the same weight. Is it possible, however, that one morpheme is more important than another and should therefore be given greater weight? Weight adjustment refers to the above-mentioned process of multiplying the frequency counts of a feature by a number (coefficient) so as to diminish or boost its contribution to the calculation of distances. If one does not adjust a feature, then the coefficient is simply 1, and the feature vector is kept “raw”. As mentioned above, the list of weights for all the features is called the weight vector. So, the weight vector for raw features is a list of 1’s. I have shown how weight adjustment can remedy the tapering off effect. However, there are various other reasons why one may want to adjust the weights, and here are three. First, note that the behavior of a higher-frequency feature can have greater statistical significance than a medium-frequency one. he frequency of a feature that occurs a relatively small number of times in a group is more susceptible to chance oscillations and may thus be statistically less significant and more prone to sampling error. To muffle such noise, it may be fitting to give more weight to the most frequent features. 49 he standard deviation of a morpheme frequency count is a measure of how much, on average, the frequency of the morpheme differs from its mean frequency. A small value for the standard deviation means that the morpheme frequency tends to stay about the same in the different groups, while a large value indicates greater diversity. 256 B. Sadeghi / Arabica 58 (2011) 210-299 Second, there are cases where two features tend to go together in a text for reasons of linguistics. For example, in the Qur ān the verb “to be” (morphemes: kāna, yakūn, yakun, etc.) correlates strongly with the morphemes representing accusative case endings. A large fraction of the accusatives in the Qur ān are predicates of the verb “to be”; and conversely, the verb “to be” is typically followed by an accusative. his correlation simply reflects the grammar of the language. So, to give equal weight to “to be” and the accusative case endings would be to count twice what is essentially one grammatical phenomenon, thus possibly hurting precision. One may deal with this by reducing the weights of such elements. hird, phenomena that are suited to statistical description will typically produce outliers. here will thus be a small number of morphemes whose behavior will be highly exceptional. To improve precision, it would be appropriate to remove these, which is to give them weights of zero. From this discussion, it must be clear that feature weight adjustment, if based on a priori reasoning, can be a highly complex and time-consuming process. Such complexity probably explains why in practice weighting is usually limited to feature normalization or standardization. here is, however, a simple and practical way out that makes weighting automatic. It involves replacing all a priori reasoning about weights with a posteriori optimization of weights. his technique can achieve higher accuracies than either normalization or standardization. Weight Optimization All features are not equally effective discriminators of chronology. Some will be more important than others. We wish to assign each morpheme a weight reflecting its degree of effectiveness. But how can one measure the effectiveness of different features? Rather than speculating in an a priori fashion, it is possible to determine this empirically. One can measure the effectiveness of a morpheme by seeing how well it performs at the task of assigning to the same cluster texts that we already know belong together. he basic idea is simple, and can be illustrated with the following hypothetical example: suppose we take a long, coherent, and self-contained sūra that we think represents a unified text from one time period. Let us divide it into two halves: say, we divide the sūra in the middle—or, say, we combine its odd verses into one text, the “First Half ”, and combine its even verses into another text, the “Second Half ”. Now we have two texts that we know belong together. We then pick two morphemes, say wa and fa, and ask how effective they are as discriminators of style. Looking at the frequencies of wa and fa in these two texts, suppose one finds that wa makes up respectively 6.1 % and 6.2 % of the morphemes in the B. Sadeghi / Arabica 58 (2011) 210-299 257 two halves, while fa forms respectively 2 % and 14 % of the morphemes in them. Note that the percentages of wa in the two halves are close to each other, as compared to those of fa. One concludes that wa is a better indicator of chronology than fa, since two passages that date from the same time have similar frequencies of wa but not similar frequencies of fa. herefore, from now on, when one computes the distances between passages by summing up the differences of morpheme frequencies, one assigns more significance to the differences in the frequency of wa. One does so by multiplying that difference by a relatively large number before adding it to other morpheme differences. So, different morphemes are weighted differently as their contributions are added up to obtain the overall stylistic distance between two passages. If one takes a number of unified texts, and divides each text into two halves, then one may measure the success of different weighting schemes in indicating the affinity of the pairs of halves. A weighting scheme is more successful if it does a better job of clustering together halves that truly belong together. In fact, one may try many different weighting schemes and pick the most successful one. his gives rise to the following optimization problem for finding the best weighting scheme: find a weight vector that achieves the best clustering success. To perform this optimization, one needs two things: a set of training texts and a measure of clustering success. Training texts (see Table 7) are bisected unified texts used in the process of optimizing feature weights. hat is, the weights are chosen for superior performance specifically on the training texts. It is important that the training texts be representative of the whole corpus; otherwise good performance on the training texts may not lead to good performance in general. Normally, the more numerous the training texts are, the more representative they will be of the larger corpus. Moreover, medium and large texts usually yield better optimized weights than small ones. To compile the eighteen training texts used here, I started with all twelve intact sūras of over 570 words, then added five large contiguous passages, each unified in Bazargan’s reckoning and each forming part of a sūra.50 Finally, I combined several smaller passages into one text (text number 17), having determined from old-fashioned stylistic analysis that they probably belong together.51 50 hese five texts I chose from six that I had picked randomly, though with a view to choosing large passages. Of the six texts, I removed the one that displayed the largest differences in the styles of its two halves. his relative imbalance could be a sign of one of two things. It could be that the halves belong to different times, or it could be that they are contemporaneous but unusual in their difference. In either case, the exclusion is not problematic. 51 I examined the occurrences of unusual phrases, words, and themes to arrive at the hypothesis that sūras 48, 58, 63, 66, and sūra 9, verses 71-96 (respectively Blocks 170, 181, 186, 152, and part of 169) are close in time. If I am wrong about this, the error will not be fatal, since each 258 B. Sadeghi / Arabica 58 (2011) 210-299 Table 7. Training Texts. One can test a method by seeing how well it joins together texts that belong together. To that end, eighteen texts are divided into halves and used alternately for optimizing (“training”) the weight vector and for testing it. Texts 1, 3, 5, 6, 8, 9, 10, 11, 12, 13, 15, and 16 are complete sūras—namely, sūras 37, 36, 21, 25, 27, 11, 29, 12, 42, 8, 57, 13. Texts 2, 4, 7, 14, and 18 are parts of sūras. Text 17 is a composite of several probably relatively contemporaneous passages, namely all of sūras 48, 58, 63, and 66, and part of sūra 9. Text no. Halves definition words (sūra) block: verses Text no. Halves definition (sūra) block: verses words 1 (a) (37) 58: 1-87. 432 10 (j) (29) 126: 1-35. 497 (37) 58: 88-182. 433 (29) 126: 36-69. 480 (26) 65: 52-139. 399 (12) 130: 1-55. 890 (26) 65: 140-227. 417 (12) 130: 56-111. 903 3 (c) (36) 88: 1-43. (36) 88: 44-83. 366 363 12 (l) (42) 138: 1-23. (42) 138: 24-53. 417 444 4 (d) (23) 90: 12-62. 502 13 (m) (08) 143: 1-41. 628 (23) 90: 63-118. 493 (08) 143: 42-75. 615 (21) 94: 1-55. 591 (10) 160: 1-30. 609 (21) 94: 56-112. 583 (10) 160: 31-70. 624 (25) 102: 1-37. 452 (57) 161: 1-16. 292 (25) 102: 38-77. 444 (57) 161: 17-29. 282 (20) 103: 53-94. 493 (13) 168: 1-20. 422 (20) 103: 95-135. 475 (13) 168: 21-43. 431 (27) 114: 1-45. 581 (27) 114: 46-93. 577 (11) 118: 1-57. 969 (11) 118: 58-123. 976 2 (b) 5 (e) 6 (f ) 7 (g) 8 (h) 9 (i) 11 (k) 14 (n) 15 (o) 16 (p) 17(q) (63) 152: 1- 5. (9) 159: 38-54. (9) part of 169: 1305 71-84. (48) 170: 1-17. (58) 181: 1-9. (66) 186: 1-6. (63) 152: 6-11. (9) 159: 55-70. (9) part of 169: 1263 85-96. (48) 170: 18-29. (58) 181: 10-22. (66) 186: 7-12. 18 (r) (09) 177: 1-20. 361 (09) 177: 21-37. 371 half-text contains half of each of these passages, so that the two half-texts should be assigned together anyway even if the individual passages belong to different times. After I had inferred the contemporaneity of these passages, I consulted the chronological sequence of sūras ascribed to B. Sadeghi / Arabica 58 (2011) 210-299 259 As a measure of how well a given weighting scheme assigns together the two halves of a text, I introduce the quantity called Individual Clustering Quality (ICQ). Given the two halves of a bisected text among a collection of such texts, the ICQ of a weighting scheme with respect to this text is defined thus: Individual Clustering Quality = Distance of the text from the nearest m neighbors he distance between the two halves of the test where normally one chooses m = 1 or m = 2, representing one or two neighbors. he idea behind the denominator, which one may call the “intra-text distance”, is simple enough: a scheme that places the two halves closer to each other, thus resulting in a smaller denominator, has a higher ICQ. (As for how to calculate this distance, recall that each half is represented by a weighted feature vector. hus, the distance between two halves is simply the city-block distance between their respective weighted feature vectors.) On the other hand, the numerator, which one may call the “inter-text distance”, reflects how well the clustering procedure distinguishes this text from other texts. It measures the “distance” of the text at hand (both halves of it) from the nearest text(s) (represented by their respective halves). he greater the distance from the nearest texts is, the higher the success of the clustering method in distinguishing this pair from the other pairs. his concept turns on the notion of the “distance” between two pairs of halves, which I define as the distance between their respective midpoints, where the midpoint of a pair of halves is obtained by averaging their feature vectors. For example, if the frequency of wa is 5 % in the first half and 6 % in the second half, the midpoint vector will have 5.5 % for its frequency for wa. Such averaging is used also to obtain all the other feature frequencies, thus defining a full feature vector representing the midpoint. With the midpoint feature vectors in hand, one may obtain the distances between the midpoint of the text at hand and the m nearest midpoints (i.e. those midpoints with the smallest distances from the midpoint at hand). he average of these distances forms the numerator. To help with the comprehension of these ideas, Figure 13 illustrates these definitions graphically. Having defined the notion of Clustering Quality for an individual text, one may now extend the concept to a collection of training texts. I calculate the Ğa far al-Ṣādiq, and noted that they are also located near each other within this fragment of his sequence: 63, 58, 49, 66, 61, 62, 64, 48, 9. (For an analysis of the al-Ṣādiq sequence and those ascribed via different isnāds to Ibn Abbās, see Bāzargān, Sayr, vol. II, p. 192-203 / 557-69. For a discussion of the sources in which these sequences and some others are found, see Rāmyār, Tārīḫ-i Qur ān, p. 661-7.) 260 B. Sadeghi / Arabica 58 (2011) 210-299 Text B first half Text B midpoint Text B second half Text C first half Text C midpoint Text C second half Text A second half Text A midpoint Text A first half Figure 13. Individual Clustering Quality (ICQ) is a measure of how well the two halves of text A cluster together. he closer the two halves of Text A are together, and the farther apart Text A is from other texts, the higher the ICQ for text A. How does one calculate the ICQ for Text A? Suppose the two nearest neighbors are texts B and C as shown. he denominator of ICQ is the distance between the two halves of Text A, represented here by the length of the line segment connecting them. he numerator depends on the choice of the number of neighbors to consider, m. If m = 1, then the numerator is the distance from the midpoint of Text A to the midpoint of the nearest text, namely Text C. his distance is represented by the line segment connecting the two midpoints. If m = 2, then it is the average of two distances, namely (1) the distance from the midpoint of text A to the midpoint of text B, and (2) the distance from the midpoint of text A to the midpoint of text C. hese two distances are represented by the line segments connecting the midpoint of Text A to those of Text B and Text C. B. Sadeghi / Arabica 58 (2011) 210-299 261 inverse of the ICQ for each member of the set, take the average of these values, and invert this average to obtain the Total Clustering Quality (TCQ). his is basically a form of averaging, but one involving taking inverses before and after. One may now pose the optimization problem as follows: find a vector of weights, i.e. a list of coefficients for the features, such that it results in the largest Total Clustering Quality attainable. his is tantamount to finding the weights that do the “best job” of putting together training texts that belong together and setting apart those that do not. “Best job”, of course, requires a measure. To that end, I defined the TCQ. A higher value of TCQ means that pairs of half-texts are closer to each other and father apart from other texts. “Weight optimization”, therefore, means finding weights that maximize the TCQ. his optimization can be performed within a software environment for technical computing such as Matlab. A solution to this problem is called an optimal, or optimized, weight vector. he optimal weight vectors for Lists A and B are given in Table 8. Table 8. Weights optimized with m=1 for feature lists A and B. he optimal weight vector for Feature List A: (91, 0.080, 0.18, 6.5, 0.24, 87, 0.03, 0.92, 50, 0.22, 35, 23, 0.15, 169, 85, 317, 0.02, 0.01, 0.03, 0.06, 181, 57, 90, 0.27, 0.06, 209, 83, 2.8) he optimal weight vector for Feature List B: (13.9, 101, 0.046, 5.7, 0.29, 0.25, 0.048, 0.41, 0.05, 0.14, 0.46, 0.23, 0.59, 0.25, 0.96, 148, 0.89, 140, 1.23, 0.48, 94, 2.3, 3.5, 4.6, 0.1, 30.5, 0.1, 30.6, 0.7, 1.1, 0.6, 208, 0.4, 1.2, 38, 1.5, 0.2, 1.2, 0.6, 371, 281, 228, 4.6, 0.1, 16.3, 0.3, 0.3, 0.3, 1.3, 3.5, 532, 9.7, 599, 1.0, 0.9, 1.7, 1.1, 570, 60, 2.4, 1.1, 1.6, 741, 504, 166, 8.2, 0.4, 0.3, 3, 21, 272, 197, 247, 15, 81, 81, 126, 66, 102, 20, 1.7, 57, 214, 0.7, 0.9, 350, 1.7, 128, 1.3, 312, 99, 322, 7.6, 204, 391, 146, 80, 62, 0.5, 723, 50, 850, 109, 0.7, 0.7, 3.4, 1.0, 5.9, 13.0, 1.0, 8.3, 15.4, 0.5, 570) How Well are the Halves Clustered? he dendrograms in Figure 14 depict the agglomerative hierarchical clustering of the eighteen pairs of half-texts with normalized features from List A (top) and with standardized features (middle). Each text is designated with a letter of the alphabet. he closest pairs of half-texts are clustered together in the first round, forming the lowest-level clusters. Next, the closest pairs of objects (half-texts or clusters) are grouped together in the second round, and so on, 262 B. Sadeghi / Arabica 58 (2011) 210-299 Hierarchical Clustering with Raw Features q q r m h n n j j mo r l p p o l a b b a c e d c d e h i i k k g g f f Hierarchical Clustering with Normalized Features q q r mm o r h i i k k h n n j j l l p p o a e b b c a d c d e f f g g Hierarchical Clustering with Standardized Features q q r r mm o h n n j j l p p o l a b b a c e d d c e h i i k k g g f f Hierarchical Clustering with Optimized Features p p o o j l l d k k h h i i n n e m q qm r r c c g g d e a a b b j f f Figure 14. Hierarchical clustering with Feature List A, with raw, normalized, standardized, and weight-optimized (m=1) features. he halves of Text 1 are designated with the letter “a”, the halves of Text 2 with the letter “b”, and so on. he length of a U-shaped line is a measure of distance. Raw features yield eight exact matches and two near matches (a, n), normalized features yield nine exact matches (i.e. with two halves in the same lowest-level cluster) and one near match (namely, Text n), while standardized features yield eight exact matches and four near ones (a, d, n, r). Where there is no match, generally pairs of half-texts that belong to each other are still part of the same mid-level clusters. hus, texts that Bazargan considers as close to one another indeed tend to be nearer one another. Optimized weights are designed to maximize the number of matches, and they do, yielding twelve exact matches and one near match (Text a). B. Sadeghi / Arabica 58 (2011) 210-299 263 until everything is subsumed in a single super-cluster. he lengths of the U-shaped lines indicate the distances between the objects (texts or clusters).52 With standardized features, we observe twelve exact or near matches in twelve out of the eighteen pairs of half-texts. An exact match is when the two halves that truly belong together are clustered at the lowest level, in the first round. Even where there is not a match, generally two half-texts that belong together inhabit the same mid-level cluster. As stylometric studies go, such level of performance is outstanding, especially considering the small sizes of the texts, the crowded field of choices (with thirty-two items), the fact that the texts are all from the same work, and the fact that some of the texts may actually be contemporaneous. In addition, it is impressive that the texts that Bazargan considers near in time tend to be placed nearer to each other. hus a is closest to b, and together they are closest to the cluster {cde}. For later texts, there is a similar tendency, but less exactly and only in a broad way. If one uses brackets to indicate the nearness of texts, the overall clustering scheme looks something like this: f, g, {{ab}{cde}}, { {{ik} h {jn}} {lop} } {mqr} he bottom dendrogram in Figure 14 depicts the clustering of the half-texts when represented by weight-optimized feature vectors. he key point here is that there are several more matches than in the previous weighting schemes. Such improvement is expected. he clustering scheme is broadly compatible with the previous cases, the most interesting difference perhaps being that now Text g is firmly joined to cde. Figure 15 depicts the dendrograms when the half-texts are represented with feature vectors from List B. In the raw, normalized, and standardized cases, the quality of clustering seems inferior to the results of List A (see the top three dendrograms). However, weight optimization helps yield performance that is nearly as good as before (bottom dendrogram). Despite its visual appeal, hierarchical clustering is not an infallible way of judging clustering success. In some situations, texts that are close to each other may end up in clusters that are far apart.53 As a more reliable guide, I have defined a clustering score. Table 9 provides the scores. As expected, optimized weights perform much better for both Lists A and B. 52 he distance between two half-texts is defined as usual. he distance between two clusters is defined here as the average distance between all pairs of half-texts in them. 53 For example, suppose A is the closest text to B, but C is closer than B to A. hen A and C will form the cluster {A, C}. In the next step, likewise, {A, C} may be joined with another item rather than B, and so on. 264 B. Sadeghi / Arabica 58 (2011) 210-299 Hierarchical Clustering with Raw Features q q o r r mm j j n p l l p o a a b d e f c c f e h n g g h k k i i d b Hierarchical Clustering with Normalized Features l l a a f f n p o r r mq q b c d e c o e h n p i i j g g m b d h k j k Hierarchical Clustering with Standardized Features q q mo o r r j n p l l a a f f b c d e p e h n i i m j g g c b d h k k Hierarchical Clustering with Optimized Weights f f b b m n a a d d r e e mk c c i i q q j p p l l o o h k h n g g j r Figure 15. Hierarchical clustering with Feature List B, with raw, normalized, standardized, and weight-optimized (m=1) features. Raw features yield six exact matches and two near matches (k, r), normalized features six exact matches and one near match (r), standardized features six exact matches and three near ones (k, o, r), and optimized weights eleven exact matches and one near match (d ). Table 9. Clustering Scores. he scores below are obtained as follows. First, for each half-text one records a score indicating the proximity of its twin half. he score is 1 if the twin half is the nearest text, ½ if it is the second nearest text, 1/3 if it’s the third, and so on. he scores for all half-texts are added up, and the total is expressed as a percentage of the maximum possible score (which would be 36). Weighting Raw Normalized Standardized Optimized Feature List A 64.9 67.3 64.8 80.9 Feature List B 53.4 44.8 56.0 77.2 B. Sadeghi / Arabica 58 (2011) 210-299 265 Optimized weights are contrived to yield superior clustering for training texts. Indeed, as seen above, they fulfill this expectation. But do they give superior results when applied to texts not used in training? One cannot know that without testing them. hat test is called cross-validation. Cross-Validation as a Test of Weight Optimization Is there any reason for believing that weight optimization performs better than other weighting schemes—better than if one used raw, normalized, or standardized features? Weight optimization is a form of learning: learning how to weight the different features. But it is learning from a limited set of texts, namely the training texts. We just saw that optimized weights perform better than other weighting schemes at clustering the training texts. But this is hardly a surprise. Optimized weights perform well on these training texts because that is precisely what they are designed to do. he real question is whether optimized weights outperform other schemes on texts on which they have not been trained. hus the true test of the merit of optimization is to see how it clusters texts not involved in training. hink of an apprentice who has been trained in the mechanics’ school to fix three specific training cars. One would like to know if the learning has made him/her better at fixing cars in general. One finds this out by testing the mechanic with a car he/she has not trained on. Now think of the eighteen texts as cars. You hide the first car, train an apprentice on all the other cars, and then test him on the first car. You then hide the second car, train an apprentice on the seventeen other cars, and then test her on the second car. And so on. hus, one excludes the first text from training and uses all the other texts combined to optimize the weights, i.e. by choosing the weights that do the best job of clustering them. One then uses the resulting optimized weights to test how well they join the halves of the first text: one calculates the clustering quality, ICQ, to see if it is higher with optimized weights than with raw, normalized, and standardized features. If ICQ is higher for the optimized weights, that means better performance than the other weighting schemes in this particular case. But maybe this text is an aberration? So, next, one excludes the second text from training instead, trains the weights on all the other texts, and sees how they perform at joining the halves of the second text compared to raw, normalized, and standardized features. One repeats this procedure for each of the texts. Figure 16 shows the results for Feature List A. Raw, normalized, and standardized features differ among themselves in their performance, but these differences are over-shadowed by the superior performance of optimized weights. Figure 17 shows a similar improvement in the case of Feature List B. Cross-validation, therefore, confirms that optimizing weights improves the 266 B. Sadeghi / Arabica 58 (2011) 210-299 CROSS VALIDATION, m = 1 Individual Clustering Quality (m= 1) 1.5 raw normalized standardized optimized 1 0.5 0 2 4 6 8 10 Text 12 14 16 18 Figure 16. Cross-validation of optimized weights for Feature List A. he plot shows the value of ICQ (m=1) for each text for raw, normalized, standardized, and weight-optimized feature vectors. A higher value of ICQ means better clustering. he weights are optimized (with m=1) with the text at hand excluded from the training set. he mean value of ICQ is 1.14 for optimized weights (with standard deviation of 0.03). It is significantly lower for the other weighing schemes, namely 0.90, 0.92, and 0.91 respectively for raw, normalized, and standardized features (with standard deviations respectively 0.22, 0.21, and 0.19). accuracy of multivariate analysis. It can also be seen, however, that optimization does not always perform better. he improvement is seen somewhat more consistently in the right half of the graph, in texts 11 and above.54 I am now in a position to address a nagging question. Could it be that some of the texts that I took to be unified actually are not, and if so, does that 54 As will become evident later, style varies more dramatically in groups 1-11. his means that more training texts from this phase may be needed than those included at present, hence the less reliable results. he problem is particularly acute for the first few groups. 267 B. Sadeghi / Arabica 58 (2011) 210-299 invalidate optimized weights? It is not entirely impossible that in a case or two the two halves of a text date from different periods. his, however, is unlikely to have been the case for more than a couple of texts, if that. Otherwise, crossvalidation would not have yielded such favorable results for optimized weights, nor would have the dendrograms exhibited clustering of such high quality. he question also arises as to whether halves that were not assigned together should be assumed to be from different times. he answer is no. In other stylometric studies where a collection of texts have been considered simultaneously, even the most effective techniques have not succeeded in clustering texts without error. If two texts are not assigned to the same cluster, that does not necessarily mean that they do not belong together. CROSS VALIDATION, m = 1 Individual Clustering Quality (m= 1) 1.2 1 0.8 0.6 0.4 raw normalized standardized optimized 0.2 0 2 4 6 8 10 Text 12 14 16 18 Figure 17. Cross-validation with Feature List B. he weights are optimized (with m=1) with the text at hand excluded from the training set. A higher value of ICQ means better clustering. he mean value of ICQ (m=1) is 1.27 for optimized weights (with standard deviation of 0.03), while mean ICQ (m=1) is 0.85, 0.93, and 0.92 respectively for raw, normalized, and standardized features (with standard deviations respectively 0.13, 0.18, and 0.15). 268 B. Sadeghi / Arabica 58 (2011) 210-299 Section Summary and Conclusion One may consider different weighting schemes. he weight of a feature is the number by which one multiplies its frequency count to boost or reduce its contribution to the calculation of distances among texts. One particular weighting scheme consists of leaving the frequency counts untouched, or “raw”. (his is equivalent to having a weight vector consisting purely of 1’s.) Two common alternatives are normalization and standardization. Finally, using weight optimization one may attempt to find weights that most accurately cluster half-texts. One may do so by finding weights that maximize Total Clustering Quality (TCQ). Once the weights are obtained in this manner, testing them using cross-validation verifies them as competitive with other weighting schemes. his section has given a glimpse of the promise of multivariate techniques as applied to frequency counts of the morphemes of the Qur ān. Particularly interesting are the results obtained from the list of the top twenty-eight morphemes. I find that for passages that are larger than a certain size—the threshold lying somewhere between 300 and 400 words—these methods, when used with normalized or standardized features, constitute a reliable means of judging the stylistic relationships of texts—reliable, but not flawless. With optimized weights, one might obtain reliable performance for somewhat smaller texts. he above investigation has implications for theories about the composition of the Qur ān. he traditional understanding, as embodied in Bazargan’s work, acknowledges that the sūras may contain passages from different periods, but it also tends to assume the chronological unity of many sūras and many pericopes. On the other hand, the scholar of the Qur ān, Richard Bell, took a very different approach. Even in the case of the twelfth sūra (Yūsuf ), which is normally regarded as unified and coherent, he viewed the sūra as a hodgepodge, a patchwork of small fragments belonging to several periods, and the outcome of an extensive process of collection, revision, and interpolation by the Prophet.55 On Bell’s approach, one would deny medium and long passages stylistic distinctiveness or temporal unity. Richard Bell’s general vision does not fit the findings in this section. What has been demonstrated is the stylistic distinctiveness and coherence of passages to a surprising degree—that is, to a degree greater than what stylometric techniques have demonstrated in some other cases in which the questions of 55 Richard Bell, A Commentary on the Qur ān, eds Bosworth and Richardson, Manchester, University of Manchester, 1991, vol. I, p. 375-406. B. Sadeghi / Arabica 58 (2011) 210-299 269 authorship and chronology are not in doubt, and where the texts are much longer and the analysis therefore less prone to error. (See especially Figure 14.) he traditional understanding is correct. he primary aim of this section has been to test and hone the techniques to be used in the rest of the essay. As an incidental bonus, however, a phenomenon has emerged that already lends some support to Bazargan’s chronology. One finds that, speaking rather broadly, the dendrogram in Figure 14 places near each other the passages that are close in time according to Bazargan’s chronology. 6. Multivariate Analysis (List A): Top Twenty-Eight Morphemes We now come to the heart of the argument. I showed how several univariate markers of style behave over Bazargan’s sequence of twenty-two groups of passages. In particular, mean verse length varies in a smooth fashion. I now compare the groups with the aid of the multivariate techniques described in the last two sections and using Feature List A. his list contains the twenty-eight most common morphemes in the Qur ān as presented in Table 5 on page 253. hus, each group is represented by twenty-eight numbers, namely the frequency counts of the morphemes of List A. he goal is to see whether style, as represented by the relative frequency counts of these twenty-eight morphemes, varies smoothly over Bazargan’s sequence of twenty-two groups. In other words, do groups that are consecutive or nearby in Bazargan’s chronology have similar stylistic profiles? To the extent that they do as judged by different markers of style, there is concurrent smoothness; and any part of the sequence that displays concurrent smoothness is confirmed. Stylistic dissimilarity is represented using the graphical techniques of MDS and PCA. Each of Bazargan’s groups is represented by a dot. he distance between two dots is a measure of the stylistic dissimilarity of the corresponding groups. hus, two groups that are stylistically similar, in the sense of using the top twenty-eight morphemes with similar frequencies, are placed near each other. he farther two groups are on the diagrams, the more different their stylistic profiles are as measured by how frequently they use the morphemes in List A. As the main question is the presence of smoothness, the aim is to check for relative proximity of consecutive groups. Figure 18 presents the PCA plot of the twenty-two groups as represented with normalized features from List A. his two-dimensional representation accounts for 64 % of the variance in the data. It gives a broad idea of the true shape of the data. Figure 19 presents graphs obtained using four different methods of MDS. 270 B. Sadeghi / Arabica 58 (2011) 210-299 PCA 6 5 11 16 4 10 3 7 2nd Principal Component 18 19 15 13 17 14 22 12 9 8 2 21 1 1st Principal Component Figure 18. PCA plot of Bazargan’s groups as represented by normalized features of List A. Consecutive groups are connected with dotted lines. he two principal components account for 64 % of variance in the data. he MDS method named “Sammon mapping” yields a “stress value” of 5.4 %. Stress is a measure of how well the distances in the two-dimensional representation match those in the original space of dimension twenty-eight. Stress values of 2.5 %, 5 %, and 10 % would be considered respectively excellent, good, and fair reproductions of the distances.56 hus, Sammon mapping gives a close approximation of the distances. he PCA and MDS plots obtained using standardized weights are shown in Figure 20. Table 10 in the appendix provides the actual distances among the twenty-two groups (normalized case). Table 12 in the appendix helps with the interpretation of the distance matrices. 56 B.S. Everitt and G. Dunn, Advanced Methods of Data Exploration and Modeling, Exeter, New Hampshire, Heinemann Educational Books, 1983, p. 65. 271 B. Sadeghi / Arabica 58 (2011) 210-299 MDS – Metric Stress MDS – Kruskal Stress 13 2221 20 15 19 1214 1817 16 798 10 1 2 1 2 11 6 3 1 13 21 15 22 12 14 19 17 8 18 20 9 10 16 2 3 4 11 5 3 45 MDS – Sammon Mapping 13 21 12 15 22 14 19 17 8 18 20 9 10 16 7 4 11 6 5 1 MDS – Classical 21 13 22 15 17 19 1214 20 18 2 89 16 7 10 11 7 6 3 4 5 6 Figure 19. MDS graphs of the twenty-two groups as represented by normalized features of List A. Four different methods of MDS are used. For Sammon mapping, the stress value is 5.4 %, indicating good reproduction of the original interpoint distances, while the others have higher (worse) stress values. One may now examine the graphs for smoothness. he question is: are groups that are near each other in Bazargan’s sequence also near each other in the graphs? his is another way of asking: do groups that have similar verse lengths use the most common morphemes with similar frequencies? In a broad way, that appears to be the case. One observes a progression in three regimes: (I) Groups 1-6, (II) Groups 7-11, (III) Groups 12-22. (Regime III is the Medinan period in Bazargan’s reckoning.) In Bazargan’s sequence, as well, these regimes appear in order. Within Regime I, one observes not only great smoothness, but also a clear progression. Within the other regimes, there is no clear progression, and only a limited tendency for consecutive groups to be near each other. hus, we have the following clusters of consecutive groups that lie near one another: {8, 9, 10}, {12, 13, 14, 15}, and {16,17, 18, 19}. In the normalized case, Groups 12 and 16 occupy intermediate positions between regimes II and III. In the case of Group 12, there is nothing odd about this. he location of Group 16, however, could be considered aberrant. Group 11 appears a bit out of place as well. 272 B. Sadeghi / Arabica 58 (2011) 210-299 MDS – Kruskal Stress MDS – Sammon Mapping 1 12 13 21 22 15 9 2 17 19 14 1 16 8 12 18 20 11 22 13 21 15 17 14 10 16 20 7 6 3 4 19 18 98 5 10 PCA 2nd Principal Component 5 6 711 10 9 16 8 20 4 3 2 11 6 2 1812 19 22 15 4 5 1 1st Principal Component 7 3 Figure 20. PCA and MDS, List A, Standardized. he first two components in PCA (inset, bottom) explain only 56 % of the variance. MDS yields stress values of 5.8 % (good) and 8.7 % (fair) respectively for Sammon and Kruskal. Figure 21 provides PCA and MDS graphs of the groups as represented by weight-optimized features. PCA accounts for 67 % of the variance in the data, while MDS yields a good reproduction of inter-point distances (stress=4.7 %). Table 11 in the appendix provides the pairwise distances among the twentytwo groups, and Table 12 in the appendix helps with the interpretation of the distance data. he first two groups, not shown in the plot, are far away from where one would expect them to appear. his is due to the inadequacy of optimized weights when it comes to the first two groups. his inaccuracy results from the fact that the training set lacks any texts from the first two groups. Now, if the first two groups had been stylistically close to the others, this absence might not have been detrimental. But as the normalized and standardized cases demonstrate, they stand far apart from the rest. his separation means that the 273 B. Sadeghi / Arabica 58 (2011) 210-299 PCA MDS — Sammon Mapping 17 7 2nd Principal Component 6 15 10 13 3 20 16 5 18 4 19 14 22 21 13 17 16 9 4 9 22 12 11 8 19 14 18 8 10 11 3 21 20 5 15 7 6 1st Principal Component Figure 21. PCA and MDS (Sammon Mapping) representations of Groups 3-22 with optimized weights from List A. he PCA plot explains only 67 % of the variance, while MDS provides a good representation (stress=4.6 %). optimization process will miss the importance of the morphemes that are distinctive of the first two groups, giving such features weights that are zero or close to zero. So, the first two groups end up being judged on the basis of morphemes that are suited specifically to the other groups.57 To put all this in less technical terms, for accurate results the training texts must be representative; but in fact they are not representative of the first two groups. Optimized weights yield a somewhat different picture. he three regimes remain valid. However, the boundary of the first two regimes is now blurred as Group 7 joins Group 6. More interestingly, one now observes hints of smoothness internally in what I called above Regimes II and III. Certain consecutive groups happen to be neighbors: {9, 10, 11}, {12, 13, 14}, {15, 16, 17}, {18, 19, 20}, and {21, 22}. his does not mean that there is either strict smoothness or a steady progression within regime III: note the jump from Group 14 to 15 and especially that from 17 to 18. Unlike in the normalized case, Group 16 does not occupy an aberrant position. he upshot is this: Consider the following sequence of eight phases: {1} {2} {3} {4} {5} {6} {7-11} {12-22}. Style, as represented by the frequencies of morphemes in List A, varies in a smooth fashion over this sequence of phases. However, it does not vary smoothly within these phases. What this means for 57 he groups 21-2 are not represented in the training set. herefore, the results in their cases cannot be fully trusted either. However, they are stylistically close to groups that are well represented in the training set. herefore, the results can be expected to be not too far off. 274 B. Sadeghi / Arabica 58 (2011) 210-299 concurrent smoothness, and hence chronology, will be discussed after two other multivariate markers of style are considered. 7. Multivariate Analysis (List B): 114 Frequent Morphemes his section is based on the 114 features that make up List B, which is provided above in Table 6 on page 253. Since this list does not overlap with List A, it can be used for an independent assessment of style. Admittedly, there may be a few features in List B that correlate strongly with some from List A for linguistic reasons, but the effects of such correlations on the final results are negligible, and the two lists can be considered practically independent. Indeed, the overall results in this and the last section are not sensitive to the inclusion or deletion of a few features. hus if one were to eliminate the most highly correlated features, there would be no important change in the overall results. Each group is now represented as a vector of 114 relative frequency counts, namely those of the features in List B. Just as was done above for List A, the aim is to check for smoothness over Bazargan’s sequence of twenty-two groups. I have included PCA and MDS graphs for normalized, standardized, and weight-optimized features. Let us examine the graphs. In the standardized (Figure 22, Figure 23) and normalized cases (Figure 24, Figure 25), the three regimes from the last section reappear more or less. he key change is that now the last regime (Groups 12-22) is divided into two parts: {12-19}, {20-22}. hese two clusters line up to create a progression. hus, the groups that for Bazargan start and end the revelation form precisely the extremities of the MDS and PCA representations, with the clusters between them lining up broadly in the expected way. Group 16 occupies an idiosyncratic position that is already familiar from List A. In most graphs, Group 12 occupies an intermediate position between two clusters. In the graphs of the weight-optimized case (Figure 26), Groups 1 and 2 must be ignored as before. One of the Groups 3, 4, or 5 has an aberrant position, but it is difficult to tell which. hereafter, there is a great deal of smoothness, with consecutive groups near each other. Groups 16 and 11 occupy aberrant positions. Group 12 occupies an intermediate position. here is a surprisingly smooth progression within Groups 14-22. he upshot is this: Consider the following sequence of seven phases: {1} {2} {3} {4-5} {6-11} {12-19} {20-22}. Style, as represented by the frequencies of morphemes in List B, varies in a smooth fashion over this sequence of phases. However, it does not vary smoothly within these phases. 275 B. Sadeghi / Arabica 58 (2011) 210-299 PCA 1 22 21 20 2nd Principal Component 19 18 17 15 13 14 12 16 2 8 7 9 6 3 10 11 4 5 1st Principal Component Figure 22. PCA, Groups 1-22, standardized feature vectors, List B. his captures only 36 % of the variance in the data. MDS – Kruskal Stress MDS – Sammon Mapping 22 21 1 14 19 15 17 16 9 6 3 4 18 15 10 8 7 2 20 18 13 12 1 2122 20 11 14 17 19 13 5 12 16 MDS – Classical 1 21 22 18 20 1915 17 14 12 16 2 78 10 9 6 2 8 7 9 6 10 11 3 45 11 3 5 4 Figure 23. MDS graphs of Groups 1-22, as represented with standardized features from List B. hree methods are used, involving Sammon Mapping (inset, top), Classical MDS (inset, bottom), and Kruskal stress. he stress values are 7.6 % for Kruskal and 8.4 % for Sammon, indicating that the representations are fair. Note how consecutive groups tend to be located near each other. Furthermore, there is an overall progression, with Groups 1 and 22 appearing at the two extremes. Groups 11 and 16 appear at somewhat aberrant locations. 276 B. Sadeghi / Arabica 58 (2011) 210-299 PCA 2nd Principal Component 1 14 2 15 7 8 9 3 20 13 19 18 12 16 21 22 17 10 4 6 11 5 1st Principal Component Figure 24. PCA plot of Groups 1-22, as represented with normalized features from List B. he two principal components account for only 36 % of the variance in the data. MDS – Kruskal Stress 1 MDS – Sammon Mapping 20 20 21 22 21 1 18 13 17 19 14 15 7 2 9 17 19 10 8 3 18 13 15 16 12 22 11 14 16 12 6 4 7 8 5 9 10 3 6 2 11 4 5 Figure 25. MDS graphs of Groups 1-22, as represented with normalized features from List B. Two methods are used, involving Sammon mapping (inset) and Kruskal stress. he stress values are respectively 8 % and 10 %, indicating that the representations are fair. Other MDS methods lead to similar images. 277 B. Sadeghi / Arabica 58 (2011) 210-299 MDS – Kruskal Stress MDS – Sammon Mapping 13 18 21 19 15 9 22 3 18 4 2 17 16 11 5 10 PCA 21 13 1520 19 2214 2 18 17 168 3 10 9 6 7 4 12 11 2nd Principal Component 15 14 19 20 7 6 12 1 10 16 8 1 13 14 17 22 20 21 8 9 12 6 7 3 11 5 4 5 2 1 1st Principal Component Figure 26. PCA and MDS, Groups 1-22, List B, Optimized weights. he PCA plot (inset, bottom) explains only 56 % of the variance. Sammon mapping (inset top, left) performs fairly well (stress=6.2 %), and MDS-Kruskal performs fairly (stress=12.4 %). Points representing Groups 1-2 must be ignored. 8. Multivariate Analysis (List C): he Generalized Smoking Gun Technique he multivariate methods employed thus far rely on highly frequent features. Traditional approaches of stylistic analysis, too, may rely on such features, but they do so only in the most extreme cases, e.g. where a feature appears in a text with such conspicuous regularity that counting is not necessary. As an example, one may mention the observation of the pre-modern scholars that Meccan verses in the Qur ān have significantly shorter verse lengths.58 Stylometry extends this approach, by use of counting, to features whose behavior changes from text to text in a less drastic and noticeable fashion. 58 For another example, see Behnam Sadeghi, “he Authenticity of Two 2nd/8th-Century Ḥ anafī Legal Texts”, cited above in footnote 18. 278 B. Sadeghi / Arabica 58 (2011) 210-299 However, more typically, traditional stylistic analysis does not involve the most frequent features. Traditionally, it is more common to rely on the least frequent features. hat is, if two texts display the same idiosyncrasies, odd spellings, peculiar habits of punctuation, rare words, and unusual phrases, then they are considered to be close in style. his is known as the smoking gun technique. he reason behind its popularity is precisely that it does not involve much computation; a specialist can spot with relative ease, say, any unusual phrases. he main limitation of this method is that many passages may lack a sufficient number of smoking guns. he smoking gun technique may be transformed using counting and multivariate analysis into a potentially powerful method that one may call Generalized Smoking Gun. he generalization involves going beyond rarities, which is the extent of the traditional practice, to include all relatively low-frequency words, say, all words that occur fewer than twenty times in the corpus. his may appear problematic at first. After all, precisely because an infrequent word is infrequent, the variation of its frequency count in two different texts can largely be explained as due to chance. he point, however, is that when one considers several thousand infrequent features simultaneously, the overall patterns may well be highly significant. As the number of features increases, the random oscillations of individual features will tend to cancel each other out, bringing any genuine patterns into bold relief. An advantage of this approach is that it may be applied to passages that are lacking in smoking guns. Generalized Smoking Gun, however, does have a drawback. It reflects not only style, but also subject matter. In what proportion each factor is represented is unknown to me, requiring tedious analysis to verify, although my sense is that style is no less a factor than subject matter. he same concept may be spoken about using a variety words, and this method captures patterns of word usage by linking passages that tend to use similar vocabulary. Because subject matter also plays a role, one may choose to treat this method as experimental, ignoring it where it contradicts the approaches of the last two sections. But where it agrees with them, there is independent corroboration, as it is highly unlikely for the agreement to be coincidental. My analysis is based on the list, called List C, of all morphemes that occur in the Qur ān more than once and fewer than twenty times. here are 3693 such features, and their distribution is shown in Figure 27. he lion’s share belongs to morphemes that occur exactly twice in the entire Qur ān. here is no overlap between this feature list and Lists A and B. So, this feature list offers a genuinely independent marker of style. Each group is represented with a vector of 3693 relative frequency counts—one number for each morpheme. As usual, the task is to investigate the similarities and distances among these vectors. B. Sadeghi / Arabica 58 (2011) 210-299 279 1600 Number of distinct morphemes 1400 1200 1000 800 600 400 200 0 0 5 10 15 Number of occurrences in the Qurʾān 20 Figure 27. Graph of the number of distinct morphemes that occur a given number of times in the Qur ān. hus, 1373 morphemes occur twice, 648 occur three times, and so on. Not included are the morphemes occurring fewer than twice or more than nineteen times. Figure 28 depicts PCA. he first two principal components explain only fourteen percent of the variance. Normally, this would be considered extremely poor. However, it is different in this case. he dataset is an extremely “noisy” one, being based on thousands of features that, having low frequencies, are highly susceptible to random oscillations in their frequency counts from group to group. In PCA, the noise is accounted for by the principal components that come after the first several ones. hus much of the variance that the first two components fail to explain is attributable to the large amount of noise that needs to be filtered out anyway. So, unexplained variance is no guide to the quality of the results; but one may fall back on the above-mentioned principle which does provide a measure of objectivity: where the results agree with those from Lists A and B, this has to be significant. Turning to the PCA graph in Figure 28, one finds impressive agreement with previous results. hat is also true of the MDS graphs depicted in Figure 29, which respectively do a “poor” and “fair” job in approximating the distances among the groups. One can see the three regimes familiar from before in both PCA and MDS. Interestingly, the final three groups, 20-22, cluster together at the end of the trajectory as was the case in List B. Group 16 occupies an aberrant position as seen in some previous cases. he main difference with previous cases is that the “leg” corresponding to Regime I (Groups 1-6) 280 B. Sadeghi / Arabica 58 (2011) 210-299 PCA 2 3 2nd Principal Component 21 20 4 22 1 15 18 13 17 19 14 5 6 89 16 10 12 7 11 1st Principal Component Figure 28. PCA for Groups 1-22, as represented with 3693 infrequent morphemes (those occurring fewer than twenty times in the Qur ān). he first two principal components explain only 14 % of the variance in the data. he feature counts have been standardized. he normalized case looks more or less the same. has now folded upon the rest of the trajectory, with Group 5 in the position of the hinge. Group 1 occupies a central position, and it appears closer to Groups 9 and 14 than Group 2. hat means that if Group 1 were joined with these others, we would have a smooth trajectory. But the 1-2-3 sequence, too, looks smooth. he difference is that the latter sequence agrees with the results from Lists A and B and is thus more in keeping with concurrent smoothness. I have experimented with different ways of weighting the features, for example by giving greater weight to less frequent features, though I have not shown the results here. he results remain about the same, with the main difference lying in the degree to which the “Regime-I leg” folds upon the rest of the trajectory. 281 B. Sadeghi / Arabica 58 (2011) 210-299 MDS – Kruskal Stress 11 7 12 6 16 10 5 19 15 14 8 18 9 1 2 MDS – Sammon Mapping 7 4 11 12 18 22 20 17 1 8 21 15 14 5 3 19 16 10 6 22 17 13 13 9 20 2 4 3 21 Figure 29. MDS for Groups 1-22, as represented with 3693 infrequent morphemes (those occurring between two to nineteen times in the Qur ān). he stress values for Sammon (inset) and Kruskal are respectively 12 % (fair to poor) and 20 % (poor). he feature counts have been standardized. he normalized case looks more or less the same. 9. Conclusion Results he correlation of the style of the Qur ān with chronology was recognized in the pre-modern period. Scholars noted that Meccan sūras have shorter verse lengths and employ somewhat different vocabulary and phraseology.59 Modern scholars have sought to extend this insight beyond the classical binary 59 See above, footnote 5; cf. Abū l-Faḍl Mīr Muḥammadī, Buḥ ūt̠ fī tārīḫ al-Qur ān wa- ulūmihi, Beirut, Dār al-Ta āruf, 1980, p. 327-36, where he questions the utility of style as evidence for chronology. 282 B. Sadeghi / Arabica 58 (2011) 210-299 Meccan-Medinan division. he Weil-Nöldeke periodization, which was intended as broad and approximate, was based not only on historical and semantic information, but also on style, which was assumed to change monotonically. Despite its limitations, Bazargan’s work remains arguably the most impressive modern attempt at proposing a chronology. Although his criterion of ordering is purely style-based, his corroboration of the resulting sequence came from historical reports and considerations of meaning. My research has provided corroboration for a modified version of his proposed sequence, this time on purely stylistic grounds. he stylometric results of this essay, however, do not merely confirm what we knew already from historical traditions. hese traditions, which are not without uncertainties of their own, pin down the dates of a limited number of passages.60 By contrast, stylometric methods apply to all texts of a certain size, albeit in a statistical sense. hey thus provide a broader confirmation than other strands of evidence possibly could. he following sequence of seven clusters or phases has been corroborated as chronological: {Group 2}, { Group 3}, { Group 4}, { Group 5}, { Groups 6-11}, { Groups 12-19}, { Groups 20-22}. his is the Modified Bazargan chronology (Figure 30). If one reduces Bazargan’s twenty-two groups in this way, then one observes a phenomenon that cannot be due to chance: several different, independent markers of style vary over these phases in a smooth fashion, adjacent clusters having relatively similar stylistic profiles. Such smoothness is observed with four markers of style: mean verse length and three powerful, independent Bazargan Modified Bazargan 1 234 5 12 34 6-11 12-19 20-22 5 6 7 Figure 30. he Modified Bazargan Chronology. 60 Bāzargān, Sayr, vol. I, p. 127-34 / 149-55 / 160-6; Robinson, Discovering the Qur ān, p. 37-44. In addition, the premodern chronological sequences (which can be traced back to a list ascribed to Ibn Abbās) probably constitute early, informed scholarly conclusions and are deserving of serious consideration, but are not necessarily infallible. Moreover, they do not divide sūras into blocks. See e.g. Bāzargān, Sayr, vol. II, p. 192-203 / 557-69; Robinson, Discovering the Qur ān, p. 284. B. Sadeghi / Arabica 58 (2011) 210-299 283 multivariate markers. he only discernable explanation for the observed concurrent smoothness is chronological development. One who denies this conclusion has the burden of explaining the pattern in some other way. here are two exceptions to the consensus of the markers: the distinction between clusters {4} and {5} and that between clusters {12-19} and {20-22} are confirmed by only three of the four markers. he first pair of clusters are blurred together if one considers List B, and the second pair of clusters are lumped together in the case of List A. his makes the distinctions between these pairs of clusters not as well-corroborated as the rest of the sequence. Bazargan’s Group 1 is excluded and no corroboration is claimed for it for three reasons: (1) at 415 words it is not clear whether it is large enough for the chosen markers of style to characterize it meaningfully; (2) its initial position makes it difficult to assess smoothness—and (3) for an important reason to be mentioned later below. Further work is needed on the blocks in Group 1 to determine whether they should be kept together or joined with other blocks/ groups. Univariate markers other than verse length are less powerful, but they nonetheless confirm the order of the initial clusters, as well as the fact that they should appear to the left side of the other clusters. Granting that the Modified Bazargan sequence reflects chronology, how does one know the direction of the arrow of time? Could it be that the Qur ān began at the right side of the chain with cluster {20-22} and progressed in reverse of the above sequence, ending at cluster {2}? Actually, stylistic evidence does not settle this issue at all. Instead, the question is answered by considerations of meaning. For example, Block 110 is an exegetical insertion in sūra 74, explaining a point in Block 27. As such, Block 110 (in Group 7) presupposes the existence of Block 27 (Group 2) and is therefore later.61 Such an argument does not take meaning into consideration except in the minimal manner of noting that a sentence obviously clarifies and presupposes another one. But if one were willing to go deeper into meaning, a variety of other indications could be marshalled to support the same direction. For example, passages referring to oppression as something of the past must be placed after passages responding to ongoing oppression. Now that we know head from tail, it is of interest to comment on the rate of stylistic change. If one accepts the broad outlines of the traditional reckoning of chronology and the division into Meccan and Medinan periods, and if 61 On this insertion, see Bāzargān, Sayr, vol. II, p. 150-2 / 543; Angelika Neuwirth, Studien zur Komposition der mekkanischen Suren, Berlin, Walter de Gruyter, 1981, p. 215; Tilman Nagel, Medinensische Einschübe in mekkanischen Suren, Göttingen,Vandenhoeck & Ruprecht, 1995, p. 89. For the definitions of the blocks, see above, Table 1 and Table 2. 284 B. Sadeghi / Arabica 58 (2011) 210-299 one makes the heuristic assumption that the text was disseminated at a roughly even rate, then from both univariate and multivariate markers, one discerns that style changed rapidly at the beginning. he pace of change slowed gradually. he initially more rapid pace grants style greater discriminatory power in the earlier phases. It is for this reason that even the relatively weaker univariate methods have no problem detecting the initial eruption. It is also for this reason that the Meccan period claimed three of the four phases of Weil’s chronology. After {Group 5}, however, univariate markers at best give a vague sense of continuity, failing to identify the patterns detected by multivariate methods. Incidentally, the greater discriminatory power of style for the Meccan period is fortunate, as in this period historical indications of chronology are sparse by comparison. Meccan texts contain fewer explicit references to “current events” than Medinan ones. It should be stressed that this chronology should not be treated in a rigid way. he fact that cluster {6-11} comes after {4} does not mean that every passage in {4} is in fact older than everything in {6-11}. he sequence is valid in an average sense only. Deviations from averages, as well as outlier behavior, are typical for phenomena complex enough to merit statistical analysis. One can expect that to be the case here as well. Of course, a common goal in statistics is estimating the likelihood of deviations of a certain size from the average. Establishing such confidence intervals and increasing precision are long-term goals of Qur ānic stylometry. What about the internal chronologies of the three clusters with more than one group: {6-11}, {12-19}, and {20-22}? Because I have failed to establish concurrent smoothness internally within these clusters, their internal chronologies are indeterminate. Is Bazargan right that Group 19 came after 18, which came after 17, and so on through Group 12? he possibilities are threefold: (i) He is right, but in this period style did not in reality change over time concurrently smoothly, or (ii) it did, and he is right, but the methods employed in this essay are not sensitive enough to reveal it consistently, or (iii) Bazargan’s internal chronology of these clusters is not correct. here are some indications for iii, such as the fact that independent multivariate markers of style consistently assign Group 16 to an earlier time than Bazargan does, thus indicating that here a sequence different from that of Bazargan is likely to satisfy concurrent smoothness. (Perhaps not coincidentally, the largest texts in Group 16 are Blocks 155 and 157, both of which are traditionally considered as Meccan.) A combination of these three possibilities could hold as well. More research is needed for better answers to these questions. If the reader is disappointed with the limitations thus far, it might cheer him or her to know that, as stylistic studies of chronology go, the Qur ān B. Sadeghi / Arabica 58 (2011) 210-299 285 F o u r t h D i m e n s i o n First Dimension Texts published in 1830s Texts published in 1840s Texts published in 1850s Texts published in 1860s Figure 31. Tabata’s application of a method called ANACOR, which is analogous to PCA, to twenty-three works of Charles Dickens. hree consecutive chronological clusters emerge along the first and the fourth dimensions (analogous to the first and fourth principal components in PCA), arranged vertically in the diagram. Tabata concludes, “with a sustained change in style from the 1830s through the 1840s, Dickens seems to have established his style by about 1850”. (Tabata, “Investigating stylistic variation in Dickens”, p. 178.) yields impressive results. For example, after over a century of stylometric studies, Plato’s dialogues have not offered conclusions of similar force, clarity, and precision. As another example, it is worthwhile to reflect on Tomoji Tabata’s elegant study of Charles Dickens.62 Figure 31 depicts twenty-three works of Dickens spanning four decades using multivariate analysis of thirty-four word-class 62 Tabata, “Investigating Stylistic Variation in Dickens”, cited in footnote 18. 286 B. Sadeghi / Arabica 58 (2011) 210-299 variables (e.g. singular nouns, possessive nouns, etc.). One observes no more than three consecutive clusters. he first two clusters correspond respectively to the first and second decades of Dickens’ carrier while the third cluster covers the remaining two decades. he level of discrimination achieved in the study of the Qur ān is all the more remarkable considering several serious actual and potential sources of error: First, the division of sūras into blocks will always include an element of conjecture, making it likely that there are errors. Such errors cause the attrition of smoothness, thus reducing the number of phases that can be obtained that display concurrent smoothness. Second, the blocks are very small. Ninety blocks contain fewer than 300 words apiece! To obtain larger samples, one must combine many blocks into groups. his may be done according to different criteria, such as verse length profiles as in Bazargan’s work. Regardless of the method used to combine blocks, however, one can expect that some blocks will be assigned together that are not in reality from the same period. his again will diminish smoothness and reduce the number of phases that can exhibit concurrent smoothness. hird, the Qur ān as a whole constitutes a relatively small corpus. he twenty-two groups add up to approximately 78,000 word-tokens. By comparison, the above-mentioned Dickensian texts contain about 4.7 million word-tokens. All but three of the individual works in that corpus are larger than the entire Qur ān. A smaller text typically means lesser statistical significance and smaller chances for devising experimental controls. Fourth, while the Qur ān displays continuous generic evolution, there may also be room for speaking of discrete genres in the Qur ān, specifically with regard to legal vs. non-legal material. Studies have shown that discrete genres can affect style heavily. For better results, the variables of time and discrete genre must be studied together. his omission, by the way, does not invalidate the degree of discrimination that has been achieved. he observed patterns cannot be explained in terms of differences in the proportions of discrete styles (genres) used.63 A plausible 63 Against my conclusions, one might argue that the stylistic variation over the seven clusters merely reflects that each cluster has material of different discrete stylistic profiles (e.g. genres) in differing proportions. For example, {2} would have no legal content, {3} would have a little, {4} would have a little more, and so on. Although the gradual decline in the amount of eschatological material in the Meccan phase (shown in the graphs in Bāzargān, Sayr, vol. I, p. 177-8 / 201-2 / missing) may have contributed somewhat to such an effect, on the whole what one observes is genuinely gradual change in style rather than the mixing of discrete styles in gradually changing proportions; thus eschatological passages experience stylistic evolution as well. Gener- B. Sadeghi / Arabica 58 (2011) 210-299 287 exception to this would be in the case of Groups 21 and 22, which have a much higher proportion of legal material than other groups (up to a third). One must investigate (i ) whether legal material indeed comprises a very distinct stylistic profile, and (ii ) if yes, how much this has contributed to the clustering of Groups 20-22. A similar case might be made for Group 1, which largely consists of what may be called “introductory sections”, i.e. the beginning portions of sūras. It is possible that Meccan introductory sections are characterized by a distinctive stylistic register. his is the third reason for my having excluded Group 1. Finally, I need to address a potential problem regarding the possibility of discontinuity in style. Could it be that at some point in time Qur ānic revelation breaks with its current style, reverts to a much earlier style for a while, and then leaps forward to resume where it left off ? If so, what would that imply for stylistic analysis? Note that this scenario does not merely envisage stylistic discontinuity. In general, discontinuity per se would not undermine my approach: it would simply mean that one cannot observe concurrent smoothness, leading one to conclude nothing (as opposed to concluding something that is wrong). Rather, this scenario poses a particular form of discontinuity, one involving a complete reversion to a previous style, comprising all markers (vocabulary, grammatical structures, verse length, word length, etc.). If this sort of thing did happen, then the outcome would be two passages from different periods with exactly the same style. Not able to differentiate between them, stylistic analysis could incorrectly assign them to the same time. Crucially, this would not entail a perturbation of concurrent smoothness. Research into whether this happened is needed. Even after such research, the scenario probably cannot be eliminated completely and could still apply to some passages. To the extent that this possibility remains open, I accept it as a potential source of error in my results. Nevertheless, I do not think that this scenario is probable, or that it could have happened on a large scale. It is odd to think that the style could leap back to that of a specific bygone year in all its particulars rather than just in some features. Furthermore, the scenario would have been more plausible if stylistic evolution in the Qur ān were characterized by the employment of discrete styles (e.g. corresponding to discrete genres). hen it would not be odd if the Qur ān switched to a formerly more common discrete style (corresponding to a specific genre) and then back. In reality, however, style tends to form a continuum.64 Moreover, it does so even within a genre. Specific genres (such as eschatological and legal) evince evolution in style as well, although more research should be done on this point. ally, the Qur ān’s style comprises a continuum rather than sharp, discrete categories. 64 See the last footnote. 288 B. Sadeghi / Arabica 58 (2011) 210-299 Additionally, tests performed in Section 5 support the centrality of time as a variable. Implications Literary sources and manuscript evidence indicate that the Prophet Muḥammad disseminated the contents of the Qur ān, and that the Caliph Ut ̠mān dispatched master copies of the scripture to several cities.65 As a thought experiment, however, let us unlearn what we know, imagining that we had come across the Qur ān not knowing where it came from or who disseminated it. In fact, let us even overlook the semantic contents of the text. What could one conclude about the Qur ān’s composition just from the formal-stylistic patterns observed? One would conclude that style backs the hypothesis of one author. For the sake of argument, suppose there were two authors: let’s say A wrote Groups 1-11, and B wrote 12-22. hen one would have to say that the style of A moved along a trajectory towards that of B, or that B picked up where A stopped, with respect to not only verse length, but also frequencies of the most common morphemes and frequencies of uncommon words. It is much easier to imagine a single author. Furthermore, if one assumed three, four, or more authors instead of two, the improbability would increase exponentially. Imagine three authors: author A being responsible for Groups {1-5}, B for {6-11}, and C for {12-22}. Again, one would have to explain why A’s style gravitated over time towards that of B, not only in terms of verse length, but also in terms of vocabulary usage for high-, medium-, and low-frequency morphemes. Moreover, one would need to explain why the style of B forms an intermediate stage between those of A and C in terms of the various markers of style that have been considered. If one imagined seven authors, one would have to explain why each person’s style is between two others not only with respect to verse length, but also with respect to frequent morphemes, medium-frequency morphemes, and unusual words. he study reveals the stylistic continuity and distinctiveness of the text as a whole. As far as this point is concerned, the present study makes palpable what we knew already: no competent and seasoned scholar of the Qur ān, while aware of the stylistic variation in the text, could lose sight of its underlying unity. Also established now is the general integrity of many passages of long and medium size. hat goes against Richard Bell’s instincts. His thoughtprovoking vision, while not implausible historically, now appears to have been 65 For a discussion of some of the evidence, see Behnam Sadeghi, “he Codex of a Companion of the Prophet and the Qur ān of the Prophet”, cited above in footnote 4. B. Sadeghi / Arabica 58 (2011) 210-299 289 misguided. Not only do eighteen test cases examined display a surprising degree of stylistic unity (Section 5), but also on his viewpoint one would be hard-pressed to explain the degree of concurrent smoothness observed.66 Moreover, one would expect revisions of earlier passages to dilute their stylistic distinctiveness. In fact, while it cannot be denied that the Prophet revised some passages over time, the present study shows that such revisions could not have been extensive. here are also implications for the Sīra literature. It has always been evident that there is a fit between the Prophet’s biography in the Sīra and the Qur ān’s style and contents. At the broadest level, the hiǧra divides the Prophet’s career into two different periods, a nice fit with the fact that stylistically these periods are distinct. More particularly, there are apparent links between the major events of the Prophet’s career and specific passages. he connections are noteworthy enough for the Sīra to offer a “plausible chronological framework for the revelations”.67 hese connections have normally been exploited to shed light on the chronology of the Qur ān. For example, Bazargan uses them to test and calibrate his chronology. hus, the flow of information has been more often from the Sīra to the Qur ān, although one could also have argued for mutual corroboration between the Sīra and the Qur ān. Now, however, that one is able in some measure to evaluate chronologies of the Qur ān without resort to historical reports, one can reverse the direction of information flow: to the extent that a style-supported chronology fits the Sīra’s outline, there is independent substantiation of the Sīra. Directions for Future Research he main way in which Bazargan’s religious faith affected his scholarship was to make it more critical and rigorous. Conscious of the gravity of mischaracterizing the “Words of God”, he reasoned cautiously, emphasized the fallible nature of his work, describing it as a research program “in its infancy”, and invited other scholars to critique, correct, and refine his findings.68 In a similar spirit, I have done my best to highlight the known limitations and potential shortcomings of my approach. However, at least one thing is amply clear by now: the utility of stylistic analysis as an effective means of determining the 66 Using methods different from mine, Angelika Neuwirth and Neal Robinson, too, reach conclusions that are incompatible with Bell’s approach (Neuwirth, Studien; Robinson, Discovering the Qur ān, p. 94-5, 162, 177-84, 187, 191). Although Bell is almost always wrong, his observations on particular points are sometimes worth thinking about. 67 Robinson, Discovering the Qur ān, p. 37. 68 Bāzargān, Sayr Mutammim, p. 409; Sayr, vol. II, p. v / 195-7. 290 B. Sadeghi / Arabica 58 (2011) 210-299 chronology of the Qur ān cannot be denied. Scholars would ignore style at their peril. Much more remains to be done on chronology in several different areas of research: the information in ḥ adīt̠s and the Sīra, traditional stylistic analysis, and stylometry. As far as stylometry is concerned, what has been done here scratches the surface of what can be attempted. It may be possible to develop yet more effective markers of style, particularly by combining the frequency of syntactical and morphological features with the different types of features used so far. Furthermore, one may refine the method of weight optimization by using more training texts and performing separate optimizations for different phases (e.g. Meccan and Medinan). Studying the variable of genre and its interaction with the variable of time remains a desideratum. Most significantly, constructing a better chronology is within reach. As a discriminator of chronology, verse length loses much of its efficacy in the Medinan period.69 One must therefore also use other markers of style to divide and reorder the text. he new sequences obtained should be examined for concurrent smoothness. he ultimate goal is to maximize the number of phases that display concurrent smoothness. Research in that area is in progress. 69 See the analysis in “Univariate Marker of Style: Mean Verse Length”, above in Section 3, especially Table 4, Figure 5, and Figure 6. 11. Appendix List A Table 10. Distance matrix for Groups 1-22, Feature List A, normalized: table of pairwise distances among the groups. hese are city-block distances, rescaled so that the maximum distance is 1,000. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 649 759 780 865 966 927 831 853 956 897 872 912 916 933 1000 943 976 954 982 877 916 649 0 488 477 529 477 433 436 494 596 489 573 616 548 663 686 677 574 602 559 618 599 759 488 0 301 370 484 498 469 471 456 453 596 662 585 584 560 600 626 639 666 631 647 780 477 301 0 231 360 430 369 365 381 386 548 564 516 552 475 579 534 536 560 592 629 865 529 370 231 0 324 443 442 369 346 412 501 591 500 531 452 523 556 529 545 600 609 966 477 484 360 324 0 282 330 369 309 287 456 528 458 504 389 473 369 429 417 554 514 927 433 498 430 443 282 0 265 301 331 332 364 455 366 481 376 440 355 395 363 445 387 831 436 469 369 442 330 265 0 298 273 375 360 391 289 370 307 337 315 280 360 409 415 853 494 471 365 369 369 301 298 0 255 286 252 381 294 363 323 335 365 392 391 375 326 956 596 456 381 346 309 331 273 255 0 325 286 367 318 298 223 326 287 314 407 406 394 897 489 453 386 412 287 332 375 286 325 0 388 526 473 470 423 494 447 480 497 510 418 872 573 596 548 501 456 364 360 252 286 388 0 348 285 282 292 328 345 337 406 370 286 912 616 662 564 591 528 455 391 381 367 526 348 0 253 243 341 354 344 247 301 223 304 916 548 585 516 500 458 366 289 294 318 473 285 253 0 239 327 281 266 217 322 297 314 933 663 584 552 531 504 481 370 363 298 470 282 243 239 0 292 297 259 242 351 278 299 1000 686 560 475 452 389 376 307 323 223 423 292 341 327 292 0 303 311 265 401 356 378 943 677 600 579 523 473 440 337 335 326 494 328 354 281 297 303 0 285 290 332 259 278 976 574 626 534 556 369 355 315 365 287 447 345 344 266 259 311 285 0 231 264 298 317 954 602 639 536 529 429 395 280 392 314 480 337 247 217 242 265 290 231 0 286 262 290 982 559 666 560 545 417 363 360 391 407 497 406 301 322 351 401 332 264 286 0 240 292 877 618 631 592 600 554 445 409 375 406 510 370 223 297 278 356 259 298 262 240 0 196 916 599 647 629 609 514 387 415 326 394 418 286 304 314 299 378 278 317 290 292 196 0 291 2 B. Sadeghi / Arabica 58 (2011) 210-299 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 406 540 784 712 642 580 549 522 665 680 637 629 641 707 1000 823 963 826 820 406 0 409 547 621 440 394 601 381 615 611 463 585 530 709 758 625 718 716 718 540 409 0 534 547 497 408 384 478 526 655 495 596 535 597 788 618 730 742 709 784 547 534 0 312 444 461 541 436 524 631 535 600 479 607 440 495 427 744 607 712 621 547 312 0 419 427 607 493 419 593 480 753 524 613 629 577 417 678 554 642 440 497 444 419 0 249 413 452 273 451 275 419 332 409 497 312 448 533 471 580 394 408 461 427 249 0 388 340 221 369 245 411 288 361 624 347 555 450 383 549 601 384 541 607 413 388 0 401 335 563 550 390 359 385 602 484 712 641 543 522 381 478 436 493 452 340 401 0 424 499 457 461 284 581 623 515 646 587 494 665 615 526 524 419 273 221 335 424 0 346 299 362 252 286 570 310 532 403 283 680 611 655 631 593 451 369 563 499 346 0 270 306 407 423 605 323 419 214 246 637 463 495 535 480 275 245 550 457 299 270 0 356 357 416 533 234 419 388 334 629 585 596 600 753 419 411 390 461 362 306 356 0 296 368 453 249 502 382 319 641 530 535 479 524 332 288 359 284 252 407 357 296 0 347 482 312 495 414 349 707 709 597 607 613 409 361 385 581 286 423 416 368 347 0 498 320 581 425 422 1000 758 788 440 629 497 624 602 623 570 605 533 453 482 498 0 345 356 509 454 823 625 618 495 577 312 347 484 515 310 323 234 249 312 320 345 0 349 344 262 963 718 730 427 417 448 555 712 646 532 419 419 502 495 581 356 349 0 370 339 826 716 742 744 678 533 450 641 587 403 214 388 382 414 425 509 344 370 0 191 820 718 709 607 554 471 383 543 494 283 246 334 319 349 422 454 262 339 191 0 B. Sadeghi / Arabica 58 (2011) 210-299 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 3 292 Table 11. Distance matrix for Groups 3-22, Feature List A, optimized weights. hese are city-block distances, rescaled so that the maximum distance is 1,000. 293 B. Sadeghi / Arabica 58 (2011) 210-299 Table 12. Nearest Neighbors, List A. hese tables were devised using the distance matrices and offer a convenient way of interpreting them. he groups are listed consecutively in the leftmost columns. he second column gives the nearest groups to the leftmost ones, the third column gives the second nearest groups to the leftmost ones, and so on. To gain an idea of the neighborhood of a group, two steps must be taken. First, find the group in the leftmost column and look up its nearest neighbors. Second, find the group in the other columns and look up the two leftmost groups. Normalized features 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 7 4 5 4 7 8 7 12 16 9 9 21 19 14 10 21 19 14 21 22 21 3 8 5 3 6 11 6 10 10 9 6 15 15 15 19 19 22 15 18 18 13 17 4 4 11 6 10 10 9 19 11 8 10 14 19 13 13 12 14 20 15 19 20 12 8 6 10 9 9 5 10 14 14 12 7 10 14 18 18 15 18 14 13 22 17 19 Standardized features 9 3 8 8 3 8 11 9 8 18 8 22 20 17 21 17 19 17 21 13 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 8 4 5 4 10 8 10 10 8 9 15 15 15 19 19 18 19 18 21 22 21 3 7 5 3 10 7 9 19 12 16 7 9 21 19 13 10 19 15 15 18 13 19 4 3 11 9 9 11 11 7 11 9 6 16 19 8 14 15 21 17 16 19 20 13 12 11 10 8 3 8 6 14 14 18 10 14 14 13 12 8 22 20 14 13 19 20 Weight-optimized features 21 6 9 6 6 5 10 16 7 6 8 10 22 9 18 12 15 10 21 22 15 17 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 4 11 10 7 6 9 12 12 16 9 21 19 19 12 12 19 14 22 22 21 11 9 9 20 20 12 14 16 9 16 22 9 16 11 19 20 15 19 13 13 5 3 4 11 8 14 8 5 4 8 14 13 13 9 16 6 22 18 19 19 10 5 11 18 12 19 16 17 10 22 15 8 22 15 9 15 12 21 20 12 9 8 14 8 9 16 11 9 12 17 19 12 14 19 15 22 8 7 15 15 294 List B Table 13. Distance Matrix, Feature List B, standardized: table of pairwise distances among the groups. hese are city-block distances, rescaled so that the maximum distance is 1,000. 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 858 776 607 613 603 534 487 530 573 542 599 498 0 502 455 408 533 446 434 547 529 605 846 696 579 598 623 537 493 475 523 526 577 452 502 0 435 408 522 438 491 601 552 510 880 732 614 588 633 533 539 497 529 540 639 439 455 435 0 492 490 468 451 537 529 529 842 708 582 557 529 483 423 456 501 433 550 459 408 408 492 0 498 446 433 519 534 531 885 741 622 640 616 587 564 555 611 583 615 576 533 522 490 498 0 562 527 584 522 560 820 779 631 644 621 515 507 497 522 531 616 459 446 438 468 446 562 0 408 486 544 466 780 724 640 581 614 533 468 479 542 509 592 475 434 491 451 433 527 408 0 513 529 525 922 831 681 643 689 621 573 606 638 613 681 548 547 601 537 519 584 486 513 0 493 507 918 761 616 708 686 630 598 595 637 664 609 531 529 552 529 534 522 544 529 493 0 467 933 820 716 674 669 647 573 568 613 623 666 540 605 510 529 531 560 466 525 507 467 0 B. Sadeghi / Arabica 58 (2011) 210-299 1 0 689 807 873 841 859 845 856 971 937 1000 804 2 689 0 562 690 741 667 651 703 750 749 838 646 3 807 562 0 580 520 597 520 567 598 568 646 560 4 873 690 580 0 473 519 519 485 511 494 533 493 5 841 741 520 473 0 450 474 474 517 506 581 516 6 859 667 597 519 450 0 461 457 492 504 533 476 7 845 651 520 519 474 461 0 424 474 465 528 435 8 856 703 567 485 474 457 424 0 457 441 523 442 9 971 750 598 511 517 492 474 457 0 474 576 461 10 937 749 568 494 506 504 465 441 474 0 535 460 11 1000 838 646 533 581 533 528 523 576 535 0 579 12 804 646 560 493 516 476 435 442 461 460 579 0 13 858 776 607 613 603 534 487 530 573 542 599 498 14 846 696 579 598 623 537 493 475 523 526 577 452 15 880 732 614 588 633 533 539 497 529 540 639 439 16 842 708 582 557 529 483 423 456 501 433 550 459 17 885 741 622 640 616 587 564 555 611 583 615 576 18 820 779 631 644 621 515 507 497 522 531 616 459 19 780 724 640 581 614 533 468 479 542 509 592 475 20 922 831 681 643 689 621 573 606 638 613 681 548 21 918 761 616 708 686 630 598 595 637 664 609 531 22 933 820 716 674 669 647 573 568 613 623 666 540 13 Table 14. Distance matrix for Groups 3-22, Feature List B, optimized weights. hese are city-block distances, rescaled so that the maximum distance is 1,000. 3 0 791 617 766 754 719 900 800 689 759 804 768 836 812 794 952 885 909 765 797 5 791 617 0 773 773 0 675 745 621 873 647 824 756 940 513 861 522 726 645 838 723 923 593 846 687 892 699 829 733 815 857 1000 668 893 790 958 856 950 694 839 6 7 8 9 10 11 12 13 14 15 16 766 675 745 0 620 676 534 779 599 531 530 694 511 687 572 567 576 616 737 650 754 621 873 620 0 477 651 638 594 482 533 532 694 583 807 618 482 619 767 541 719 647 824 676 477 0 706 617 529 576 605 549 774 607 857 760 665 761 706 652 900 756 940 534 651 706 0 744 751 586 693 675 735 608 801 750 694 797 824 685 800 513 861 779 638 617 744 0 640 714 784 624 779 575 857 837 629 819 973 749 689 522 726 599 594 529 751 640 0 629 645 554 697 670 671 762 631 780 753 605 759 645 838 531 482 576 586 714 629 0 406 513 520 505 651 596 459 563 575 574 804 723 923 530 533 605 693 784 645 406 0 546 513 570 666 597 533 644 652 702 768 593 846 694 532 549 675 624 554 513 546 0 770 459 731 622 515 678 675 572 836 687 892 511 694 774 735 779 697 520 513 770 0 813 559 781 620 718 781 707 812 699 829 687 583 607 608 575 670 505 570 459 813 0 785 624 495 649 706 613 17 18 19 20 21 22 794 952 733 857 815 1000 572 567 807 618 857 760 801 750 857 837 671 762 651 596 666 597 731 622 559 781 785 624 0 731 731 0 681 474 737 494 746 764 632 565 885 668 893 576 482 665 694 629 631 459 533 515 620 495 681 474 0 565 805 563 909 790 958 616 619 761 797 819 780 563 644 678 718 649 737 494 565 0 828 622 765 856 950 737 767 706 824 973 753 575 652 675 781 706 746 764 805 828 0 539 797 694 839 650 541 652 685 749 605 574 702 572 707 613 632 565 563 622 539 0 B. Sadeghi / Arabica 58 (2011) 210-299 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 4 295 296 B. Sadeghi / Arabica 58 (2011) 210-299 Table 15. Nearest Neighbors, List B. hese tables were drawn up using the distance matrices and offer a convenient way of interpreting them. Just follow the instructions in the caption for Table 12. Normalized features 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 19 3 7 8 4 8 8 7 12 8 8 16 16 16 12 14 16 19 18 21 22 21 14 7 8 12 7 7 16 12 8 16 7 7 19 12 14 13 21 13 16 18 20 18 12 14 5 5 6 5 12 9 7 12 10 19 18 15 19 19 19 12 13 19 17 16 18 12 12 7 8 12 19 10 10 7 4 14 7 8 13 7 14 16 12 16 19 19 Standardized features 2 4 14 9 12 9 9 16 16 9 6 9 12 18 8 12 22 14 7 22 18 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 3 5 5 6 5 16 7 8 16 8 7 16 16 14 13 15 19 18 18 22 18 19 12 7 8 4 8 8 10 12 8 7 15 19 15 12 14 16 14 16 21 20 21 12 7 12 12 7 7 12 12 7 12 4 8 18 18 19 7 14 13 13 22 17 20 3 6 2 10 8 12 6 16 10 7 6 14 15 12 13 10 21 16 15 19 13 14 18 1 8 9 10 16 10 6 6 9 10 16 7 8 18 19 19 12 7 16 15 19 Weight-optimized features 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 5 10 3 15 8 7 6 4 4 13 12 16 6 14 15 19 12 18 22 21 11 11 11 13 12 11 12 16 8 19 15 12 13 19 6 20 18 12 12 7 8 14 6 12 19 14 16 8 14 7 6 19 12 12 22 22 7 19 13 19 7 7 4 9 14 12 7 14 7 16 7 7 17 13 12 6 16 6 14 18 12 12 17 18 13 13 14 19 6 14 19 13 19 10 13 12 14 7 8 14 List C Table 16. Distance Matrix, Feature List C, standardized: table of pairwise distances among the groups. hese are city-block distances, rescaled so that the maximum distance is 1,000. 0 465 660 660 681 596 681 629 585 608 648 620 574 511 611 605 586 594 609 665 636 667 465 0 789 802 828 774 835 795 770 796 828 803 761 694 800 784 775 774 790 857 815 851 3 4 660 660 789 802 0 866 866 0 865 868 880 869 953 923 869 887 870 837 877 873 920 918 921 903 878 879 854 854 923 922 912 884 908 925 886 901 922 934 973 983 958 951 975 1000 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 681 828 865 868 0 828 884 864 848 856 910 895 890 843 919 874 919 880 918 988 962 991 596 774 880 869 828 0 835 813 804 785 806 811 821 744 819 795 816 810 824 865 864 892 681 835 953 923 884 835 0 849 846 810 875 881 845 805 885 832 886 869 865 944 931 919 629 795 869 887 864 813 849 0 777 784 853 829 792 761 831 805 817 795 830 893 875 911 585 770 870 837 848 804 846 777 0 735 831 783 771 707 835 762 793 782 820 870 846 875 608 796 877 873 856 785 810 784 735 0 800 819 785 730 801 768 799 776 814 863 864 888 648 828 920 918 910 806 875 853 831 800 0 842 838 792 869 833 860 833 862 927 910 921 620 803 921 903 895 811 881 829 783 819 842 0 763 753 814 797 818 805 797 878 867 875 574 761 878 879 890 821 845 792 771 785 838 763 0 701 742 752 765 757 758 830 795 831 511 694 854 854 843 744 805 761 707 730 792 753 701 0 752 742 736 731 730 816 776 800 611 800 923 922 919 819 885 831 835 801 869 814 742 752 0 793 743 784 789 849 802 848 605 784 912 884 874 795 832 805 762 768 833 797 752 742 793 0 772 747 786 851 831 868 586 775 908 925 919 816 886 817 793 799 860 818 765 736 743 772 0 763 793 839 790 828 594 774 886 901 880 810 869 795 782 776 833 805 757 731 784 747 763 0 793 827 822 815 609 790 922 934 918 824 865 830 820 814 862 797 758 730 789 786 793 793 0 852 817 840 665 857 973 983 988 865 944 893 870 863 927 878 830 816 849 851 839 827 852 0 810 863 21 22 636 667 815 851 958 975 951 1000 962 991 864 892 931 919 875 911 846 875 864 888 910 921 867 875 795 831 776 800 802 848 831 868 790 828 822 815 817 840 810 863 0 845 845 0 297 2 B. Sadeghi / Arabica 58 (2011) 210-299 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 298 B. Sadeghi / Arabica 58 (2011) 210-299 Table 17. Nearest Neighbors, List C. hese tables were drawn up using the distance matrices. See the caption for Table 12. Normalized features 14 2 13 9 1 14 13 9 1 2 14 9 1 14 9 2 1 14 9 6 1 14 10 16 1 14 10 16 1 14 9 10 1 14 10 16 1 14 9 16 1 14 10 6 1 14 13 9 1 14 16 15 1 13 9 2 1 14 13 17 1 14 13 18 1 14 13 15 1 14 16 13 1 14 13 16 1 14 13 18 1 14 13 17 1 14 18 13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 16 16 8 13 10 9 9 13 13 13 16 16 9 10 16 9 16 17 15 21 15 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Standardized features 2 14 13 9 1 14 13 9 1 2 14 5 1 2 9 14 1 2 6 14 1 14 2 10 1 14 10 16 1 14 9 10 1 14 10 16 1 14 9 16 1 14 10 6 1 14 13 9 1 14 15 16 1 2 13 9 1 13 17 14 1 14 18 13 1 14 15 18 1 14 16 13 1 14 13 16 1 21 14 18 1 14 17 13 1 14 18 17 17 6 4 3 9 16 2 13 2 18 2 16 18 10 18 9 13 17 15 13 15 13 Precise Definition for Smoothness Given an ordered sequence of texts P1, P2, . . ., Pn, a measure of smoothness is provided by the sum of the stylistic differences between consecutive texts. If the difference between Pi+1 and Pi is represented by the symbol ||Pi – Pi+1||, then this sum is ||P1 – P2|| + ||P2 – P3||+ . . .+||Pn–1 – Pn||, or in more compact notation: Σ||P – P n–1 J= i || i+1 i=1 Now suppose we change the order of the texts, thus reassigning the labels Pi to the texts in a different way. he new permutation may yield a new value for J since the value of J depends on the ordering. A permutation of texts that B. Sadeghi / Arabica 58 (2011) 210-299 299 reduces J increases smoothness. Smoothness is thus a relative concept, but one can define an absolute version. If J is “significantly” lower than the value of J averaged over all possible permutations, then we have a smooth trajectory. Of course, one may formalize “significantly” in terms of confidence bands. he quantity ||Pi – Pi+1|| is most straightforward to calculate for univariate data, as Pi is represented by a number (say, average verse length or the frequency of one morpheme). In that case, ||Pi – Pi+1|| is simply the magnitude of the difference between the two numbers. Graphically, it’s the difference in height of two consecutive columns. hus J is the sum of the differences in heights of consecutive columns. J is minimized, and smoothness is maximized, if the columns are arranged in decreasing order of height, or in the reverse order of that. However, there are many smooth trajectories that fall short of maximal smoothness. In the multivariate case, Pi are represented as vectors (rows) of morpheme frequencies. Arrange Pi as rows on top of one another. One gets a matrix in which each row represents a text (there being n texts) and each column a morpheme (out of m morphemes). At the i’th row and j’th column, we have Fij, the relative frequency count of the j’th morpheme in the i’th text, Pi . he cityblock distance between consecutive texts Pi and Pi+1 is obtained by summing the differences of their morpheme frequencies: ||Pi – Pi+1|| = Σ|F m ij j=1 – F(i+1)j| As before, a measure of smoothness is the sum of such distances between consecutive texts: ||Pi – Pi+1|| = n–1 J= Σ i=1 ΣΣ|F n–1 m ij – F(i+1)j| i=1 j=1 As in the univariate case, smoothness is inversely related to J. Incidentally, to reduce or filter out noise or reduce the effects of outlier texts, one may sum a moving average, for example J= Σ||P – P i ||+ ||Pi – Pi+2||+ ||Pi – Pi+3|| i+1 where and are constants between 0 and 1, and where the summand is defined appropriately for the boundary cases. Obviously, variations on the summand are possible.