The ancient science of harmonics investigates the arrangements of pitched sounds which form the basis of musical melody, and the principles which govern them. It was the most important branch of Greek musical theory, studied by philosophers, mathematicians and astronomers as well as by musical specialists. This 2007 book examines its development during the period when its central ideas and rival schools of thought were established, laying the foundations for the speculations of later antiquity, the Middle Ages and the Renaissance. (...) It concentrates particularly on the theorists' methods and purposes and the controversies that their various approaches to the subject provoked. It also seeks to locate the discipline within the broader cultural environment of the period; and it investigates, sometimes with surprising results, the ways in which the theorists' work draws on and in some cases influences that of philosophers and other intellectuals. (shrink)

For over 40 years, Andrew Barker has been studying the ways in which ancient Greek philosophers, scientists and others analysed and discussed the structures underlying musical compositions; he has focused, in particular, on their methodologies, the conceptual frameworks within which their analyses were formed, and the various philosophical commitments they brought to their work. This volume contains a selection of the essays that Barker has published on these and related topics. The essays are preceded by an English version (...) of his book on 'psychomusicology', previously published only in Italian. It examines Greek writers' diverse and often influential speculations about correlations between musical structures and their counterparts in the soul, speculations which led them to striking conclusions about music's practical value in human affairs, and to recommendations for appropriate ways of employing it in such fields as, for example, moral education, psychotherapy, and the organization and governance of a healthy society. (shrink)

The science called 'harmonics' was one of the major intellectual enterprises of Greek antiquity. Ptolemy's treatise seeks to invest it with new scientific rigour; its consistently sophisticated procedural self-awareness marks it as a key text in the history of science. This book is a sustained methodological exploration of Ptolemy's project. After an analysis of his explicit pronouncements on the science's aims and the methods appropriate to it, it examines Ptolemy's conduct of his investigation in detail, concluding that despite occasional uncertainties, (...) the declared procedure is followed with remarkable fidelity. Ptolemy pursues tenaciously his novel objective of integrating closely the project's theoretical and empirical phases and shows astonishing mastery of the concept, the design and the conduct of controlled experimental tests. By opening up this neglected text to historians of science, the book aims to provide a point of departure for wider studies of Greek scientific method. (shrink)

The tattered remains of a few paragraphs of a work on harmonic theory were published in 1986 as P. Oxy. LIII.3706, with a careful commentary by M. W. Haslam. There are six fragments. Four of them are too small for any substantial sense to be recovered; and while fr. 2 and the second column of fr. 1 allow us to pick out significant words and phrases here and there, the remnants of these columns are very narrow, and the line of (...) reasoning seems inaccessible. Musicological analysis must focus on the first column of fr. 1. There is little enough even of that, and in attempting a relatively detailed interpretation I shall have to be rather less cautious than Haslam quite properly was. But I think that something can be made of it without stretching speculation too far, and if I am right the piece is of some genuine interest. Here are the two versions of the text that Haslam prints. The first records what is decipherable on the papyrus itself, while the second represents a partial reconstruction, restoring word-divisions and some of the missing letters. (shrink)

Porphyry's Commentary, the only surviving ancient commentary on a technical text, is not merely a study of Ptolemy's Harmonics. It includes virtually free-standing philosophical essays on epistemology, metaphysics, scientific methodology, aspects of the Aristotelian categories and the relations between Aristotle's views and Plato's, and a host of briefer comments on other matters of wide philosophical interest. For musicologists it is widely recognised as a treasury of quotations from earlier treatises, many of them otherwise unknown; but Porphyry's own reflections on musical (...) concepts and his snapshots of contemporary music-making have been undeservedly neglected. This volume presents the first English translation and a revised Greek text of the Commentary, with an introduction and notes designed to assist readers in engaging with this important and intricate work. (shrink)

Little is known about the harmonic theorist Archestratus (probably early 3rd century BC). Our only substantial information comes from Porphyry, who quotes a brief comment by a certain Didymus on his epistemological stance, and seeks to justify it through reflection on a rather startling technical doctrine which Archestratus propounded; and from Philodemus, who comments scathingly on his view of the relation between harmonic theory and philosophy. Neither passage is easy to interpret; this paper tries to make sense of them, and (...) to set them in an intelligible relation to one another. It argues that the doctrine recorded by Porphyry becomes comprehensible when it is placed against the background of Aristoxenus' work in harmonics, and it discusses Porphyry's inferences about the way in which his epistemological position diverged from that of Aristoxenus. It argues that Philodemus' report gives evidence of Archestratus' interest in issues of central concern to philosophy and in particular of an engagement with Aristotelian thought; it tries to identify some specific questions which attracted his attention, and to explain how he seems to have answered them, and why. It suggests that the two reports can be brought together as elements in a single, though fragmentary picture, and finally that Archestratus can be assigned an interesting though minor role in the history of Peripatetic philosophy and science. (shrink)

In his Harmonics, Ptolemy constructs a complex set of theoretically ‘correct’ forms of musical scale, represented as sequences of ratios, on the basis of mathematical principles and reasoning. But he insists that their credentials will not have been established until they have been submitted to the judgement of the ear. They cannot be audibly instantiated with the necessary accuracy without the help of specially designed instruments, which Ptolemy describes in detail, discussing the uses to which each can be put and (...) cataloguing its limitations. The best known of these instruments is the monochord, but there are several more complex devices. This paper discusses one such instrument which is known from no other source, ancient or modern, whose design was prompted by the geometrical construction known as the helikôn. It has several remarkable peculiarities. I examine its design, its purposes, and the merits and shortcomings which Ptolemy attributes to it. An appendix describes an instrument I have built to Ptolemy’s specifications, in an attempt to find out how satisfactorily such a bizarre contraption will work; and it explains how various practical problems can be resolved. Keywords: Ptolemy; Greek harmonics; Scientific instruments; Helikôn; Experiment; Mathematics. (shrink)

Editors and translators have found this paragraph troublesome. Though its general tenor is fairly clear it is not easy to interpret in detail, and the task is complicated by three points of uncertainty about the text, Bury conjectured that in 5 is misplaced, and should stand in 3 after . After in 5, the second hand of Ven. 189 adds modern editors have often accepted this addition, In 6, has been thought incomprehensible: Badham offered instead, and this suggestion too has (...) found some favour. (shrink)

Since Epigonus spent most of his life in Sicyon, it seems likely that Lysander was himself one of the associates of Epigonus that the passage mentions. This would place him in the latter part of the sixth century. But we have no further information about Lysander, and nothing of what is known of Epigonus is any help in the interpretation of the present account. Some innovations in kithara-playing are being credited to Lysander, but what they are is far from clear.

Aristides Quintilianus' dates are not known, but he can hardly be earlier than the first century A.D. or later than the third. Several passages in the early pages of his de Musica1 purport to record facts about the practice of much older theorists, in contexts which make it clear that his references are to the period before Aristoxenus. Since our knowledge of music theory in that period is extremely sketchy, it is obviously worth trying to assess the reliability of Aristides' (...) information. Two of his references have often been recognized as being of special interest, and there is a third, to which, I shall argue, the other two are intimately related. The first records two systems of notation, alleged by Aristides to have been used by oί ρχαîoι. The second is the famous, or notorious, account of certain ‘divisions of the tetrachord’ which were employed by oί πνυαλαιότατoι πρς ρμoνίας. It is these, Aristides tells us, which are mentioned by Plato in the Republic.2 The remaining passage is superficially rather less exciting: it records the names and initial notes of the ρμoνίαι, or forms of octave scale, said to have been distinguished by oί παλαιoί, and says something about a method by which the πoιότης of each can be made clear. The information given here about the nature of the ρμoνίαι is familiar: it is to be found, for example, in Cleonides Eisagoge 19. 4 ff., where rather more detail is given, and where the names of the ρμoνίαι are again ascribed to oί ρχαîoι . I shall suggest, however, that Aristides' version has independent interest. What he tells us in the first two passages is found nowhere else. (shrink)

This book is one of the Macmillan ‘Sources in Philosophy’ series, under the general editorship of Lewis White Beck. According to the publisher’s blurb, each volume ‘contains a substantial original essay and selected readings in a major segment of philosophy. The introductory essay for each volume provides an original point of view from which the coherence and unity of the diverse selections can be appreciated’. Now the preparation of such collections involves major difficulties, especially when, as in this series, the (...) total space available is so small. It is hard to avoid the over-simplification of issues, except at the expense of clarity; it is impossible to do justice to different sides in a controversy without trivialising the issues; lack of space in which to state particular points of view in full may lead to their presentation in an unbalanced and half-argued form, sacrificing both interest and cogency; and little can be made of the intricate network of interconnections relating the major issues, so that their relevance to one another and their general importance in the field may remain obscure. (shrink)

Athenaeus (637a) records these lines from the dithyrambHymenaios, along with a number of other snippets of poetry, in the course of an inconclusive discussion about the characteristics of the instrument (if it is an instrument) called the magadis. Athenaeus had good reasons for being puzzled; the wordfirst appeared in Greek, so far as we know, in the seventh centuryb.c., and its sense was already a matter of some doubt in the fourth. As to this particular fragment, even Telestes' original audience (...) might be forgiven some bafflement in the face of the third line, on which I shall initially focus here. What can be meant by ‘the five-rodded joining of the strings’? (shrink)

It would be doing this book an injustice, no doubt, to suggest that it is of value primarily for its bibliography—eighty-five well chosen titles from the recent literature on the philosophy of science. Nevertheless, it is a point which might be argued. Few short ‘collections’ provide much scope for reviewers, particularly considering the extent to which they have been proliferating recently. Anyone who is likely to read the review will in all probability have read already most of the essays collected, (...) and it is unusual to find an example in which the editor’s introduction, or the principles on which the articles are arranged, are of enough interest to demand extended treatment themselves. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The End of Argument: Knowledge and the InternetSimon Barker1. Fermat's last videoModern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science.(Jaffe and Quinn 1993, 1)The above passage illustrates how mathematicians have come to esteem rigorous argument as the most important feature of their subject. (...) In turn, the example of mathematics has, over a long period, made rigorous argument an ideal in many other areas of inquiry—from the sciences to theology. How interesting it is, then, that some mathematicians have begun to question that ideal. Gian-Carlo Rota, for example, has said that the axiomatic method of presentation in mathematics has now reached "the zenith of fanaticism" and that the identification of mathematics with this method "is having a corrosive effect on the way mathematics is viewed by scientists in other disciplines" (1997, 189). William Thurston has blamed the formality of contemporary mathematics for "the whole system of generally poor communication" within the subject. According to Thurston, ordinary conversation between mathematicians can often convey a new solution in a matter of minutes, whereas a formal proof delivered in a seminar will take an hour, and a written article may take several hours or even days to understand. In Thurston's view, this disparity arises precisely because conversationalists go far beyond formal mathematical language:They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention. With these channels of communication, they are in a much better position to convey what's going on, not just in their logical and linguistic facilities, but in their other mental facilities as well. [End Page 154](Thurston 1994, 166)Sadly, however, mathematicians can only converse like this in private. To reach the rest of their audience, they have to publish in journals. Journal articles are limited to formal language and so must convey results more slowly. Formality, though less efficient, is more publishable.Yet there is hope. Another mathematician, Andrew Odlyzko, has suggested that the unfortunate limitations of printed journals can now be overcome with computers. Computers can allow mathematicians to distribute something more than the traditional style of printed proof. Odlyzko suggests an electronic scrapbook that would include a video of each author lecturing about their results (Odlyzko 1995, 90-91). Fermat's Last Video? The point is that if these suggestions come to pass, publication of mathematical results will no longer involve stripping away informal features and replacing them with formal language, as if rigorous argument was all that mattered. Instead, the informal features that help mathematicians to understand new results so quickly in ordinary conversation will be given equal air time. The ideal, for publication at least, will be less formal.Is this just a gimmick? Is it something that will go away, if we shut our eyes, leaving the centuries old ideal of formal proof intact? What I wish to discuss here are some of the reasons why mathematics, as well as other branches of knowledge, may adopt greater informality and may cease to idealise rigorous argument.2. From parents to premissesI will start by examining how and why rigorous, formal argument became an ideal in the first place. This is a long story, going back to the origins of Western literature; so I am forced to be brief.1 I will start with the matter of how argument became an ideal. The first thing we should recognise is that even in the West's earliest literary works—the Iliad and the Odyssey—there are plenty of rudimentary arguments, particularly in the speeches put into the mouths of the major characters. Nevertheless, for sustained or rigorous argument, we must look a little harder. Havelock has put forward the following passage as an example of unusual rigour in early literature (Havelock 1966b, 209). The passage is taken from the start of Hesiod's poem, Works and Days, a piece of wisdom literature almost as old as... (shrink)

This paper analyzes sluicing as anaphora to an anti-constituent (a continuation), that is, to the semantic remnant of a clause from which a subconstituent has been removed. For instance, in Mary said that [John saw someone yesterday], but she didn’t say who, the antecedent clause is John saw someone yesterday, the subconstituent targeted for removal is someone, and the ellipsis site following who is anaphoric to the scope remnant John saw ___ yesterday. I provide a compositional syntax and semantics on (...) which the relationship between the targeted subconstituent and the rest of the antecedent clause is one of scopability, not movement or binding. This correctly predicts that sluicing should be sensitive to scope islands, but not to syntactic islands. Unlike the currently dominant approaches to sluicing, there is no need to posit syntactic structure internal to the ellipsis site, nor is there any need for a semantic mutual-entailment requirement. Nevertheless, the fragment handles phenomena usually taken to suggest a close syntactic correspondence between the antecedent and the sluice, including case matching, voice matching, and verbal argument structure matching. In addition, the analysis handles phenomena exhibiting antecedent/sluice mismatches, including examples such as John remembers meeting someone, but he doesn’t remember who, , and especially so-called sprouting examples such as John left, but I don’t know when, in which there is no overt subconstituent to target for removal. In Sect. 5, I show how the analysis accounts for Andrews Amalgams such as Sally ate [I don’t know what] today, in which the antecedent surrounds the sluiced clause. Finally, in Sect. 6, I propose a new semantic constraint on sluicing: the Answer Ban, which says that the antecedent clause must not resolve, or even partially resolve, the issue raised by the sluiced interrogative. (shrink)

Restatements of the law are usually considered a uniquely American phenomenon, explained by the complexities and uncertainties of a multi-jurisdictional common law system. They have also been subject to the accusation from legal realists that they are misleading, conservative and formalistic exercises. This review interrogates the role of the restatement in a jurisdiction with a singular common law tradition, focusing on Andrew Burrows’ recent Restatement of the English law of Unjust Enrichment. It compares and contrasts his restatement with previous American (...) restatements and with other types of ‘public statement’ of private Law, such as codes and legal treatises. It concludes that there may be good (and particular) reasons for restating unjust enrichment law in England and that Burrows’ work is unique in its form and purposes. This article’s central thesis is that a restatement can provide an important public platform for intellectual debate and a powerful centripetal force addressing legal fragmentation and uncertainty even in a singular common law jurisdiction. It further suggests that modernist doctrinal enterprises of this type may be more sympathetic to the critical insights of legal realism than is sometimes suggested. (shrink)

Events and intellectual fashion have conspired to bring the once neglected topics of nationalism and Staatslehre to the forefront of political-philosophical debate in this country, a development which makes David Runciman’s intellectual-historical study quite timely. The history Runciman tells is that of the early twentieth century English movement of political pluralism, whose central figures were Frederic W. Maitland, John N. Figgis, Ernest Barker, George D. H. Cole, and Harold Laski. To tell their story Runciman quite properly recognizes that he (...) must also discuss Thomas Hobbes and Otto von Gierke. For the story of political pluralism is the story of a debate over the concept of group personality, the problems of which are originally and decisively formulated by Hobbes, but which are received by the English pluralists by way of Gierke. Runciman’s approach here is “somewhat dialectical”, in that the discussions of Hobbes and Gierke are meant to shed light upon the English pluralists even as the discussions of the latter are intended to illuminate them. Runciman characterizes this as a movement between philosophy and history. (shrink)

ANDREW BARKER, Scientific Method in Ptolemy's Harmonics. Cambridge: Cambridge University Press, 2000. viii+281 pp. $70. ISBN 0‐521‐55372‐5. The quotation is taken from the preface of the treatise,...

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.1 (2003) 123-124 [Access article in PDF] Andrew Barker. Scientific Method in Ptolemy's Harmonics. New York: Cambridge University Press, 2001. Pp. viii + 281. Cloth, $69.95. Ptolemy's Harmonics is an important source not only for the history of music, but also for the history and philosophy of science. Two recent monographs, by J. Solomon, and A. Barker, now provide a basis (...) for a more widespread discussion of this much neglected work. Barker focuses on the part that is concerned with musical science proper (I, 1-III, 3), and more specifically on the scientific method it displays.Barker uses an analysis of Ptolemy's programmatic statements in I, 1/2, as a key to a reading of Ptolemy's treatise. His exposition has the format of a commentary, with special emphasis on the methodologically relevant passages. For the most part, he follows the order of Ptolemy's text closely, providing historical background information in excursus form. One needs to read a complete text of Ptolemy's treatise along with Barker's exposition in order to understand some of his arguments. It is preferable to also use Düring's [End Page 123] edition of the Greek text. One is rewarded with a detailed and in-depth analysis of the passages that helps one understand Ptolemy's method in harmonics as it unfolds in his treatise. This is so even if one disagrees with Barker's claims concerning Ptolemy's methodology.These claims are outlined in chapter 2, on the basis of I, 1/2 of the Harmonics. The methodological program Barker filters out of the text remains vague in a number of important respects. One misses, for example, a clear exposition on what will count as a sufficient integration of "empirical and theoretical phases," and on what will count as "empirical." This is a serious drawback, since Barker claims that Ptolemy instantiates a complex scientific methodology with a decisive, fully integrated empiricist trait. The drawback is aggravated by the fact that the already vague terms of the postulated methodology have to be watered down considerably in the subsequent chapters to make them fit with Ptolemy's actual procedures. Ptolemy does not seem to deliver even on the vague terms. One may very well remain unconvinced both by Barker's initial arguments concerning Ptolemy's methodology, and by his attempts to show that Ptolemy's procedure in the Harmonics can count as a satisfactory instantiation. Perhaps Ptolemy's intentions are less ambitious. When he states (5.13 ff. Düring, ed.) that "the aim of the student of harmonics must be to preserve the hupotheseis..., as never in any way conflicting with the perceptions that correspond to most people's estimation," he may just mean that he intends to vindicate an essentially Pythagorean (locally modified) account by showing it does not conflict with the phenomena. In my view, he certainly delivers on this less ambitious program. In fact, it seems to me that the understanding of Ptolemy's argumentative path is impeded by Barker's interpretation of Ptolemy's goal of "saving the hupotheseis" as a commitment to the integration of a genuine empiricist trait.Chapters 7 and 8 give an exposition of Ptolemy's tetrachord divisions. For the reader interested in an evaluation of the theoretical aspect of Ptolemy's scientific method, these are the central chapters of Barker's book. Barker provides a detailed analysis, together with an exposition of the historical and technical information needed for a non-specialist to follow both his and Ptolemy's argument. Barker's analysis reveals inter alia that a considerable part of Ptolemy's theory has no counterpart in the realm of phenomena perceived as musical, and that another part, concerning the single most important division of musical practice, is not justified, and not justifiable, by derivation from the principles. It is difficult to see how Ptolemy's account can nevertheless be viewed as a genuine integration of empirical and theoretical aspects. It appears to be an attempt at showing that an essentially Pythagorean theory... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 125.1 (2004) 140-144 [Access article in PDF] C. J. Tuplin and T. E. Rihll, eds. Science and Mathematics in Ancient Greek Culture. Foreword by Lewis Wolpert. Oxford: Oxford University Press, 2002. xvi + 379 pp. 21 black-and white ills. 3 tables. Cloth, $80. It has become something of a truism to say that, whatever their ambitions for abstraction, scientists remain profoundly caught up in the (...) life of their culture. Yet, that is a truism only at a certain level. No one is surprised to hear that Euclid had to eat and drink just like everyone else and relied on royal patronage for the leisure to pursue his studies, but it may occasion surprise to learn that the shape of his number theory should owe something to the particular way in which his eye fell upon a multiplication table. That Galen was influenced by his teachers is not news; but that his commitment to a particular school of philosophy could cause him to see and not see certain things when he was conducting dissections—here is something of interest. The papers in this volume, which were first delivered at a conference in Liverpool in 1996, have it as their common task to show how ancient forms of life influenced the pursuit and acquisition of knowledge on the part of scientists like Euclid and Galen.Heading the collection is a study by Andrew Barker of some idiosyncratic features of Greek harmonic theory. Archytas, Aristotle, and Ptolemy each attempted to characterize the physical basis of the two opposite sound qualities, oxusand barus. As Aristotle himself observed, the terms are metaphors drawn from the tactile realm, and trouble arose because in that realm they are heterogeneous opposites, i.e., the proper opposites of "sharp" and "heavy" are "dull" and "light." Barker uncovers a corresponding uncertainty in the texts about whether these sound qualities should be analyzed as something more like weight or something more like sharpness. Thus, an excessive trust in the descriptive adequacy of ordinary language terms led these thinkers to frustration.Harry Hine's paper on earthquakes and vulcanology deals with ancient knowledge of two phenomena that elicited the attention of scientist and non-scientist alike. Like Barker, Hine draws our attention to a terminological deficiency: although there was a fairly elaborate vocabulary for seismoi, neither Greek nor Latin possessed a generic noun that could properly be translated as [End Page 140] "volcano." This deficiency seems to be correlated with the fact that earthquakes form a regular part of the meteorological curriculum, while volcanoes usually received independent treatment in works such as the Aetna. Particularly intriguing is Hine's suggestion that the opinion of locals, especially sailors, may have been the source of certain scientific theses about volcanoes, e.g., that Aetna's behavior was closely correlated with the weather.The difference between ancient astronomy as practiced and as it was popularly believed to operate is the subject of Alan Bowen's paper. His central question is, when did astronomers first set out to make predictions of eclipses? The surprising answer is that, despite popular beliefs to the contrary, such prediction was never practiced by astronomers in a serious way before Ptolemy. Along with scrutiny of the misunderstanding in Herodotus, Diodorus, and Pliny, the paper offers thorough analysis of the most well-documented "prediction," that of a lunar eclipse that took place the night before the battle of Pydna in 167 B.C.E., which was supposedly foreseen by the Roman (!) general C. Sulpicius Gallus. Bowen casts doubts upon the entire tradition, showing how adherence to an ideology of the well-rounded commander led the sources to inflate his achievement: while some reports have Gallus simply assuaging his troops' fears at the event, others have him explaining its causes or even foretelling it. Bowen's skepticism is well founded, and one must regard these supposed feats of eclipse-prediction as a kind of science fiction, prescient yet false all the same.R. Hanna's subject is the format and function of the parapegma calendar first devised by... (shrink)