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)

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)

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)

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)

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)

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)

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.

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.

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)

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 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)