Book contents
- Frontmatter
- Contents
- List of figures
- Preface
- Part I PRELIMINARIES
- Part II EMPIRICAL HARMONICS
- Chapter 2 Empirical harmonics before Aristoxenus
- Chapter 3 The early empiricists in their cultural and intellectual contexts
- Chapter 4 Interlude on Aristotle's account of a science and its methods
- Chapter 5 Aristoxenus: the composition of the Elementa harmonica
- Chapter 6 Aristoxenus: concepts and methods in Elementa harmonica Book i
- Chapter 7 Elementa harmonica Books II–III: the science reconsidered
- Chapter 8 Elementa harmonica Book iii and its missing sequel
- Chapter 9 Contexts and purposes of Aristoxenus' harmonics
- Part III MATHEMATICAL HARMONICS
- Postscript: the later centuries
- Bibliography
- Index of proper names
- General index
Chapter 8 - Elementa harmonica Book iii and its missing sequel
Published online by Cambridge University Press: 22 September 2009
- Frontmatter
- Contents
- List of figures
- Preface
- Part I PRELIMINARIES
- Part II EMPIRICAL HARMONICS
- Chapter 2 Empirical harmonics before Aristoxenus
- Chapter 3 The early empiricists in their cultural and intellectual contexts
- Chapter 4 Interlude on Aristotle's account of a science and its methods
- Chapter 5 Aristoxenus: the composition of the Elementa harmonica
- Chapter 6 Aristoxenus: concepts and methods in Elementa harmonica Book i
- Chapter 7 Elementa harmonica Books II–III: the science reconsidered
- Chapter 8 Elementa harmonica Book iii and its missing sequel
- Chapter 9 Contexts and purposes of Aristoxenus' harmonics
- Part III MATHEMATICAL HARMONICS
- Postscript: the later centuries
- Bibliography
- Index of proper names
- General index
Summary
The third book of the El. harm. consists almost entirely of a set of twenty-three demonstrative proofs of propositions about melodic sequences (62.34–74.8). Before them come four preliminary arguments establishing points on which the proofs will rely (58.14–62.33); and they are followed by the beginning of an argument designed to show that there are three and only three melodically acceptable ways of arranging the constituent intervals of a perfect fourth (74.9–25). At this point our manuscripts break off.
Here and there in the course of his exposition Aristoxenus pauses to examine a methodological issue or to clarify theses which his hearers, so he says, have previously found obscure. The bulk of the text, however, is devoted to the logical derivation of rules about melodic sequences, one at a time, from axiomatic principles. Though the arguments are set out with various degrees of formality, all are presented within the conventions familiar to us from Greek mathematical works, notably from Euclid's Elements. A proposition is stated; there follows an argument deriving it directly or indirectly from the agreed axioms; finally, in many cases but not all, the proposition is restated as an established conclusion. The unadorned rigour of the mathematical treatises is reflected also in Aristoxenus' style of writing in this book; it is severe and concentrated, admitting few of the rhetorical flourishes and none of the barbed allusions to other theorists which enliven Books i and ii.
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- The Science of Harmonics in Classical Greece , pp. 197 - 228Publisher: Cambridge University PressPrint publication year: 2007