Here and there B presents A’s theses in a different and more helpful

order. In A, remarks and digressions that are tangential to the main themes relate to historical and aesthetic issues, and to the vocal and instrumental resources of musical practice; their counterparts in B are more abstract, ‘philosophical’ in a recognisable sense of that elastic expression, calling for intellectual reflection rather than for observation and experience. In B, but not in A, these digressions are cast as responses to problems raised previously by Aristoxenus’ audiences, so adding to the impression that Book ii is the sequel to an earlier account. Finally, the two principles enunciated in B

provide a much smoother and more economical transition to the theorems of Book iii than does the untidy collection of seven in A, which can only with difficulty be given any coherence at all. The case is not proved, but I think it has much in its favour.

t h e ev i d e n c e o f p o r ph y ry

Thoughts prompted by allusions in a later writer will make an appropriate appendix to the main business of this chapter. Porphyry’s commentary on Ptolemy’s Harmonics, written some six centuries after Aristoxenus, is peppered with quotations from his work, along with looser paraphrases and other references, over twenty of which reflect passages in the Elementa

harmonica as we now have it. Four are of interest here, since they give the

titles, as Porphyry knew them, of the writings from which they were taken. At In Ptol. Harm. 80.22–7 he quotes a passage of our Book i (El. harm. 14.18–28), not quite ‘mot pour mot’, as B´elis puts it (1986: 29), but with only minor verbal variations; and he tells us that it comes from the first book of the treatise Peri arch¯on (On Principles). At 81.23–5 he comments on the fact that Aristoxenus puts the genera first in the sequence of topics which harmonics should consider – which he does only in Book ii – and assigns this to a work called Harmonika stoicheia (Harmonic Elements). A passage from the middle of Book ii, El. harm. 45.3–46.1, is quoted fairly accurately at 124.15–125.8, and comes, we are told, from the first book of the Harmonika stoicheia; and Porphyry’s reference (quoting Didymus) at 28.22–3 to ‘the introduction to the first book of the Harmonika stoicheia’ seems to allude to a discussion near the beginning of El. harm. Book ii, starting at 32.10.

Porphyry therefore thought of the contents of our Elementa harmonica as belonging to two separate works, the Peri arch¯on and the Harmonika

stoicheia, each of which was divided into at least two ‘books’. The first

book of the Peri arch¯on included at least part of El. harm. Book i, and much if not all of El. harm. Book ii was in the first book of the Harmonika

stoicheia. Our texts of Aristoxenus refer often enough to archai, sometimes

in the sense ‘principles’ and sometimes simply ‘beginnings’, but none of these references makes it probable that Peri arch¯on was Aristoxenus’ own name for the treatise including Book i, or that he thought of it in the way this title implies.28 _{A better case can be made for assigning the title}

28_{Relevant passages are cited in B´elis1986, 30–2. I do not think that they justify the conclusions she}

Harmonika stoicheia to the work that included Books ii–iii. I mentioned

earlier a remark near the end of Book i, rounding off its preliminary sketch of the concepts of melodic continuity and succession. ‘How they come about, and which interval is placed after which and which is not, will be shown in the elements [stoicheia]’ (28.33–29.1). In our surviving texts this promise is fulfilled in Book iii, and Book iii, I have argued, is a revision of the original continuation to Book i. We cannot tell whether the statement I have quoted was present in Book i from the outset and pointed to the original continuation, or whether it was inserted later as a signpost to Book iii itself. But in either case the expression ‘the elements’ could be read as the title of a work, or a section of a work, devoted to the detailed, argumentative demonstration of propositions in theorematic style.

A similar expression occurs in Book ii, in a context that generates an intriguing contrast. It comes right at the end of the book’s preliminary discussion, just before Aristoxenus begins his exposition of factual data in the passage that overlaps with Book i. ‘These, more or less, are the things one might say by way of preface to the study of harmonics (t¯es harmonik¯es

pragmateias); but those who are setting off to address the study of the ele-

ments (t¯ei peri ta stoicheia pragmatei¯ai) must understand in advance things of the following sort’ (43.25–30). These latter ‘things’ turn out to be logical and methodological principles governing the procedure of demonstration, and are heavily indebted to Aristotle.29_{Once again the study of ‘elements’}

must at least include the theorems of Book iii, though it might also extend to the contents of Book ii, as Porphyry’s use of the title suggests, in so far as the latter can be thought of as establishing the premises on which those theorems will depend. What is particularly interesting is that Aristoxenus’ antithesis in the passage I have quoted is not, like Porphyry’s, between ‘elements’ and ‘principles’, but between the study (pragmateia) of the ele- ments and the pragmateia of harmonics as a whole. These are apparently two quite different projects, calling for different kinds of preface. Aristox- enus’ students are about to be introduced to the second. His remarks in the preceding pages of Book ii, reviewing the ‘things one might say by way of preface’ to the first, serve partly to remind them of the wider context within which their specialised investigations will take place. As we shall shortly dis- cover, however, they are designed also to provide a new and much more sophisticated perspective on the science as a whole.

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