Ptolemy's musical models for mind-maps and star-maps

Abstract

The Greek science called harmonics was concerned above all with the analysis of the structures underlying musical melodythe patterns of pitches and intervals constituting scales or attunements, the substructures out of which these scales were composed, the larger structures of which they themselves formed parts, and so on and of the ways in which those various structures could be systematically interrelated and transformed. In the tradition of harmonic theorising to which Ptolemys treatise (with some qualifications) belongs, the terms in which these analyses are set and the principles of order which govern the constructions are mathematical, drawn primarily from the mathematics of ratio. Unlike many of his predecessors, however, Ptolemy insists that the harmonic scientists task is not just to spin pretty patterns with numbers, regardless of their relation to real musical practice. It is to reveal the intelligible, mathematical form of the structures which the ear perceives as musical, and thus to show that the beauty we perceive in them is a reflection of rational order. There must therefore be no conflict between the findings of mathematical reasoning and the aesthetic judgements of the musical ear; and the results of his mathematical analyses and constructions cannot be accepted as correct until they have been empirically tested and confirmed. Hence they must be transposed, in all their detail, into a form in which the ear can assess them. This is done by transferring the mathematical structure onto special experimental instruments, translating arithmetical ratios into ratios between lengths of the instruments strings. Ptolemys careful description of these pieces of gadgetry, of the principles on which they work, of procedures for testing their accuracy and for controlling irrelevant variables, of the ways in which they can and cannot reliably be used, and so on, are among the most fascinating parts of his treatise; and the overall methodology which he articulates and pursues, combining mathematically sophisticated modes of rational derivation with rigorous empirical test-procedures, is as impressive and compelling as any to be found in our record of the ancient sciences.3 The project I have been lightly sketching occupies only the first 91 of the works 111 large pages in Drings edition. At this point Ptolemy announces that the task he had set himself at the outset is complete. It seems to me, then, that we have demonstrated accurately and in several ways that the nature of musical attunement possesses its own proper ratios, all the way down to the melodic intervals,4 and that we have shown which ratio belongs to each of them, in such a way that those who have set themselves, with keen enthusiasm, both to grasp the reasoning involved in the propositions we have set out and to undertake their assessment in practice, according to the methods of using the kann5 which we have expounded, will be left in no doubt, since they will recognise through all the species [of scales and similar structures] their agreement with what we accept on the basis of our perceptions (Harm. III.3, 91.22-92.1).6 The remainder of the Harmonics (III.3-16, of which some later parts were lost at an early stage of transmission and were reconstructed by Byzantine editors) is therefore an appendix to its main agenda; but it is with this appendix that our present business lies. Ptolemy begins by setting the results of his harmonic enquiries in a wider context. The structures he has revealed are to be understood as products of a special sort of reason or rationality which he calls harmonic reason. It is this that makes correct the ordering that exists among things that are heard. It works towards three interconnected goals: the theoretical discovery of the proportions by means of intelligence; their practical exhibition by means of skill;7 and the development, through practice and habituation, of a mode of experience which follows them, that is, in which they are recognised and their status is appreciated (III.3, 92.30-93.4). Ptolemy continues: When we consider that reason in general also discovers what is good, establishes in practice what it has understood, and brings the underlying matter into conformity with this by habituation, it is to be expected that the science which embraces all the species of science that rely on reason, and which has the special name mathematics, is not limited by only the theoretical grasp of beautiful things, as some people would suppose, but includes at the same time the exhibition of them, and the dedication to them which arises from habituation in following them. (III.3, 93.4-10) Mathematics, then, is a science whose goal is the analysis and exhibition of beauty, together with the development of a trained capacity to recognise it empirically and a disposition to embrace it. Beauty is a property whose real nature lies in rationally ordered structure, and harmonics is the branch of mathematics which is devoted to the study of beauty in the domain of sound. This inspiring conception of mathematics leaves open the possibility that the forms of order underlying the beauty of things in other domains are different from those at work in music; but Ptolemys reflections in the remainder of III.3 point in the opposite direction. Sight and hearing, alone among the senses, assess their objects not only by the standard of pleasure but also by that of beauty (93.13-14). Our eyes and ears, but no other sense-organs, are therefore capable of cooperating with one another and of serving as the allies of reason in pursuit of a single, shared objective, that of penetrating progressively into what is beautiful and valuable (94.12). An astronomers study of visible beauty in the movements of the heavens and a prises whose ultimate subject matter is one and the same. It is the mathematically integrated complex of structures which, as a whole or in one or another of its constituent patterns, underlies the beauty accessible to our two privileged senses wherever it is to be found. In III.4 Ptolemy specifies the kinds of entity, outside the realm of music itself, which possess the faculty of attunement. In the light of the previous chapters discussion, this must mean that they have the capacity to grasp, to construct and to conform themselves to harmonious and beautiful patterns of order. Such a faculty must exist to some degree, he says, in all things that have in themselves a source of movement,8 but especially and to the greatest extent in those that share in a more complete and rational nature. Only in these can this faculty bring to light, and preserve fully and clearly, so far as that is possible, the likeness of the ratios which create appropriateness and attunement (95.4-10). A few lines later the possessors of such natures are identified. They are those whose movements and changes can be specified as manifestations of intelligible formal structures. These movements, as we said, are those of things that are the most perfect and rational in their natures, as among divine things are the movements of the heavenly bodies, and among mortal things those of human souls, most particularly, since it is only to each of these that there belong not only the primary and complete mode of movement, that is, movement in place, but also the characteristic of rationality. The faculty of harmonic reason reveals and displays in them, so far as a human being can grasp it, the pattern of organisation that corresponds to the harmonic ratios of the notes, as we can see if we analyse each of these kinds in turn. (95.20-27) Ptolemy thus construes the beautiful dynamic ordering of the stars and planets, and of the elements of the human psyche, as arising from the operations of a rational faculty which each of them possesses. Through the operations of this faculty they organise their own movements and components into patterns corresponding to the systems of ratios which define the structures of musical attunement. The project which Ptolemy announces at the end of III.4, that of analysing each of these kinds in turn to show that they do indeed display musical forms of structure, occupies the remainder of Book III. It is here that we finally arrive at the issues about scientific models and their functions which I announced at the start. Ptolemy is proposing to elucidate the patterns of organisation manifested in the travels of the stars and planets and in the workings of the human soul, by mapping them onto the model provided, in the main body of the Harmonics, by his complex analyses of musical structures. He tackles human psychology first (III.5-7) and stellar phenomena second (III.8-16). We shall reverse this order, since the application of his musical model to the heavens is more straightforward than its psychological counterpart, and will provide a reasonably secure basis for comparison. © 2006 Springer.