Polyphonic Music and Classical Physics: The Origin of Newtonian Time

Plato , Timaeus, 47b (transl. by Jowett B. ).

It is useful for later reference to spell out exactly in what sense the word “measurement” is used here. Measuring a quantity always means the determination of its magnitude by comparison with a standard (or unit) for that quantity. There are no measurements without units or standards nor without the process of comparison.

These words are quoted often enough. Their meaning, however, appears to be clearer when the whole paragraph is read: “Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: Relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.” Newton I. , Principia, 3rd edn (London, 1726), 46, transl. by Cajori F. (Berkeley, 1934), 6.

Mumford Lewis , Technics and civilization (New York, 1934), 14.

ibid., 15.

Zerubavel Eviatar , Hidden rhythms (Chicago and London, 1981). Contains extensive references.

Clagett Marshall , The science of mechanics in the Middle Ages (Madison and Oxford, 1959).

Gillispie C. C. , The edge of objectivity (Princeton, 1960), 90.

ibid., 42.

Jerison Harry , Evolution of the brain and intelligence (New York, 1973). For a less technical summary of the theory by the same author: Scientific American, ccxxxiv (1976), 90–101.

The translation of the temporally extended patterns into a spatial map was necessary because only the latter can answer the relevant questions in animal life: “How far is the moving thing?” “How fast does it move?” The purely temporal features of received information were, in other words, relatively unimportant in themselves until and unless they could be translated into such spatial features as distance and speed. For the use of temporal informations in animal life see, however, Gibbon J. and Allan L. G. (eds), Time and time perception (Annals of the New York Academy of Sciences, cdxxiii (New York, 1984)).

Piaget Jean , Psychology and epistemology (New York, 1971), 110.

Ibid.

Indeed, the idea that the mechanism of the solar system was such that an exact calendar was impossible in principle had been proposed repeatedly. The best known speculations in this direction are due to Nicole Oresme, the most important mathematician and astronomer of the fourteenth century. He — ironically — was also the inventor of the metaphor of the “heavenly clockwork”. (See Grant E. , Nicole Oresme and the kinematics of circular motion (Madison, 1971), 64.)

To appreciate the significance of this circumstance, one only has to imagine a world where all external objects, as well as our own bodies, arms, legs, etc., are changing their lengths and volumes. Some change regularly, others haphazardly, while the laws which regulate these changes are also subject to changes. Assume, in addition, that we had no reliable ways to estimate spatial measures by our senses. It is difficult to believe that in such a world the meaning of exact length standards could have arisen spontaneously.

Boal Ellen TeSelle , “Timepieces, time and musical tempo before 1700”, Ph.D. thesis, Washington University, Saint Louis, Missouri, 1983.

Rohr R. J. , Sundials: History, theory and practice, trans. by Godin G. (Toronto, 1965), 95.

Neugebauer O. , The exact sciences in Antiquity (Providence, 1957), 86.

Boal , op. cit. (ref. 16).

Needham J. Ling W. and de Solla Price D. J. , Heavenly clockwork (Cambridge, 1960).

de Solla Price Derek J. , Science since Babylon (New Haven and London, 1961), 29.

ibid., 30.

Bedini Silvio A. , The scent of time (Transactions of the American Philosophical Society, liii, Philadelphia, 1963).

Mumford , op. cit. (ref. 4), 12, and Zerubavel , op. cit. (ref. 6), 35. See also North J. D. , “Monasticism and the first mechanical clocks”, in Fraser J. T. and Lawrence N. (eds), The study of time (New York, 1975), 381–8.

Plato , Timaeus, 39. For a discussion, see Cornford F. M. , Plato's cosmology (London, 1948), 97.

Aristotle , Physics, Book IV, 218b ff.

Whitrow G. J. , The natural philosophy of time (Oxford, 1980).

Augustine St , Confessionum libri XIII (Cambridge, Mass., 1912), Liber XI, Cap. XXIII–XXIV.

ibid., Cap. XXVI.

In Clagett's study (op. cit., ref. 7), which contains extensive translations of the most important medieval texts on mechanics, St Augustine's name does not figure in the index. See also the rich collection of medieval texts in A source book in medieval science, ed. by Grant Edward (Cambridge, Mass., 1975). Augustine's time ideas are mentioned exactly once and then in a context which is irrelevant to our purposes.

Clagett , op. cit. (ref. 7), 311.

This definition avoids the hair-splitting arguments about whether polyphony can be found in all civilizations since, for example, in group singing, men and women sang an octave apart, or groups could sing perhaps in parallel fifths or thirds.

For a clear and non-technical description of the problem, including the necessity of the invention of the musical metre, see, for example, Zuckerkandl V. , Sound and symbol (Princeton, 1969), 158.

Bukofzer M. , Studies in medieval and Renaissance music (New York, 1950), 176.

Since the ability to sing polyphonic parts commanded extra renumeration it is likely that singing of such parts demanded extra preparation. Centuries later, the baroque masters of the keyboard, Bach, Handel, Domenico Scarlatti among them, were able to improvise, say, fugues in concert performances. Another couple of centuries saw jazz players improvising several simultaneous rhythmic patterns. But it sounds unlikely that anything resembling this was possible at the beginnings of polyphony.

The idea of mensural notations was discussed, it seems, earlier by Arab and Persian music theorists. But since music remained essentially monophonic in all non-Western, including Islamic, civilizations, there never had been any real need for the development of a consistent system of symbolic time value notations. Accordingly, none has developed. For a short history, see The new Grove dictionary of music and musicians ed. by Sadie S. (London, 1980), under “Notations”, vol. xiii, 333.

Anonymous IV, trans. by Dittner L. (Institute of Medieval Music, New York, 1959), 9.

The adjectives are of no particular significance. They merely comply with the medieval idée fixe that three was a more perfect number than two. For more details, see, for example, Caldwell John , Medieval music (Bloomington and London, 1978).

Anonymous IV (ref. 37), 47.

A representative selection of the ideas of time in the thinking of the most important medieval philosophers is given in Duhem P. , Medieval cosmology (Chicago and London, 1985).

See, for example, Howe R. H. , Music through sources and documents (Englewood Cliffs, N.J., 1978), 71.

Also known as John of Murs (c. 1300–50). He dedicated one of his works to Philipe with typical medieval modesty as “the one person in the world more estimable than this work”.

Also known as Gersonides or Leo Hebraeus (c. 1288–1344). At Philipe's request he wrote a book entitled De numeris harmonicis which dealt with the mathematical consistency of the simultaneous use of triple and double divisions.

Seidel W. , Rhythmus: Eine Begriffsbestimmung (Darmstadt, 1976), chaps. 1–2. See also: Hoppin R. H. , Medieval music (New York, 1978), chaps. 14–18.

Clagett , op. cit. (ref. 7), 331. They used this idea to give a geometric proof for a simple kinematic theorem relating to uniform acceleration.

It is a cliché but none the less true that three important developments characterized the early evolution of western civilization: Polyphonic music, perspective painting and experimental science. These developments were, of course, quite different from each other in many fundamental aspects. But a common feature of all of them is that they evolved through struggling with the same technical problem: How to create numerical order, how to find mathematical laws in perceivable time and/or space. A more elaborate discussion of this topic is given in Szamosi Geza , The twin dimensions: Inventing time and space (New York, 1986), chaps. 5 and 6.

The study of the laws of music was, together with arithmetic, geometry and astronomy, part of the quadrivium, and so a compulsory part of the university curriculum. While the theory taught was the ancient “Pythagorean-Boethian” theory, its role in the curriculum helped nevertheless to maintain the perception that music was an important model of a lawful universe.

See The new Grove (ref. 36) under “Notations”, vol. xiii, 362.

Not surprisingly, St Augustine's rhythmic ideas became influential again in the fifteenth century, Seidel , op. cit. (ref. 44), 23.

Palisca Claude V. , Scientific empiricism in musical thought in seventeenth century science and art, Rhys H. H. (Princeton, 1961), 91–137.

Sachs Curt , Our musical heritage (Englewood Cliffs, N.J., 1955), 164.