Music as a Model in Early Science

Research for this paper was made possible by the generous financial assistance of the Australian Research Grants Committee and by the encouragement and assistance of Dr D. R. Oldroyd. I am grateful to the School of History and Philosophy of Science, University of New South Wales, Kensington, for welcoming a musicologist into their midst and for making available their facilities. Acknowledgment is due to Father Barry Brundell for his translation of two Latin texts.

This is the definition of Turbayne C. M. , The myth of metaphor (rev. edn, Columbia, South Carolina, 1970).

Berlin I. , “The purpose of philosophy”, Cogito, i (1966–67), 3–11, p. 11.

Ibid.

On this point see Kassler J. C. , The science of music in Britain, 1714–1830…(2 vols, New York & London, 1979), vol. 1, pp. xix, xxv–lxii.

E.g., Neugebauer O. , A history of ancient mathematical astronomy (3 pts. New York & Berlin, 1975), 933.

E.g., Crombie A. C. , “Mathematics, music and medical science”, XII^e congrès international d'histoire des sciences (Paris, 1968; Actes, Tome I B Discours et conférences colloques discussion des rapports, Paris, 1971), 295–310; and Knobloch E. , “Musurgia universalis: Unknown combinatorial studies in the age of baroque absolutism”, History of science, xvii (1979), 258–75.

E.g., Drake S. , “Renaissance music and experimental science”, Journal of the history of ideas, xxxi (1970), 483–500.

For the fragments of the early Pythagoreans, see Freeman K. , Ancilla to the pre-Socratic philosophers: A complete translation of the fragments in Diels, Fragmente der Vorsokratiker (Oxford, 1952), 73–77 (especially Fragments 6, 10 and 11), and 78–81 (especially Fragments 1 and 2).

Burketx W. , Lore and science in ancient Pythagoreanism , translated by Minar E. L. Jr (Cambridge, 1972), 400, who also treats the numerology of the Pythagoreans and provides extensive references to the secondary literature on the Pythagorean tradition. Surveys of Pythagorean notions concerning music from antiquity to the twentieth century may be found in Haar J. , “Pythagorean harmony of the universe”, Dictionary of the history of ideas , ed. Wiener P. , iv (New York, 1973), 38–42, and Haase R. , Geschichte des Harmonikalen Pythagoreismus (Vienna, 1969). Two special studies also deserve to be cited: Crocker R. L. , “Pythagorean music and mathematics”, Journal of aesthetics and art criticism, xxii (1963–64), 189–198, 325–35 and Münxelhaus B. , Pythagoras musicus: Zur Rezeption der pythagoreischen Musiktheorie als quadrivialer Wissenschaft in lateinischen Mittelalter (Bonn - Bad Godesberg, 1976). The last study includes important iconographic evidence.

Freeman K. , op. cit. (ref. 9), 79–80. Taking a and b to represent integers, b being larger than a, the formula for finding each mean is usually given as follows: (1) arithmetic ½(a + b) (2) geometric √(ab) (3) harmonic 2ab/(a + b).

Nicomachus of Gerasa, Introduction to arithmetic, translated by D'Ooge M. L. Robbins F. E. and Karpinsky L. C. , in Hutchins R. M. (ed.), Great books of the Western world , 11 (Chicago, London & Toronto, 1952), 848. See also Levin F. R. , “Nicomachus of Gerasa, Manual of harmonics: Translation and commentary” (Columbia University doctoral dissertation, 1967), 191–3. A similar sentiment is to be found in the medieval philosopher, Bacon R. , The Opus majus…, a translation by R. B. Burke (2 vols, New York, 1962), i, 118, 120, who stated that all philosophy is reducible to the mathematical means and that “whatever is worthy of consideration in the category of relation is the property of quantity, such as proportions and proportionalities, the geometrical, arithmetical, and harmonic means, and the kinds of greater and lesser inequality”.

Freeman K. , op. cit. (ref. 9), 24–34, especially Fragments 8. 51 and 54. Jones W. H. S. , “Heracleitus on the universe”, Hippocrates with an English translation (4 vols, London & Cambridge, Massachusetts, 1957–59), iv, 450–509 gives also the testimonies in translation.

Freeman K. , op. cit. (ref. 9), 51–64, especially Fragments 17, 22, 23 and 96. There is little evidence to suggest that Empedocles arrived at his theory of harmony through musical insights; but he could have done so by observing the performance of a musician, whose sung recitations had to accord with his accompaniment on the lyre. That is, voice and instrument two polarities—had to be in tune. The Greek reliance on music may stem from that subject being the oldest of the arts (along with gymnastic or dance) and from its embracing a much wider field than we now understand by the term ‘music’. See Kassler J. C. , op. cit. (ref. 5). vol. 1. p. xxvi, and Lippman E. , Musical thought in ancient Greece (New York. 1975: First issued, 1964).

For examples, see Heath T. , A history of Greek mathematics (2 vols, Oxford, 1921): Clagett M. The science of mechanics in the middle ages (Madison & London. 1959); and Grant E. (ed.). A source book in medieval science (Cambridge. Massachusetts, 1974).

According to Sambursky S. , Physics of the Stoics (London, 1971). vii–viii. the essential feature of Stoic theory was the dynamic notion of the concept of continuity, which led “the older Stoa…to a first grasp of the modern mathematical notions of the function and the limit and so constitutes the first break through of the barriers of the merely static contemplation of mathematical quantities”. The acoustical theories of the Stoics, which are surveyed by Sambursky, provide an apt illustration of their dynamic conception of the pneuma: But the musical implications of their theories have been little noticed. For example, tonos, as in pneumatikos tonos (tension of the pneuma), was also a musical term which could mean pitch, interval, key (i.e., locus of the voice), and even harmonia. See Michaelides S. , The music of ancient Greece: An encyclopaedia (London, 1978), 335–40. Musical implications are also absent in Sambursky's interesting study, “Harmony and wholeness in Greek scientific thought”, in Braudel F. (ed.), Mélanges Alexandre Koyré… (2 vols, Paris, 1964), ii, 442–57.

For the range of fields in which the Empedoclean conception of harmony was applied, see Hall T. S. , Ideas of life and matter: Studies in the history of general physiology 600 B.C. – 1900 A.D. (2 vols, Chicago & London, 1969). The index to vol. i of this work includes the term ‘harmony’ with references to only five pages. However, in his paraphrase of many writers, Hall inadvertently uses a number of musical terms present in the original sources (e.g., ‘attune’, ‘well-tempered’) so that the page references to harmonic conceptions are incomplete. The Empedoclean conception of harmony was also applied to the ‘physiology’ of music, an example of which is to be found in Bragard R. (ed.), Iacobi Leodiensis speculum musicae… (7 vols, Rome, 1955–73), iv, 92ff., 103ff. and passim.

I have used Cornford F. M. , Plato's cosmology: The Timaeus of Plato translated with a running commentary (London, 1971).

ibid., 27–97.

The different kinds of harmonic order are detailed by Lippman E. , op. cit. (ref. 14).

Spitzer L. , Classical and Christian ideas of world harmony: Prolegomena to an interpretation of the word ‘Stimmung’… (Baltimore, Maryland, 1963, revised and expanded from articles published first in the 1940s in Traditio), 14–15.

Fowler H. N. , “Phaedo”, Plato with an English translation (London & Cambridge, Massachusetts, 1960), 315–29.

Cornford F. M. , op. cit. (ref. 18), 27–28 and passim. The fundamental idea in Plato's philosophy is summarized by Dijksterhuis E. J. , The mechanization of the world picture , translated by Dikshoorn C. (London, Oxford & New York, 1969), 13, thus: “…things perceived by us are only imperfect copies, imitations or reflections of ideal forms…which in a suprasensible world beyond space and time lead to an independent existence that can only be expressed by pure thought”. Thus, Plato's analogies or ‘geometrical equalities’ may be represented by the formula, a:b:: C:d. Three of these terms refer to the conception, “ideas are to things as things are to reflections or mirrors in water”. The fourth term is found in mathematical ideas, that is, in the ultimate harmony of things which are harmoniously ordered according to reason.

Bochner S. , The role of mathematics in the rise of science (Princeton, 1966), 149.

Tracy T. J. , Physiological theory and the doctrine of the mean in Plato and Aristotle (The Hague & Paris, 1969), 344–6.

Burrell D. , Analogy and philosophical language (New Haven, 1973), 32.

The three types of music were distinguished by Boethius in his De institutione musica. See Bower C. M. , “Boethius' The principles of music: An introduction, translation and commentary” (George Peabody College for Teachers doctoral dissertation, 1967). Although Boethius seems to have limited musica instrumentalis to the music of instruments (according to the translation of Bower), he also included music produced by instruments governed by breath. Hence, by extension musica instrumentalis could denote the music of voices as well as of instruments; indeed, it could be used more inclusively to signify practical or applied music. This is how the term was used by Hugh of St. Victor, The Didascalicon… a medieval guide to the arts , translated from the Latin with an introduction and notes by Taylor J. (New York & London, c1961), 69–70.

The reconstruction of the science of harmonics is based on my own research and may need some modification as further studies are undertaken into the neglected history of this discipline. Two pioneering works, however, that treat facets of the science of harmonics should be cited in this place: Hunt F. V. , Origins in acoustics: The science of sound from Antiquity to the age of Newton (New Haven & London, 1978), and Truesdell C. A. , The rational mechanics of flexible or elastic bodies 1638–1788… (Zurich, 1960). Both authors, however, seem to be unaware that the ‘parent’ discipline was harmonics, of which acoustics and rational mechanics were ‘offspring’.

A principal source for this view is De institutione musica of Boethius, in which the author argued that judgment ought to be based on reason, not sense. According to Boethius, the “sense focusses its power on something confused but yet approximate to that which it senses. But reason judges truth, and searches out the ultimate difference. Thus the sense finds something confused but close to the truth, but one accepts the truth from reason”. For Boethius, reason was the province of a ‘real’ musician, one who—through a liberal education—had acquired the ability to exercise judgment. Performers and composers could never aspire to the definition of a real musician, since they were merely mechanically trained artisans, the one studying music to exhibit skill (manual or vocal dexterity) and the other, to display natural instinct. See Bower C. M. , op. cit. (ref. 27), 44–48, 57–59, 101–104, 295–300 and passim. The low opinion of the mental capacities of practical musicians continues to this present day. For an example of such a view expressed in the nineteenth century, see Kassler J. C. , op. cit. (ref. 5), ii, 1002.

See Lippman E. , “The place of music in the system of the liberal arts”, LaRue J. (ed.), Aspects of medieval and renaissance music… (New York, 1966), 545–59. Historians of science tend to overlook the implications of the fluctuating position of music in the curriculum of the seven liberal arts or to appreciate that music had a variable position.

For example, Roger Bacon focussed on the metric and rhythmic aspects of music, since, like Augustine, he was concerned to employ musical doctrine as a tool for investigating the language of Scripture. Thus, he held that “grammar depends causatively on music” as does logic, for the “whole utility” of logic “is drawn from the relations of all logical arguments…and therefore since they depend on the arguments of music, necessarily logic must depend on the power of music”. See Bacon R. , op. cit. (ref. 12), i, 118–19.

Scholars are not in agreement about the early history of the mathematical doctrine of music. For example, Szabó Á. , The beginnings of Greek mathematics (Dordrecht & Boston, 1978), provides evidence based on linguistic analysis that all the important terms of the theory of proportion had their origin in the theory of music. An earlier version of this argument was challenged by Knorr W. R. , The evolution of the Euclidean Elements: A study of the theory of incommensurable magnitudes and its significance for early Greek geometry (Dordrecht & Boston, 1975), 242, who states that “music theory from its very origins was an adjunct of the study of integers and integral ratios, and the adoption of geometric representations in that field…occurs only because the same practice had become fixed in arithmetic theory proper”. Knorr's position is supported by Burnyeat M. F. “The philosophical sense of Theaetetus' mathematics”, Isis, lxix (1978), 489–513.

Henderson I. , “Ancient Greek music”, in Wellesz E. (ed.), New Oxford history of music (London, 1957), i, 341–2 and Lippman E. , op. cit. (ref. 14).

Burkert W. , op. cit. (ref. 10); Crocker R. L. , op. cit. (ref. 10); and Knorr W. R. , op. cit. (ref. 32).

Henderson R. , op. cit. (ref. 33).

The principal treatises representing the view of the Canonists are the Euclidean Sectio canonis and the Harmonica of Ptolemy. See Mathieson T. J. , “An annotated translation of Euclid's division of a monochord”, Journal of music theory, xix (1975), 236–58 and Düring I. (ed.), Die Harmonielehre des Klaudios Ptolemaios… (Göteborg, 1932). To the best of my knowledge, no theoretical treatise representing the position of the Harmonists is extant, but some of the doctrines of that school were detailed by the thirteenth century writer, Bryennius M. , The harmonics , edited with translation, notes, introduction and index of words by Jonker G. H. (Gronigen, 1970). It is possible that the different methods of these two schools correspond to the different methods of astrologers and astronomers up to and including the time of Ptolemy. See, for example, Neugebauer O. , The exact sciences in Antiquity (2nd ed., Providence, Rhode Island, 1957), 157–8, who describes the ‘linear methods’ of the astrologers and the ‘arithmetical methods’ of the astronomers.

The harmonics of Aristoxenus, edited with translation, notes, introduction and index of words by Macran H. S. (Oxford, 1902). The selected quotations given by Macran from Aristoxenus's Rhythmic elements suggest that the same procedure was employed in that treatise.

Crocker R. L. , “Aristoxenus and Greek mathematics”, in La Rue J. (ed.). Aspects of medieval and renaissance music… (New York, 1966), 96–110.

Crocker R. L. , op. cit. (ref. 38), 100.

Henderson I. , op. cit. (ref. 33), 343–4.

Crocker R. L. , op. cit. (ref. 38), 101. The two different mathematical approaches, insofar as the doctrine of tuning was concerned, are summarized by Burkert W. , op. cit. (ref. 10), 369–71. In general, the Pythagoreans relied on proportional tuning systems, whereas from the time of Aristoxenus equal temperament was, in principle, possible. However, Burkert makes the important point that intervals thought of as lengths of line could correspond in modern mathematics to the logarithms of the respective ratios, so that to this extent at least the two descriptive systems of the Aristoxenians and the Pythagoreans are equally accurate and can be converted one into another. This mathematical schematicism, of course, was unknown before the seventeenth century, when logarithms were first invented; and it was not until 1653 that logarithms were first applied to music (by William Brouncker). A recent author argues that Plato's Republic is a treatise on equal temperament and that Plato's basic material was common property of the major Middle Eastern civilizations for the preceding three thousand years. However, these claims have yet to be critically evaluated. See McClain E. G. , The myth of invariance: The origin of the gods, mathematics and music from the RgVeda to Plato (New York, 1976), and The Pythagorean Plato: Prelude to the song itself (Stony Brook, N.Y., 1978).

For example, the Manual of harmonics, written in the second century by Nicomachus of Gerasa, exhibits certain features of both Aristoxenian and Pythagorean schools. In his assignment of number and numerical ratios to notes and intervals, in his recognition of the indivisibility of the octave and whole tone, and in his treatment of musical consonances as being either in multiple or superparticular ratios, Nicomachus adopted Pythagorean views. However, he attempted to validate some of these views by reference to the measurements of lengths of strings, so that his treatment of consonances, musical genera, note and interval are Aristoxenian. Despite the presence of the latter features, Nichomachus's treatise may be characterized as ‘Pythagorean’, since its approach is rationalist and since the mathematical operations employed refer to whole numbers only. Thus, while, for example, he accepted Aristoxenus's definition of interval as the distance bounded by notes which have not the same pitch, he disassociated himself from the Aristoxenians when the question of measurement of intervals arose. See Levin F. R. , op. cit. (ref. 12), 253–6, 260–5.

In an earlier version of this paper, I included examples of the application of the musical model in the natural sciences of medicine, alchemy and music. This part, which has been excised, will be published elsewhere. For some illustrations of harmonic conceptions and their musical expression in fields not covered in this paper, see Daly J. , Cosmic harmony and political thinking in early Stuart England (Philadelphia, 1979); Finney G. L. , Musical backgrounds for English literature: 1580–1650 (New Brunswick, New Jersey, 1962); Hollander J. , The untuning of the sky: Ideas of music in English poetry 1500–1700 (Princeton, 1961); Walker D. P. , Spiritual and demonic magic from Ficino to Campanella (London, 1958); and Wittkower R. , Architectural principles in the age of humanism (2nd ed., London, 1952). All these books, however, are limited to the period of the Renaissance, and a more synoptic view is offered by Spitzer L. , op. cit. (ref. 21).

E.g., Aquinas Thomas , whose theory of analogy is treated by Burrell D. , op. cit. (ref. 26), 119–70 and passim. For an elaborate example of the musical model applied to Scripture, see Chamberlain D. S. , “Wolbero of Cologne (d. 1167): A zenith of musical imagery”, Mediaeval studies, xxxiii (1971), 114–26.

The theme of harmony continues right up to the present day, of course, but after the seventeenth century it is rarely expressed musically. Exceptions to this rule occur in the domain of religion, the occult and the fine arts, one of the most interesting examples of the application of a musical model to art being found in the work of Wassily Kandinsky.

It was not my purpose to elucidate the different uses of the musical model; indeed, much research remains to be done by philosophers of science on this problem.

Summaries of astronomical and astrological doctrines may be found in Dijksterhuis E. J. , op. cit. (ref. 23), 54ff., 84ff., 154ff., 209ff. The Roman Catholic Church tolerated astrology as applied astronomy but not as divination. Therefore, Hugh of St Victor, Roger Bacon and other scholastics classed astronomy as an instructional science (mathesis) but allocated divination to the ‘vain’ or ‘false’ sciences (matesis).

Ptolemy C. , Tetrabiblos , edited and translated into English by Robbins F. E. (London & Cambridge, Massachusetts, 1956), 73–75. For a study of geometrical forms, including the aspects, see the article by de Solla Price D. on geometrical and scientific talismans and symbolisms, in Teich M. and Young R. (eds.), Changing perspectives in the history of science… (London, 1973), 250–64.

Oresme N. , Le livre du ciel et du monde , edited by Menut A. D. and Denomy A.J. ; translated with an introduction by Menut A. D. (Madison, Milwaukee & London, 1968), 485–6.

The first extant allusion to, and exposition of, the doctrine of music of the spheres occurs in Plato's Republic (Books VII and X). For the later allusions see Heath T. , Aristarchus of Samos: The ancient Copernicus… (Oxford, 1959), 107–15 and Meyer-Baer K. , Music of the spheres and the dance of death: Studies in musical iconology (Princeton, 1970). It is important to note that the conception of cosmic music was not bound to any particular astronomical system.

Even though Ptolemy did not class conjunction as an aspect, he treated it as such throughout the Tetrabiblos.

Oresme N. , op. cit. (ref. 49), 485, where the translator has rendered medietas sesquitercie as half-sesquitertial or one-half the proportion between 4 and 3; medietas duple as double proportion or 2:1; duple as the proportion is double; medietas sesquialtere as half the sesquialtera or 3:2; medietas tripla as half the proportion called triple; and medietas duple as half the double. However, Clagett M. , op. cit. (ref. 15), 464, points out that when “the schoolmen spoke of one proportion being ‘double’ another, they meant that it was the ‘square’ of that proportion. The words ‘triple’, ‘quadruple’, etc., when applied to the comparison of proportions meant ‘cubed’, ‘raised to the fourth power’, etc. On the other hand, when the expression a ‘double proportion’ was used, it meant a proportion of 2:1. Similarly a ‘triple proportion’ was 3:1, etc”.

Clagett M. , “Nicole Oresme”, Dictionary of scientific biography, xiv (1976), 224.

Grant E. , Nicole Oresme and the kinematics of circular motion: Tractatus de commensurabilitate vel incommensurabilitate motum celi, edited with an introduction, English translation, and commentary (Madison, Milwaukee & London, 1971). Oresme dealt with the subject of the kinematics of circular motion in an earlier treatise. See Oresme N. , De proportionibus proportionum and Ad pauca respicientes, edited with introductions, English translations, and critical notes by Grant E. (Madison & London, 1966), in which Grant provides an interesting survey (83ff.) of the few scientists before Oresme who held that celestial motions probably were incommensurable. In particular, Grant states that “specific utterances concerning the incommensurability of two or more celestial motions have not yet been found earlier than the twelfth century” and that there were only a few mentions of the topic between the twelfth and mid-fourteenth centuries. In view of Oresme's contributions to mathematics and his apparent interest in the doctrine of music, it is highly likely that he drew his ideas about commensurability and incommensurability from both theoretical and practical harmonics. For the mathematical foundation of Oresme's argument in favour of incommensurability, see Clagett M. , Nicole Oresme and the medieval geometry of qualities and motions: A treatise on the uniformity and difformity of intensities known as Tractatus de configurationibus qualitatum et motum, edited with an introduction, English translation, and commentary (Madison, Milwaukee & London, 1968), 10–11.

Grant E. , op. cit. (ref. 54), 19–20.

ibid., 67–77.

ibid., 317.

Coopland G. W. , Nicole Orēsme and the astrologers: A study of his Libre de divinacions (Liverpool, 1952), 25–48. Oresme's influence, of course, was in the area of mathematics.

Copernicus , On the revolutions of the heavenly spheres , translated by Wallis C. G. , in Hutchins R. M. (ed.), Great books of the Western world, 16 (Chicago, London & Toronto, 1952), 525 n. 1. This work also includes a concise explanation of the doctrine of chords (532ff).

The collected works of Kepler have been edited by Caspar M. ; only Book V of the Harmonices mundi libri V has been translated into English. See Kepler J. , The harmonies of the world , translated by Wallis C. G. , in Hutchins R. M. (ed.), Great books of the Western world , 16 (Chicago, London & Toronto, 1952), 1005–85. See also Dickreiter M. , Der Musiktheoretiker Johannes Kepler (Bern & Munich, 1974), and Rogers J. and Ruff W. , “Kepler's harmony of the world: A realization for the ear”, American scientist, lxvii (1979), 282–92.

Godwin J. , Robert Fludd: Hermetic philosopher and surveyor of two worlds… (London, 1979).

Intimations of this movement are to be found in Three books of occult philosophy…translated out of the Latin into the English tongue, by J. F. (London, 1651; first Latin edition, 1533), 74, by the physician, soldier, lawyer, theologian, philosopher and magician, Heinrich Cornelius Agrippa von Nettesheim.

The controversy is summarised by Amman P. J. , “The musical theories and philosophy of Robert Fludd”, Journal of the Warburg and Courtauld Institutes, xxx (1967), 198–227, and Wightman W. P. D. , Science in a renaissance society (London, 1972).

Some of these innovations are surveyed by Lowinsky E. E. , “Music in the culture of the Renaissance”, Journal of the history of ideas, xv (1954), 509–53.

See Hutton J. , “Some English poems in praise of music”, English miscellany, ii (1951), 1–63.

It is important to note that the dislocation of the music semantic field occurred also because of significant changes in the doctrine of music, some of which are detailed by Kassler J. C. , op. cit. (ref. 5).

Cochlaeus J. , Tetrachordum musices, introduction, translation and transcription by Miller C. A. (Rome, 1970), 19–21. Translation copyright © 1970 by Armen Carapetyan of the American Institute of Musicology.

Murdoch J. E. , “Music and natural philosophy: Hitherto unnoticed Ouestiones by Blasius of Parma (?)”, Manuscripta, xx (1976), 119–36.

Kilwardby R. , De ortu scientiarum, edited by Judy A. G. (Toronto, 1976), 58–59 (English translation by Brundell B. ).

May H. G. and Metzger B. M. (eds.), The new Oxford annotated Bible with the Apocrypha… (New York, 1973), 115.

Cusanus N. , Of learned ignorance, translated by Heron G. (London, 1954), 118–19.

Kassler J. C. and Oldroyd D. R. , “Robert Hooke's Trinity College ‘Musick Scripts’, his music theory and the role of music in his cosmology” (forthcoming).

Tatarkiewicz W. , “Form in the history of aesthetics”, Dictionary of the history of ideas , ed. Wiener P. , ii (New York, 1973), 216–25. The historical representative of proportion (temperare) was usually Pythagoras holding a monochord or a balance. Thus, he symbolized not only proportion but also proportionate mixture (including temperature, temperament and temperance). This point is completely missed by White L. Jr , “The iconography of Temperantia and the virtuousness of technology”, in Rabb T. K. and Siegel J. E. (eds), Action and conviction in early modern Europe… (Princeton, 1969), 197–219. The symbol for both music and light was Apollo.

Other consonances were gradually accepted, but these were classed as ‘imperfect’. See, for example, the classifications of Iacobi Leodiensis in Bragard R. , op. cit. (ref. 17).

There is no work on the transmission of Aristoxenus's thought comparable to the numerous books and monographs on Pythagoreanism. See, for example, the bibliographies listed in Vetter W. , “Aristoxenos”, Musik in Geschichte und Gegenwart, i (1949–51), cols 646–654, and Winnington-Ingram R. P. , “Aristoxenus”, Dictionary of scientific biography, i (1970). 281–3. In addition to translations and commentaries of Aristoxenus's treatises made from the time of the Renaissance. his thought was disseminated by such disciples as Aristides Quintilianus and Cleonides, and by Vitruvius's summary (somewhat inaccurate) in De architectura. Copies of Aristoxenus's works, of course, circulated in private hands. See, for example, Halliwell J. O. (ed.), The private diary of Dr. John Dee, and the catalogue of his library of manuscripts… (Cambridge, 1842).

Some suggestive studies on artifacts of the Middle Ages and Renaissance are Lowinsky's E. E. Secret chromatic art in the Netherlands motet , translated from the German by Buchman C. (New York, 1967; 1st publd 1946) and James's J. The contractors of Chartres (Dooralong, 1979), the latter of whom surveyed Chartres Cathedral. Of importance also is Victor's S. K. Practical geometry in the high middle ages… (Philadelphia, 1979), who deals not with artifacts but with hitherto unstudied practical geometry texts.

Lowinsky E. E. , op. cit. (ref. 64), 543. Interpreting the term ‘mathematician’ in its proper sense to include musicians, Lowinsky is surely correct in stating: “The fact that Johannes de Muris evolved his notational method before Nicholas of Oresme presented a consistently elaborated theory and mathematical notation of power development suggests that this topic had occupied French mathematicians for some time. Nicholas of Oresme was probably the man who summarized and brought to a conclusion the mathematical thought of a whole generation”.

Oresme N. , “An algorism of ratios: Manipulation of rational exponents…”, translated and annotated by Grant E. , in Grant E. , op. cit. (ref. 15), 150–7 (translation of the first of three parts only). This treatise is considered to be the first extant systematic attempt to describe operational rules for multiplication and division of ratios involving integral and fractional exponents. Its date is unsettled, and historians of science seem to be unaware of another treatise that has bearing on the subject, the De numeris harmonicis of Levi ben Gershon (Gersonides, Leon de Bagnols, Magister Leo Hebraeus). This work, which is discussed by Werner E. , “The mathematical foundation of Philippe de Vitri's Ars nova”, Journal of the American Musicological Society , ix (1956), 128–32, contains the following statement: In the year 1342 of the incarnation of Christ, my work on mathematics having been completed, I was requested by the noted master of musical theory, Master Philippe de Vitri, to demonstrate a certain postulate of that science. The postulate is this: All pairs of harmonic numbers are mathematically distinguishable except the following: 1 and 2, 2 and 3, 3 and 4, 8 and 9. We mean by a harmonic number every number that is itself divisible and subdivisible down to a unity not only by 1 but also by either 2 or 3 or by any combination of 2 and 3 (and whose factors are similarly divisible and subdivisible down to unity). Examples of harmonic numbers are 2, 4, 8, 9, 12, 27, 18, 24, etc.

See, for example, Needham J. , Science and civilisation in China … (Cambridge, 1954-), esp. iv, pt 1, 126–228; Farmer H. G. , “The music of Islam”, in Wellesz E. (ed.), New Oxford history of music (London, 1957), i, 421–77; Nasr S. H. , Islamic science (London, 1976); Job of Edessa, Encyclopaedia of philosophical and natural sciences as taught in Baghdad about AD. 817 , Syriac text edited and translated with a critical apparatus by Mingana A. (Cambridge, 1935); and Tagore S. M. , Universal history of music…with various original notes on Hindu music (Calcutta, 1896), appendix and addenda separately paginated, i.xiv, i-xx.

For example, in 1600 Willliam Gilbert stated that “a body in touch with another body by peculiar radiation of effluvia makes of the two one: United, the two come into most intimate harmony, and that is what is meant by attraction” (“unita confluunt in coniunctissimam convenientiam, quae attractio vulgo dicitur“). See Gilbert W. , De magnete, translated by Mottelay P. F. (New York, 1958), 91.

The fact of sympathetic vibration was well known to the ancients and was one of the harmonic problems in the Aristotelian Problemata. See Hett W. S. , Aristotle: Problems I and II , with an English translation (2 vols, London & Cambridge, Massachusetts, 1961–62), 392–3, 407–9. However, the law of vibrating strings was not discovered until the sixteenth century, when it was enunciated by Benedetti G. B. (1585) and rediscovered in the seventeenth century by Beeckman I. (1614–15), Mersenne M. (1623) and Galilei G. (by 1636). In solving the problem of sympathetic vibration, Galileo formulated the law of vibrating strings that the ratios representing musical intervals were not immediately determined either by the length, size or tension of strings but rather by the ratio of their frequencies. This achievement may have been stimulated by an increased interest in the problem of sympathetic vibration during the period of the Renaissance, when, for example, Girolamo Fracastoro described sympathetic vibration as follows: A unison moves another unison, for strings that are similarly tightened are by nature fitted to make and to receive similar air waves. But strings that are tightened unequally are not apt to be moved by the same circulators, for the one circulation hinders the other of the string [so that] the motion is a composite of two motions: Of one by which the string is driven on, that is, towards the circulations of the air, and of another which goes in reverse as the string returns to its proper place. If, therefore, one string is to be moved itself and to move another, it is required that the second string be in such proportion that the waves and circulations of the air which impel it and produce the forward motion do not hinder the return of the string. Only [those] strings which happen to have unlike tension do not move each other, for while the second motion takes place—that is, the return of the string—the second circulation opposes it and [the motions] hinder each other; whence there is no motion apart from the first impulse, which is imperceptible. See Fracastoro G. , “De sympathia et antipathia liber I”, Opera omnia (Venice, 1555), 90 verso (English translation by Brundell B. ).

See especially Truesdell C. A. , op. cit. (ref. 28), and Drake S. , op. cit. (ref. 8).

Digby K. , Two treatises: In the one of which, the nature of bodies: In the other, the nature of mans soule, is looked into… (London, 1645; first issued, Paris, 1644). 349; see also 301–11, 335, 336, 339–40, 344 and passim. Digby is said to be the first writer to describe folie à deux, a phenomenon which he thought was analogous to sympathetic vibration. See Hunter R. and Macalpine I. , Three hundred years of psychiatry 1535–1860 (London, 1970), 124–7.

Hunt F. V. , op. cit. (ref. 28), 46. Plato, of course, early recognized that music could disturb and even destroy existing political orders, and this notion is implicit in many papal bulls on sacred music. See, for example, Hayburn R. F. , Papal legislation on sacred music 95 A.D. to 1977 A.D. (Collegeville. Minnesota. 1979). The papal bull relating to the licences of the Ars nova is given at pp. 20–21, where is stated: “…we are prepared to take effective action to prohibit, cast out, and banish such things from the church of God” Such a strong reaction to the Ars nova arose not merely from liturgical considerations but also from a perceived danger to existing cosmological views.