Abstract
The scientific revolution saw a shift from the natural philosophy of music to the science of acoustics. Joseph Sauveur (1653–1716), an early pioneer in acoustics, determined that science as understood in the eighteenth century could not address the fundamental problems of music, particularly the problem of consonance. Building on Descartes, Mersenne, and Huygens especially, Sauveur drew a sharp divide between sound and music, recognizing the former as a physical phenomenon obeying mechanical and mathematical principles and the latter as an inescapably subjective and unquantifiable perception. While acoustics grew prominent in the Académie des sciences, music largely fell out of the scientific discourse, becoming primarily practiced art rather than natural philosophy. This study illuminates what was considered proper science at the dawn of the Enlightenment and why one particular branch of natural philosophy—music—did not make the cut.
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Notes
Kuhn refers to the classical science of music as “harmonics,” although to scholars both ancient and early modern it was simply musica.
The question of why certain musical intervals sound pleasant and pleasing while others are painful and jarring is perhaps the oldest problem in the history of music, originating in the apocryphal story of Pythagoras happening upon four (or five depending on which version of the myth we believe) variously sized hammers—each ringing out a different tone—at the forge. From this, Pythagoras supposedly made the first connection between consonance and whole number ratios.
Within the senario, the octave had a ratio of 2:1, the fifth was 3:2, the fourth was 4:3, and so on. With the exception of the 8:5 ratio of the minor sixth (which inspired numerous ad hoc explanations over the centuries), this convention accounted for all traditionally accepted consonances in medieval music. By the seventeenth century, the physico-empirical coincidence theory replaced Zarlino’s numerological explanation.
Temperament is a system in which intervals are deliberately out of tune from their “pure” Pythagorean ratios (such as 9:8 for a whole tone). Equal temperament tempers intervals such that all semitones (the smallest interval) form identical ratios.
This is what musicians today call tone color or timbre.
Newton himself cited Sauveur’s calculation of the frequency of organ pipes in the second edition of the Principia. Newton computed of the wavelength (“lengths of the pulses”) produced by organ pipes as roughly twice the length of the pipe. Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, trans. I. Bernard Cohen and Anne Miller Whitman (Berkeley: University of California Press, 1999), scholium in book II, sec. 8, 778.
I have presented Sauveur’s mathematics in its original notation. Note that Sauveur writes nn in a time before the exponential notation n 2 became common. He eventually determines the relation \(f = \frac{cnn}{8ap}\) in which n is length, p is tension, \(\frac{c}{a}\) represents the weight per unit length, and f is the flèche or sag in the string. We can only speculate whether Sauveur intentionally named the sag after La Flèche, the village of his birth. His use of Leibnizian differentials to find the vertical component of tension made Sauveur’s allegiance to the modernist camp of analysts during the calculus dispute clear.
Hence, \(P = \frac{4}{5}f = \frac{cnn}{10ap}\).
In modern form, \(F \approx \frac{1}{2\pi }\sqrt{\frac{g}{L}}\) for frequency F, acceleration due to gravity g, and length of string L.
Synthesis refers to the process of changing timbre by adding sine waves together in a particular manner. Organs since the fifteenth century were built with a choice of stops such that different combinations of pipes could produce a single pitch with different timbres. Sauveur likened this process to painters mixing colors to create new shades.
References
Thomas S. Kuhn, “Mathematical vs. Experimental Traditions in the Development of Physical Science,” The Journal of Interdisciplinary History 7 (1976), 1–31, on 9; H. Floris Cohen, Quantifying Music: The Science of Music at the First Stage of the Scientific Revolution, 1580–1650 (Dordrecht: D. Reidel, 1984).
Joseph Sauveur, “Système général des intervalles des sons,” in Collected Writings on Musical Acoustics: (Paris 1700–1713), ed. Rudolf Rasch (Utrecht: The Diapason Press, 1984), 16. The term “acoustic,” however, had been used earlier in a similar sense. See Narcissus Marsh, “An Introductory Essay to the Doctrine of Sounds, Containing Some Proposals for the Improvement of Acousticks,” Philosophical Transactions 14 (1684), 472–88.
Bernard de Fontenelle, Éloges des académiciens de l’Académie royale des sciences, morts depuis l’an 1699 (Paris: Libraires Associés, 1766).
Sauveur’s definition of music more or less held sway until twentieth-century composers, most obviously John Cage, took issue with the “agreeable” part. See Joseph Sauveur, Joseph Sauveur’s “Treatise on the Theory of Music”: A Study, Diplomatic Transcription and Annotated Translation, trans. Richard Semmens, Studies in Music from the University of Western Ontario 11 (London, Ont., Canada: University of Western Ontario, Faculty of Music, Dept. of Music History, 1987), 89.
Sauveur, “Système général” (ref. 2), 99.
Ofer Gal and Raz Chen-Morris, Baroque Science (Chicago: The University of Chicago Press, 2013), 26.
The most accessible enunciation of this idea is in the Treatise on Man. See René Descartes, The Philosophical Writings of Descartes: Volume 1, trans. John Cottingham, Robert Stoothoff, and Dugald Murdoch (Cambridge: Cambridge University Press, 1985), 105.
Isaac Beeckman and Cornelis de Waard, Journal tenu par Isaac Beeckman de 1604 à 1634 (La Haye: M. Nijhoff, 1939), 1:313. Discussed in Cohen, Quantifying Music (ref. 1), 140.
Marin Mersenne, Harmonie universelle, Livre premier des consonances, prop. 33. Quoted in Cohen, Quantifying Music (ref. 1), 107.
Christiaan Huygens, Oeuvres complètes de Christiaan Huygens, ed. Johan Adriaan Vollgraff and Hollandsche Maatschappij der Wetenschappen (La Haye: M. Nijhoff, 1888), 19:364.
Although rumors portraying Sauveur as the deaf scientist of sound (the “Beethoven of science”) are certainly exaggerations if not outright myths, he was indeed born with a severe speech impediment that only marginally improved later in life. This myth was likely born when Fontenelle recalled that he “had neither voice nor ear.” What little we know of Sauveur’s personality also comes directly from Fontenelle, who in his Éloges painted a picture of a shy, gentle, humorless, and modest individual—a plainspoken man of exceptional talent but simple tastes and conventional faith.
For Sauveur’s personality, see Fontenelle, Éloges (ref. 3), 424; Sauveur, Collected Writings (ref. 2), 8. For his musical training, see Rudolf Rasch, “Sauveur on Music and Acoustics,” in Sauveur, Collected Writings (ref. 2), 14.
Sauveur, Treatise (ref. 4), 89.
Johannes Kepler, The Harmony of the World, trans. E. J. Aiton, A. M. Duncan, and J. V. Field (Philadelphia: American Philosophical Society, 1997), 447–8.
Music in the medieval and Renaissance eras was principally a mathematical discipline and not exactly part of what was then considered “natural philosophy.” Like its fellow quadrivium discipline astronomy, music gradually entered the philosophical discourse of the seventeenth century as it acquired a physical dimension (finding harmony in nature) and natural philosophy grew increasingly mathematized. When I call music part of natural philosophy, I mean that Galileo, Mersenne, Huygens, and others studied it alongside other physico-mathematical problems of their day.
See Cohen, Quantifying Music (ref. 1); V. Coelho, ed., Music and Science in the Age of Galileo (Dordrecht: Springer, 1992); Paolo Gozza, Number to Sound: The Musical Way to the Scientific Revolution (Dordrecht: Kluwer Academic Publishers, 2000); Peter Pesic, Music and the Making of Modern Science (Cambridge, MA: MIT Press, 2014).
Cohen uses the term “coincidence theory” for the empirical observation that the vibrations of consonant intervals will coincide more often than dissonant intervals. An octave, whose two notes have frequencies of f and 2f, will coincide every other vibration of the higher frequency, whereas a much less consonant minor sixth, with frequencies 5f and 8f, coincide only one out of eight times. This theory, despite its many flaws, became the dominant explanation of consonance during the seventeenth century.
Sauveur, Treatise (ref. 4), 90–92, 104. Semmens translates âme as “soul” in the Cartesian sense, roughly equivalent to “mind” or “consciousness.” For the coincidence theory, see Cohen, Quantifying Music (ref. 1), 90.
René Descartes, The Philosophical Writings of Descartes: Volume 1, trans. John Cottingham, Robert Stoothoff, and Dugald Murdoch (Cambridge: Cambridge University Press, 1985), 152.
“Système général,” in Robert Eugene Maxham, “The Contributions of Joseph Sauveur (1653–1716) to Acoustics” (PhD diss., University of Rochester, 1976), part II, 1. I have used Maxham’s translations where needed. All other references to the “Système général” are from Rasch.
Penelope Gouk presents an intriguingly alternate rendition of Newton as a “Pythagorean magus.” Penelope Gouk, Music, Science, and Natural Magic in Seventeenth-Century England (New Haven: Yale University Press, 1999), 254–7.
Huygens, Oeuvres complètes (ref. 10), 19:364.
Sauveur, Treatise (ref. 4), 28. Sauveur’s ruler example is on page 125. Early advocates of the coincidence theory recognized similar scales of consonance long before Sauveur.
René Descartes, Compendium of Music: (“Compendium Musicae”), trans. Walter Robert, ed. Charles Kent (Rome: American Institute of Musicology, 1961), 13; Marin Mersenne, Harmonie Universelle contenant la théorie et la pratique de la musique (Paris: CNRS, 1963), Livre premier des consonances, prop. 33, coroll. 3. Discussed in Cohen, Quantifying Music (ref. 1), 109.
For references to Mersenne, see Sauveur, Collected Writings (ref. 2), 128, 159, 191, among others. For the role personal taste in music see Sauveur, Treatise (ref. 4), 121.
Sauveur, Treatise (ref. 4), 120.
Ibid., 31.
Though in the “Système général” Sauveur himself avowedly studied sounds “in order to infer from them the causes of the agreement and disagreement of sounds which serve as the objects of music and harmony,” this is exactly the sort of preamble one would expect for a treatise addressed to an Académie in which practical music played a growing role. The remainder of his treatise certainly did not back up this claim.
Maxham, “The Contributions of Joseph Sauveur,” (ref. 20), part II, 2.
Ibid., part I, 144.
Sauveur, Treatise (ref. 4), 187.
Richard Semmens, “An Early Eighteenth-Century Discussion of Musical Acoustics by Étienne Loulié,” Canadian University Music Review 2 (1981), 194.
Nodes are points on a vibrating string that remain relatively still compared to points immediately to the left and right. Huygens first observed that one could generate higher intervals by gently placing and releasing one’s finger on these points. Huygens, Oeuvres complètes (ref. 10), 19:366–7. Mersenne also employed the phrase petit son in Harmonie universelle. For Sauveur’s work see Sauveur, Collected Writings (ref. 2), 100–103. Sauveur did not coin the word “overtone.” The term originated as a loan translation of the German oberton, first used by Helmholtz as a contraction of overpartialton, the “upper partial tone.”
Thomas Christensen and Ian Bent, Rameau and Musical Thought in the Enlightenment (Cambridge: Cambridge University Press, 2004), 137.
Descartes, Compendium of Music (ref. 24), 18.
Aristotle, The Complete Works of Aristotle, ed. Jonathan Barnes (Princeton: Princeton University Press, 1984), 2:142. The Problems is not considered an authentic work of Aristotle, by many scholars and was likely compiled by the Peripatetic school over several centuries.
Marin Mersenne, Correspondance du P. Marin Mersenne, ed. Bernard Rochot, Cornelis de Waard, and Marie Paul Tannery (Paris: CNRS, 1932), 3:458.
Both Descartes and Mersenne maintained Beeckman’s corpuscular theory and viewed sound not as waves but as pulsations or “globules” of air. This conception almost singlehandedly precluded the explanation of overtones—as interference patters in air—reached in the eighteenth century.
Marin Mersenne, Harmonie Universelle: The Books on Instruments, trans. Roger E. Chapman (The Hague: M. Nijhoff, 1957), 269–71, 321.
Cohen, Quantifying Music (ref. 1), 102.
“Système général,” in Maxham, “The Contributions of Joseph Sauveur,” (ref. 20), part II, 70; Burdette Lamar Green, “The Harmonic Series from Mersenne to Rameau: An Historical Study of Circumstances Leading to Its Recognition and Application to Music” (Ph.D. diss., The Ohio State University, 1969), 407–17.
Why he heard them better at night remains a mystery. Maxham, “The Contributions of Joseph Sauveur,” (ref. 20), part II, 3.
Isaac Newton, Opticks, or, A Treatise of the Reflections, Refractions, Inflections & Colours of Light (New York: Dover Publications, 1952), 280.
Sauveur, Collected Writings (ref. 2), 180, 256.
Another much more extensive table in his “Application des sons harmoniques”—listing all harmonics, up to the 1024th term, that corresponded to musical intervals—further testifies to Sauveur’s painstaking attention to detail but also shows that he was never above presenting his studies of sound in a form actual musicians (organists in this case) might find useful. In the “Application,” unlike the mostly speculative “Système général,” he emphasized the sons harmoniques that produced octaves, thirds, and fifths, a practical choice for a practical text. Sauveur, Collected Writings (ref. 2), see “plate 5” insert.
Ibid., 150.
Mersenne, Harmonie universelle (ref. 24), prop. 10, coroll. 3. Discussed in Cohen, Quantifying Music (ref. 1), 109.
Huygens, Oeuvres complètes (ref. 10), 20:161–2.
Daniel Bernoulli’s proposal of the superposition principle (1753) partially solved this mystery by interpreting a single vibrating system as the sum total, or superposition, of many proper vibrations occurring simultaneously. A string could vibrate in many modes at once that all constructively interfered to build the fundamental mode. Joseph Fourier’s 1807 “Mémoire sur la propagation de la chaleur dans les corps solides” introduced the concept of Fourier analysis: any function could be modeled by a trigonometric series. For sound, this meant the series of sine functions whose wavelengths formed ratios matching the harmonic series could express the fundamental mode. Fourier thus gave rigorous mathematical expression to Bernoulli’s physical principle.
“Rapport des sons des cordes d’instruments des musique (1713),” in Sauveur, Collected Writings (ref. 2), 242–68 and “plate 7” insert.
An early proof of this relation appeared in Huygens’ “De vi centrifuga” of 1659; see Huygens, Oeuvres complètes (ref. 10), 16:289–90, 320–1.
Sauveur determined the sound (measured by the double frequency in modern terms) as \(S = \frac{\sqrt{\frac{1}{P}}}{\sqrt{\frac{1}{q} }} = \frac{\sqrt{q}}{\sqrt{P}} = \frac{\sqrt{q}}{\sqrt{\frac{cnn}{10ap}}} = \frac{\sqrt{10apq}}{\sqrt{cnn}}\). Technically, this gives a ratio to the frequency of the second pendulum. As Cannon and Dostrovsky noted, this differs from today’s calculation only by a factor of \(\frac{{\sqrt {10} }}{\pi } \approx 1.00658\). The π comes from the fact that \(q = \frac{g}{{\pi^{2} }}\) by the laws for a simple pendulum. Modern calculations incorporate Bernoulli’s superposition principle and do not treat strings as simple, rigid pendulums the way Sauveur did. John T. Cannon and Sigalia Dostrovsky, The Evolution of Dynamics (New York: Springer-Verlag, 1981), 25.
This is the result detailed in the “Système général” that Newton referenced. See Sauveur, Collected Writings (ref. 2), 159–62. For a deeper examination of Sauveur’s mathematics, see Cannon and Dostrovsky, The Evolution of Dynamics (ref. 52), 23–7. Sauveur improved upon Huygens’s 1673 derivation in which he compared the string to a cycloidal pendulum with equal restoring force; see Huygens, Oeuvres complètes (ref. 10), 18:489–494.
Bernard de Fontenelle, Histoire de l’Académie royale des sciences, 1700 (Paris: Imprimerie de Du Pont, 1702).
Albert Cohen, Music in the French Royal Academy of Sciences: A Study in the Evolution of Musical Thought (Princeton, NJ: Princeton University Press, 1981), 18–24.
Sauveur, “Système général,” (ref. 2), 117–22 and “plate 2” insert.
Mersenne, Harmonie Universelle (ref. 47), 136.
John Hawkins, A General History of the Science and Practice of Music (London: T. Payne and Son, 1776), 2:604.
When Sauveur’s speaks of “harmony that nature observes,” he refers to the mathematics of sons harmoniques and not to any Mersennesque belief that nature inherently favors harmony.
I draw this dichotomy between theory and practice with some reservations. As Jean-François Gauvin has noted, for example, an organum—defined as “an abstract or material tool designed for the sake of and end”—in the early modern world increasingly functioned as a terminus a quo: a beginning, not an end, of the search for scientific truth. Gauvin emphasizes the role habitus, the special training needed to utilize instruments, played in experimental science from Boyle’s air-pump to Mersenne’s musical instruments. Since Aristotle, musical instruments were the most common example of habitus; in this sense science and practical music were intimately intertwined all throughout the scientific revolution. For example Huygens, an occasional composer himself, envisioned his thirty-one-tone scale in terms of a special harpsichord, with either split black keys or a movable keyboard. Sauveur examined the construction of both string and pipe instruments in his studies of overtones and, having “no ear,” hired musicians to help him distinguish the excruciatingly small divisions of the octave in his “Système général.” Like his predecessors, he used tacit knowledge of musica practica as a tool towards scientific discoveries. Jean-François Gauvin, “Instruments of Knowledge,” in The Oxford Handbook of Philosophy in Early Modern Europe, ed. Desmond M. Clarke and Catherine Wilson (Oxford: Oxford University Press, 2013), 315–37. For Huygens’ scale and harpsichord, see Huygens, Oeuvres complètes (ref. 10), 20:141–73.
Rameau is the first person to whom we can attribute a “music theory” in the modern sense. Jean-Philippe Rameau, Treatise on Harmony, trans. Phillip Gossett, Reprint edition (New York: Dover Publications, 1971), xxxv.
To be called “the Newton” of one’s discipline was the highest honor a scholar in the eighteenth century could receive. Hawkins, Science and Practice of Music (ref. 58), 901.
Jean le Rond D’Alembert, Eléments de musique théorique et pratique suivant les principes de M. Rameau (Paris, 1752), quoted in Christensen, “Music Theory as Scientific Propaganda: The Case of D’Alembert’s Eléments de Musique,” Journal of the History of Ideas 50 (1989), 409–427, on 420.
Fontenelle, “Sur l’application des sons harmoniques aux jeux d’orgues,” Histoire de l’Académie royale des sciences, 1702 (Paris, 1704), 92. Also in Sauveur, Collected Writings (ref. 4), 170.
Rameau, Treatise on Harmony (ref. 61), 1, 7–13.
Acknowledgments
I would like to thank Peter Pesic, Robert Crease, and Joseph Martin for their help in editing this paper. I am thankful to Dr. Pesic in particular for his support and encouragement, as well as for writing the book that got me interested in the history of the science of music in the first place. I am also grateful to Jennifer Alexander for her comments. Finally, I thank my adviser Victor Boantza for his continued help in writing and revising this paper.
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Adam Fix is a graduate student in the Program in the History of Science, Technology, and Medicine at the University of Minnesota.
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Fix, A. A Science Superior to Music: Joseph Sauveur and the Estrangement between Music and Acoustics. Phys. Perspect. 17, 173–197 (2015). https://doi.org/10.1007/s00016-015-0164-x
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DOI: https://doi.org/10.1007/s00016-015-0164-x