SAUVEUR, JOSEPH

(b. La Flèche, France, 24 March 1653: d. Paris, France, 9 July 1716)

physics.

Sauveur worked on early problems in the physics of sound, especially beats, harmonics, and the determination of absolute frequency; and he was influential as a teacher of practical mathematics.

Sauveur was the son of Louis Sauveur, a notary, and Renée des Hayes. Born with a voice defect, he did not begin to speak until the age of seven and retained a lifelong difficulty with his speech. He first attended the famous Jesuit school of La Flèche, where arithmetic intrigued him. Hoping to learn science, Sauveur went to Paris in 1670, where he studied mathematics and medicine and attended the physics lectures of Jacques Rohault. Despite his speech problem, Sauveur became well known as a good teacher and was a tutor at the court of Louis XIV. Interested primarily in practical mathematics, he prepared tables for simplifying calculations and for converting weights and measures. He also worked on problems of engineering and in 1681 conducted hydraulic experiments with Mariotte at Chantilly. In 1691 he visited the town of Mons while it was under active siege, in connection with his plan to write a treatise on fortification. In 1703 he replaced Vauban as examiner for the Engineering Corps. When he obtained the chair of mathematics at the Collège Royal in 1686, Sauveur was sufficiently well known that he dared to read the required public speech (earlier, he had dropped a plan to apply for the post because the speech seemed too difficult for him). In 1696 he became a member of the Paris Academy of Sciences.

Like Mersenne and others in the seventeenth century, Sauveur used musical experience to obtain information on sound and vibration. According to Fontenelle, Sauveur was fascinated by music, even though he had no ear for it, and consulted frequently with musicians. Despite the musical foundation of his work, Sauveur proposed the development of a new subject, which he named acoustique¹ dealing with sound in general rather than with the son agréable of music.

Sauveur began his work in acoustics by developing a method of classifying temperaments of the musical scale. He divided the octave into forty-three equal intervals, or merides, each of which was divided into seven eptamerides.² Sauveur’s intention was to indicate the size of any musical interval, at least to a reasonable approximation with respect to the ability of the ear to discriminate pitch, in terms of an integral number of eptamerides. These divisions made it simple to describe and compare different tuning systems. For example, the fifths and thirds given by an integral number of merides approximate those of the one-fifth comma meantone tuning used in the sixteenth and seventeenth centuries.³

Sauveur’s first work on the physics of vibration, originally presented to the Paris Academy in 1700, concerned the determination of absolute frequency. The problem of pitch standardization was a natural successor to the problem of temperament standardization. Sauveur wanted to use a son fixe of 100 cycles per second. Pitch had been identified with frequency early in the seventeenth century, and the ratio of the frequencies of a pair of tones was known, ultimately, from the inverse relative string lengths.⁴ Sauveur was the first to use beats to determine the frequency difference and was therefore able to calculate the absolute frequencies. Since he correctly interpreted beats, it appears that Sauveur may have been the first to have an understanding of superposition.

To determine absolute frequency, Sauveur used a pair of organ pipes a small half-tone apart in just intonation (frequency ratio 25:24). This interval is sufficiently small that the beats can be counted, for low pitches. Furthermore, the interval can be obtained accurately by tuning through thirds and perfect fifths (for example, by tuning up two major thirds and then down a fifth). As a result of experiments done with Deslandes, an organ builder, Sauveur found that the frequency of an open organ pipe five Paris feet long was between 100 and 102 cps. Sauveur claimed to have obtained consistent results from experiments done with other pipes. Newton made a rough check of Sauveur’s results; knowing that the velocity of sound is the product of frequency and wavelength, he knew the wavelength of a tone of 100 cps to be just over twice the length of Sauveur’s pipe.⁵ Since the exact dimensions of the pipe are not known, there is no way to determine how accurate Sauveur was in his determination of its frequency. However, its pitch was an A of the time, in agreement with a later determination of the frequency of an eighteenth-century Paris tuning fork.⁶ If each of the three tunings made in the process of obtaining the small halftone is accurate to within half a cent (1/200 of an equal tempered semitone), it would be possible, in principle, to find the absolute frequency to within 2.5 percent.

Later, in work presented in 1713, Sauveur derived the frequency of a string theoretically. He treated the string, stretched horizontally and hanging in a curve because of the gravitational field, as a compound pendulum and he found the frequency of the swinging motion, assumed to have small amplitude. His result agrees with the modern one except for a factor of

In 1701, at a lively meeting of the Paris Academy, Sauveur explained the basic properties of harmonics. Harmonics had seemed paradoxical ever since the early seventeenth century, when the identification of pitch with frequency had implied the seemingly impossible phenomenon of a single object vibrating simultaneously at several frequencies. Sauveur argued that a string can vibrate at additional, higher frequencies, which he called sons harmoniques, by dividing up into the appropriate number of equal shorter lengths separated by stationary points, which he called noeuds. Sauveur apparently did not know of the paper rider demonstration, reported by Wallis (1677)⁷ and Robartes (1692),⁸ nodes are associated with higher modes. However, Wallis’ paper was mentioned by a member of the audience, and Sauveur’s argument for the existence of nodes culminated with Wallis’ demonstration. In a discussion of the organ, presented in 1702, Sauveur stated explicitly (he was the first to do so) that harmonics are components of all musical sound and that they affect the timbre of a tone.

Among Sauveur’s interests, the subject of harmonics proved the most important for later developments—in mathematics, physics, and music. In the eighteenth century, analysis of the vibrating string was inspired, in part, by knowledge of the higher vibrational modes. The composer and theorist Rameau used harmonics to provide a physical basis for his theory of harmony, and a century later Helmholtz emphasized the effect of harmonics on timbre. It was through Sauveur and the Paris Academy that ideas about harmonics became well known in the early eighteenth century.⁹ Sauveur’s terminology, including “harmonics” and “node,” was adopted and is still current.

NOTES

1. The word had already been used occasionally in connection with sound.

2. Other systems of multiple division were already in use; the most important, developed by Huygens, was the division of the octave into 31 intervals.

3. J. Murray Barbour, Tuning and Temperament, an Historical Survey (East Lansing, Mich., 1953), 122.

4. Mersenne made the first known estimate of vibrational frequency by extrapolating from the countable vibrations of a long string to the frequency of a short section of it; see Harmonie universelle (Paris, 1636; facs. ed., 1963), l, Bk. 3, Prop. VI, 169–170.

5. In the 2nd and 3rd eds. of the Principia, Bk. 2, Prop. L; cf. Isaac Newton, Mathematical Principles of Natural Philosophy, F. Cajori, ed. (Berkeley, Calif., 1962), I, 383–384.

6. Alexander J. Ellis, “On the History of Musical Pitch,” in Journal of the Society of Arts, 28 (1879–1880), 318, col. 1.

7. John Wallis, “On the Trembling of Consonant Strings, a New Musical Discovery,” in Philosophical Transactions of the Royal Society, 12 (1677), 839–842.

8. Francis Robartes, “A Discourse Concerning the Musical Notes of the Trumpet, and the Trumpet Marine, and of Defects of the Same,” ibid., 17 (1692), 559–563.

9. In 1809, when Chladni’s experimental demonstration of nodal lines inspired the Academy competition for theoretical analysis of vibrating surfaces, Prony remarked that Sauveur’s work had led to important research on the vibrating string; see Procès-verbaux de l’Académie des sciences, IV (Hendaye, 1913), 175.

BIBLIOGRAPHY

Sauveur’s papers include “Système général des intervalles des sons,” in Mémoires de l’Académie royale des sciences, 1701 (Paris, 1704), 297–460; “Application des sons harmoniques à la composition des jeux d’orgues,” ibid., 1702 (Paris, 1704), 308–328: “Rapport des sons des cordes d’instruments de musique aux flèches des cordes; et nouvelle détermination des sons fixes,” ibid., 1713 (Paris, 1716), 324–348.

Fontenelle is the main source of information on Sauveur’s life and the reception of his work. His éloge is in Histoire de l’Académie royale des sciences, 1716 (Paris, 1718), 79–87. His discussions of Sauveur’s work include “Sur la détermination d’un son fixe,” ibid., 1700 (Paris, 1703), 131–140: “Sur un nouveau systéme de musique,” ibid., 1701 (Paris, 1704), 123–139; “Sur l’application des sons harmoniques aux jeux d’orgues,” ibid., 1702 (Paris, 1704), 90–92; and “Sur les cordes sonores, et sur une nouvelle détermination du son fixe,” ibid., 1713 (Paris, 1716), 68–75.

See also Léon Auger, “Les apports de J. Sauveur (1653–1716) à la création de l’acoustique,” in Revue d’histoire des sciences et de leurs applications, 1 (1948), 323–336; and the article on Sauveur by F. Winckel in F. Blume, ed., Die Musik in Geschichte und Gegenwart, XI (Kassel, 1963), cols. 1437–1438.

Sigalia Dostrovsky