Early Vibration Theory: Physics and Music in the Seventeenth Century

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1 Early Vibration Theory: Physics and Music in the Seventeenth Century SIGALIA DOSTROVSKY Communicated by I.B. COHEN Contents 1. Introduction Frequency 2.1. The Arithmetic of Pitch The Interest in a Physics of Pitch The Idea of Frequency The Fundamental mode 3.1. The Vibrating String: The Dependence of Frequency on Length MERSENNE'S Laws for the Vibrating String Seventeenth Century Derivations of MERSENNE'S Laws The Vibrating Air Column The Puzzle of Overtones and Higher Modes 4.1. Overtones The Trumpet Marine Wind Instruments A Final Example Absolute Frequency 5.1. MERSENNE HOOKE and HtJYGENS SAUVEVR Higher Modes 6.1. The First Discussions of Nodes: WALLIS and ROBARTES SAUVEtrR NEWTON'S Analysis of Sound Waves NEWTON'S Model for a Pressure Wave l 7.3. The Velocity of Sound Introduction The basic ideas of vibration theory were formulated during the seventeenth century. BENEDETTI, VINCENZO and GALILEO GALILEI, BEECKMAN, DESCARTES, MERSENNE, HUYGENS, WALLIS, NEWTON, SAUVEUR, and others sought to under-

2 170 S. DOSTROVSKY stand vibration and waves in physical and mathematical terms. As a result of their interest, by the end of the century there was a recognition of vibrational frequency, a law for predicting the frequencies of the vibrational modes of a string, a quantitative theory of sound waves in air, and an heuristic understanding of superposition. This paper describes the development of vibration theory in the seventeenth century and it is concerned with showing that, except for NEWTON'S analysis of the propagation of sound, the subject developed from problems and information that arose in the context of music. That pitch can be identified with frequency was a major discovery of the seventeenth century, and this identification made possible very precise measurements of relative frequencies. A periodic vibration can be heard as a tone and two tones make a consonant interval when the ratio of their frequencies is a ratio of small integers. This correspondence between intervals and ratios, well known since the PYTHAGOREANS, provided the essential quantification for discovering and measuring frequencies. Thus musial experience was crucial for the development of vibration theory: It trained the ear to make precise determination of consonant intervals and it provided the classification that helped to analyze them. There were other interactions between musical experience and the emerging ideas about vibration. For example, with training, one can hear overtones in most muscial sounds. In the seventeenth century, this suggested the existence of higher modes and made possible a study of them. Also, the beats produced by tones of close, but not identical, frequencies were familiar from musical instruments. Beats were important for learning about the interference of waves and, specifically, for determining frequency absolutely. Not only was it more or less necessary that the study of vibration originate in the context of an interest in musical sound; it was also quite natural for the seventeenth century to study musical phenomena in connection with the new physics. Music theory had traditionally been included with mathematical subjects. Also, music had always inspired a sense of cosmic structure. Although this obviously had no direct connection with the analysis of vibration, it seems to have encouraged a confidence in the general relevance of musical problems. On the more practical level of empiricism, musical instruments posed questions and provided a wealth of information on the nature of vibration. A study of the development of vibration theory in the seventeenth century shows, not only that problems of vibration and sound were being considered by 1700, but that there were specific challenges to improve the physical understanding and the mathematical analysis of these problems in continuum mechanics. Thus this paper also provides some background for the work of TAYLOR, JOHANN (I) and D. BERNOULLI, D'ALEMBERT, EULER, LAGRANGE and LAPLACE on the vibrating string and the propagation of sound. Some of the material in the present paper appeared, in an earlier form, in the author's dissertation.1 Aspects of early vibration theory have been studied by C. PALISCA and by C. TRUESDELL. PALISCA has analyzed the ways in which scientific modes of thought affected seventeenth century musical composition 1 SIGALIA DOSTROVSKY, "The Origins of Vibration Theory: The Scientific Revolution and the Nature of Music" (Princeton University, Ph.D. Dissertation, 1969).

3 Early Vibration Theory 171 and music theory. 1 Two results of PALISCA'S work that are especially important for the history of physics are his discovery that VINCENZO GALILEI did physical experiments on vibration and his demonstration that in the late sixteenth and early seventeenth conturies some people concerned with music theory included ideas about the physics of vibration in their recommendations for a freer use of musical sound. S. DRAKE has suggested that this musically oriented interest in the physics of vibration may constitute the source of the controlled experiment that characterizes the scientific revolution of the seventeenth century. 2 TRUESDELL has given a basic and valuable survey of early vibration theory by considering the analyses of specific problems, especially those of the vibrating string and the propagation of waves.a 2. Frequency 2.1. The Arithmetic of Pitch. The musical notions of pitch and interval were important for learning about the physics of vibration. This is due to the fact that the sensation of musical pitch can be identified with the frequency of audible vibrations. 4 For this reason vibrational frequencies were compared quantitatively and correctly long before there was a physical concept of vibration. In fact, the art of music provided an experience with pitches and intervals, and the theory of music provided a sophisticated analysis of them. Music theory has been at least in part arithmetical since the ancient Greeks. The arithmetic is based on the ratios that characterize the musical intervals. Until the seventeenth century, the ratios were defined as the length ratios of pairs of similar strings under the same tension sounding at the appropriate pitches. Ever since the seventeenth century, when the connection between pitch and frequency was discovered, the ratios have been defined in terms of the relative 1 frequencies. (Since v~:~, frequency ratios are the inverse of length ratios.) Con- sonant intervals are characterized by the simplest ratios: The frequency ratio for the octave is 2: l, for the fifth, 3:2, and for the fourth, 4:3. 5 The correlation between consonances and simple ratios was presumably noticed by many players of stringed instruments, but it was the Pythagoreans who emphasized it and used it, not only as the basis for music theory, but also as the inspiration for an entire cosmology. The Pythagoreans attributed the discovery of the consonant ratios to PYTHA- 1 CLAUDE V. PALISCA, "Scientific Empiricism in Musical Thought," in Seventeenth Century Science and the Arts, ed. HEDLEY H. RHYS (Princeton: Princeton University Press, 1961), STILLMAN DRAKE, "Renaissance Music and Experimental Science," Journal of the History of Ideas, XXXI (1970), ; "Vincenzio and Galileo Galilei," in Galileo Studies (Ann Arbor: University of Michigan Press, 1970), C.A. TRUESDELL, "The Rational Mechanics of Flexible or Elastic Bodies, ," in Euleri Opera Omnia, 2 nd Series, XI Part 2 (Ztirich, 1960), ; "The Theory of Aerial Sound, ," in Euleri Opera Omnia, 2 nd Series, XIII (Lausanne, 1955), pp. XIX-LXXII. 4 The word "identification" is used because the sensation of pitch is appropriately quantified by frequency. 5 The names octave, fifth, etc. refer to the number of notes of the scale spanned by the interval.

4 172 S. DOSTROVSKY GORAS himself. Various writers referred to experiments with strings, but they also included in their accounts many details that are obviously wrong. NICOMA- CHUS (end of first century A.D.) told how PYTHAGORAS was "considering whether it could be possible to devise some kind of instrumental aid for the ears which would be firm and unerring, such as sight obtains through the compass and the ruler or the surveyor's instrument; or touch obtains with the balance or the device of measures. While thus engaged, he walked by a smithy and, by divine chance, heard the hammers beating out iron on the anvil and mixedly giving off sounds which were most harmonious with one another, except for one combination. He recognized in these sounds the consonance of the octave, the fifth and the fourth... Delighted, therefore, since it was as if his purpose was being divinely accomplished, he ran into the smithy and found by various experiments that the difference of sound arose from the [variation in the] weight of the hammers, but not from the force of the blows, nor from the shapes of the hammers, nor from the alteration of the iron being forged. 1 (The weight of the hammer striking the plate will not, in itself, affect the plate's frequencies, but one can obtain different frequencies by hitting the same plate, supported in the same way, in different places.) According to NICOMACHUS, PYTHAGORAS was motivated to experiment with strings: "he suspended four strings of the same material and made of an equal number of strands, equal in thickness and equally twisted. To each string he fastened a [different] weight which he suspended from the lower extremity. When he had arranged that the lengths of the strings should be exactly equal, he alternately struck two strings simultaneously and found the aforementioned consonances, a different consonance being produced by a different pair of strings.-2 The specific cases described (for example, strings with weights of 12 and 6 sounding an octave apart) imply that frequency is proportional to the tension, instead of the square root of the tension. THEON OF SMYRNA (first part of second century A.D.) wrote that PYTHAGORAS "investigated these ratios [octave, fifth, fourth, etc.] on the basis of the length and thickness of strings, and also on the basis of the tension obtained by turning the pegs or by the more familiar method of suspending weights from the strings. And in the case of wind instruments the basis was the diameter of the bore, or the greater or lesser intensity of the breadth. Also the bulk and weight of disks and vessels were examined. Now whichever of these criteria is chosen in connection with any one of the aforesaid ratios, 1 NICOMACHUS OF GERASA, Manual of Harmonics, translation and commentary by F.R. LEVIN (Columbia University, Ph.D. Dissertation, 1967), Ibid. p. 29.

5 Early Vibration Theory 173 other conditions being equal, the consonance which corresponds to the ratio selected will be produced. "1 Much later, in an account similar to NICOMACHUS', BOETHIUS (C ) explicitly associated the ratios of the weights of the hammers at the blacksmith's with the ratios of the musical intervals. BOETHIUS explained that since PYTHAGORAS "had no faith in a human ear" and "no confidence in musical instruments," he wanted "a method by which he might learn the fixed and unalterable measurements of consonances?' PYTHAGORAS compared the weights of the five hammers: "those two which gave the consonance of an octave were found to weigh in the ratio 2:1. He took that one which was double the other and found that its weight was four-thirds the weight of a hammer with which it gave the consonance of a fourth. Again he found that this same hammer was three-halves the weight of a hammer with which it gave the consonance of a fifth." BOETHIUS then mentioned some experiments "on those proportions" with weights on strings, lengths of reeds, weights in vessels and, finally, on lengths and thicknesses of strings. 2 Most of the observations reported above could not have been made. Although the ratios of the intervals correspond to lengths of strings and (approximately) to lengths of pipes, they are not equal to the ratios of arbitrary pairs of variables. 3 According to the various accounts, PYTHAGORAS' discovery sounds like a legend. 4 But the fact that the legend exists suggests that the importance of the musical ratios was perceived long before they were understood. The repetition of errors in the descriptions indicates that it was mainly the arithmetical relationships between pitches that were of interest, and not the physics of sound vibrations. In the seventeenth century, however, the physics of sound vibration would be a topic of primary interest, so MERSENNE and others would be horrified at the casual misrepresentation of the basic facts of vibration in the accounts of PYTHAGORAS' discovery. MERSENNE would say that he was "astonished" that various historians and music theorists had been "so negligent that they did not perform a single experiment to discover the truth and undeceive the world." 5 He would explain that the ratios could not have been found with the weights on strings, or with the lengths of flutes, or with the weights of hammers; he would refer to his own experiments showing what proportions are necessary to produce the consonant intervals in these different cases. 6 Even a generation later, the bizarre account of PYTHAGORAS' discovery would still hold interest. In 1661, HUYG~NS would remark that pairs of hammers could not have produced intervals corresponding to the ratio of their weights: "The proportion of their MORRIS R. COHEN & I.E. DRABKIN, A Source Book in Greek Science (Cambridge, Massachusetts: Harvard University Press, 1948), De Institutione Musica; selections in COHEN & DRABKIN, COHEN & DRABI~rN make this observation. Ibid. pp. 295n2, 298n3, 298nl, 298n2. Ibid. p M. MERSENNE, Traitd de l'harrnonie Universelle (Paris, 1627), M. MERSENNE, Nouvelles Observations Physiques et Mathdrnatiques, in Harrnonie Universelle (Paris, 1636; all references are to the facsimile edition, 3 vols.. Paris: Centre National de la Recherche Scientifique, 1963), III,

6 174 S. DOSTROVSKY weights with regard to their tones could not have been that which the authors of this story tell." 1 HUYGENS would criticize the report that stretching a string with twice the tension produces the octave: "[if they] had taken the trouble to perform an experiment, they would have found that four times as great a weight is necessary in order to make the pitch of a string rise to the octave.., and that universally the ratio of the weight should be the square of that which determines the consonances by the parts of a stretched string." HUYGENS would assume, reasonably, that PYTHAGORAS must have found the ratios of the consonances by comparing strings of different lengths, not different tensions. 2 The predominant early aesthetics of music were based on the arithmetic of pitch. Not all music theorists in ancient Greece were PYTHAGOREAN. For example, the followers of ARISTOXENUS, a student of ARISTOTLE, criticized the PYTHAGOR- EANS' mathematical analysis and used an empirical aesthetics of music. However, the PYTHAGOREAN approach dominated music theory throughout the middle ages and the early Renaissance. Music was associated with arithmetic, geometry, and astronomy in ancient Greece, and it belonged with these subjects in the medieval quadrivium of learning. A ninth century theorist taught that music is the" daughter of arithmetic," and that since" it is joined throughout to the theory of numbers,... so it is through numbers that we must understand it. "3 Thus consonance and dissonance in musical sound was explained in terms of number. According to medieval theorists, an interval characterized by a ratio of small integers was more consonant than an interval characterized by a ratio of larger integers. Some of the problems associated with this aesthetics of music were to be important, much later, in motivating an interest in the physics of vibration The Interest in a Physics of Pitch. In the late sixteenth and early seventeenth centuries, BENEDETTI, V. GALILEI, DESCARTES, and others sought to understand the ratios that characterize musical intervals in terms of physical characteristics. This interest in the physics of pitch was often associated with criticism of a dogmatic arithmetical definition of consonance. It constituted the basis for the eventual association of pitch with vibrational frequency. This section describes some instances of the new attention to physical characteristics of musical sound. J. B. BENEDETTI, known in the history of physics primarily for his work on free fall, explained degrees of consonance with an analysis that invoked the physics of musical sound. Around 1563, BENEDETTI corresponded with Cn'RIANO DE RORE, a composer, in connection with problems of tuning and the need for temperament. C. PALISCA has analyzed these letters and has emphasized BENEDETTI'S attention to the physical origin of musical intervals. 4 BENEDETTI 1 CHRISTIAAN HUYOENS, Oeuvres Complbtes (22 Vols., The Hague, Martinus Nijhoff, ), XIX, Ibid. p Scholia enchiriadis; Selection translated in OLIVER STRUNK, Source Readings in Music History (New York: W.W. Norton & Co., Inc., 1965), I, PALISCA, "Scientific Empiricism in Musical Thought,"

7 Early Vibration Theory 175 discussed the well known fact that when a bridge divides a string at one third of its length, the sections sound an octave apart. He said that the longer section takes twice as long to complete a cycle as the shorter one. PALISCA presents B~NEDETTI'S explanation as follows: "Consonances arise, he says, from a certain equalization of percussion, or from an equal concurrence of air waves, or from their co-termination. The unison, he continues, is the first and most agreeable consonance, and after it comes the diapason or octave, then the fifth. Now, these preferences may be shown to be the result of the ' order of agreement of the terminations of the percussions of the air waves, by which the sound is generated.'"1 (This is basically the explanation that was to be used throughout the seventeenth century; it will be discussed further in connection with GALILEO in Section 2.3.) BENED~TTFS own understanding did not seem to affect general knowledge, z VINCENZO GALILEI (GALILEO'S father) is the first person known to have studied explicitly the variety of ways in which the physical properties of a source of vibration determine the pitch of the sound, and he did so by performing experiments. VINCENZO'S early acoustical studies do not seem to have been known until certain of his manuscripts were found and discussed by PALISCA. 3 VINCENZO was a music theorist and also a composer, lutanist and philosopher of music. He admired Greek monody; he was an important member of the Florentine "camerata," Count BARDI'S group devoted to reviving Greek styles in music and drama; and he did not approve of contemporary polyphony. He recommended greater freedom from formal arithmetical considerations in music composition; he translated ARISa'OXENUS; and he argued with ZARLINO, a more conservative theorist, about problems of temperament. (Since the late fifteenth century, some theorists had been advocating greater dependence on the "judgment of the ear" in analyzing musical intervals, instead of the dogmatic medieval emphasis on simple ratios. In 1477, for example, TINCTORIS had rejected the fourth because it sounds, to the ear, like a discord. He had referred to the invervals i.hat his contemporaries defined as consonances as "concords which have, in the tradition of Aristoxenus, been tested by the judgment of the ears." 4) VINCENZO did his work on the physics of vibration very much in the context of these musical and philosophical interests, because he wanted to find facts that would be useful in arguing against an overly strong adherence to formal rules. His experiments, done in 1589 or 1590, are described in two manuscripts: In "A particular discourse concerning the ratios of the octave," VINCENZO showed, from experiments with strings, pipes, weights, and coins, that "the ratio" of the octave is not unique; he stated that although the ratio is 2 : 1 for lengths, it is 4: 1 for tensions and 8:1 for volumes (such as those of organ pipes); In "A particular 1 Ibid. p TRUESDELL, "The Rational Mechanics of Flexible or Elastic Bodies, ," PALISCA discusses Vineenzo and his work in "Scientific Empiricism in Musical Thought" and in the article "V. Galilei" in Die Musik in Gesehiehte und Gegenwart, ed. F. BLUME, IV (Kassel- Basel, 1955). 4 JOHANNES TINCTORIS, The Art of Counterpoint in "Musicological Studies and Documents" 5 (Rome, American Institute of Musicology, 1961),

8 176 S. DOSTROVSKY discourse concerning the unison," he described experiments showing that, to be at the same pitch, strings must be alike not only in length but also in material, tension, and thickness. 1 When GALILEO, his son, wrote the Two New Sciences (1638), he had SAGREDO mention the perplexing variety of different ratios. FRANCIS BACON, as to be expected, was critical of the arithmetical analysis of pitch which, however, he did not properly understand. He made some perceptive observations on problems in understanding sound. 2 He noticed that although music theory was a highly sophisticated subject, it had not been studied in physical terms. He therefore made long and careful lists of many properties of sound, often referring to musical instruments; in fact, he recommended (as MERSENNE would) that information be obtained from instrument makers. 3 He was puzzled by the consonance of some intervals, and he was confused as to how the character of an interval could be understood by a number. 4 He suggested, like VINCENZO, that intervals be studied in terms of the physical properties of the objects that produce them. However, on making some correlations himself, BACON became confused by the complexity of the phenomena. He could not separate frequency from the mixture of pitch, volume, and quality that is perceived in real tones. To give just one example out of many: BACON noticed that thick strings produce lower tones than thin strings and wide pipes, lower tones than narrow ones, so he suggested that pitch is associated with the quantity of air being "percussed. "5 KEPLER was not directly concerned with physical analyses of pitch and consonance; he explained consonance in the same way that he explained astrological influences, with the idea of archetypes that are common to the universe and to man's soul. 6 However, he criticized the traditional arithmetic of pitch. D.P. WALKER has shown how KEPLER'S appreciation of the music of his time-kepler felt that contemporary polyphony had reached a peak of musical perfection- and his empirical aesthetics of music affected his decision as to which intervals to call "consonances. ''7 Like many contemporary musicians, KEPLER included thirds and sixths among the consonances because they sounded good. In the Harmonice Mundi (1619) KEPLER wrote: "the Pythagoreans were so addicted to... philosophizing in numbers, that they failed to keep to the judgment of their ears, though it was by means of this that they had initially been brought to this philosophy; they defined solely by their numbers what is a melodic interval and what is not, what 1 PALISCA, "Scientific Empiricism in Musical Thought," z BACON wrote about sound in the "Sylva soni et auditus" and in two sections of the Sylva Sylvarum. References here are to paragraphs of the Silva Sylvarum in F. BACON, Works, eds. J. SVEDDING, R.L. ELLIS & D.D. HEATh (Boston, 1860), IV. 3 Between paragraphs 183 and Paragraphs Paragraph KEPLER'S explanation for consonance is discussed in W. PAULI, "The Influence of Archetypal Ideas on the Scientific Theories of Kepler," in C.G. JUNG, The Interpretation of Nature and the Psyche (New York: Pantheon Books, 1955). 7 D.P. WALKER, "Kepler's Celestial Music," Journal of the Warburg and Courtauld Institutes XXX (1967),

9 Early Vibration Theory 177 is consonant and what is dissonant, thus doing violence to the natural instinc-,1 tive judgment of the ear. Thus KEPLER disapproved of the dogmatic PYTHAGOREAN classification of consonances, even though he was inspired by the idea of the cosmological importance of the mathematical harmonies. KEPLER gave a geometric derivation of the consonances (by inscribing polygons in circles), made more difficult by the inclusion of thirds and sixths. As in his astronomy, KEPLER, combined observation (judgment of the ear, in this case) with geometric analysis. Indirectly, KEPLER'S work touched upon the notions of pitch and frequency in one instance. Trying to relate the harmonic ratios to the solar system, Kepler examined the planets' relative periods of revolution. He showed that the ratios of almost all the angular velocities at aphelion and at perihelion, both for a single planet and for pairs of planets, are close to the ratios of consonant intervals. 2 To describe the results in actual musical terms, KEPLER assigned pitches (whose frequencies were) proportional to these angular velocities. DESCARTES' Compendium Musicae, a book on music written in 1618, provides another example of the interest in physical attributes of pitch that developed in connection with comparing musical intervals. The arithmetical foundations of music theory may have inspired DESCARTES in his search for a "method" for understanding natural phenomena. 3 At any rate, the Compendium Musicae was his very first book, and it treated musical intervals and the ratios used to characterize them. DESCARTES wrote the book as a gift for ISAAC BEECKMAN, 4 with whom he had talked about problems in music and sound when they were both in the army of the Dutch prince (BEECKMAN will be discussed in Sections 2.3 and 3.1). Although DESCARTES explained the musical ratios primarily abstractly (in terms of various divisions of a line segment), he also justified some basic ideas by considering physical characteristics of sound. For example: To define the nature of consonant intervals, DESCARTES suggested that they involve intervals in which the lower pitch "contains" the higher; he explained this by referring to the resonance that occurs between strings whose pitches differ by an octave or a fifth (with the lower pitch producing the higher)? To justify 1 Ibid. p ALEXANDER KOYRI~, La Revolution Astronomique (Paris, 1961), In the Discourse on Method DESCARTES explained that he had noticed that in all the mathematical sciences the study of ratio and proportion was important. In the Rules Jbr the Direction of the Mind he referred to all four subjects of the quadrivium as having shown him what a "universal mathematics" might be. 4 DESCARTES asked BEECKMAN not to show the manuscript to anyone, but it became known. BEECKMAN had it copied in 1627 (returning the original to DESCARTES in 1629), and CONSTANTHN HUYGENS made a copy in 1637 (ISAAC BEECKMAN, Journal, ed. C. DEWAARD (The Hague: Martinus Nijhoff, ), I, p. XXVIII); the treatise was published in 1650, after DESCARTES' death, by the Dutch firm of Elzevier. W. BROUNCKER published an English translation, Excellent Compendium ~?f Music ; with Necessary and Judicious Animadversions Thereupon (London, 1653); NICOLAS JOSEPH POISSON published a French translation in 1668, available in DESCARTES' Oeuvres, ed. V. COUSIN (Paris, 1824) V, A recent translation is R. DESCARTES, Compendium of Music, trans. WALTER ROBERT, Musicological Studies and Documents, VIII (American Institute of Musicology, 1961). 5 R. DESCARTES, Oeuvres, ed. V. COUSIN (Paris, 1824)V, 454.

10 178 S. DOSTROVSKY classifying the octave as the first consonance, DESCARTES remarked that the octave often accompanies other tones and that it is the first interval produced when overblowing a flute: "Cela se confirme par l'exp&ience des flfites, qui 6tant embouch~es et remplies de vent plus qu'a l'ordinaire, passent d'un ton grave ~ un autre plus aigu d'une octave enti&re. Or il n'y a pas de raison pourquoi on passe tout d'un coup/t l'octave, et non pas ~ la quinte et aux autres consonances, sinon parceque l'octave est la premiere de toutes et qui diff~re le moins de l'unisson.,1 DESCARTES did explicitly associate pitch with frequency once, in the Compendium, in connection with a classification of the intervals of a third (5:4), a tenth (5:2), and a seventeenth (5:1). 2 DESCARTES included a physical reason for calling the last of these thirds the "most perfect" (it had been so classified in earlier music theory, for numerical reasons). He may have been motivated to find a physical justification because he did notice that when a string is tuned to a seventeenth above a tone it resonates more readily than if it is tuned to a tenth or a third above (which it should, since the seventeenth is a harmonic). At any rate, DESCARTES compared the intervals "supposing that sound strikes [frappe] the ears with many pulses [coups], so much the more promptly as the sound is higher." For the interval of a seventeenth, every pulse associated with the tone of lower pitch coincides with one from the higher tone, whereas for the interval of a tenth, only every second pulse from the lower tone coincides with one from the higher. DESCARTES, GALILEO, and others used this idea of coinciding pulses later, in explaining consonance and dissonance (to be discussed in the next section). Some fifteen years later, in L'Homme and in letters to MERSENNE, DESCARTES' ideas on musical sound would be developed further (these later remarks will be mentioned mainly in Sections 2.3 and 3.2). VINCENZO criticized arithmetical music theory and emphasized the variety of physical characteristics associated with sources of tones; KEPLER criticized the PYTHAGOREAN approach; and DESCARTES, although writing a rather traditional book, had physics in the background. Once frequency was related to pitch, number would return to the analysis of musical intervals. But to discover that vibrational frequency determines musical pitch, one had to emphasize physics, not arithmetic The Idea of Frequency. Following his father VINCENZO, GALILEO was critical of the traditional use of the ratios based on string lengths to describe musical intervals. He went on to connect pitch with frequency and to explain dearly that frequency ratios correspond to intervals. Thus he found the quantity that refers directly to the sound as propagated and not merely to properties of the vibrating object producing the sound. In discussing pitch and frequency, GALILEO also described some general properties of waves and vibration. GALILEO had some knowledge of music, and he was living at home at the time of VINCENZO'S 1 Ibid. p, 456. z Ibid. pp

11 Early Vibration Theory 179 experiments. He played lute and keyboard, 1 he corresponded with G.B. DONI, a theorist and historian, on ancient Greek music, 2 and he had books on music in his library. 3 In The Assayer (1623), GALILEO described the variety of musical instruments and asserted that one can learn about nature from them. 4 GALILEO discussed sound in the Discourse on Two New Sciences (1638), 5 when he extended his analysis of the pendulum to clarify certain "musical problems. "6 SALVIATI, in the dialogue, offers to tell about his "thoughts pertaining to music-a most noble subject" and to give "reasons for marvelous things in the matter of sounds." SAGREDO, who describes himself as" one who is delighted by all musical instruments," is most interested. He has three specific questions: "Having philosophized much about the consonances, I have always remained puzzled and perplexed by them, in as much as one pleases and delights me far more than another, while some not only fail to delight, but actually offend me. Then there is the old problem of the two strings tuned in unison, of which one moves and audibly resounds to the sound of the other. I am unresolved about this, as I am also unclear about the forms [ratios] of the consonances, and other particulars." 7 GALILEO first discussed resonance. Explaining how a pendulum's period depends on its length, he emphasized the idea of characteristic frequency: "it is necessary to note that each pendulum has its own time of vibration, so limited and fixed in advance that it is impossible to move it in any other period than its own unique and natural one... On the other hand, we confer motion on any pendulum, though heavy and at rest [he later gives the example of a bell], by merely blowing on it," provided that the "puffs" are "given at the right time.-8 By analogy, GALILEO then explained "that remarkable problem of the zither - or harpsicord- string that moves and even resounds, and not only with one in unison and concord, but also with its octave and fifth. The cord struck begins and continues its vibrations during the whole time that its sound is heard; these vibrations make the air near it vibrate and shake; the tremors and waves extend through a wide space and strike on all the strings of the same instrument as well as on 1 GALILEO GALILEI, Opere, National Edition, Dir. A. FAVARO (20 Vols., Florence, ), XIX, Ibid. XV, A. FAVARO, "La Libreria di Galileo Galilei," Bulletino di Bibliografia e di Storia delle Scienze matematiche efisiche, XIX (1886), 219-2; * Selections in Discoveries and Opinions of Galileo, trans. S. DRAKE (New York : Doubleday, 1957), GALILEO GALILEI, Two New Sciences, trans. H. CREW 8 A. DE SALVIO (New York: Dover Publications), ; new translation by STILLMAN DRAKE (Madison: The University of Wisconsin Press, 1974), All quotations are from the DRAKE translation. 6 Ibid. p Ibid. pp s Ibid. p. 99.

12 180 S. DOSTROVSKY those of any others nearby. A string tuned in unison with the one struck, being disposed to make its vibrations in the same times, commences at the first impulse to be moved a little; the second, third, twentieth, and many more [impulses] being added, all in exact periodic times, it finally receives the same tremor as that originally struck, and its vibrations are seen to go widening until they are as spacious as those of the wave." 1 GALILEO next discussed an illustrative experiment to help one see what happens when the pitch of a tone changes. A goblet of thin glass can be made to sound by resonance or by friction around its rim. As GALILEO explained, one can see that the goblet's vibrations extend all around because water inside (or outside, if the goblet is placed in a large container of water) becomes rippled. The crux of the matter is that "sometimes it happens that the tone of the goblet jumps one octave higher, at which moment I have seen each of the waves divided in two; an event that very clearly proves the form of the octave to be the double [ratio].-2 Thus GALILEO provided a reason, more general than that of relative string lengths, for assigning the ratio 2:1 to the octave. It would actually be rather unusual to have a glass with modes an octave apart; also, the wavelength for capillary surface waves is not inversely proportional to frequency. Since GALILEO was obviously thinking about sound waves in air, for which frequency is inversely proportional to wavelength, he was not incorrect about this characteristic; but he made a mistake in choosing water to illustrate it. Approaching the idea of vibrational frequency for sound waves, GALILEO pointed out that it would be helpful to count the waves. Thus GALILEO suggested counting the marks that a sharp object leaves on a metal plate when it is quickly drawn across it (producing a squeak); GALILEO explained that having heard ' the plate emit a rather strong and clear note," he noticed "a long row of thin lines parallel to one another and at exactly equal distances apart... The marks made during the shriller tone were closer together, and those during the lower tone less so. Sometimes also, according as the stroke itself was made faster at the end than the beginning, the sound was heard to rise in pitch and the lines were seen to increase in frequency, though always marked with extreme neatness and absolutely parallel." 3 Miraculously, he heard two squeaks separated by a fifth, and he counted 45 marks in a unit length of one track and 30, in the other. But it was in his discussion of the problem of consonance and dissonance 4 that GALILEO identified pitch with frequency explicitly. Presumably GALILEO imagined sound as involving smooth waves (since antiquity, sound had been corn- i Ibid. pp Ibid. p Ibid. pp Ibid. pp

13 Early Vibration Theory 181 pared with water waves1), but having no quantitative description of waves, he explained a tone as a succession of pulses. The rate of the pulses, which do not interact with the pulses of other tones, determines the pitch: "I say that the length of strings is not the direct and immediate reason behind the forms [ratios] of musical intervals, nor is their tension, nor their thickness, but rather, the ratio of the numbers of vibrations and impacts of air waves that go to strike our eardrum, which likewise vibrates according to the same measure of times. This point established, we may perhaps assign a very congruous reason why it comes about that among sounds differing in pitch, some pairs are received in our sensorium with great delight, others with less, and some strike us with great irritation; we may thus arrive at the reason behind perfect consonances, and imperfect, and dissonances.,,2 With this idea, he first explained that the consonance of an interval is essentially determined by the proportion of pulses of the higher tone that coincide with pulses of the lower tone. He said this by giving specific examples: ' Hence the first and most welcome consonance is the octave, in which for every impact that the lower string delivers to the eardrum, the higher gives two, and both go to strike unitedly in alternate vibrations of the high string, so that one-half of the total number of impacts agree in beating together... The fifth also gives pleasure, inasmuch as for every two pulsations of the low string, the high string gives three, whence it follows that counting the vibrations of the high string, one third of all [pulses] agree in beating together..." 3 Next comes the fourth, but for the whole tone, "only one in every nine pulsations comes to strike in agreement with that of the lower string" GALILEO ommitted the major third (5:4), no doubt because it was, by then, regarded in some ways as more consonant than the fourth. 4 GALILEO'S explanation orders the consonances in the same way that the purely arithmetical PYTHAGOREAN scheme orders them (later in the century, HUYGENS tried to explain why the third sounds ' better" than the fourth; see Section 5.2). 5 GALILEO further pointed out that two tones produce a sequence of pulses 6 that form a i For example: according to BOETHIUS, "in the case of sounds something of the same sort takes place as when a stone is thrown out and falls into a pool or other calm water.., when air is struck and produces a sound, it impels other air next to it and in a certain way sets a rounded wave of air in motion" (COHEN & DRABKIN, ). 2 Ibid. p Ibid. pp Since the late fifteenth century, a fourth appearing as the lowest interval in a chord had been treated as a dissonance. s HELMHOLTZ' theory of consonance (based on beats between harmonics) and more modern extensions of it (for example, R. PLOMP & W. J. M. LEVELT, "Tonal Consonance and Critical Bandwidth," Journal of the American Acoustical Society, 38 (1965), ) do not succeed in putting the third ahead of the fourth, even though it is often classified as a more "pleasant" interval. The nature of an interval may be determined, to a great extent, by the musical context in which it is perceived; thus the third exists in the triad. 6 In GALILEO'S example of a vibrating string, pulses are given off twice each cycle, at the points of maximum displacement. This would make the propagated sound waves have twice the frequency of the string, not a usual situation.

14 182 S. DOSTROVSKY characteristic pattern in time. 7 Thus in GALILEO'S explanation, an interval is a characteristic rhythm repeating at high speed. For example, a fifth is a rhythm of two against three: FJr,3 3 3 GALILEO does not seem to have been concerned about the likelihood that the two sequences of pulses will not be synchronized to produce coincidences. MERSENNE, the French friar and natural philosopher who encouraged extensive interest in the physics of musical sound during the first half of the seventeenth century (he will be discussed in Sections 3.2, 3.4, 4.1, 4.2, 4.3 and 5.1), was anxious to learn about GALILEO'S explanation of consonance. Although he wrote to GALILEO a number of times, he never received an answer, 1 but also other people were explaining the degree of consonance in terms of the proportion of coincidences between pulses from different sources. MERSENNE received this explanation from BEECKMAN 2 and from DESCARTES 3 and he included it in his Harmonie Universelle (1636). 4 GALILEO'S explanation for the source of musical consonance could be called a first "modern" one because it was based on physical models both for sound and for its effect on the ear. More important, however, is the fact that GALILEO'S and others' interest in consonance and dissonance inspired some understanding of vibration. A number of GALILEO'S contemporaries recognized the central importance of frequency even though they did not discuss it as clearly and as explicitly as he did. ISAAC BEECKMAN had associated pitch with frequency already by 1615, when he had tried to derive the inverse proportionality between frequency and length of a vibrating string (to be discussed in Section 3.1). Although DESCARTES had not emphasized frequency in his early Compendium Musieae, in L'Homme, written in 1632, he referred to it as the source of pitch: "les petites secousses composeront un son que l'fime jugera.., plus aigu ou plus grave, selon qu'elles seront plus promptes fi s'entresuivre, ou plus tardives.-5 Accordingly, he explained concord and discord in terms of the degree of rapport between the vibrations of different tones. 6 MERSE~E often dwelt on reasons for the musical 7 DRAKE has drawn attention to this aspect of GALILEO'S explanation (Galileo Studies, pp ), which he explains differently. 1 MARIN MERSENNE, Correspondance, ed. CORNELIS DE WAARD (12 Vols., Paris ), I, 603; II, Ibid. I, Ibid. III, MERSENNE, Harmonie Universelle, II, prop. XVIII, R. DESCARTES, Oeuvres, ed. V. Cousin (Paris, 1824) IV, Ibid. pp

15 Early Vibration Theory 183 ratios. In the Harmonie Universelle he gave a variety of possible ratios for the octave and a variety of reasons for the particular ratio 2 : 1. Finally, he wrote that it is "entirely necessary" to use this ratio: "sound being nothing other than the movement of the air, and this movement finding itself always double in the octave, and never quadruple or octuple, it follows that the two sounds of the octave are in the same ratio as these movements. "1 On another occasion he remarked that : "the sharpness [aigu] of the sound does not come from the faster motion of a body or of air, but only from the frequency or the velocity of the returns or reflections of this air, or of the body that beats the air and divides it. This is perhaps why one says that the object of music is the sonorous number.,2 To summarize: according to the picture that was around by the end of the first third of the seventeenth century, sound consisted of a succession of pulses (percosse for GALILEO, secousses for DESCARTES, battemens for MERSENNE and, later, HUYGENS). The frequency of these pulses was the same-except for an occasional factor of two 3 _ as that of the vibrating object that initiated them. The various trains of pulses travelled without affecting each other and the proportion of coincidences, at the ear, determined an interval's degree of consonance. The identification of pitch with frequency became categorically accepted, in spite of the fact that it was only possible to measure relative frequencies and these only by using the identification itself. 4 The problem of determining frequency absolutely was to challenge natural philosophers throughout the seventeenth century. (This will be discussed in Chapter 5.) 3. The Fundamental Mode 3.1. The Vibrating String: The Dependence of Frequency on Length. Since a string's fundamental frequency of vibration at a given tension is related simply to its length voc~, experience with musical strings led, in ancient Greece, to the quantification of pitch and thus to the PYTHAGOREAN ratios. In the middle ages, music theorists and teachers used the monochord, a single string stretched over a sound box, to display the ratios. The following extract from a dialogue by Ono OE CLUNY (10 th century) provides an example of the medieval attitude to the monochord: (Disciple) What is music? (Master) The science of singing truly and the easy road to perfection in singing. (Disciple) How so? 1 M. MERSENNE, Harmonie Universelle, II, Book 1, prop. X, M. MERS~NNE, Harmonie Universelle, I, Book 1, prop. XIII, M. MERSENNE, Harmonie Universelle, I, 169; III, 141; and p. 181 note 6, above. 4 The word "frequency" was used only occasionally (it was much more common throughout the seventeenth century to refer to the pitch of the sound or to the "number of vibrations"). As late as 1894, RAYLEIGH felt it necessary to justify his use of the term "frequency", and he referred to YOUNG'S use of the word (J.W.S. RAYLEIGH, The Theory of Sound, Dover reprint, 7).

16 184 S. DOSTROVSKY (Master) As the teacher first shows you all the letters, in a table, so the musician introduces all the sounds of melody on the monochord. (Disciple) What is the monochord? (Master) It is a long rectangular wooden chest, hollow within like a cithara; upon it is mounted a string, by the sounding of which you easily understand the varieties of sounds. (Disciple) How can it be that a string teaches more than a man? (Master) A man sings as he will or can, but the string is divided with such art by very learned men..., that if it is diligently observed or considered, it cannot mislead. (Disciple) What is this art, I inquire. (Master) The measurement of the monochord, for if it is well measured, it never deceives. 1 The monochord was still familiar in the seventeenth century: MERSENNE stated that" "one calls it a harmonic or canonic ruler, because it serves to measure the low and the high in sound, in the way that the ordinary ruler of the geometers serves to measure straight lines and the compass to describe circles."2 In fact, the monochord quantified pitch so well that it may have made the discovery of frequency more difficult (for example, as in the need for VINCENZO'S experiments). But once frequency was discovered, its dependence on length would be well known. 1 BEECKMAN tried to derive voc~ for the vibrating string. His argument, as presented and analyzed by TRUESDEEL, 3 was as follows (see Figure 1): ab and cb are similar and equally tense strings that differ in length by a factor of two. BEECKMAN assumed that when the strings are plucked, they have similar triangular forms, so hc = 2lm. (The choice of a particular amplitude for the string's vibration does not limit the analysis since BEECKMAN knew, as will be shown below, that the vibrations of a string are isochronous.) Since the strings are under equal tensions, they receive, in BEECKMAN'S argument, equal velocities when they are released. BEECKMAN assumed that all points of the strings return all the way to the equilibrium position with constant velocities This assumption is only correct for the midpoint, but in any case it does imply that the time required for a string to return to its equilibrium position is proportional to its length, h a c m b Fig. 1 i O. STRUNK, Source Readings in Music History, I, I05. 2 M. MERSENNE, Harmonie Universelle, III, Book 1, TRUESDELL, "The Rational Mechanics of Flexible or Elastic Bodies, ,"

17 Early Vibration Theory 185 BEECKMAN did not publish the argument himself, but MERSENNE included it in the Harmonie Universelle (1636). a That the vibrations of a string are isochronous seems to have been readily known from the fact that tones produced by musical strings retain the same pitch even when they change in volume. In 1618 BEECKMAN wrote : "Since the string comes to rest at last, we must believe that the space through which it moves at the second stroke is shorter than that at the first stroke; and thus that the spaces of the strokes diminish. But, since to the ears all sounds seem the same up to the end, it is necessary that all the strokes are always distant from one another by an equal interval of time, and therefore the following motions move more slowly..., since the string crosses a little space in the same time it formerly used to cross a greater one. ''2 In 1629, answering a question of MERSENNE concerning changes in amplitude in the vibrations of strings, BE~CKMAN made a similar statement. 3 Also corresponding with MERSENNE on this question, DESCARTES seemed to think that a string's amplitude and velocity would, in principle, both decrease in the same proportion, so that the period of vibration would remain constant, but that actual strings might be affected by the air they move in. So he suggested checking this "experimentally with the ear in examining whether the sound of a string... is higher or lower at the end than at the beginning, because if it is lower, it means that the air retards it, if it is higher, that is because the air moves it more quickly." ~ About a month later, DESCARTES emphasized that in contrast with the (non-simple) pendulum, the string is isochronous. He explained the isochronism with the idea that the force making the string return to equilibrium is proportional to the displacement, s In the Harmonie Universelle (1636) MER- SENNE emphasized this isochronism: "since all the returns of the string continue the same sound, and since the two thousandth return of the string is no flatter or sharper than the first or the second, it follows that all these returns joined together produce only the unison." Mersenne's Laws for the Vibrating String. The quantitative description of the fundamental mode of a vibrating string, voct//~ (v is the frequency, l is the length, F is the tension, a is the cross sectional area), for materials of the same density, is generally named after M~RSENNE. Although others knew correctly how frequency depends on length, tension, and cross section, it was MERSENNE who performed detailed experiments to determine the dependence precisely. MERSEYNE studied the vibrating string in the context of his work on musical sound, which was of special interest to him. More than anyone else 1 I~ TRUESOZLL, "The Rational Mechanics &Flexible or Elastic Bodies, ," Ibid. p. 29. MERS~NNE, Correspondance, II, Ibid. p MERSENNE, Harmonie Universelle, II, Book 1, 9.

18 186 S. DOSTROVSKY in the seventeenth century (or since) MERSENNE wanted to develop a comprehensive science of music. The range of his interest within this subject was immense. To give just a few examples : he worked on musical aesthetics, ancient and non-western music, the characteristics of a musician, the properties of musical instruments, and the nature of sound. "The industrious Mersenne," as BOYLE called him, 1 wrote many treatises, among which the Harmnonicorum Libri (1635) and the Harmonie Universelle (1636) are the most important for the physics of sound. His correspondence; which was immense, stimulated many people to think about the physics of musical sound. By the end of the sixteenth century there was a great variety of instruments - some, like the organ, of great complexity - and MERSENNE perceived that a great deal could be learned about sound from them. He wrote that "it is impossible to explain the diversity of all the sounds without speaking about the diversity of the instruments that produce them. "2 Like BACON, he felt that by preparing lists of the structural and musical properties of instruments, "the manufacturers can aid philosophy.-3 For his study of instruments, MERSENNE often asked his correspondents to obtain specific information. For example, he sent CORNIER (a lawyer) to learn from organists about certain details of pipes, 4 and PEIRESC, to obtain a drawing of some cymbals. PEIRESC wrote to GASSENDI that it was difficult "to do it with the exactness which is required. "5 MERSENNE gave a detailed exposition of his laws in the Harmonie Universelle, in a proposition on tuning stringed instruments: "Un homme sourd peut accorder le Luth, le Viole, l'epinette, et les autres instrumens a chorde, et trouver tels sons qu'il voudra, s'il cognoist la longeur, et la grosseur des chordes: de la vient la Tablature des Sourds."6. He considered different variables or pairs of variables separately and indicated what their proportions should be to produce particular musical intervals (generally unisons or octaves) between the two strings. The laws did not appear in the above algebraic form but unfolded themselves as follows: According to the first rule, if strings of the same material, length and cross-sectional area (the French word is grosseur; from the context it is clear that it is being used to designate area, except in the final table) are to sound an octave apart, then if one of them is stretched by a weight of one pound, the other must be stretched by a weight of four pounds, "since the weights are in double ratio to the harmonic intervals." (This rule indicates that voc V ~, for constant l, a.) The second rule gives a correction factor (to be applied generally) which will be discussed below. According to the third rule, if strings of equal cross section and unequal lengths are to be in unison, then the weights that stretch them must be in a ratio equal to that of the squares of the lengths, with the smaller weight on the shorter string (this rule, in conjunction with the first, indicates that roe/l/~ for constant 0-). According to the fourth rule, if 1 ROBERT BOYLE, Works, ed. THOMAS BIRCH (London, 1744), I, MERSENNE, Harmonie Universelle, III, Book 1, prop, I, 1. 3 Ibid. Book 6, prop. XIX, MERSENNE, Correspondance, I, 416, MERSENNE, Correspondance, III, MERSENNE, Harmonie Universelle, III, Book 3, prop. VII, ; also in Harmonie Universelle, The Books on Instruments, trans. ROGER E. CHAPMAN (The Hague: Martinus Nijhoff, 1957)

19 Early Vibration Theory 187 strings of equal length and unequal cross section are to be in unison, the ratio of the weights must equal the ratio of the cross sections. (This rule, in conjunction 1 with the first and third, indicates that voct~. ) Four more rules give other combinations Although the three rules described above give the correct law for the fundamental mode, MERSENNE indicated that an adjustment to these rules is necessary in reality. He claimed (in his second rule) that to tune strings that are equal in length and cross section an octave apart, one must use weights in the ratio 4~: 1 (instead of 4: 1). It appears from the discussion that MERSENNE intended this to be a general correction, to be applied to the tension in all cases. This "correction" is obviously empirical. Deviations due to the stiffness of the strings would not require so large a correction. 1 It seems that the correction was needed because of experimental error in determining the tensions of the strings: MERSENNE probably used horizontal monochords with strings passing over pulleys; in which case friction in the pulleys could have made such corrections necessary. 2 There could also be uncertainty as to the effective length of the string, depending on how rigidly its "ends" are held against the bridges. It is interesting that MERSENNE divided his empirical result into two components: a "law" (rule 1) and a "correction" (rule 2). What he chose to call the law happens to have been correct, but one wonders how he knew. Was it physical intuition or a PYTHAGOREAN confidence in the importance of small whole numbers? It seems that in all seventeenth century references, "Mersenne's laws" appeared without "corrections." When comparing vibrational frequencies of strings made of different materials, 1 the density, Q, must be included: v oc 7 Fl/~a. MERSENNE knew from experience that strings made of different materials (with equal dimensions and tensions) sound at different pitches, but he did not quantitatively relate this difference in frequency to the difference in density. Nevertheless, in the ninth rule of the proposition discussed above, he indicated exactly how to adjust the tensions of four strings made of different materials so that they sound at the same pitch. (On this occasion he did not mention density at all; in the Harrnonicorurn Libri, however, he said that density affects the sound, but not proportionally.)3 MEg- 1 For a reasonable wire, taking stiffness into account, the ratio of the tensions for the octave (vl/vz=2) is F1/F2~4.08, while the ratio on the basis of MERSENNE'S correction is F1/Fzg4.25. The fundamental frequency for a stiff string is given by the approximation v~(l/2l)}/f/~a(l +2//) 1 / ~ where Q is YOUNG'S modulus, Q is the density, a is the cross sectional area, and K is the radius of gyration (for a circular cross section of radius r, K=r/2); see, for example, PHILIP M. MORSE, Vibration and Sound (New York: McGraw Hill, 1948), 169. Calculating F 1 (for vl/v2=2) for a steel wire (0=7.7 grams per cm 3, Q=2 x 1012 dynes per cm2), with F~= 106 dynes, l=50 cm, r=2 x 10-z cm, (v 2 ~ 100 cps), we obtain F1 ~4.08 x 106 dynes so F1/Fz ~ On making casual observations one does have deviations of the same order of magnitude as MERSENNE'S, To reduce the error due to friction, W.C. S~INE recommended that, in determining the frequency produced by a given weight, one use the average of results obtained when the weight has reached its equilibrium position both from above and from below (A Student's Manual of a Laboratory Course in Physical Measurements, Boston, 1893, p. 35); one can also use a vertical monochord. 3 TRUFSDELL, "The Rational Mechanics of Flexible or Elastic Bodies, ," 29.

20 188 S. DOSTROVSKY SENNE gave the following data on the pitches of strings equal in length and cross section but made of different metals: 1 gold 2886 E: silver 2160 A; copper 2025 B-flat; steel 1920 B. He explained that the pitch of the gold is a fourth below that of the silver and a fifth below that of the steel, the pitch of the silver a whole tone below that of the steel, etc. That is, the numbers given above are proportional to the inverse frequencies. According to MERSENNE, strings of equal length and cross section are to be brought into unison by stretching them with weights proportional to the squares of the numbers listed. For example, he stated that the weights on the gold and the steel strings should be in the ratio 6~,3:3 (not including the correction factor). Since 2886/1920 = 3/2, the ratio of the weights is 9 : 4 = 6}43: 3. GALILEO, who knew MERSENNE'S laws (in part, at least, from his father), perceived the role of density directly: "of the three ways in which pitch may be raised, that which you assigned to thinness of string should more properly be attributed to the weight... if I should want to form the octave between a brass string and one of gut, it would be done not by thickening [the lower] one four times, but by making it four times as heavy. As to thickness, the metal string would not be four times as thick at all, but four times as heavy, and in some cases this would even be thinner than the corresponding gut an octave higher in pitch. So it comes about that stringing one harpsichord with gold strings, and another with brass strings of the same length, tension, and thickness, the first tuning comes out about a fifth lower, since gold is about twice as heavy. "2 In fact, if ~g=2ob, then v o / % = ~ = 1/]f2, so the pitches of the two instruments will differ by a tritone, a diminished fifth. In a letter to MERSENNE, DESCARTES criticized GALILEO'S argument; he said that the pitch of the gold string was lower not because it is heavier, but because it is softer. 3 To indicate the "thickness" of strings, MERSENNE, GALILEO, and others in the seventeenth century used the cross-sectional area; frequency was thus inversely proportional to the square root of the "thickness." However, if the string's diameter (known directly from measurements) had been used instead, the dependence of frequency on thickness (inversely proportional to the diameter) would have had exactly the same form as the dependence on length. Considering the traditional interest in the musical ratios, it seems that the proportionality in terms of diameters would have been attractive. In fact, observing that a string's frequency is inversely proportional to the square root of the thickness (defined by the cross-sectional area), MERSENNE wrote that this fact "is marvellous, since it seems that the string double in thickness should sound an octave below, as does the string double in length.'4 The use of cross-sectional area may have been the result of a sense that, since it is directly proportional to the mass per unit length, it is physically the more relevant quantity. 1 Harmonie Universelle, IIL Two New Sciences, p G. GALILEI, Opere, VIII, Harmonie Universelle, I, 174.

21 Early Vibration Theory Seventeenth Century Derivations of Mersenne's Laws. MERSENNE wanted to know why the variables of the string are related according to the laws that he found. In 1645 he suggested to TORRICELLI that he try to prove v oc ]/F mechanically. 1 In 1646 he wrote to CHRISTIAAN HUYGENS who, although only seventeen years old, already had a reputation for his talent in mathematics and physics. MERSENNE asked him why a string's tension must be increased by a factor of four to make its pitch rise by an octave whereas its length needs to be decreased only by a factor of two.2 HUYGENS replied that he had often speculated about this, without success, and that it must be a difficult question, otherwise "it would not have been ignored by so many brave souls until now.-3 Nevertheless, HUYGENS worked on the problem of the vibrating string some- time around He derived the vibrational frequency of a string, v = ~//]/F~, under the simplifying assumption that its mass is concentrated at its midpoint, by finding the cycloidal pendulum with the same restoring force. This agrees with MERSENNE'S laws except that the constant is not appropriate for a uniform string. He also made a preliminary attempt to analyze the vibration of a string with the mass concentrated at a number of discrete points. S (Other work of HUYGENS on vibration will be discussed in Sections 3.4, 4.4, and 5.2.) In 1713, JOSEPH SAUVEUR derived the absolute vibrational frequency of a string. 6 On the one hand, SAUVEUR used the fact that the gravitational field has only a negligible effect on the string's vibrational frequency while, on the other hand, he used it to put the string in a special dynamic form. The string is stretched horizontally and, because of the gravitational field, it hangs in a curve. The amplitude of the vibrations can be assumed small, compared with the string's sag, and horizontal. In this situation, the fundamental vibration is a swinging motion. SAUVEUR'S assumption was that the string performs this swinging motion as a rigid body. SAUVEUR found the length of the flbche, f, the distance from the lowest point on the string to the horizontal axis: where W is the total weight of the string, l is its length, and F is its tension. SAUVEUR made his geometrical calculation for the flbche in the limit that the curve is an arc of a circle or a parabola, but the result is correct in general. SAUVEUR used HUYGENS' result that the length, p, of a simple pendulum isochronous with a compound pendulum is [ y2 du P-'Sydu, 1 TRUESDELL "The Rational Mechanics of Flexible and Elastic Bodies, ," 47, n3. 2 HUYGENS, Oeuvres, I, Ibid. p HUYGENS, Oeuvres, XVIII, 489~494. s TRUI?SDELL, "The Rational Mechanics of Flexible and Elastic Bodies, ," "Rapport des sons des cordes d'instruments de musique aux fl6ches des cordes; et nouvelle determination des sons fixes," Memoires de l'acaddmie Royale des Sciences (1713), (Paris, 1716); FONTENELLE'S summary is: "Sur les cordes sonores, et sur une nouvelle determination du son fixe," Histoire de l'acaddmie Royale des Sciences (1713), (Paris, 1716). W1 8F'

22 190 S. DOSTROVSKY where du is an element of mass and y is its distance from the axis. For the string, treated as a compound pendulum, SAUVEUR found 1 that So the string's frequency is p 4 =~f. =~[/4~- ~ 21 ' which would be the modern result for the ideal string if l/~ were replaced rc by 1. Thus SAUVELrR'S result is off by less than ~-% (but in the right direction to accommodate some stiffness!). SAUVEUR himself checked it by comparing it with his own experimental determination of absolute frequencies (to be discussed in Section 5.3). In the same year (1713), BROOK TAYLOR, then twenty-eight years old, presented his celebrated analysis of the fundamental mode of the vibrating string, 2 which looks forward to the mathematical physics of the eighteenth century. In this work, TAYLOR obtained the modern result. But SAUVEUR presented his analysis when he was seventy years old, after having worked on many of the central problems of seventeenth century vibration and sound. (His other work will be discussed later in this paper.) Both in its context and its style-involving an ingenious use of the relevant pendulum-satjvet:r's derivation belongs firmly in the seventeenth century The Vibrating Air Column. In the first half of the seventeenth century, the vibrations of air columns posed a much more difficult problem than the vibrations of strings. In the context of a theoretical analysis (after the eighteenth century), the ideal string and the ideal air column are quite comparable. The wave equation (with appropriate constants) describes them both and only the boundary conditions vary. For the fundamental modes, the frequency of the string is V=Vs/2l (the velocity of propagation of a wave along a string is vs= F ~ and the frequency of a cylindrical pipe open at both ends is v=va/21 (the velocity of sound in air is va=~, where (TP) is the adiabatic elasticity and 0a is the density). However, in the context of an empirical analysis, the string and the air column present themselves in very different ways. For the vibrating string, a correlation of the "obvious" variables successfully yielded the correct proportionalities for the fundamental, without any need for a model of the vibrations. For the vibrating air column, although wind instruments provided a variety of pipes for observation, the "obvious" variables 1 That is, for the string z=o:x 2 with endpoints at x= +]/~, gyt~ S (f-c~x2)2]/a+4 :2xgdx o 4 P VYT~ ~gf' (f- ax2)]/1 +4o~2xZdx 0 where the approximation is good since a and f are small. (There is equality if ~ by one.) 2 BROOK TAYLOR, "De motu nervi tensi," Phil. Trans. 28 (1713), (1714). is replaced

23 Early Vibration Theory 191 (such as width) could be misleading and confusing. Even the dependence of frequency on length, simple in an idealized situation, is not exactly observable with real pipes because of the ambiguity of the boundary conditions. The one advantage of pipes over strings is that they can indicate the wavelength of sound approximately, but a model of longitudinal vibrations is necessary to find this. In the first half of the century such a model did not exist, and the intuitive ideas about analogies with water waves were not helpful in this case. In the first half of the seventeenth century, only MERSENNE was sufficiently bold (or naive!) to work on the physics of pipes. He tried hard to find laws describing the connections between the pitches of pipes and their various properties - length, width, shape, material, blowing pressure. This is particularly evident in his discussion of organs in the Harmonie Universelle. The great variety of pipes in use in organs (especially convenient because each pitch has its own pipe or set of pipes) provided much material for observation; in addition, MER- SENNE experimented with specifically chosen organ pipes. Superficially, a pipe's length appeared to determine its pitch in a straightforward way. Discussing the effect of length on the properties of a telescope in The Assayer (1623), GAEmEO had referred to the trombone and to organ pipes as familiar examples of a situation in which the length is a crucial factor. 1 From rough observations, it appeared (as expected) that increasing the length of a pipe by a factor of two, without changing its shape and other properties, caused the pitch to descend by an octave. However, as MERSENNE found in his careful study of organ pipes, even lengths (in real pipes) do not bear a simple relation to pitches. MERSENNE found that the interval formed by two large pipes whose lengths differed by a factor of two was smaller than an octave by an interval of half a tone to a tone. 2 Faced with this situation, MERSENNE commented that since reason alone cannot find the laws of sound vibrations, "it is necessary to consult experience. "3 The effect of a pipe's width was even more puzzling. MERSENNE expected that increasing the cross-sectional area of a pipe would have the same effect on its frequency as increasing its length, since the change in volume is the same in both cases. 4 MERSENNE was presumably thinking that, as in strings, the total amount of material being moved was the important factor. To explain why pitches tend to be lower in bigger pipes, MERSENNE suggested, on another occasion, that the air cannot "beat" as frequently because it meets a larger volume of air which it must "chase out of the tube" or which it must "shake" before producing sound. 5 This is reminiscent of DESCARTES' remark, made in 1630 in a letter to MERSENNE, that lower tones are associated with the motion of a larger quantity of air and that this motion is slower because of the greater resistance. 6 MERSENNE found, however, that width had a much smaller effect (than length). The pitches of five pipes whose diameters were successively smaller pp. 250~51. 2 Harmonie Universelle, III, Book 6, prop. XIII, Ibid. p Ibid. prop. XII, Ibid. prop. 24, MERSENNE, Correspondance, II, 419.

24 192 S. DOSTROVSKY in the ratio 2 : 1, were separated by minor thirds (except for the narrowest pair, where the interval was a whole tone). MERSENNE indicated that the total interval between the widest and the narrowest (diameters differing by a factor of 16) was only a seventh, and that to obtain the octave without other adjustments (such as blowing pressure), diameters differing by a factor of 24 were necessary. Finding that in a rank of organ pipes there were always variations in both length and width, MERSENNE tried to find a rule to relate these dimensions; ~ he indicated, however, that organ builders did not claim the ratio was crucial. 2 MERSENNE knew about two striking situations in which a pipe's pitch changed drastically without any change in its length being involved: the lowering of the pitch by an octave when the end is closed; and the change of pitch (by discrete intervals) in overblowing. The second of these surprised MERSENNE greatly and he had no explanation; this will be discussed in Section 4.3. The first situation did not surprise him because he was able to account for it to his satisfaction. His explanation was based on the idea that in a closed pipe the air can leave only by the opening through which it entered, so that after travelling down the pipe the air is reflected and travels up it again. 3 This makes the distance travelled by the air the same in a closed pipe as in an open pipe of twice the length. MERSENNE used this idea in a quaint attempt to explain why the tone quality of closed pipes is different from that of open pipes: the tone is "sweeter" because the air, returning up the (closed) pipe, "weakens the impetuosity" of the entering air. (MERSENNE was very curious about the variety of tone qualities produced by different organ stops; he observed that the material out of which a pipe is made seemed to affect its tone quality, not the pitch, 4 but he admitted that tone quality was difficult to understand.) MERSENNE studied also smaller effects on pitch, such as the effects of various tuning devices used by organ builders 5 and he reported on the finding of his friend CORNU (a surveyor) on the relative dimensions required in cylindrical and rectangular pipes (of the same height) so that they will sound in unison. 6 The nature of a wave in air and the way in which it "fits" into a pipe became clear only later, towards the end of the seventeenth century. It appears (although only a brief indication is available) that HUYGENS understood the connection between wavelength of sound and pipe length. In a note written in 1682 or later, HUYGENS calculated the wavelength (he does not call it wavelength) of a tone from measurements of the velocity of sound and of absolute frequency, and he verified that its pitch was close to that of a tone produced by an open pipe whose length was half the wavelength 7 It was NEWTON who stated explicitly the relation between wavelength and pipe length: "it is probable that the wavelengths [latitudines pulsuum], in the sounds of all open pipes, are equal to twice the lengths of the pipes."s He Harmonie Universelle, III, Book 6, prop. XIV, Ibid. prop. III, Ibid. prop. XX, Ibid. props. XVIII, XX. 5 Ibid. prop. XI, o Ibid. prop. XVIII, 346. HUYGENS, Oeuvres, XIX, 376. s ISAAC NEWTON, Philosophiae Naturalis Principia Mathematica (London, 1687), 372.

25 Early Vibration Theory 193 used this to make an extra check on the velocity of sound that he had obtained from his mathematical analysis of the propagation of a wave. Knowing that velocity is the product of frequency and wavelength, NEWTON made a rough (implicit) calculation of velocity from measurements of the absolute frequency of the tone produced by an organ pipe of known length (see Section 7.3 for details). 4. The Puzzle of Overtones and Higher Modes 4.1. Overtones. Tones are generally linear superpositions of fundamentals and overtones. Musical tones are generally periodic vibrations; so the frequencies of their overtones are integral multiples of the fundamental frequency; that is, they are harmonic. 1 Although the natural modes of a freely vibrating object are not generally harmonic, the vibrating string does have harmonic modes: In the case of a string of length l, cross section a, density Q, held with a tension F between fixed points, the modes are ngx y,=a, sintcos2~v,t n with the frequencies v n =~ ]/~a. Therefore, a vibrating string of given length, whether bowed or plucked, can produce various harmonic pitches, and it can also produce the same pitches simultaneously. 2 No doubt, overtones have always been heard; and they presented a problem even in antiquity. 3 But to MERSENNE and others in the early seventeenth century they presented a bothering problem because, with the newly achieved identification of pitch with frequency, they implied that an object could vibrate with more than one frequency at once. At the time this seemed paradoxical. Harmonics were understood towards the end of the century when they came to be associated with the existence of nodal points on a vibrating string. Although he never understood harmonics, MERSENNE was aware of the importance of the problem that they posed, and he frequently urged his correspondents, including DESCARTES 4 and CONSTANTIJN HUYGENS (the father of CHRISTIAN HUYGENS), 5 to search for a solution. MERSENNE discussed the problem in the Harmonie Universelle (1636). 6 He stated that an open string "makes at least five tones at the same time." He explained that although it is helpful to listen 1 The name "harmonic" was not used in this context until SAUVEUR proposed it in A wind instrument produces harmonic overtones even though its (undriven) modes are only roughly harmonic because it is driven, rather than left to vibrate freely. Bells, drums, and plucked strings are freely vibrating, but generally only the strings (if they are not too thick) have harmonic overtones. 3 In the ARISTOTELIAN Problems it was asked "why is it that in the octave the concord of the upper note exists in the lower, but not vice versa?" (XIX, 13). 4 MERSENNE, Correspondance, Ill, HUYGENS, Oeuvres, I, For example, in III, Book 4, prop. IX, 208~11. All quotations in the following discussion are from this proposition.

26 194 S. DOSTROVSKY under quiet conditions (at night), to concentrate well, and to have had some experience, he and many musicians had no trouble hearing these tones. He identified the (harmonic) pitches: "these tones follow the ratio of the numbers 1,2, 3, 4, 5 because one hears four tones different from the natural one, of which the first is at the octave above, the second at the twelfth [3 : 1], the third at the fifteenth [4:1], and the fourth at the major seventeenth [5:1]." MERSZNNE sometimes heard also a tone at the twentieth (20:3), but he realized that it did not fit with the set mentioned above. Presumably this was a misjudging of the seventh harmonic, which does not coincide with a note of the scale. MERSENNE was confident that the tones were, indeed, all coming from the same string (and not, by resonance, from other strings) because he had heard them "more than a hundred times," not only from the strings of musical instruments but also from the single string of the monochord. MERSENN~ realized that the relative pitches of the overtones were the same as those of the trumpet and the trumpet marine (to be discussed below). But comparison with other sets of overtones revealed confusing differences: in an organ tone he heard only the twelfth; and in the tones of bells he heard the octave, major tenth, and twelfth 1 (which, with the "natural" tone, form the sequence of relative frequencies 1, 2, 2½, 3). 2 MERSENNE was curious about many details: He wondered why the string produced only certain overtones and why some were stronger than others, but, most of all, he was puzzled by the fact that many tones were produced simultaneously. That the same string could have a variety of frequencies seemed impossible: "[Since the string] produces five or six tones..., it seems that it is entirely necessary that it beat the air five, four, three, and two times at the same time, which is impossible to imagine, unless one says that half the string beats it twice while the whole string beats it once and that, in the same time, third, quarter, and fifth parts beat it three, four, and five times, a situation that is against experience, which shows clearly that all parts of the string make the same number of returns in the same time, because the continuous string has a single motion, even though parts near the bridge move more slowly.,,3 MERSENNE wrote that some people thought that the string was made of concentric layers, each of which produced a different frequency. He did not think this was very likely and suggested, among various possibilities, that the air acquired extra frequencies- by interacting with the string in some way The Trumpet Marine. Since the superposition of different frequencies was so perplexing, one might expect that situations in which higher modes are produced separately would have revealed the nature of harmonics more clearly. Brass instruments do make explicit use of the higher modes to obtain a scale of notes, but the air column and the nature of its vibrations cannot be seen IbM. III, Book 7, prop. XVIII. 2 These would be harmonic only if there were a missing fundamental, but presumably the bell was tuned to give these overtones. 3 Ibid. III, Book 4, prop. IX, 210.

27 Early Vibration Theory 195 easily. In addition, their higher modes deviate from the harmonics and, in any case, they are manipulated by the player. However, the trumpet marine was an appropriate instrument for demonstrating higher modes. The trumpet marine is a bowed-string instrument. It has a single string, about six feet long, stretched over a resonating cavity. 1 In contrast with all other stringed instruments, where a string is pressed firmly against the fingerboard to produce a vibrating section of any chosen length, the string of the trumpet marine is touched lightly at the nodal points only. Instead of a continuum of pitches (supposing the instrument to be unfretted), the trumpet marine, like the trumpet, sounds only a discrete set of pitches that correspond to its natural modes. Early forms of the instrument existed already in the thirteenth and fourteenth centuries. The instrument was developed in the late fifteenth century, when its trumpet quality was enhanced by using an asymmetric bridge. One foot of the bridge was held firmly, but the other foot was left free to rattle against the soundbox. This invention may have been due to the French musician MARIN (and hence the name, trompette marine). Because its tones were limited to harmonics, the trumpet marine aroused curiosity. M~RSENNE and his correspondents, followed by others throughout the seventeenth century, attempted to understand the trumpet marine physically. MERSENNE learned about the instrument in 1634 from CHRISTOPH DE VILLIERS, 2 with whom he had already corresponded on problems in the physics of musicincluding the problem of overtones. Although he had not yet seen the trumpet marine at that time, VILLIERS sent MERSENNE a description. Later that year, he sent more details about its structure and dimensions, and about the manner in which it was played, a Unable to explain why the trumpet marine produces only certain pitches, VILLIERS remarked that "geniuses are needed to explain these secrets.,,4 Soon after, MERSENNE himself had a chance to see and hear the trumpet marine in performance, 5 and he wrote to Descartes about it. 6 MERSENNE described the tumpet marine and its "very remarkable" properties in the Harmonie Universelle.7 He stated that the points at which one must touch the string to "imitate" the trumpet tones divide it according to "the first, second, and third bisection. "8 At all other points, he said, the instrument produced an ugly tone that is "difficult to endure". This surprised him, "since the bow touching a viol string makes good sounds no matter where the left hand stops the strings." For MERSENNE, the trumpet marine differed from the viol only in its trembling bridge, but he realized that even with a regular bridge i The trumpet marine and its history are discussed by CECIL ADKINS, in Grove's Dictionary of Music and Musicians, sixth edition (to appear), and by F.W. GALPIN, "Monsieur Prin and his Trumpet Marine," Music and Letters 14 (1933), 18~9. 2 M. MERSENNE, Correspondance, IV, 59. Ibid. pp Ibid. p Ibid. p Ibid. p Harmonie Universelle, III, Book 4, props. XII, XIII, All quotations in the following discussion are from these propositions. 8 If l is the length of the string, these are the points l/n from the nut end.

28 196 S. DOSTROVSKY the trumpet marine would produce its characteristic set of tones. He was apparently not aware of-or did not dwell upon-the facts that the string of the trumpet marine is not pressed firmly against the fingerboard (in fact, it must be touched very lightly) and that no matter what pitch is sounding, the entire string vibrates. It is surprising that he did not know these things, but he probably had not played the instrument himself. Although MERSENNE never understood the trumpet marine, he did draw attention to the fact that its relative pitches are the same as those of a string's overtones, and he hoped that his discussion would at least "give good minds a chance to explain it properly." The mystery of the trumpet marine would be solved towards the end of the century, as part of understanding the connection between nodes and higher vibrational modes. (This will be discussed in Section 6.1.) 4.3. Wind Instruments. Wind instruments were perplexing in the early seventeenth century partly because different modes are used in playing them. (In flutes and other instruments with tone holes, the first three or four modes are used; in the trumpet and other instruments of fixed length, more than a dozen.) This contrasts with stringed instruments in which, except for the trumpet marine, the fundamentals are used almost exclusively. M~RSENNE was puzzled by the trumpet and often asked why it can produce only a specific set of pitches. In the Harmonie Universelle he gave two obscure reasons. These are worth mentioning, if only to indicate how far MERSENNE was from understanding higher modes. In one of the explanations, MERSENNE suggested that the trumpet changes from its lowest frequency to another twice as great before it changes to any other frequency because nature takes the shortest route; 1 in the other, he said that with two sources of sound, one of them would have to vibrate 8 times and the other 9, to produce an interval of a whole tone (9 : 8), whereas it is sufficient for one source to vibrate once and the other, twice, to produce the octave, so the octave is obtained before the whole tone. 2 Organ pipes were even more puzzling to MERSENNE because there was no systematic set of higher modes. 3 Observing this variety, MERSENNE said that "it is difficult to explain why the pipes do not produce the same intervals as the trumpets. "4 Experimenting himself with organ pipes, MERSENNE found that open pipes overblow at the octave, and stopped pipes at the fifth [?] or the twelfth. However, he knew that in some pipes increasing the air pressure raised the pitch by only half a tone, that in short pipes (width approximately equal to height) the pitch did not rise at all, that some reed pipes overblew by a major third [?] and that one of his long pipes could produce the octave, the twelfth, and the fifteenth. It was on observing this confusing variety that MERSENNE proposed, as mentioned above, that manufacturers of musical instruments help in the study of musical sound- for example, by correlating various properties of organ pipes with their overblowing characteristics. 1 III, Book 5, prop. XII. 2 III, Book 5, prop. XVIII. 3 III, Book 6, prop. XIX. 4 Ibid.

29 Early Vibration Theory A Final Example. HUYGENS' apparent confusion about harmonics, forty years after MERSENNE'S Harmonie Universelle, provides a final example of the puzzle posed by overtones and higher modes. HUYGENS' attention may have been drawn to harmonics quite early when, as mentioned in Section 4.1, MERSENNE wrote to HUYGENS' father about them. In 1674, on reading MERSENNE'S Harmonie Universelle, HUYGENS wrote notes on harmonics and on the tones of the trumpet marine and the trumpet. 1 The only available indication of HUYGENS' thinking about harmonics consists of a few notes he wrote in 1675, about a way to observe the" tremblements entremeslez" of a string. 2 Naturally, these notes should not be taken as HUYGENS' final statement on the subject of harmonics; nevertheless, they indicate that even for someone with a good sense of some properties of waves, the nature of higher modes of vibration was not obvious. Presumably to make the harmonics louder than they usually are, HUYGENS recommended a particular procedure for setting the string in motion: the string is touched at the ½ point for the first instant while the shorter section is plucked; he claimed that tones a fifth and a twelfth above the fundamental are then heard. Indeed, if the ½ point were thus made a node, the string would vibrate in its third mode, sounding the third harmonic which is a twelfth above the fundamental. But HUYGENS' idea that a fifth would be heard is strange because the fifth is not a harmonic. If the point was held too firmly to be a proper node at the instant of plucking, but there was still enough looseness in the contact that energy could get to the longer section of the string, then the two sections could vibrate more or less independently producing the fundamental pitches appropriate to their lengths (a fifth and a twelfth higher than the pitch of the entire string). However, this situation would apply only for the first instant, while the string was being touched. Nevertheless, since the string, on being released, would vibrate in the third mode, one might think that one still heard a fifth, because of hearing the twelfth (an octave above). If HUYGENS actually tried the experiment, this might have been the source of some confusion. At any rate, he seems to have reached the erroneous conclusion that the short section oscillates twice for each oscillation of the long section even after the finger is removed. He wrote that to be convinced that the shorter section is going through two cycles while the longer section goes through one, it is necessary actually to see this happening. He suggested, therefore, that lead wire be wrapped around the string to slow down the vibration. 5. Absolute Frequency 5.1. Mersenne. In connection with identifying pitch with frequency, early in the seventeenth century, MERSENNE and others became concerned with the possibility of measuring actual frequencies. MERSENNE made the first determination of the absolute vibrational frequency of a tone by assuming that the frequency- 1 length rule, v oc~, holds even for a string so long that its inaudible vibrations can be counted visually. Although MERSENNE was interested in the frequencies i HUYGENS, Oeuvres, XX, HUYGENS, Oeuvres, XIX,

30 198 S. DOSTROVSKY of tones whether or not produced by vibrating strings, he dealt specifically with the frequencies of vibrating strings. MERSENNE published the following data and results: In the Harmonie Universelle (1636), he wrote that a string 17-} French feet long, stretched by a weight of 8 pounds, vibrates at 8 cps; and that a section 20pouces long (12 pouces ~1 French foot), at the same tension, is in unison with a four foot open pipe pitched at the ton de chapelle. As MERSENNE wrote, it follows from this that the frequency of the short section is 84 cps. 1 In another part of the Harmonie Universelle MERSENNE listed the frequency associated with a four foot pipe as 96 cps. 2 In the Harmonicorum Libri XII (1648), MERS~NNE described the results indicating that a string in unison with a one foot closed pipe vibrated at 200 cps. 3 Since one cannot know the actual dimensions or even the length of a pipe without familiarity with the actual organ, and since there are many influences on the pitch produced, one cannot immediately know how close MERSENNE'S results were to the actual frequencies. However, for the one foot closed pipe mentioned above, MERSENNE gave measured length and width. Estimating the pipe's thickness. ALEXANDER ELLIS made an open pipe of similar measurements and found its frequency to be about 477 cps. 4 Thus MERSENNE'S pipe probably sounded at about 224 cps. In this case, MERSENNE'S determination was off by about 10%. According to ELLIS, it is likely that the four foot pipe mentioned in the Harmonie Universelle sounded an octave below this one, so its frequency might have actually been 112 cps and MERSENNE'S measurements for this would have been even further off. However, it is really inappropriate to analyze MERSENNE'S measurements too closely, because MERSENI~ himself seems to have been quite casual about the actual results. He was mainly interested in the idea of finding absolute vibrational frequencies. He never had a use for precise results, and he invited his readers to experiment for themselvesfl The values published are, therefore, not necessarily the best that could have been obtained using his method at the time. (In 1860, a refined version of MERSENNE'S experiment measured frequencies to an accuracy of about 1%.)6 Nevertheless, despite the inaccuracy of MERSENNffS results, they were probably the only estimates of frequency available. Fifty years after the Harmonie Universelle, NEWTON referred to MERSENNE'S measurements 7 when he calculated the length of a sound wave (to be discussed in Section 7.3). 1 I, Book 3, prop. VI, III, Book 3, prop. XVII, The frequency is actually listed as 48 cps, but, noting the discrepancy with the 84 cps result himself, MERSENNE explained that he had sometimes used the higher and sometimes the lower octave (I, 171). Since the organist's convention is to characterize all the notes in a given octave by the same approximate length, to distinguish them from notes in other octaves, one can see how such a confusion would be possible. 3 M. MEI~.SENNE, Harmonicorum Libri XII (Paris, 1648; facsimile reprint, Geneva, 1972), Book II, prop. XXI, MERSENNE wrote 150 for the frequency, but this is obviously an arithmetical error. 4 ALEXANDER J. ELLIS, "On the History of Musical Pitch," Journal of the Society o farts XXVIII ( ), ; pp Harmonie Universelle, I, ELLTS, ISAAC NEWTON, Principia (London, 1687; facsimile reprint, London, 1960), 372. NEWTON used 104 cps for the frequency of a four foot pipe and referred to the Harmonicorum Libri, Book I,

31 Early Vibration Theory Hooke and Huygens. A generation after MERSENNE, ROBERT HOOKE was interested in estimating absolute frequencies of tones. In the Micrographia (1665), 1 HOOKE asserted that "'tis very probable (from the sound [a fly makes], if it be compar'd with the vibration of a musical string, tun'd unison to it) [the fly's wings make] many hundreds, if not some thousands of vibrations in a second..." SAMUEL PEPYS recorded a conversation with him on the subject a couple of years later: "... discoursed with Mr. Hooke a little, whom we met in the street, about the nature of sounds, and he did make me understand the nature of Musical Sounds made by Strings, mighty prettily; and told me that having come to a certain Number of Vibrations proper to make any tone, he is able to tell how many strokes a fly makes with her wings (these flies that hum in their flying) by the note that it answers to in Musique during their flying." 2 There does not appear to be any explicit evidence that HOOKE made his own determination of absolute frequency. He could have based his estimates on MER- SENNE'S results. However, it is unbelievable that he did not make some estimates of his own. In 1676, he told Sir CHRISTOPHER WREN that "sound was nothing but strokes within a determinate degree of velocity" and that he "would make all tones by strokes of a hammer. "3 In 1681, he showed "a way of making musical and other sounds by the striking of the teeth of several brass wheels, proportionally cut as to their numbers [relative frequencies], and turned very fast round, in which it was observable that the equal or proportional strokes of the teeth, that is 2 to 1, 4 to 3, etc., made the musical notes [intervals].,,4 HOOKE'S wheel s demonstrated experimentally, essentially for the first time, that which GagmEO had demonstrated conceptually, namely, the correctness of identifying pitch with frequency or, more precisely, interval with relative frequency. Furthermore, it provided a natural and obvious means of determining absolute frequency. But the accuracy of the method would be limited, in particular, by the fact that the wheel produces a very impure tone. Around 1682, CHRISTIAAN HUYGENS made a frequency measurement. 6 This arose naturally in the context of his strong interest in music. The son of a prop. IV; although there is no discussion of absolute frequency in that place, one gets a frequency of 100 from the discussion in the Harmonicorum Libri mentioned above. I have not come across any mention of the frequency 104 for the four foot pipe. 1 R. HOOKE, Micrographia (London, 1665; Dover reprint, 1961), 173. z The Diary of Samuel Pepys, ed. R. LATHAM & W. MATTH~WS (University of California Press, 1972), VII, The Diary of Robert Hooke, ed. H.W. ROBINSON & W. ADAMS (London, 1935), RICHARD WALLER, The Life of Robert Hooke, in ROBERT T. GUNTHER, Early Science in Oxford, VI (Oxford, 1930), 1-68, p It is inappropriately known as SAVART'S wheel, after FELIX SAVART ( ), who worked in acoustics. 6 CHI{[STIAAN HUYGENS, Oeuvres Complktes (22 Vols., The Hague: Martinus Nijhoff, ), XIX,

32 200 S. DOSTROVSKY musician and composer, HUYGENS sang, played viola da gamba, lute, harpsichord 1 and probably also flute. 2 He read many books, both ancient and modern, on music, including MERSENNE'S Harmonie Universelle. He published a discussion of a system of temperament dividing the octave into 31 equal intervals. 3 His numerous and fascinating notes (which he did not publish) include many observations, experiments, reflections, and hypotheses on a variety of musical and acoustical topics that intrigued him. 4 The following are some examples of HUYGENS' interests. He measured the velocity of sound 5 and experimented with the effect of medium density on propagation (he did an experiment, reminiscent of BOYLE'S, with an alarm clock in an evacuated jar). 6 He wondered whether the pitch of a flute is affected by changes in the air pressure. 7 He derived MERSENNE'S laws for the vibrating string (see Section 3.3), he speculated on the nature of harmonics (see Section 4.4), and he was concerned with the problem of consonance and dissonance, s HUYGENS appreciated GALILEO'S work on frequency and consonance. 9 Like GALILEO, he stated that, except for the unison, consonances are more "pleasing" when a greater proportion of pulses (battemens) coincide. 1 HUYGENS asserted, like DESCARTES, that the seventeenth is more beautiful than the tenth which, in turn, is more beautiful than the third, because the seventeenth has the greatest proportion of coinciding pulses and the third, the least. He applied this comparison to the problem, avoided by GALILEO, of the relative consonance of the fourth and the major third, HUYGENS wrote that "comparing the fourth with the third according to this maxim [the proportion of coinciding pulses], one would say that the latter should be less agreeable than the fourth... However, the fourth seems the less good of the two." He suggested that since the seventeenth is more consonant than the fourth (it has a greater proportion of coinciding pulses), and since the seventeenth "replicates" the third (it spans a third plus two octaves), it was natural that the third would sound more consonant than the fourth. He justified the attention to the "replicated" intervals on the basis of the octave and double octave overtones that are present in tones. He hinted that this analysis would explain also the consonance of the sixth (even though a slight mistuning is involved). HUYGENS remarked that taste is always involved in judging degree of consonance and he referred to the "ancients" who included neither the thirds nor the sixths among the consonances as an example of this. He suggested looking for other, undiscovered, 1 H.J.M. BOS, Dictionary of Scientific Biography, VI, HUYGENS referred to his flute in connection with a number of acoustical observations : Oeuvres XIX, 377; XX, HUVGENS first worked on this system of temperament in 1661 (his notes are in Oeuvres, XX, ), and it was published in 1691 (reprinted in Oeuvres, X, ). HUYGENS' writings on musical sound, with notes and background material, are in C. HUYGENS, Oeuvres Complktes, primarily in Vols. XIX, and XX, s Oeuvres, XIX, Oeuvres, XIX, v Oeuvres, XIX, Oeuvres, XX, Oeuvres, XIX, lo Oeuvres, XX,

33 Early Vibration Theory 201 consonances "because we might make the same mistake as the ancients." HUYGENS also warned against searching for an ultimate cause of consonance: "C'est cette percussion ordonne~ de l'air qui, agissant dans notre oreille produit le plaisir des consonances-non pas toute fois que l'esprit puisse aucunement compter ni discerner ces battemens [pulses here, not beats] ni contempler leur commensurabilit&.. Et je croy que ce seroit en vain de vouloir chercher la cause de ces plaisirs plus loin. "1 To measure frequency, HUYGENS produced a tone by rotating a large wheel connected by a driving belt to a small wheel with a prong that struck a piece of metal on each revolution. 2 Knowing the ratio of radii for the wheels, HUYGENS calculated that the tone (in unison with the D on his harpsichord) had a frequency of 547 cps. HVYGENS calculated the wavelength of the tone (from its frequency and the velocity of sound) and obtained rough agreement between its pitch and the pitch produced by an open organ pipe of half that length. As mentioned in Section 3.4, HUYGENS was aware of the relation between wavelength and pipe length for a vibrating air column, and he made essentially the same calculation that NEWTON made in the Principia (see Section 7.3). There is no indication that HUY~ENS' estimate was known. Among HuYCENS' notes there is a sketch of something that might be a siren 3 -this would have provided another way to measure frequencies Sauveur. With JOSEPH SAUVEUR'S measurements in 1700, 4 frequencies became known to within a few percent. SAVVEVR was initially concerned mainly with practical mathematics and engineering, s He was a tutor at the court of Louis XIV, and he held the chair of mathematics at the CollOge Royal. But he was fascinated by music. Although he does not seem to have had musical training, he consulted with musicians and became quite interested in the arithmetic and the physics of music. Like his predecessors throughout the seventeenth century, SAVVEVR made use of musical experience to obtain information on sound and vibration. However, he also proposed the development of a new subject which he named acoustique, 6 the study of sound in general and not just the son agr~able of music. SAUVEUR worked on a variety of problems which included, in addition to frequency measurements, temperament of the 1 Oeuvres, XIX, 364. HUYGENS, Oeuvres, XIX, HUYGENS. Oeuvres, XX, 105. Described by FONTENEI.LE in "Sur la determination d'un son fixe," Histoire de l'acaddmie Royale des Sciences (1700), (Paris, 1703) and by SAUVEUR in "Mani+re de trouver le son fixe," in "Syst~me General des intervalles de Sons," MOmoires de l'acad~mie Royale des Sciences (1701), (Paris, 1704). 5 For a biography of SAVVEUR, see FONTENELLE'S kloge in Histoire de l'acadbmie Royale des Sciences (1716), (Paris, 1718). 6 M~moires (1701), 294. The word acoustics had already been used in connection with sound. For example: as the title of the second volume of GASPAR SCHOTT, Magia Universalis (1658) which included discussion of the voice and hearing, music, echoes, and sound conduction; in the title of the paper on sound amplification by N. MARSH, "An introductory essay to the doctrine of sounds, containing some proposals for the improvement of acoustics," Phil. Trans. XIV (1684),

34 202 S. DOSTROVSKY musical scale, beats, harmonics, and a theoretical derivation of MERSENNE'S Laws (already discussed in Section 3.3.). At the time that SAUVEUR did his work on vibration, FONTENELLE was the secrktaireperp~tuel of the Paris Academy. FONTENELLE, one of the most famous of the secretaires, was extremely skillful at writing the yearly reports for the Histoire of the Academy and at explaining technical matters. He seems to have been enthusiastic about SAUVEUR'S discoveries and he helped to make them known. FONTENELLE'S reports provide some information on the background to SAUVEUR'S interest in sound and on the impression his ideas made at the time. SAUVEUR began his work in acoustics by developing a method of classifying temperaments of the musical scale, a He divided the octave into 43 equal intervals, merides, each of which was divided into seven eptamerides. SAUWUR's intention was to indicate the size of any musical interval, at least to a reasonable approximation with respect to the ear's ability to discriminate frequency, in terms of an integral number of eptamerides. These divisions make it simple to describe and to compare different tuning systems. For example, the fifths and thirds given by an integral number of merides approximate those of the ½ comma meantone tuning in use in the sixteenth and seventeenth centuries. 2 SAUVEUR'S method of determining absolute frequency, demonstrated before the Paris Academy in 1700, represented his first work on the physics of vibration. SAUVEURused beats to determine frequencies. Beats consist of periodic fluctuations of loudness produced by the superposition of tones of dose, but not identical, frequencies. Of course, beats have always been familiar to musicians, who generally avoid them as a matter of good intonation. But there is no indication that beats were understood before SAUVEUR. MERSENNE, for example, was perplexed about the cause of beats between mistuned organ pipes. 3 SAUVEUR knew that the frequency of beats produced by a pair of tones is equal to their frequency difference. He seems to have understood that the total sound produced by different sources is louder when the vibrations are in phase (although he did not use this terminology). As FONTENELLE explained in his summary of SAUVEUR'S lecture, "when two pipes are tuned so that they beat only six times [per second], that means that their vibrations come simultaneously [se recontroient] only six times in a second. ''4 This explanation of beats, understood in terms of coinciding pulses, did not require going beyond the traditional seventeenth century picture of sound as a succession of pulses. (The idea of coinciding pulses had previously been extensively used, less correctly, in explanations of consonance, but only to account for a pleasing sound, not for a greater amplitude.) SAUVEUR may thus have been the first to have had an understanding of the superposition principle. 5 1 "Syst+me General des Intervalles des Sons," M~moires de l'academie Royale des Sciences (1701), (Paris 1704). 2 j. MURRAY B~BOUR, Tuning and Temperament, an Historieal Survey (East Lansing, 1953), Harmonie Universelle, III, Book 6, prop. XXVIII, Histoire (1700), 139. s Superposition was an implicit theme in SauwuR's acoustics; it appeared also in his studies of the vibrating string and of tone color (see Section 6.2).

35 Early Vibration Theory 203 The absolute frequencies of a pair of tones can be calculated from their frequency difference (given by the beat rate) and their frequency ratio. 1 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 perfect thirds and fifths (for example: tune up two major thirds and then down a fifth). As a results of experiments done with DESLANDES, an organ builder, SAUVEUI~ found that the frequency of an organ pipe five French feet long was betwen 100 and 102 cps. SAUVEUR claimed to have obtained consistent results from experiments done with other pipes.2 In the second and third editions of the Principia, NEWTON used SAUVEUR'S result for his check on the velocity of sound instead of MERSENNE'S result, which he had used in the first edition (see Section 7.3). Since the exact dimensions of the pipe are not known, there is no direct way to determine how accurately SAUVEUR found 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. 3 If each of the three tunings made in the process of obtaining the small half-tone is accurate to within half a cent, it would be possible, in principle, to find the absolute frequency to within 2½%. 4 SAUVEUR was motivated to measure absolute frequencies by the practical need for pitch standardization. For this reason, he arranged his experimental procedure so as to obtain directly the length of a string that vibrates at 100 cps, his choice for the son fixe, a standard frequency. SAUVEUR provided a special f2 I and.f2-,fl = n. So we have.fl = For frequencies.)~ and.f;, )11 = " 1"- t 2 "Rapport des sons des cordes," Mdrnoires de l'acaddmie Royale des Sciences (1713), (Paris, 1716), p ALEXANDER J. ELLIS, "On the History of Musical Pitch," Journal of the Society of Arts, 28( ), , p The number of cents, c, in an interval of frequency ratio r is given by c (log 2/logr). For example, a semitone in the equal tempered scale contains 100 cents. If an interval is out of tune by 0.5 cents, its frequency ratio is wrong by a factor of 6 ~ An interval obtained by tuning three intervals will deviate (at worst) by a factor of 53~ Since the frequency, f, to be determined is given by f=n/(r-1), the ratio of the exact frequency, f, to the experimentally determined frequency, f*, will be (25/24) f/f* =(r53-1)/(r- 1) /24-1 The choice of half a cent is somewhat arbitrary, but piano tuners can tune the middle octave to this accuracy. In tuning a tempered scale on the piano, various helpful cross-checks are available. However, organ pipes have the advantage of sustained tone and, probably, of less inharmonicity. As SAUVEUR warned, a steady flow of air to the pipes is essential for constant pitch, but fluctuations would presumably affect the intervals less than the pitches. Although interval discrimination is more difficult for lower pitches (SAuVEUR had to work at about two octaves below concert A), the experiment can be done at higher frequencies by including extra pitches within the small half-tone and obtaining the beat rate as the sum of the rates between each pair of adjacent pitches. This was the procedure used by A. CAVAILLE-COLL, the famous organ builder, when he repeated SAUVEUR'S experiments in 1868 (A. CAVAILLI~-COLL, "Note sur la d&ermination du nombre absolu des vibrations par ia m&hode des battements, '~ Bulletin Hebdomadaire de, l'association Seientifique de France, IV (1868), )

36 204 S. DOSTROVSKY "ruler" for measuring the length of the pendulum used in timing beats. This ruler indicated the adjustment necessary to the length of a string, in unison with the higher of the two tones, to make it vibrate at the standard frequency. Probably because he paid attention to beats in connection with determining absolute frequency, SAUVEUR thought of using them also as a basis for solving the problem of consonance and dissonance. He suggested that beats are involved in causing dissonance, but he did not develop the idea. In particular, he did not say anything about beats between harmonics (in spite of his interest in harmonics). 6. Higher Modes 6.1. The First Discussions of Nodes: Wallis and Robartes. Harmonics began to be understood in the last quarter of the seventeenth century, when it was seen that a vibrating string producing a harmonic without the fundamental has certain stationary points (nodes). It appears that in this period some English musicians knew explicitly about the connection between nodes and harmonics. JOHN WALLIS, who translated PTOLEMY'S Harmonics and other Greek works on music theory j and who wrote some articles on temperament, 2 heard about certain demonstrations of nodes from N. MAnSH of Dublin. 3 WALHS had the impression that the observations had initially been made around 1674, and that by 1677 they were "commonly known" to the Oxford musicians. WALLIS experimented himself and described his observations of nodes in his paper "On the trembling of consonant strings, a new musical discovery" (1677). 4 WALHS placed small rings of paper (riders) around a vibrating string to provide a visual demonstration of nodes (riders shake around vigorously almost everywhere along the string, but they remain more or less at rest at the nodes). When he tuned two strings an octave apart, be found that exciting the higher pitched string caused the lower one to resonate and to divide into two equal sections separated by a stationary point; when the strings were tuned a twelfth apart, the lower pitched string divided into three sections. In the more complicated case that the strings were tuned a fifth apart, exciting the higher string caused the lower one to divide into three, and exciting the lower string caused the higher one to divide into two. WALHS remarked that "the like will hold in lesser concords; but the less remarkably as the number of divisions increases.,,s In connection with these observations WALHS was interested to find that when 1 These translations are published in JOHN WALLIS, Opera Mathematica (Oxford, ), III. "A question in music lately proposed to Dr. Wallis, concerning the division of the monochord, or section of the musical canon, with his answer to it," Phil. Trans. XX (1698), 80-84; "A letter of Dr. John Wallis to Samuel Pepys Esquire, relating to some supposed imperfections in an organ," Ibid. pp MARSH later wrote a paper on acoustic amplification: "An introductory essay to the doctrine of sounds, containing some proposals for the improvement of acoustics," Phil. Trans. XIV (1684), Philosophical Transactions, XII (1677), WALLIS included a similar discussion in his De Algebra (in Opera Mathematica, II, ). s Phil. Trans. XII (1677), 840.

37 Early Vibration Theory 205 a string was plucked at a potential nodal point, it produced an unusually "rough" and "confused" tone. He explained that when a string is" struck at the respective points of divisions, the sound is incongruous, by reason that the point is disturbed which should be at rest." 1 WALLIS did not explicitly assert that higher modes sound simultaneously with the fundemental and that the composite quality is changed when any is absent. But he did conclude from the observations that a string can vibrate in sections-in "unison parts." He suggested that a similar situation must exist in wind instruments. The experiments described by WALLIS were performed shortly after the trumpet marine (discussed in Section 4.2) became known and popular in England. Inside the instrument there were, by this time, numerous (as many as fifty) resonating strings. One can easily imagine that after hearing and seeing the trumpet marine, some musicians began to experiment with the harmonics of strings on other instruments and that WALLIS subsequently heard about their observations. At any rate, an explicit connection between WALLIS' experiments and the trumpet marine was made by FRANCIS ROBARTES, an English composer (and member of Parliament), in ROBARTES wanted to explain why the trumpet produces only a discrete set of pitches and why some of these pitches are "imperfect." (The seventh harmonic and other, higher, harmonics, do not coincide with pitches of the diatonic scale beginning with the fundamental.) To understand the trumpet, ROBARTES turned to the trumpet marine: "In this matter we may receive some light from the trumpet marine, an instrument though as unlike as possible to the trumpet in its frame (one being a wind instrument, the other a monochord) yet has a wonderful agreement with its effect." To understand the trumpet marine, ROBARTES considered experiments involving resonance between strings of different pitches. The procedure and observations were the same as those described by WALLIS. ROBARTES stated that a string can vibrate in equal sections provided that the length of the sections is an integral fraction of the entire string length: "This experiment holds when any note is struck which is a unison to some aliquot part of the string.,,a Thus the trumpet marine sounds only when the thumb, touching lightly, divides the string at a point that "makes the upper part of the string an aliquot of the whole." (ROBARTES' picture of what happens if the thumb is not at the right spot is a little vague : "The vibrations of the parts will cross one another, and make a sound suitable to their motion, altogether confused.,,)4 RO~ARTES showed that the relative pitches of a monochord vibrating in different numbers of sections are the same as those of the trumpet marine. From the similarity of the trumpet's intervals to those of the trumpet marine, ROBARrES suggested that the air in the trumpet breaks up into separate "vibrations"" "It is reasonable to imagine that the strongest blast raises the sound by breaking the air within the tube into the shortest vibrations, but that 1 Ihid. p FRANCIS ROBARTES (also spelled ROBERTS) "A discourse concerning the musical notes of the trumpet, and the trumpet marine, and of defects of the same," Philosophical Transactions XVll (1692), Ibid. p ~ Ibid. p. 561.

38 206 S. DOSTROVSKY no musical sound will arise unless they are suted to some aliquot part, and so by reduplication exactly measure out the whole length of the instrument. " Sauveur. In France, harmonics were discussed by JOSEPH SAUVEUR in 1701 and SAUVEUR explained the connection between harmonics and nodes in a way that was similar to, but independent of, WALLIS' explanation. In addition, he incorporated the explanation of harmonics into a more general description of sound. SAUVEUR'S originality was in his recognition of the importance of harmonics; for he came to realize that they are not only curiosities produced in certain situations but that they are components of all musical sound. SAUVEUR'S ideas on harmonics became quite well known in the Paris Academy, and some of the terminology that he introduced is still used today. 2 At a lively meeting of the Paris Academy in 1701, SAUVEUR explained the basic properties of harmonics. 3 He considered the following "ph~nom~ne si bizarre':4 A point on a vibrating string is maintained at rest, either by touching it lightly (as in the trumpet marine) or by placing a light obstacle on it; whichever section of the string is excited, the same pitch (higher than the characteristic frequency of the whole length) is produced. As FONa'ENELI~E remarked, "the marvel is that [the obstacle] gives unequal lengths the same pitch. "5 (SAUVEUR was considering the situation in which a node is induced directly, causing the string to vibrate in a higher mode, rather than the situation, described by WALLIS, in which the string vibrates in a higher mode because it is resonating to a higher pitch.) SAUVEtYa argued that a string can vibrate at additional, higher, frequencies which he called sons harmoniques by dividing it up into the appropriate number of equal shorter iengths separated by stationary points. He defined the son harmonique as "a sound which makes a number of vibrations while the fundamental sound makes only one." 6 He called the points that do not move n~euds (the word was taken from astronomy) and the points of maximum motion, ventres. It seems that SAUVEUR did not know of the paper rider demonstration of nodes, reported by WALLIS and ROBARTES. However, WALLIS' paper was mentioned by a member of the audience, and SAUWUR'S argument that nodes exist was culminated by WALLIS' demonstration. In the context of the mechanics of the time, the idea of nodes was not trivial. As FONTENELLE observed, "What philosopher would have believed that a body put into motion in such a way that all its parts should move will, nevertheless, leave some parts immobile, or maybe make some immobile by a certain distribution of the motion. ''7 It is not surprising, therefore, that SAUVEUR tried to make 1 Ibid. p acoustique, harmonique, nceud, ventre. 3 FONTENELLE discussed SAUVEUR'S analysis in "Sur un nouveau Syst~rne de musique," Histoire de l'acaddmie Royale des Sciences (1701), , (Paris, 1704), pp ; SAUVEUR discussed his ideas in "Systeme General des Intervalles des Sons," M~moires de l'acaddmie Royale des Sciences (1701), , (Paris, 1704), pp Histoire (1701), 133. s Histoire (1701), MOmoires (1701), Histoire (1701),

39 Early Vibration Theory 207 the situation more intuitive. He considered the case of an obstacle at a point one-fifth of the distance along a string. He explained that since the string cannot vibrate right at the obstacle, it divides into two sections. The shorter section (~ of total) vibrates with the frequency appropriate to its length, a frequency five times that of the fundamental. The longer section (~ of total) "attempts" to vibrate at the (lower) frequency appropriate to its length, but the more frequent vibrations of the adjoining section make it move more frequently until, finally, a piece of it (½ of total) acquires this higher frequency. This piece affects the nearby section in a similar manner and so forth, until the whole string is divided into five equal sections, each vibrating at a frequency five times that of the fundamental and sounding a pitch two octaves plus a major third above the pitch of the fundamental. SAVVEUR mentioned some generalizations and applications of the basic idea of harmonic vibrations: A given harmonic is always excited when a light obstacle is placed on any of the associated nodes; it is possible to subdivide sections and produce harmonics of harmonics; a higher mode is obtained not only by placing a light obstacle on the string (to induce a node) but also by resonance with a string that has its fundamental or one of its harmonics at that pitch (the procedure described by WALLIS); higher harmonics tend to be weaker; by sliding an obstacle lightly along a vibrating string, an entire sequence of harmonics (" un gazouillement de sons harmoniques'l) is heard. SAUVEUR assumed that air columns vibrate in harmonic modes, too, He wrote that if an ondulation of air, extending from the mouthpiece to the first open tone hole, "is forced to move more quickly, it divides into two equal undulations, then three, four." He did not discuss the nature of a sound wave in air, but he remarked that "to discover all the properties of wind instruments, a special study is necessary. "2 Finally, SAUVEUR mentioned that his theory explained why the trumpet, the trumpet marine, and certain other instruments have only discrete sets of pitches. Referring to the success of the theory of harmonics in explaining these traditionally puzzling instruments, SAUVEUR concluded with the hope that harmonics would lead to yet other discoveries, "for the perfection of acoustics and even for finding acoustical instruments corresponding to the most admired optical ones." 3 As mentioned earlier, two problems were associated with harmonics. The simpler of these was to understand how an object can vibrate at a higher frequency; this was the problem associated, for example, with understanding the pitches of the trumpet and the trumpet marine. It was clarified by the demonstration of nodes. The second and more complex problem concerned the simultaneous existence of different modes. WALLIS had hinted that this was possible. At first, SAUVEUR referred to this only briefly and rather vaguely: "Experience shows that long strings, when they are good or harmonious, make one hear the first harmonic sounds, especially those that do not make an octave with others; bells and other resonant and harmonious objects have the same effect." 4 1 M~rnoires (1701), 353. z Mdmoires (1701), M~moires (1701), 354. MOmoires (1701), 353.

40 208 S. DOSTRO SKY Later, however, SAUVEUR emphasized the fact that any musical sound contains a mixture of harmonics. The idea was brought out in connection with his investigation of the organ. 1 The organ is an ideal instrument for learning, not only about the harmonic basis of musical sound, but also about the relationship between harmonic structure and timbre. In fact, since the fifteenth century, the organ has been built with a choice of stops so that different combinations of pipes can be activated to produce a single pitch with different tone colors. (The stops of an organ are, specifically, the levers that control which ranks of pipes are being blown, but the word is also used, more generally, to refer to the combinations of pipes themselves.) The mutation stops (which add a particular higher harmonic to the basic pitch) and the mixture stops (which combine various foundation and mutation ranks) are especially relevant for the physics of harmonics. As a result of experience and intuition, organ builders made use of higher modes long before there was any understanding of their nature or even a sense of their existence. SAUVEUR examined organ stops with DESLANDES, the organ builder who had assisted him in the experiments involving beats; SAUVEUR was interested in the relative pitches of pipes that sound simultaneously to produce an organ tone. For the tones associated with different stops, he noted the relative pitches of the pipes; and he showed that the tones were composed of a fundamental and some of its harmonics. For SAUVEUR, this discovery of the phenomenon that was the basis for organ design was the primary result of this investigation. Realizing that harmonics had been "discovered" twice, first by the organ builders and then by the philosophers, FONTENELLE indicated that SAUVEUR had united the two modes of understanding: "It seems that whenever Nature, by herself, so to speak, makes a system of music, she uses only these kinds of sounds [the harmonics], yet they remained unknown to the theory of musicians until the present. When they were heard, they were treated as bizarre and irregular... Not that Nature did not sometimes have the strength to make musicians fall into the system of harmonic sounds, but they fell into it without knowing it, led only by their ear and their experience. Sauveur has given a very remarkable example of this in the structure of the organ, for he has shown that it is based entirely on this principle, however unknown. "2 SAUVEUR wrote enthusiastically about the connection between harmonic content and timbre : "Organists begin by knowing their organ, that is, the stops and the particular effect of the mixtures of those stops; because although mixtures of the same stops produce almost the same effect, there is always some difference that challenges the organist to mix them almost the way the painters mix 1 FONTENELLE'S discussion is ' Sur l'application des sons harmoniques auxjeux d'orgues," Histoire de l'acaddmie Royale des Sciences (1702), (Paris, 1704); SAUWUR'S paper is "Application des sons harmoniques a la composition des jeux d'orgues," M~moires de l'acadomie Royale des Sciences (1702), (Paris, 1704). 2 Histoire (1702), 92.

41 Early Vibration Theory 209 colors, and each has his own taste. However, there are general rules that determine these mixtures. The first is that in all these mixtures the sounds of organ pipes activated by the same key are harmonic; so that one must regard a departure from this a form of dissonance. The second is that one does not use all the stops which give harmonic sounds indifferently, but one pays attention to the nature of the piece being played and to the taste and whim of the organist who, like the chefs, likes stews milder or spicier." 1 Although SAUVEUR discussed the effect of harmonics on tone color in the context of the organ, he implied that this is more generally true since "by the mixture of its stops, the organ is only imitating the harmony that nature observes in sonorous objects."2 To make a more specific and quantitative connection between harmonic content and timbre would have been (and still is) very difficult. Even 150 years later, HELMHOLTZ (to whom the idea of harmonic content as a source of tone color is generally attributed) made only a tentative beginning. In his summary of SAUVEUR'S study of the organ, FONTENELLE referred to the physics of superposition even more explicitly. Explaining the fact that harmonics sound simultaneously with the fundamental, FONTEN~LLE wrote that "each half, each third, each quarter of the string of an instrument makes its partial vibrations, while the total vibration of the entire string continues. ''3 Having a familiarity with the traditional arithmetic of pitch, FONTENELLE was prepared to appreciate SAUVEUR'S analysis of musical sound which used the series ~ ~'2,~,z[.-. 1 i (to become known as the harmonic series). He considered the analysis to be appropriate for all music, since it could describe even "music provided by nature herself, without the help of art." With SAUVEUR'S work, FONTENELLE'S enthusiasm, and the prestige of the Paris Academy, it is not surprising that ideas about harmonics and higher vibrational modes were being discussed in the early eighteenth century. Studying the vibrating string, D. BERNOULLI and EULER began to develop solutions that took account of the higher modes, thereby founding the subject of harmonic analysis. In music, RAMEAU based his theory of harmony on harmonics Newton's Analysis of Sound Waves 7.1. Introduction. Although there was an intuitive understanding of sound waves in the seventeenth century, GALILEO and others were able to be quantitative only to the extent of counting the "pulses." But in the Principia (1687), NEWTON gave a mathematical and physical analysis of (pressure) waves in a compressible medium. 5 Already before the Princip&, NEWTON had given some qualitative descriptions of longitudinal waves. In 1675, influenced by HOOKE, he had presented an 1 Mdmoires (1702), Ibid. p Histoire (1702), JEAN-PHILLIPPE RAMEAU, Nouveau Systbme de Musique Theorique (Paris, 1726), iii, iv, 17. s Book II, props. XLVII-L.

42 210 S. DOSTROVSKY hypothesis on the waves associated with light, and he had compared them with sound waves : "the agitated parts of bodies, according to their several sizes figures, and motions, do excite vibrations in the aether of various depths of bignesses... if by any means those of unequal bigness be separated from one another, the largest beget a sensation of a red colour; the least or shortest, of a deep violet; and the intermediate ones, of intermediate colours, much after the manner that bodies, according to their several sizes, shapes, and motions, excite vibrations in the air of various bignesses, which according to those bignesses, make several tones in sound." 1 Perhaps it was because of the optical context of his thinking that NEWTON compared vibrations in terms of wavelengths ("bigness")2 rather than the more customary (at least for sound), frequencies. NEWTON had made a rough estimate of wavelength: "It is to be supposed, that the aether is a vibrating medium like air, only the vibrations far more swift and minute; those of air, made by a man's ordinary voice, succeeding one another at more than half a foot or a foot distance; but those of aether at a less distance than the hundred thousandth part of an inch." 3 (Later, in the Principia, NEWTON gave a good estimate of sound wavelength; see Sections 3.4 and 7.3.) NEWTON had also stated that the different wavelengths of sound travel at the same velocity: "as in air the vibrations are some larger than others, but yet all equally swift (for in a ring of bells the sound of every tone is heard at two or three miles distance, in the same order that the bells are struck;) so, I suppose, the aetherial vibrations differ in bigness, but not in swiftness.,,4 In a more speculative spirit, NEWTON had wondered whether the eye's response to light is similar to the ear's response to sound, "for the analogy of nature is to be observed." s In this case, "as the harmony and discord of sounds proceed from the proportions of the aereal vibrations, so may the harmony of some colors, as of golden and blue, and the discord of others, as of red and blue, proceed from the proportions of the aetherial." He had wondered whether the similarity might be quantitative, in the sense that the relative distances occupied by the seven colors of the spectrum (in a display on a screen) would be the same as the relative lengths of the sections (on a string, from one end to the midpoint) between the points at which it would be stopped to produce the seven intervals of the octave: "possibly light may be distinguished into its principal degrees, red, orange, yellow, green, blue, indigo, and deep violet, on the same 1 I. BERNARD COHEN, ed., Isaac Newton "s Papers and Letters on Natural Philosophy (Cambridge: Harvard University Press, 1958), For a demonstration of the fact that by "bigness" NEWTON meant "wavelength" see A.I. SABRA, "Newton and the 'bigness' of vibrations," Isis 54 (1963), NEWTON'S Papers and Letters, Ibid. s Ibid. p. 192.

43 Early Vibration Theory 211 ground that sound within an eighth [octave] is graduated into tones." 1 His observations seemed to indicate this correspondence between color and pitch, and he assumed it in some calculations in the Opticks. z It thus appears that although NEWTON had not dealt explicitly with musical sound, he had shared with others of the time the sense that the relations to be found in music have more general relevance. Later, he referred to the PYTHAGOREANS' interest in the musical string and their doctrine of celestial harmony as implying that they knew about the inverse square law of gravitation, a (MERSENNE'S law, vocll]/~, which NEWTON assumed to have been known to the PrrnAGOREANS, does hold for a given planet in a given orbit provided that one takes l to be the planet's distance from the sun, F to be the gravitational attraction on it, and furthermore that one takes v to be proportional to the planet's angular velocity, as KEPLER did (section 2.2).) 7.2. Newton's Modal for a Pressure Wave. To describe the propagation of vibrations in a compressible elastic medium mathematically, NEWTON considered the problem in one dimension. The length of a "segment" of fluid changes periodically. Although the physical picture is straightforward, the style of the analysis is quite obscure. LAGRANGE wrote that mathematical physicists had never been satisfied with the analysis. He quoted D'ALEMBERT'S statement that NEWTON'S discussion of the propagation of sound was ' possibly the most obscure and the most difficult" part of the Principia and his comment that JOHN II BERNOULLI had said that "he would not dare to pride himself on understanding [it]". 4 The analysis is confusing even today, because NEWTON performed differentiation geometrically and expressed properties of simple harmonic motion by reference to appropriately chosen pendulums. Therefore, a rather detailed exposition that shows what is happening at each step will be given here. This involves considering three versions of NEWTON'S analysis: a version (A) using trigonometric functions, partial differentiation, and the second law of dynamics; an intermediate version (B), closer in form to NEWTON'S but using trigonometric functions, to clarify the relation between the first version and NEWTON'S geometric analysis; and, finally, an outline (C) of NEW- TON'S geometric analysis as it appeared in the Principia. Version (A), which essentially follows TRVESDELL, 5 is based on EULER'S exposition of NEWTON'S argument. The segment of fluid (and its extreme position) are labelled by x. The segment oscillates between x and x+2r, and its position at any time t is given by x + y (x, t) (1) y(x, t)=r [1-cosco (t-x)]. 1 Ibid. z I. NEWTON, Opticks (New York: Dover Publications, 1952), 154, 212, 225, 284, j. R. McGumE & P. N. RATTANSI, '" Newton and the 'Pipes of Pan,'" Notes and Records of the Royal Society of London XXI (1966), , j. L. LAGRANGE, "Recherches sur la nature et la propagation du son" (1759), in Oeuvres (Paris, 1867), I, s "The Theory of Aerial Sound, ," pp. XXXII-XXXIII.

44 212 S. DOSTROVSKY Fig. 2 The motion begins from the left extremum (see Figure 2). r is the radius (proportional to amplitude), co is the angular frequency, and v is the velocity of propagation. The "volume" between the particles labelled by "x" and "'x + 6x" at time t is [x +6x + y(x +6x, t)] --(x + y(x, t))=6x(1 +~y(x, O/ax). (Here and in the following, equalities are written since 6x and 6t can be treated as infinitesimals.) Let x--o, for convenience. At t=o, the volume is 6x. If 6m is the mass of fluid between the "0" particle and the "6 x" particle, the density at any general time t is given by (2) et=6x(l+sy(o,t)/ox--i+?~y/ex-d D 1+ sinco t- \ dx! v ' where D is the density when the particle is in its extreme position (t=0); the third equality is really an approximation valid for the small amplitude r. (In the approximation (2), D is also the mean density.) By BOYLE'S law, the pressure is given by [to (3) P=kQ=kD l+--sinco t-. v Thus, the pressure differential across the length of the segment causes a resultant force (4) 6F= --~x 6x=kD--c sc v2 t-- 6x. On the other hand, from NEWTON'S "Second law", 02 Y ~Y2 = D ~ x r co2 cos co ( t _~ ) (5) 6F= Q6x~t2=D6x Ot Since (4) and (5) must be equal, v =l/k; since the mean pressure, Po = kd, = /Po/D.

45 Early Vibration Theory 213 The argument can be generalized easily. 1 One can consider the "excess" pressure and density, Pe and ~e = D --sin co t-, so P = Po + P~ and ~ = D + Qe" Thus, t,=~~=d. Assuming, for example, LAPLACE'S adiabatic relation, Poc~ ~, where ), is the ratio of specific heats, %/%, one has v =]/7(Po/D). For the intermediate version (B), consider the two particles of fluid, x = 0 and x = fix. After a time t, their displacements are y(0, t) and y(6x, t), and after a time (t+&), they are y(o,t+3t) and y(3x, t+6t). To avoid partial differentiation, choose 3t such that (6) y(bx, t) =y(0, t + &). Since y = r [1 - cos co (t - x/v)], one must have cos co (t - 3 x/v) = cos co (t + c5 t). Thus (7) & = - fix - - v To calculate the change in the length ("volume") of a segment, consider the length at time t, l,=[bx+y(6x, t)]-y(o, t). By use of (6), lt=6x+y(o,t+6t )- y(0, t); It y(o, t+&)-y(o, t) 1 d /o =1-~ (~x = l -v dt Y(O' t)' by (7). Finally, (8) It 1 co.... r sincot. lo The force on the segment extending between x = 0 and x = 3x is caused by the difference between the pressures at the end points. The pressure is inversely proportional to the length of a segment (" BOYLE'S law" in one dimension): P = Po loft, where Po is the original pressure. Thus the force is F=-Po ( l -71o) (9) 1 = - Po 1 - (r co~v) sin co (t + b t) 1 - (rco/v) sin cot = -Por~(sinco(t+3t)-sincot)=Por~]6xcoscot, which is the same as (4). For version (C), 2 consider the sequence of pulses, A, B, C, D, in Figure 3 (from the Principia). The areas marked with dashed lines represent positions of successive pulses, so the distance between them corresponds to the wavelength. The "pieces" of fluid oscillate across distances that are much smaller than the wavelength; "E", for example, moves through e to e and then back to E. (In the diagram it is not apparent that this distance is much smaller than the wavelength.) 1 TRUESDELL has stated that LAGRANGE was the first to do so (ibid. p. LIII). z This follows NEWTON'S analysis in the third edition of the Principia (London, 1726),

46 214 S. DOSTROVSKY Ilflll~ tlllblltl IIII1~, IIIIIIII g f e IIIIl~ { [ Fig. 3 The symbols E, F, G, represent both the particles and their extreme positions to one side. To analyze the motion, NEWTON used the circle shown in Figure 4. The diameter PS is proportional to Ee, the distance between the endpoints of the vibration, and the circumference is proportional to the time of a complete cycle. The points H, I, K, are marked on the circle such that HI = IK; perpendiculars H P Fig. 4

47 Early Vibration Theory 215 connect these points with the diameter. The projection, PL, of the arc PI! is proportional to Ee, the distance covered by E in the time represented by Ptl. NEWTON chose the arc length KIt so that (lo) KH_ E6 p BC where p is the circumference of the circle; note that BC is the wavelength. This corresponds to the condition put on fit in (6). (In the notation of (A) and (B): [KH/2nrl=[rco&/2nrl=~l&l, and E = [(6x)/v[. NEWTON'S& is negative if fix is positive.) NEWTON stated that LN IM (11) KH -OP ~ To see this, draw the chord HK, the radius OI, and KT (a line parallel and equal to LN). The triangles HKTand OIM are similar, and the arc KH is approximately equal to the chord. This relation shows how to differentiate (in the notation of (A) and (g)): dlcoscot = y(o,t)-y(o,t+&)~o6t =K-H=opLN IM = rsine)t =]sincot[. Let the original length between particles E and G be l o =EG. When the particle E is at e, the length is l = e? = E 7 - Ee = (EG + G7)- Ee = EG + PN- PL = EG - LN. (Ee = PL, but G~, = PN because its length equals the length that Ee had at the earlier time corresponding to PK.) So, as in (8), l EG-LN LN 1 o EG 1 EG" IM IM V =BC From (10) and (11), LN_IM_. P =2re =--, where Therefore, EG OP BC BC V -2re" 1 V-IM (12) 1 o V NEWTON assumed that the elastic force (the pressure) of a segment is inversely proportional to its length. (As stated in (B), this is BOYLE'S law in one dimension.) For convenience, call NEWTON'S elastic force "pressure" and use the symbols Po and P for the pressures associated with the lengths 1 o and I, respectively. From (12), P=eo =Co V-/M' NEWTON assumed that this pressure was appropriate for the midpoint F of the V segment. He wrote the corresponding expressions for the endpoints: P= Po V-HL V and P~ = Po V-K~" The difference between these pressures causes the resultant

48 216 S. DOSTROVSKY force; as in (9), This can be rewritten" =eov V-HL V-KN HL-KN P~-P~=Po VV2 V. KN_V. HL+HL.KN. NEWTON eliminated some of the terms by observing that both HL and KN are much smaller than V (that is, the amplitude is small, as in (2)). Therefore, HL- KN (13) force = Po V ' which is the last equality in (9). To see this in the notation of (B), note that HL = r sine)t, KN=r sin co (t + b t) and V=(BC)/2~z=v/co, and HL -KN co Po v - p0 r-- v sin e)(t + &)_ sine)t (..0 2 = ~ r U (6x) cos cot. Having found the force, NEWTON could have used his second law, as in (A), to find the velocity However, NEWTON analyzed the motion in terms of appropriately chosen pendulums. This brings in the acceleration of gravity, which is irrelevant, but in the mechanics of the time real cycloidal pendulums were more familiar than abstract simple harmonic motion. Since the period in simple harmonic motion is (14) Tl= 2re ]/~----, where M is the mass, R is the amplitude, and Fex t is the absolute value of the extremum force, the cycloidal pendulum having the same period, (15) has length (16) MR l=g ee t" To find the period T of the wave, 1 NEWTON introduced two pendulums: a pendulum of period T., whose mass, m=d(eg), equals that of the segment of fluid, whose amplitude is r, the amplitude of the wave, and whose extremum force equals its weight, rag, so its length is also equal to r, by (15); and a pendulum of period T A whose length is (171 A_-e- Dg a The notation for the various periods is introduced here for clarity.

49 Early Vibration Theory 217 where P0 is the atmospheric pressure (equal to the extremum pressure, as mentioned in (A)) and D is the density. For the propagating wave, the force, given by (13), has the extremum value Po (H K)/V. (Since HL - KN = sin co t - sin co (t + 8 t), for t ~ O, HL -- KN ~ co 8 t = H K.) From (10), (16), (17), and the definition of V,, this extremum force is (18) mgra fext-. By (15), T,.= TA]/~2~A, and by (14), T = T,.~. Thus, by (18), (19) BC _2rcA T TA The left hand side is the velocity of propagation. The right hand side can be evaluated directly from (15). But NEWTON used the fact that a distance 2~A is traversed in a time equal to the period of a pendulum of length A if the velocity is equal to the velocity acquired in free fall from a height A/2. Thus 7.3. The Velocity of Sound. Maybe NEWTON'S inspiration for analyzing sound waves came from the challenge of calculating the velocity of sound; or maybe the velocity of sound provided a crucial experiment for checking the validity of his model. In any case, he found a general model which he believed applied both to light and to sound in air: "Because they result from vibrating bodies, sounds are nothing other than pulses of air... This is confirmed by the vibrations which sounds, if they are strong and low, like those of drums, excite in nearby bodies; for shorter and more rapid vibrations are excited with more difficulty. But it is well known that any sounds reaching strings that are in unison with the sonorous bodies excite vibrations [in them]. "1 In various notes and editions of the Principia there is an involved sequence of comparisons and modifications of theoretical and experimental values for the velocity of sound. 2 Although the calculated result, 968 ft/second for air at a pressure of 30 inches of mercury (its density taken as 1/870 that of water), was lower than most available measurements, it was within the limits of some measurements of the velocity of sound that NEWTON made himself (by sychronizing a pendulum with successive echos). However, by the time of the second edition of the Principia, DERHAM'S excellent measurements indicated a mean velocity of 1142 ft/second. NEWTON therefore corrected his theoretical result (which was then 979 ft/second, because of a slightly different value for the density of air), rather arbitrarily, by allowing for dust and water vapor in the air. Most 1 Ibid. p I.B. COHEN & A. KOYRI~, eds., ISAAC NEWTON'S Philosophiae Naturalis Principia Matkematica, the third edition with variant readings (Harvard University Press, 1972) I,

50 218 S. DOSTROVSKY of NEWTON'S comparisons of his velocity with measured velocities and his correction of his result have been discussed elsewhere by various authors.1 However, not much attention has been given to one experimental determination: Knowing that velocity is the product of frequency and wavelength, and being reasonably confident that the wavelength of a tone produced by an open pipe is twice the length of the pipe, NEWTON was able to make a rough (implicit) calculation of velocity from measurements of frequency. He calculated the wavelength of an organ tone of known frequency and compared it with twice the length of the pipe. He obtained the following results: In the first edition, he used a velocity of 968 English feet/second (the calculated velocity) and 104 cps as the frequency of the tone produced by a four (French) foot pipe (MERSENNE'S result; see Section 5.1) and he obtained a wavelength of 9~ feet. 2 NEWTON was evidently concerned about this result (which is, indeed, large, even considering end effects) and made a number of changes in his own copies of the book. 3 In the second (1713) and third (1726) editions, he used a velocity of 1070 French feet/second (his corrected result, in agreement with DERHAM'S measurements) and 100 cps as the frequency of a five foot pipe (SAUVEUR'S result; see Section 5.3), and he obtained a wavelength of 10.7 French feet, a more satisfactory result. As mentioned above, NEWTON'S analysis does generalize to give v =]/dff/do, which gives the correct velocity when the pressure as a function of density is defined appropriately. As is well known, NEWTON used BOYLE'S law, so he obtained a result that was low by 20%. The discrepancy between theoretical and measured velocities, which was removed only when LAPLACE introduced the adiabatic relation, challenged mathematical physicists throughout the eighteenth century. As LAPLACE said, "Newton's theory, although imperfect, is a monument of his genius." 4 Acknowledgements. I am grateful to CHARLES C. GILLISPIE for suggesting the interaction between music and physics in the seventeenth century as a topic for my dissertation. I would like to thank JOHN T. CANNON for many stimulating and helpful conversations about the physics and mathematics of vibration, especially in connection with NEWTON'S analysis of the propagation of a wave. I would also like to thank CECIL ADKINS, STILLMAN DRAKE, MARK LINDLEY, CLAUDE V. PALISCA, THOMAS SETTLE, CLIFFORD TRUESDELL and especially ARTHUR H. BENADE and JEAN LE CORBEILLER for helpful comments and suggestions on earlier versions of this paper. 1 BERNARD S. FINN, "Laplace and the Speed of Sound," Isis 55 (1964), 7-19, pp. 7-9; CLIFFORD TRUESDELL, "The Theory of Aerial Sound, ," pp. XXIII-XXIV; RICHARD S. WESTFALL, "Newton and the Fudge Factor," Science 179 (1973), , pp NEWTON was casual with the arithmetic here and does not seem to have converted English to French feet. 3 For example: he changed the velocity to 1023 and to "about 1050" feet/second (968 English feet are about 1030 French feet), he changed the wavelength to 9~ and to "about ten," and he even questioned the conclusion that wavelength is about twice the pipe length. 4 P.S, LAPLACE, Oeuvres Complktes V (Paris, 1882), History of Physics Laboratory Department of Physics Barnard College, Columbia University New York, N.Y., (Received June 6, 1975)

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