Renaissance music and experimental science

UNIVERSITY OF PENNSYLVANIA PRESS Author(s): Stillman Drake Source: Journal of the History of Ideas, Vol. 31, No. 4 (Oct. - Dec., 1970), pp. 483-500 Published by: University of Pennsylvania Press Stable URL: http://www. jstor.org/stable/2708256 Accessed: 27/08/2013 05:28 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www‚jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. University of Pennsylvania Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of the History of Ideas. http://www jstor.org

RENAISSANCE MUSIC AND EXPERIMENTAL SCIENCE The influence of early modern science on musical theory and practice has not gone unnoticed by historians of music. Professor Claude Palisca has called particular attention to the impact of scientific thought on the music of the late sixteenth and early seventeenth centuries. His "Scientific Empiricism and Musical Thought" concludes with this summary: "It would be wrong to conclude from this exposition that the sensualism and freedom of early Baroque music can be ascribed mainly to the liberalizing force of scientific investigation. . . . Scientific thought did reveal, however, the falsity of some of the premises on which existing rationalizations of artistic procedures had rested.... By creating a favorable climate for experiment and the acceptance of new ideas, the scientific revolution greatly encouraged and accelerated a direction that musical art had already taken. Finally, the new acoustics replaced that elaborate conglomeration of myth, scholastic dogma, mysticism and numerology that had been the foundation of the older musical theory, giving it a less monumental but more permanent base."' The reciprocal influence of musical theory and practice upon early modern science, however, has been neglected by historians of science. This is somewhat surprising when we consider that music in medieval education held an equal place in the quadrivium with arithmetic, geometry, and astronomy as the recognized mathematical disciplines. For that reason alone, music might be expected to have remained rather closely associated with mathematics and science, and to have shared in their sudden transformations during the late Renaissance. I am convinced that that was indeed the case, and that the origins of the experimental aspect of modern science are to be sought in sixteenth-century music, just as its mathematical origins have been traced to the ancient Greek astronomers and to Archimedes. In saying this, I have in mind the emergence of physics as a distinct separate study, no longer an integral part of philosophy, after the work of Galileo. The new physics, in which mathematics and experiment replaced logic and authority, was so successful as to become a sort of model for other sciences, at least so far as method was concerned. It is, of course, possible to trace the lineage of physics as far back as one wishes. Any form of intellectual activity that exists in one 'C. Palisca's article in Seventeenth Centurl Science and the Arts, ed. H. H. Rhys (Princeton, 1961),91 137.

generation can be found to have some counterpart in the preceding generation, and it is instructive to detect and study these. On the other hand, it is no less instructive to recognize the occurrence from time to time of fundamental changes in patterns of intellectual activity. One way to do this is to select some pattern in modern times, and to move back until an essential element in the pattern disappears, or at least becomes of negligible importance in the writings of men who pursued the activity under study. It is easy to trace modern physical science back in an unbroken line to Sir Isaac Newton, who unquestionably linked experimental evidence in his work with mathematical laws, in a way that is highly characteristic of science today. Many historians of science feel that it is proper to trace this same unbroken line further back, to Galileo. Others, who would pursue it yet further, are soon confronted with definite evidence of discontinuities, to say the least. The most influential historians of science (e.g. Pierre Duhem, Alexandre Koyre) contend that the work of Galileo was merely a continuation of lines of thought laid down in the Middle Ages by men whose work subsequently fell into obscurity until very recent times. In that way a link with classical antiquity is found, through commentators on Euclid and Aristotle who flourished in the Middle Ages. But if Galileo's work was a continuation of medieval thought, there seems to have been some interruption, for it is hard to find any sixteenth-century writers on mechanics whose conception of the relation of mathematics to experimental evidence even resembles that of Galileo. Hence, if we want to go farther back than Galileo in an unbroken line, we have to proceed with caution. Obviously the unbroken line reaching back from the present will end wherever we discover the disappearance from physics of either mathematics or experiment. Now, the sixteenth century marked a notable increase of emphasis on and development of mathematics, ushered in by the publication and translation of ancient Greek works such as those of Euclid, Archimedes, and Pappus of Alexandria. And since mathematics pervades all typically sixteenth-century writings on mechanics, our quest turns automatically into a search for the origins of experiment. But here we shall need some kind of definition of experiment in its characteristically scientific sense. For such a definition we can eliminate at once the concept of experiment that is associated with the name of Sir Francis Bacon; that is, the idea that we can arrive at the true explanation of things by the systematic and patient accumulation of observed facts, without any preconceived theory. In Bacon's view, the explanation would ultimately force itself upon us as a result of this procedure. That may be true, though in many matters it is open to doubt; but whether it is true or not, it has really nothing to do with science. The mere accumu-

RENAISSANCE lation of data, useful though its results may be, does not constitute scientific knowledge as such. I shall cite a single example in illustration of this. One of Galileo's "Two New Sciences," published in 1638, was the science of strength of materials. There had been an impressive accumulation of information on that subject by earlier builders and engineers. But it was not such knowledge that gave rise to that science, or ever would have given rise to it, though that knowledge can be seen to fit into Galileo's new science. The new science was mathematical in character, and it developed not out of engineering, but out of the theoretical science of mechanics, and it was derived by Galileo from the Archimedean law of the lever. So in eliminating the Baconian definition of experiment, I am not denying the existence from time immemorial of accumulated observational data. Still less am I denying the usefulness of such activities. I mean only that it is not the Baconian type of experimentation that did, or could, account for the rise of modern science. The kind of experimentation which interests historians of physics is the deliberate manipulation of physical objects for the purpose of corroborating by their behavior a definitely preconceived mathematical rule, or for the purpose of discovering a mathematical rule applicable to their behavior. Some physical objects are not amenable to manipulation-e.g., the heavenly bodies-so in their case deliberate selections of precise observations are made, and comparisons of them serve in place of manipulation. Thus some kind of tinkering with external measurements, directly or indirectly, is involved. Such an investigation, with an exact rule in question, for the purpose of seeing whether that rule is obeyed, or of discovering some exact rule, constitutes scientific experimentation. In physics and astronomy, "exact rule" means some mathematical expression; in other sciences, a rule may be considered precise without its being mathematically expressed. But it is always the testing of exact rules by some kind of external activity that distinguishes experimental method in science from various other possible approaches to knowledge and truth. It is therefore natural for historians of science to be interested in discovering when, and under what circumstances, this approach began its uninterrupted prosecution down to the present time. It is equally natural for them to suppose that, so far as physics is concerned, experiment in the sense of manipulations to confirm or establish a law began in mechanics. Proceeding thus, it is tempting to suppose that the first physical experiments were concerned with relatively simple measurements of equilibrium and of motion; for example, with the balancing of weights and the measuring of speeds by comparing distances and times. Suppositions of that kind are so easily made, and so intrinsically plausible

that they tend to go unquestioned.It is thus that people reconstruct the invention of the wheel by supposing that it must have evolved from the use of logs or rollers in the movingof heavy objects. Fortunately for those who enjoy that kind of speculation,the inventionof the wheel goes so far back that they need not fear contradiction, though they cannot adduce the slightest positive evidence in favor of their unanimousconjecture.A historianwho may try to discoverthe origin of scientific experimentation,however, is less fortunate. In unbrokensuccession,this reallydoes not go very far back in time; not even as far back as the invention of printing, let alone of writing. Hence he may expect to see any assumptionor conjecturehe makes about the origin of scientific experimentationchallenged, and he is obliged to look for somethingthat can be supportedby some kind of recordedevidence. With these things in mind, let us take a new look at the idea that scientificexperimentationbegan in mechanics.A very simple and fundamentallaw in mechanicsis the law of the lever. Of course, that law may have been discovered by a Baconianaccumulationof observations, but then again it may not. The law is a mathematicalstatement, and it is perfectlypossible that nobodyever knew it in that form before Archimedes. Levers may have been used long and widely without their exact law having been known, and that law may have been deducedby someone who never used a lever, or at least never experimented with it. Certainly Archimedes made no use of experimental inductionin his proof of the law of the lever, nor do his postulates suggestan experimentalorigin. I say this not to suggestthat there was no scientific experimentationin antiquity,but to show that the most naturalconjecturesabout its role in the origin of mechanicsmay be quite misleading. In any event, the law relating distances and weights for the lever was both discoveredand mathematicallyformulated so long ago that it would have been simply silly to performany experimentsto verify it as late as the sixteenthcentury.Not only that, but the law of the lever is so simple and so far-reaching,once it is known, that there was also little point in performing actual experimentsto determinethe laws of any of the other simple machines. They could be much more effectively discovered and demonstrated mathematicallyfrom the law of the lever. And that, I am quite certain, is exactly the attitudethat was taken by writerson mechanicsin the sixteenthcentury. Reliance on mathematical reasoning had by then driven out of physics any feeling of need for experimentationthat may once have existed. We shall see presently what the analogous situation was in music. The reason I am quite certain of the situation in mechanics is the following. Pappusof Alexandria,writingin the fourthcenturyof our era, had

derived mathematically a law for the force required to drive or draw a heavy body up an inclined plane. His law was quite mistaken, because he had employed a false assumption,but the mathematical derivation was very ingenious and complicated,and it had the lovely aura of remote antiquityto recommendit to sixteenth-century mechanicians. From the theorem of Pappus, there easily followed a law for the equilibriumof bodies suspended on different inclined planes, also false, of course, Now, it happenedthat the correct law of equilibriumon inclined planes had been stated in the thirteenth century by Jordanus Nemorarius, who did not know the work of Pappus. In 1546, Niccolo Tartaglia publishedthe medieval theorem, with some improvementsin its purportedproof. So all later writers were confrontedwith two different laws, each of which was accompaniedby an attempted mathematicaldemonstration.If ever there was an occasionfor experimentaltest, this was it; moreover,the decidingexperimentwouldhave been very easy to perform.But that is not what happened.In 1570,GirolamoCardano,who was certainlyfamiliar with the correct medieval theorem and who in all probability also knew the work of Pappus, ignored them both and published a brand-newlaw for inclinedplanes, which had to be in error, since the medieval theorem was correct. Seven years later Guido Ubaldo del Monte publishedthe first comprehensivework on mechanics.An astute critic of his predecessors,he certainlyknewthe correcttheorem of Jordanus, and probably also knew the erroneous theorem of Cardano;yet he adopted in his own book the false (but ancient and elegant) theorem of Pappus.These events are hardlyexplicableif the idea of experimentationin mechanics,to test a preconceivedmathematical rule or to discover a new one, is assumed to have been prevalentin the sixteenthcentury. I might add that even in the later work of Galileo we findevidence of the use of experiment only to confirm a preconceived mathematical law, and not of its systematicuse to discover new laws. That step came after his time, as a logical extensionof his work. But we are interested in tracing the probable origin of Galileo's applicationof experiment (in the scientific sense) to mechanics,which means that it is time to turnto the historyof music. In the sixteenth century, music and mechanics were more obviouslyclosely relatedsciences than they are today. Tartagliawrote in his revisedpreface to the first vernaculartranslationof Euclidever published:"We know that all the other sciences, arts and disciplines need mathematics;not only the liberal arts, but all the mechanical arts as well .... And it is also certain that these mathematical sciences or disciplines are the nurses and mothers of the musical sciences, since it is with numbersand their properties,ratios, and proportions that we know the octave, or double ratio, to be made up of the ratios

4:3 and 3:2, and it is similarly that we know the former [that is, the interval of the fourth] to be composed of two tones and a [minor] semitone, and the latter [that is, the perfect fifth] to be composed of three tones and a minor semitone. And thus the octave (or double) is composed of five tones and two minor semitones; that is, a comma less than six tones; and likewise we know a tone to be more than 8 commas and less than 9. Also, by virtue of those [mathematical] disciplines, we know it to be impossible to divide the tone, or any other superparticular ratio, into two equal [rational] parts [in geometric proportion], which our Euclid demonstrates in the eighth proposition of Book VIII."2 But if, about the year 1550, the sciences of music and mechanics were alike in their purely mathematical character, the relations of those two sciences to the practical arts that bore the same names were totally different. Musical theorists were in possession of mathematical rules of harmony that they believed must be very strictly followed in practice. It would be unthinkable for musicians to depart from those rules, lest the very basis of music be destroyed. Musical theorists, moreover, received a good deal of respectful attention from musical practitioners and even had a certain authority over them. This was certainly not the case with theorists in mechanics. Musicians quite frequently studied under musical theorists, but engineers did not study under mechanical theoreticians. And if there were any theorists of mechanics who believed that their rules must be strictly followed in practice, this would have been in the sense that failure to observe the rules would result in wasteful use of materials or in the collapse of buildings, not in the ruin of architecture. There was a further difference between the arts of music and mechanics in the sixteenth century, a difference that probably has a bearing on the events which (in my opinion) led to the origin of experimental physics. This was the fact that commencing about 1450, and quite markedly after 1550, musical practice underwent fundamental changes, while mechanical practice did not. Those changes in musical practice brought about a real need to expand or alter musical theory, and with it a need for critical examination of its basis and its claims to correctness. No such need was felt by engineers. It was just as easy to test the mathematical rules of music in practice as it would have been to test those of mechanics; the difference in applying or neglecting such a test lay only in the feeling of need. Furthermore, the tests could be carried out in music with a great deal more accuracy, for no sixteenth-century mechanical measurement came even close to the precision of the trained ear of a musician. I believe that 2Euclide . . .diligentemente rasettato . . . per... Nicolb Tartalea (Venice, 1543), has a shorter and incorrect reference to musical proportions.

RENAISSANCE that may very well still be true today, if we restrict ourselves to mechanical measurement and do not bring in electronic devices. Finally, one might go so far as to say that the only possible means of detecting errors in, and ultimately of overthrowing, incorrect musical theories, was the appearance of experimental evidence against them. This was perhaps not equally true of errors in theoretical mechanics, which were in fact corrected by the detection and elimination of certain false assumptions, rather than by the revision of the whole theoretical structure of the science. But a discussion of that would lead us too far afield. The heart of ancient musical doctrine was an arithmetical theory of proportion credited to the Pythagoreans. The very existence of musical consonances and dissonances was ascribed in it to certain numbers, related as ratios. That is, the cause of consonance itself was thought to be the so-called sonorous numbers. According to the Pythagorean rule, the musical intervals of the octave, fifth, and fourth were governed by the ratios 2:1, 3:2, and 4:3. Music owed the possibility of its existence to these ratios, which are made up of the smallest integers. Here we have a marvellous example of the use and abuse of mathematics and experiment. It is said that Pythagoras discovered these ratios as a result of his having noted, in good Baconian fashion, the different tones given out by hammers of different weight when striking an anvil. The story is certainly apocryphal, but observations of string-lengths would have led to the ratios. Having obtained their simple numbers in this or some other empirical way, ancient theorists put them in the place of any further experimentation, and derived from them an elaborate theory of musical intonation. In time, the arithmetical theory was seen as transcending in authority its original source in pleasant sensation. The consequences were not serious for a very long time, but by the late sixteenth century it was no longer possible to maintain the ancient idea of simple numerical ratios as the cause of harmony. We must pause to see how that came about. Ancient Greek music, though it bore the name "harmony," was largely innocent of harmony in its present musical sense. Singing was chiefly homophonic, and when it was accompanied by instruments, they were played in unison with the voice, or, at most, in the octave. The fitting-together that was implied by the word "harmony" was not a fitting-together of different melodies or of parts sung simultaneously, as with us, but a fitting-together of succeeding notes so as to preserve consonance within the recognized tonoi, ancestral to the medieval modes, which differed in an essential way from our keys. These facts, coupled with the limited range of the human voice, and the practice of remaining within the selected mode during a given song, made it possible to decide on the intervals that were to divide the octave without imposing limitations on the composition of music.

By the thirteenth century, however, the simultaneoussinging of two or three airs had long been in vogue,and thereaftermuchingenuity was exercised in the compositionand arrangementof motets in such a way that separatevoices might interweavewithoutclashing.The use of perfect fourths and fifths exclusivelyas chief accents, or points of repose, fitted in very nicely with this purpose,and the momentaryappearanceof imperfectconsonances,as thirdsand sixths had come to be regarded,did not disturbthe ear so long as they were not carried to excess. In time, however, deliberate violation of the ancient rules began to be attractive;the ears of singersand of listenerscame to recognizea sort of general harmonic flow, though the form of composition remained polyphonic.Along with this came the developmentand multiplicationof instruments,used at first to accompanyvoices, but later coming to be played together with or without voices. Instrumentsintroduceda new element, because unlike voices, most instrumentsare incapable of producing whatever note the player desires with the exactitude of the human voice. Lutes, recorders, and viols, for example, unlike trombonesand the later violins,are limitedto definite sets of notes by frettingor by the positionsof wind-holes.Organpipes emit specific notes that cannot be varied in pitch by the organist,just as the harp and most of the keyboardinstrumentsare governed by fixed string-lengths.Inevitably,problemsarose over the propertuning of instruments,particularlywhen differentkinds of instrumentsbegan to be playedtogether,whichbringsus to the sixteenthcentury. The fretting of early stringed instrumentshad led quickly and naturallyto temperedscales for them. Octavessimplymust be in tune; even the untrainedear cannot tolerate much variationfor the octave, which is heard instinctivelymuch the same as unison. Intervalsof the fourthand fifth together must also make up the octave, and while it is nearly as easy to tune perfect fourths or fifths as it is perfect octaves, you cannot make several strings agree over a range of two or three octaves by tuning them in perfect fourths or fifths. There is that plaguey "comma"that Tartagliamentioned,and it has to be divided up in some way. For lutes and viols, this is accomplishedby tempering, usuallyby makingeach successivefret interval 17/18ths the length of the precedinglower fret, a rule advocatedby VincenzoGalilei late in the sixteenth century. Organs, however, seem to have been tuned by the mean-tonesystem from an early period, certainlyby the fifteenth century.Recorders,on the other hand,were generallytunedin C or F, depending on size, in just intonation. But temperament,mean-tone tuning, and just intonation do not make use of precisely the same intervals. The natureof the difficultythat gave rise to those variousintervals is purelymathematicalin a sense;the numbers1, 2, 3 ... are very use-

ful for counting,but they do not form a continuum.Musicalsoundsdo form a continuum,and hence the whole numbers,even when they are formed into fractions, have a limited applicationto musical sounds. The set of fractionsthat will divide a given octave into seven different tones will obviously divide a different octave into seven analogous tones; but if we give them names in the ordinaryway, those which bear the same names will not all be in unison or separatedby exact octaves. As Tartaglia remarked,and as the ancients knew, the ratio 2:1 for the octave and the ratio 3:2 for the fifth implied the ratio 9:8 for the whole tone between the fifth and the fourth. This in turn implied too small a value for the exact semitone. As Simon Stevin, the Flemish counterpartof Galileo, wrote, "The naturalnotes are not correctly hit off by such a division. And althoughthe ancientsperceived this fact, neverthelessthey took this divisionto be correctand perfect, and preferredto think that the defect was in our singing-[which is] as if one shouldsay the sun may be wrong,but not the clock. They even consideredthe sweet and lovely sounds of the minor and major third and sixth, which sounded unpleasantlyin their misdividedmelodic line, to be wrong, the more so because a dislike for inappropriate numbersmovedthem to do so. ButwhenPtolemyafterwardswantedto amendthis imperfection,he dividedthe syntonicdiatonicin a different way, making a distinctionbetween a major whole tone in the ratio 9:8 and a minorwhole tone in the ratio 10:9,a differencethat does not exist in nature, for it is obvious that all whole tones are sung as equal."3By "sung," Stevin meant "sounded"in general, and his final remarkis ratheran exaggeration. Regarding the musical string as a continuum,Stevin announced flatlythat rationalproportionshad nothingwhateverto do with music, and declared that the proper division of the octave into equal semitones was as the twelfth root of two. It is in this way that we tune pianostoday;it enablesus to play in all keys withoutseriouslydisturbing the ear in any. His conclusionwas, however,not experimentalbut purely mathematical;one might say that it is only accidentalthat it opened the door to modern harmony. I shall say no more about it here except to mentionthat Stevin, in a characteristicaside, considered the whole problem to have arisen from an inadequacyof the Greek language: ". . . the Greeks were of the most intelligent that Nature produces,but they lacked a good tool, that is, the Dutch language,withoutwhichin the most profoundmattersone can accomplish as little as a skilled carpenterwithout good temperedtools can carry on his trade."4For Galileo, the book of Nature was writtenin mathematics;for Stevin,the book of mathematicswaswrittenin Dutch. 3The Principal Works of Simon Stevin, 5, Music, ed. A. D. Fokker (Amsterdam, 1966),431-33. 4Ibid., 433.

And so, finally, we come to the sixteenth-century controversy over musical tuning that created modern music, with experimental physics as a by-product. In one corner stood the mathematical theory of antiquity, which took sonorous number as the cause of concord and asserted that number must govern string-length, or the placement of wind-holes, or the like. In the other was the human ear, with that curious taste for pleasant sounds that goes along with-or at least once went along with-the composition and performance of music. They were in conflict, as Stevin observed, and with or without the Dutch language, that conflict had to be resolved. The question was, which was to be master, numbers or sounds? The conduct of the debate is instructive, because it is symbolic in many ways of the debates that created the modern world, and of many that are still going on in it. For it was at once a battle of authority versus freedom; of theory versus practice; of purity versus beauty, and so on. Curiously enough, the issue was partly decided by antiquity; that is, by the recovery of an ancient treatise that was largely neglected by theorists before the sixteenth century. Nothing helped a good cause in that century like the discovery that some ancient writer had already thought of it. Copernicus was careful to mention some ancient writers, e.g., some Pythagoreans, who were said to have subscribed to the motion of the earth. It was of great help to musicians in their struggle for emancipation from the syntonic diatonic tuning of Ptolemy to be able to cite the view of Aristoxenus that when all was said and done, the ear of the musician must prevail. I should like to remark at this point that in a paradoxical way, the struggle to free and broaden music in our own time is a mirror image of the struggle that freed and broadened it in the seventeenth century. Electronic computer music has much in common with the program of the late Renaissance conservatives who fought to preserve the mathematical beauty of sonorous number and superparticular ratios against the mere pleasure of the human ear. The argument of Gioseffo Zarlino, roughly paraphrased for its philosophical content, was that there could be only one correct tuning, established by the mystic properties of numbers, and if that put limitations on instrumental music, then so much the worse for instruments. The voice must govern instruments, and must in turn be governed by divine proportions of numbers. It is amusing that the argument for the divine right of theory, used at one time to impose narrow bounds on music, is now used to support the abandonment of all bounds, and with them all mere sensory criteria for music. Theory before pleasure, once conservative, is now avantgarde. On purely theoretical grounds, Zarlino advocated the extension of

the Pythagoreanratios up to the number6, from the ancient 4. This would allow as consonances,in addition to the fourth and fifth, the major sixth as 5:3, the majorthird as 5:4, and the minorthird as 6:5. He was also willingto allow the minorsixth as 8:5, but that finishedthe list; no other consonanceswere admissible.The number6 was, after all, the first perfect number-that is, the lowest numberthat was equal to the sum of its factors-so the sonorous numberscould safely be extended that far, but no further. Zarlino's Harmonic Institutions, publishedin 1558, did away with all rivals of the syntonic diatonic tuning, such as temperament and the mean-tone system, as theoretically unjustifiable.His treatise on compositionthus limited music to polyphony as before, though this was given a somewhat larger range. Method had been perfected. Mathematicswas saved by the senario, as Zarlino called his six-based ratio system, but harmony was stillbornif his systemprevailed. About five years later, Zarlino'sscheme was subjectedto criticism, both mathematicaland experimental,by G. B. Benedetti. Benedetti has long been recognizedamong historiansof science as the most important by far of the Italian precursorsof Galileo in mechanics,but only recently Professor Palisca has called attention to his achievements in the physics of music. These are contained in two letters writtento Ciprianoda Rore at Venice,probablyin 1563,and published in 1585together with Benedetti'smore famouscontributionsto mathematics and physics. First, he showed by means of examples that a strict adherence by singers to Zarlino's ratios for consonancescould result in a considerablechange of pitch within a few bars. Benedetti argued from this that singers must in fact make use of some kind of tempered scale in order to preserveeven the most elemental musical orthodoxywith regardto pitchof openingandclosingnotes. Next, Benedetti proceeded to examine the problem from the standpoint of physics. Instead of relating pitch and consonance directly to numbers, that is to string-lengthsand special ratios, he relatedthem to rates of vibrationof the source of sound.Consonance, he asserted, was heard when air waves producedby different notes concurredor recurredfrequentlyin agreement;dissonance,when they recurredinfrequentlyor broke in on one another.Thus stringsvibrating in the ratio 2:1 would concur on every other vibration;in the ratio 3:2, on every sixth vibration,and so on. The direct cause of consonance was thus related to a physical phenomenon,which in turn bore a relationto certain numbers;but the numbersas such were not regardedby Benedettias the cause of the phenomenonof consonance. Pursuingthis reasoning,he suggestedan index of consonance,formed by multiplying together the terms of the ratio. The smaller this product,the greater the consonance.Viewed in this light, consonance

and dissonancewere not two separate and contradictoryqualities of sounds,but ratherthey were terms in a continuousseries withoutsharp divisions.A similarview was soon to emerge for heat and cold, speed and slowness, and other ancient physical concepts. Also, by Benedetti's index, it followed that the traditionally abhorred subminor fifth, with the ratio 7:5, was in fact a better consonancethan the minor sixth, with the ratio 8:5, which Zarlino himself had allowed even though it lay outside his senario. Obviouslythis was not just a further modificationof the old theory, but a fundamentalattack on the very basisof that theory. In framing these ideas, Benedetti had had recourse to experiment; that is, to the deliberate manipulationof physical equipment in orderto test a preexistingmathematicaltheory. The physicalequipment consistedof a monochord,on which a movablebridgepermitted the division of a single stretchedstring into any desired ratio, and allowed the measurementof the ratio to a fair degree of accuracyby measurementof the lengths of the two sections, which could then be sounded simultaneouslyby pluckingin order to determine the tonal effect. Benedetti's basis for asserting that string-lengthswere proportionalto frequenciesof vibrationis not stated; it may or may not have been experimental.But the appeal to physical results over the authority of a mathematicaltheory that had been developed over many centurieswas in any case a novel event. Experimentssimilarto Benedetti's,though probablynot directly inspiredby him because he did not publishthem, will presentlybe shown to have been intimately connectedwith Galileo's work in mechanics.First, though, I want to comment on the step-by-stepnature of the process, for Benedetti's results were in an important sense only a small start toward the modernphysicalscienceof musicalacoustics. Benedetti's brief writings on music did not remove mathematics from musical theory; they merely changed its role. For his contemporaries, as for the ancients, numberswere the cause of harmonious soundin a totally differentway from that in which Benedettisaw them as relatedto that cause. Others regardednumbersas rulingthe nature of sound; Benedetti formulateda physical theory of sound that was capableof explainingthe associationof certainnumbersor ratioswith certain tonal effects. But Benedetti himself was not a composer, and therefore did not concern himself with the possibilityor desirability of expandingthe rangeof acceptabletonal effects. He seems not even to have noticed the phenomenonof partial vibrations.These things were soon to come. What Benedetticontributedwas in the natureof a new explanationof knowneffects, ratherthan a basis for the exploration of new ground.Thus to the question,"Why is the fifth more harmoniousthan the major third?"the classicalanswerwas of this form:

"Because the ratio 3:2 containstwo sonorous numbers,and the ratio 5:4 does not." Zarlino's reply might have been: "The ancients were mistaken; both are equally harmonious,for all numbers within the senario are sonorous numbers. But since sonorous numbersare the cause of harmony,the slightestdeparturefrom the ratiosnamedmust be avoided, or dissonancewill result." Benedetti'sreply would have been, "Because harmony proceeds from agreement of vibrations, which occurs every sixth time for the fifth, and only every twentieth time for the major third." But to the question, "What can we do to widen the scope of harmoniousmusic?"no one gave an answer. To Benedetti's opponents, the question was unthinkable; and to Benedetti,the first man who wouldhave been able to answer,the question neveroccurred;he was a scientistand not a musician. It is virtuallycertain that Zarlino, who opposed any temperingof the vocal scales, never heard of Benedetti'sdemonstrationthat such temperingwas necessaryin practice and theoreticallyjustifiable. But Zarlino was opposed in print by a former pupil of his, Vincenzo Galilei, father of Galileo. In 1578 Galilei sent to Zarlino a discourse in which the departures of practicing musicians from the tuning recommendedby Zarlino were stated and defended; there was no escape from a temperedscale in the musicof the late sixteenthcentury. Zarlino paid no attention to this attempt of Galilei's to refute his theory; indeed, he appears to have attempted to suppressthe printing of his former pupil'sbook, which neverthelessappearedmuch expanded, in 1581. Galilei, who had long followed Zarlino's teaching, beganto questionit only after he had learnedfrom GirolamoMei, the best informed man in Italy on the music of the ancients,that among the ancients themselves there had been musicians who questionedthe absoluteauthorityof mathematicaltheoryover sense. It was Mei who invited Galilei to put to actual test certain doctrines of the old theory. The resultwas Galilei'sabandonmentof Zarlino'steaching and his publicationof the contraryviewin favorof temperedscales. Zarlino, far from accepting these criticisms, counterattackedin 1588 with his final book on music theory, the Sopplementimusicali. Though he did not mentionGalilei by name, he quotedfrom his book and identifiedhim as a former pupil. Galilei lost no time in replying; in 1589 he publisheda little volume which he dedicated,with obvious sarcasm,to Zarlino.The firstpart of this book is merelypolemicaland personal,but the balanceof it is of the greatestinterestwith respectto the beginningsof experimentalphysics. Galilei was, first of all, outspokenlyagainstthe acceptanceof authorityin mattersthat can be investigateddirectly. He did not even accept the idea that some musical intervalswere natural,or that any were consonantby reason of their being capableof representationby simple proportions.Any soundwas

as naturalas any other. Whetherit pleased the ear was quite another matter, and the way to determine this was to use the ear, not the numbersystem. Zarlino had extolled the humanvoice as the greatest musicalinstrument,being the naturalone, and concludedthat musical instruments,as artificialdevices, were boundby the laws of the natural instrument.Galilei repliedthat instrumentshad nothingto do with the voice, and made no attemptto imitate it; they were devisedfor certain purposes,and their excellencecould be determinedonly by the degree to whichthey successfullycarriedout those purposes.Mathematicshad no power over the senses, which in turn were the final criterionof excellencein colors, tastes, smells, andsounds. For a tuningsystem, Galilei advocatedan approximateequal temperamentas determinedby the trainedear. Here we should recallthat Stevin, about the same time, went still further;he boldlydeclaredthat ratiosandproportionshadnothingto do with musicandthat the proper and ideal intervalswere those given by the twelfth root of two. Stevin, like Benedetti,failed to publishhis discussionof music, but it is likely that he was influentialin his owncountry. It is in Galilei'sfinal refutationof Zarlinothat we find for the first time the specific experimental rejection of sonorous number as the cause of consonance.Referringto the celebratedopinionof Pythagoras that the small-numberfractionsare always associatedwith agreeable tones, Galilei stated first that he had long believedthis, and then said that he had determinedits error by means of experiment.The ratios 2:1, 3:2, 4:3 will give octaves, fifths, and fourths for strings of like materialand equaltensionbut of lengthsin these ratios,or for columns of air of similar lengths. But if the lengths are equal and the tensions are varied,then the weightsrequiredto producethe tensionsare as the squaresof these numbers.In a later, unpublishedmanuscript,he added that the cubes would have to be calculatedwhere volumesdetermined the sounds. Thus, he said, the ratio 9:4 was just as closely associated withthe fifthas the ratio3:2;andby implication,so was the ratio27:8ratios which were simple abominationsto the Pythagoreanand Ptolemaicnumerologistswho believedin sonorousnumberas a cause. In addition,Galilei experimentedwith stringsof differentmaterials and weights, discoveringthat unison cannot be consistentlyobtained betweentwo strings if they differ in any respectwhatever.Stringinga lute with strings of steel and gut, he foundthat if these were brought into agreementas open strings,they wouldnot be in perfectagreement whenstoppedat the frets. The removal of number magic from musical theory performeda doubly significantservice. First, it deprived numberof causal properties; second, and perhaps more important,it called attentionto the real significanceof numberas it referredto a specificdimension,such

as length, surface, and volume. The trouble with the older musical theory was not only that it failed to provide anything fruitful for expanding musical practice, but also that by purporting to give an explanation in mystical numerical terms, it tended to prevent the direct study of the actual application of number to the material instruments of music. A similar double error pervaded most of physics; not only did the erroneous physical principles of Aristotle fail to agree with observation, for example with regard to falling bodies, but their very existence discouraged the search for correct principles.5 The experiments of Vincenzo Galilei, like those of Benedetti, were true scientific experiments in the sense in which we have defined that term: the manipulation of physical objects for the purpose of verifying a mathematical rule preconceived as applicable to their behavior, or for the purpose of discovering a rule involved in that behavior. Whether music inspired the first such experiments in an unbroken line to the present remains to be seen. If others preceded them, it would seem that they must have been independent of them, for it is hard to see how Benedetti and Galilei might have attacked musical theory experimentally under the influence of some earlier experimental attack on some other mathematical theory. And to establish the probable continuity of their work with the application of experiment to mechanics, which has had such profound consequences for all science, we may now turn briefly to Galileo himself. Galileo was enrolled at the University of Pisa in 1581, his father having selected for him a medical career. Two years later he became deeply interested in mathematics under the guidance of a family friend, Ostilio Ricci. The chair of mathematics at the University appears to have been vacant when Galileo was there. His new interest distracted him from the study of medicine, and in 1585 he left the University without a degree. During the next five years he lived mainly in Florence, giving some private instruction in mathematics and commencing on researches that obtained for him the chair of mathematics at Pisa in 1589. Now it was precisely during those years, and particularly in 1588-89, that Vincenzo Galilei is most likely to have carried out many of the experiments in refutation of Pythagorean music theory that he never published, but that survive in manuscripts among Galileo's papers. It seems to me extremely likely that Galileo was himself involved in his father's experiments, some of which he appropriated and published many years later in his Two New Sciences. Galileo was an accomplished amateur musician, instructed by his father, and as a young mathematician he could hardly have remained indifferent to 5C. Palisca, 130; cf. V. Galilei, Discorso intorno alle opere di Gioseffo Zarlino (Venice, 1589;facs. repr. Milan, 1933), 127-28.

what his father was doing in the measurementof tunedstringsand the examinationof Pythagoreanmusicalnumerology.Thus the conception of experimentalverificationof mathematicallaws in physics,which is often illustratedin Galileo's books, may very well have been inspired by his father's work during the years in which he had just left the university and was developing his own mathematicalskills. It was precisely at this time that Galileo applied experimentto an ancient theory;he deviseda hydrostaticbalanceand reconstructedthe reasoning of Archimedesin the famed detection of the goldsmith'sfraud in makingthe crownof Hiero of Syracuse. If we consider the nature of Vincenzo Galilei's experiments,we gain also a possibleclue to Galileo's early interest in the pendulum,a device with which his future work in physics was to be intimately connected. One of the most interesting of his father's experiments concernedthe determinationof the numbersassociatedwith particular musicalintervalsunderdifferentconditionsof the productionof tones. You will recall that the celebratedratios 3/2 for the fifth, 4/3 for the fourth, and 2/1 for the octave were related to string lengths, given stringsof the same materialanddiameter.It was VincenzoGalileiwho remarkedfor the firsttime, in his finaldiatribeagainstZarlino,that the same ratios did not hold at all for the weights that must be used to stretch a given string to the equivalent pitches; here, the inverse squares of the ratios would hold, and to raise the pitch an octave, where half the lengthwouldsuffice,quadruplethe weightwas required. The experimentis not particularlydifficult,but however it is carried out, an observercan hardly escape the phenomenaof the pendulum. This is obvious if one thinks of suspendingtwo stringsof equal length and size, and weightingone with four times the weight attachedto the other. In order to set up the experiment,let alone to elicit a tone, say by plucking,the stringsandtheir weightsare boundto be set in at least a slight swingingmotion. If VincenzoGalilei used insteadthe apparatus commonlyillustratedfor the false Pythagoreanrule,the pendulum effect would also occur;here, parallelstrings are stretchedover a flat bed, weights being appliedto the ends handingdown from a terminal bridge. The applicationof differentweights to these relativelyshort verticalstrings would set them swingingand would invite attentionto the lengthsandperiodsof oscillation. Galileo's observationsof the pendulumwere already reflectedin his earliest contributionto the analysis of motion and of the law of falling bodies. About 1590 he composed a treatise on motion, which was never published. In this treatise he attempted to analyze the motions of bodies on inclined planes, and this was done in a manner that treatedinclinedplanesas tangentsto a circle. By 1602,we findhim discussingthe motions of bodies along arcs and chords of circles, and

it is quite evident that he and his patron, the MarquisGuido Ubaldo del Monte, were doing some experimental observations of those motions. Galileo's decision not to publish his first treatise on motion seems to be relatedto his discovery,by experimentaltest, that the rules he had deducedfor speeds on inclinedplaneswere incorrect.By 1609, after variousfalse starts that can be traced in his letters and notes, he had got to a point where he plannedto publisha systematictreatiseon motion, havingironed out the earlier false assumptionsby means of a combinationof mathematicalreasoningand simple experimentaltests. But in 1609 his attentionwas divertedto the newlyinventedtelescope, and as a resulthis scienceof motionwas not publisheduntilmanyyears later, in 1638. As one wouldexpect,Galileo'sfirstuse of experimentin physicswas limited to rough and simple tests to see whether the rules he had workedout mathematicallywere actuallyfollowed.As time went on, he took a greater interestin devisingexperimentsto corroborateor illustrate his science of motion.The extent to which he actuallyconducted experimentssuch as those he describedlate in life is a matterof debate, but there is no questionthat they servedas modelsto his pupilsand his readers. Such I believe to have been the origin of experimentalmethod in physics in the sense of the unbroken thread leading from the late sixteenth century to the present. The first conscious experimentsto test a preexistingmathematicaltheory were probablythe musicalexperimentsof Benedettiand VincenzoGalilei. They were extendedinto mechanicsby Galileo, whose pupilsCastelliandTorricellicarriedthem on over into hydraulicsand phenomena of air pressure; refined by Pascal and Boyle experimentled to the gas law. Boyle's law is said to have been the first scientificlaw to be experimentallydiscovered.Yet Vincenzo Galilei's discovery that the weights requiredfor producing tensions correspondingto given pitches are as the inverse squares of lengths must have been empirical. In any event, the manipulationof physicalequipmentset up to test a mathematicallaw had come much earlier than Newton, even earlier than Galileo;and it came becauseof the conflictbetweennumerologyandphysicsin the fieldof music. The desire for ever-increasingprecision in experiment, a necessary counterpartof the new methodin science that soughtcertaintyin the reconciliationof sense experienceand mathematics,was strikingly evidencedby MarinMersenne,the friendof Descartesand the spokesman of Galileo in France. Mersenneis best known for his Harmonie Universelleof 1636-37,a monumentalwork on the theoryand practice of music in which the role of the science of mechanicsis much emphasized.It was Mersennewho carriedout with great care the experiments on fallingbodies and on bodiesdescendingalong inclinedplanes

which Galileo mentionedbut only roughlydescribedin his books.0The intimate linkage of music to mechanics,mathematics,and experiment that is made in Mersenne'swork tends to increase the probabilityof my generalthesis concerningthe originof experimentalphysics. I should like to conclude with a remark about a totally different relationshipbetween Renaissance music and modern science, in an area where experimentin the sense of deliberatephysicalmanipulation is not possible;that is, in astronomy.This second musicalapproachto science found expressionin the work of Johannes Kepler, an almost exact contemporaryof Galileo. Kepler was deeply motivatedby precisely the kind of faith in the dominationof the universeby numerical harmonies in astronomy that Galileo's father fought to destroy in music. Little interested in mechanics, Kepler devoted his life to the discovery of that Pythagoreanmusic of the spheres that ought to be producedby the celestial motions. His goal turnedout to be as illusory as that of the rule of musical harmony by exclusivelysmall-number fractions. In the course of his work, however, Kepler discoveredthe mathematical laws that do describe the planetary motions. Those laws destroyedthe heavenlyspheresthemselves,replacingcircleswith ellipses and uniform motions with varying speeds, much as physical laws destroyedthe Pythagoreanmusicalratios. With varyingdistances from the sun, the planets were freed from monotony;but they could only grind out, on Kepler'smost favorablecalculations,a dreary succession of uninterestingscale-passages.Nevertheless, in his unending quest for better resultsand his devotion to precisionof measurement, Kepler establisheda new physics of the heavens as firmly in mathematics and observationas Galileo foundeda new terrestrialphysicsin the same solid base. The fountainheadof Renaissance music was thus at least partly responsiblefor the emergencenot of experimentalsciencealone, but of a whole new approach to theoretical science that we now know as mathematicalphysics. It is well knownthat Sir Isaac Newton accomplished the synthesis of Kepler's astronomical laws and Galileo's science of motion, setting the pattern of modernphysics that unifies terrestrial with cosmic events. In retrospect, it seems not to have matteredgreatly which side the scientist took, providedonly that he was deeply interestedin the musicalcontroversiesof the Renaissance. What did matter was that the older mathematicaltheories of music were capableof exact test by experimentand preciseobservation.That was a key which, in the hands of Galileo and of Kepler, opened the door to modernscience. Universityof Toronto. 6Mersenne, Harmonie Universelle (Paris, 1636-37), 1, 112; Les nouvelles pensees de Galilee (Paris, 1639), 188.