How To Understand The Sound Of A Flute

Transcription

1 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 1. Chter I. Concerning the Nture nd Progtion of Sound. 1. The exlntion of sound by the old hilosohers ws very obscure nd confused, so much cn be understood from their writings tht hve come down to us. Some were of the oinion, like Eicures [ B. C.], tht sound emnted from ulsting body rther like the flow of river; while others with the foremost of the Ltin writers, believed with Aristotle, [ B. C.] tht sounds were formed from the breking of the ir which rose from the more violent collisions of bodies. Among the more recent commentries, Honoré Fbri [ ], nd Descrtes [ ], discovered tht sound consisted of tremors or vibrtions of the ir, but their resoning concerning these vibrtions were eqully confused. Newton [ ], with the shrest of minds, considered the mtter with more cre, nd undertook to set forth n exlntion esecilly for the rogtion of sound, truly with much more success. A determined effort hs been mde [by me] to grs the difficult mtters involved in n understnding of the nture of sound, which re set out in the two chters of this disserttion. In the first chter it becomes rent, fter some creful thought, wht the nture of sound relly is, nd how it is rogted from one lce to nother. Moreover, in the following chter, three sources of sound re to be considered. 2. However, before this work on sound is undertken, certin fcts relting to ir in the genertion of sound re first to be relted. I regrd ir s consisting of smll globules, in stte of comression from the incumbent tmosheric weight, nd this comression is relieved to gret extent with elevtion, s the force of comression diminishes with height, so tht the rticles cn restore themselves to their nturl stte. Thus, the weight of the ir bove comresses the ir below, nd the ir globules re not llowed to be extended. The elstic force of comression on the ir globules is equl to the weight of the tmoshere; on ccount of which one cn mesure this force by exeriment, which truly is equl to the mximum weight of the tmoshere resent. This weight is equl to column of mercury of height 2460 scrules or thousndths of Rhenish feet [One Rhenish foot is equl to mm], nd I will lwys dhere to these mesurements in the following text; if the tmoshere hs smller weight, equl to column of mercury of height 2260 scrules, then this too cn be tken s equivlent to the elstic force of the ir [t greter ltitude]. Indeed, the weight of the ir hs been determined with the id of neumtic ums; nd the rtio of the secific grvity of quicksilver to the secific grvity of the wrmest ir hs been observed to be in the rtio to 1; while for the coldest ir the rtio is round to If we consider one of series of ir globules to be comressed more thn the rest, then tht globule will dilte ccording to the lw discussed bove, while the surrounding globules become comressed by the force cting on them from the diltion of the single globule, which in turn comress others further wy, s the globules scttered t distnce exerience little of the [originl] comression. And by this line of resoning the sound is trnsferred to other lces. But, concerning the motion by which the globule considered exndes out, fter coming to rest reltive to the others, it then returns suddenly nd is unble to be confined, s it hs been extended excessively; hence it is gin comressed with resect to the other globules, yet gin excessively. Thus ech one of the not too distnt globules is itself dilted in this wy by the trembling motion of tht first globule considered, nd in this mnner ech globule is constrined to move. [Thus, the hysicl ide of centrl source consisting of n ir globule executing n S.H.M. is resented, with time dely or hse shift for neighbouring globules ; there is no hysicl rgument resented for the relity of such globules, which re convenient figment of the imgintion.] But such vibrtion of the globules of ir nerby ought not to occur for globules of very smll size, nd which hence deend on n indefinitely short time for single oscilltion; therefore innumerble oscilltions or undultions with finite eriod re to be given out by globule in the mnner rescribed, since truly the motion of ny such globule of continully decresing size cnnot hen. Moreover, finite time is required for ercetion by our senses, nd it is not ossible for sound to consist of vibrtory motion of tht kind in the ir. 4. Then t lst the sound is roduced by the sme globule, from the force exerted on other globules, with finite intervls lced between those llowed to hve denser comressions. It is of course required in order to roduce the sound, tht the sme globule is lterntely contrcted nd relxed, nd indeed the time for

2 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 2. these oscilltions should not to be indefinitely smll, but finite, in order tht the number of these vibrtions or oscilltions for given time cn be determined. [Note: Mersenne, in his Hrmonie (1635) hd lredy set out tbles of frequencies ssocited with musicl scles, nd determined the seed of sound exerimentlly.] Of course the number of ulses rriving on the er from n orgn note in given finite time cn be exressed numericlly. 5. With the time now noted for which the sound is resent, it is esy to exlin the differences of sounds; here I will only distinguish between the rincil kinds. Generlly there re loud nd soft sounds. A sound is loud or violent when the comressions of the ir globules re stronger, nd sound is soft or smll when these comressions re weker. When the sound mde by the oscillting globule is rogted by the communiction of the comressions with ech of the globules lced round it, the number of these increses in the rtio of the squre of the distnces from the lce of origin, nd the strength of the sound decreses in the inverse squre rtio of the distnces, unless erhs the sound is ugmented from elsewhere. The distinction between notes of low nd high tones lies in the nture of the mximum durtion of the movement. The cse for low notes occurs when the vibrtions of the ir globules follow ech other in turn more slowly, or for given time the undultions re sent out less frequently. Moreover, the note is of higher tone when the vibrtions hve shorter delys lced between them, in order tht more oscilltions re crried out in the sme time. Hence the notes, with resect to low nd high notes, re in the rtio of the number of oscilltions mde in given time intervl. 7. A sound is lso either simle or comosite. A simle sound [or note] is one in which the vibrtions hve equl distnces between ech other, nd they re of equl strength. A comosite sound is constructed from mny simle sounds roduced t the sme time, nd this sets u either concordnt or discordnt sounds. Concordnt notes [or chords] re erceived s being roduced by simle sounds or notes mintining the simlest rtio between the comonents, such s two s in the octve, or s one nd hlf s in the musicl fifth, etc. On the other hnd, the dissonnt [or discordnt] sounds hve their comonents in more bstruse rtio, such s two suerimosed frctions s in three tone. [The interested but uninformed reder my wish to hone u on the hysics of music scles; good introduction is found in rther dted book : Anlyticl Exermentl Physics, by Ference, Lemon, & Stevenson (1956) U. Chicgo, Ch. 33] 8. Now we my observe the rogtion of sound with little more ttention, tht cn be done with some consistency. For the distnce which give sound cn trvel cross in given time cn be found from the theory set out bove : for the minutes nd seconds of the hour re found by observtion to be the sme for ll sounds, either loud or soft, low or high tones, to be crried through given distnce; nd in fct these sounds lwys move forwrds with the sme seed. In order tht this shll be so, this question my be sked : during the time the globule of ir is comressed, wht distnce does it get thrust forwrds? This question cn be nswered without difficulty from the rules governing the communiction of the motion nd from the nture of ir. Indeed wy cn be found, but I omit doing this, s I refer not work with imgined quntities. I ut in its lce wht results re found from hysicl mesurments. 9 In order tht I cn consider the roblem in generl wy, the secific grvity of mercury to ir is ut in the rtio n to 1; the height of mercury in the brometer is equl to k, the length of endulum is f, from which it is lesnt tsk to mesure the time tken for the sound to trvel distnce from the endulum's oscilltions. From these denominted fctors, I cn find the rtio of the time for one oscilltion of the endulum f to the time for the sound to trvel distnce to be s 1 to 4 nkf [Following Newton, we ssume the endulum is one tht follows the rc of cycloid, in which cse it executes simle hrmonic motion (shm) whtever the mlitude, nd the eriod T of this oscilltion is f relted to the length f of simle endulum by T = 2π. On the other hnd, the seed of sound g P ccording to Newton v = = ga, where A is the the height of the homogeneous tmoshere. Hence, ρ

3 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 3. the time T ir for the sound to trvel distnce is given by T ir = / v = / ga. Hence, f T / Tir = 2π ga / = 2π fa / = 2π nkf /. Here A is the height of the homogeneous g tmoshere which is ρ ir g A, or ρ Hg g k, from which A = nk. Newton ws the first erson who hd grs of the mechnicl nture of wve motion to the extend tht he ws ble to roduce formul for the seed of sound c in ir; tht this formul does not redict exctly the correct vlue for c does not detrct in the lest from the method, but only reflects the lck of understnding t the time of how het ws involved in the gs lws. Thus, the rising nd lowering of the temerture in the ir due to comressions nd rrefctions re considered dibtic s they hen so quickly. These effects were ccounted for by Llce hundred yers lter by inroducing n extr fctor γ into Newton's formul for c (γ = c /c v, the rtio of the secific hets of ir t constnt ressure nd temerture). Thus, in Euler's dy, these temerture relted effects were still unknown. We now look t Euler's formul.] 10. If nd k re mesured in scrules, but in lce of f is ut 3166 [ scrule is ~ 0.34mm; hence f ~ 1076mm, giving time of 2 seconds for comlete swing, or 1 second for swing from one side to the other], will give the vlue, the number of seconds in which the sound should be rogted nk distnce. For the length of the endulum with time of one oscilltion [i. e. hlf-swing] is indeed 3166 scru. Thus, with the distnce solved for the time, [v = /t =] /, the distnce tht sound nk trvels out in time of one second will be nk scru. [Thus v = 4 ( ) = scru. or 415 m/s. If we ut π in lce of 4 in the formul for the hlf-swing, we get 326 m/s, which is ner the correct vlue.] 11. Thus these conclusions follow on. The seed of sound remins the sme if nk remins the sme too, if the density of the ir nd the ressure re in roortion, the sounds re crried with the sme seed. For one my know tht there is no difference to the senses nor seed for sound in ir of the mximum comression thn for sound moving in ir with the mximum rrefction. Hence sound on the summits of mountins ought to trvel with the sme seed s sound within the vlleys, excet for other cuses to be exlined soon tht might be dded. 12. By incresing the fctor nk, the seed of sound should be incresed. Hence with the density of ir remining the sme or decresing little, but with the ressure incresed, then the seed of sound will be greter ; but truly from the contrry with the density of the ir incresing, nd with the ressure remining the sme or decresing little, the sound is slowed down. And hence collecting these things together, since both the density or weight nd ressure of the ir surrounding the erth re subject to vrious chnges, the seed of sound is constntly chnging lso. Hence the mximum seed of sound will be [found] on the hottest dys with cler sky, or with the most creful mesurements mde from brometers nd thermometers t the highest levels of elevtion. Truly with the hrshest cold nd the fiercest storm, the seed of sound should be minimum, tht which my come bout by mesurements from the liquids in brometers nd thermometers resent in the lowest lces. 13. Hence the mximum seed of sound is found, if is ut in lce of n nd 2460 scru., in lce of k s thus the distnce trveled by sound in time of one second is = , i. e. the mximum seed of sound following my theorem should be 1222 Rhenish feet er second. Indeed the minimum seed of sound is obtined by utting for n nd 2260 for k, s thus the distnce ssed through in one second is = , or Hence the distnce which sound sses through in one second ought to be contined between the limits 1222 nd If those numbers re brought together with exerimentl trils, they re found to be in excellent greement with those, tht confirms my theory. For FLAMSTED mde observtions [J. Flmsteed ( ) circ 1675 hd determined the seed of sound in ir in Greenwich with Ed. Hlley ( ).] nd DERHAM [W. Derhm ( ), Exeriments nd observtions concerning the motion of sound

4 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 4...., Philosohicl Trnsctions (London) 26, 1708, no. 313,. 1] with the most ccurte exeriments set u found tht sound trveled through 1108 feet in one second, which is number lced lmost midwy between the limits found. If now we consider wht NEWTON hs to sy in [I. Newton, Philosohie nturlis rincii mthemtic Book II ro. 50, scholium; 2nd Ed, Cmbridge 1713, ; see lso this website on Newton] in the Phil. Book. II, Section VIII, he found tht sound sses through distnce in one second (with his resoning reduced to our wy of tlking) of 3166nk Rhenish scrules with d denotining the rtio of the dimeter to the erihery of circle, i. e. tht is roximtely 7 : 22. And thus his exression is less thn our one, if indeed NEWTON introduced the fctor 3 to 3166 nk, however, I will dhere to 4 in lce of this number. [Newton ctully comred the times with endulum nd so did not introduce π.] 15. Hence this is not so wonderful, since the most cute NEWTON found the exceedingly smll distnce tht sound will rech in second ; tht he could not determine greter thn 947 feet, which certinly is huge discrency [relly only ~ 11% out] from tht distnce which is found by exeriment ; but he reorts in order to confirm his method, tht the descrency is shred by the imurities mixed with ure ir tht slow it down. Indeed ir is corruted by vours, but the force of the ressure is lwys equl to the tmosheric weight, nd the weight of the ir to the senses too does not chnge. It is not ossible to be lwys mintining the chnge in the seed of sound in tems of these fctors, nd neither does the size of the molecules of the ir mke ny difference to these things. d 1 7 CHAPTER II. CONCERNING THE PRODUCTION OF SOUND. 16. It is well-known requirement in the roduction of sound, for the theory set forth in the revious chter, tht the vibrtions hve to be re-lied in some mnner, in order tht the globules of ir cn hve serte lternte contrctions nd exnsions for short time intervls. I hve been ble to infer three kinds of vibrtory motion from the three wys in which sound is generted in the first lce. Whereby few words should be sid here concerning the three diverse wys in which sounds cn be roduced. Moreover, I refer to s the first kind of sound, the sound of stringed instruments, drums, bells, musicl instruments under the control of the tongue, etc., ll well-known sounds, which hve their origin in the vibrtion of solid body. To be referred to s the second kind of sound, re sounds such s thunder, bombrdment nd the sning of twigs, nd ny in which body is set in some more violent stte of motion, ll nevertheless re sound rising from the sudden restitution of comressed ir, nd s stronger ercussion of the ir. Finlly, the third kind of sound I dd re the sounds of wind instruments such s the flute, nd I will send some time in exlining the nture of these with cre, since no one u to the resent hs given nything of substnce concerning these instruments. 17. As fr s ll the sounds of the first kind re concerned, so much I know, tht clerly ll sounds initilly were referred to nd considered to be generted by the vibrtions of some solid body, nd no other kind of sound considered ossible; but the flseness of this ide will soon be shown, when I set forth the two remining wys in which sound cn be roduced. But now, concerning the mnner in which sounds re roduced, the first hs been exmined with greter cre. Indeed for the resent I need only consider how sounds of different kinds rise from strings, s the other sounds of this tye cn esily be deduced from this exmle. I exmine closely how the exct tension in the string cn be obtined by being extended round the column of the instrument in order tht the force in the string is llowed to be mesured ccurtely. 18. Before ll else, it is to be noted tht strings give rise to sounds with the sme equl rtio of low or high itch, whtever the lied ulsting force, lthough it is ossible to hve huge difference in the rtio of the strength to wekness of the force; indeed the intensity of the sound is roortionl to the seed with which the string bets the ir, nd sounds re eqully intense if the ir is struck with the sme force.

5 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 5. Wherefore, since musicl sounds of either low or high in itch should be of equl strength, in order tht lesnt hrmony is roduced, this is given ernest ttention in the mking of musicl instruments, in order tht equll sounds with the sme rtio of intensity or robustness re roduced, which should be obtined following the rules, which indeed hve been found now from much gross tril nd error rctise by the most recent crftsmen, the truth of which indeed is erfectly rent from wht follows, which hve been diligently observed. I. The lengths of the strings re in the recirocl rtio of the notes, i. e. the number of vibrtions sent out in given time. II. The thickness of the strings or the trnsverse cross-sections re lso in the recirocl rtio of the notes, if the sme mteril of course my be sid to be used; if indeed this is not so, then the rtio of the densities hs to be included with the rtio of the widths. These rules cn be lied for the construction of wind instruments such s the flutes, where in lce of the length of the strings is to be tken the length or height of the flute nd in lce of the thickness of the strings the internl width of the flute. 19. When the string is vibrting, it strikes the ir globules which re nerby nd since they unble to recede, they re comressed ; but with the enduring oscilltory motion the ir globules suffer continul new comressions, thus the note rises. And thus the ir strikes the er or the tymnic membrne of the er s often s the string returns. Indeed the number of ercussions of ny note crried to the er for given time cn be found of course by finding the numberof oscilltions of the string emitting the note in the sme time. Moreover, my solution, which with the solutions of Cl. Cl. D. D. JOH. BERNOULLI [JOH. BERNOULLI ( ), Medittiones de chordis vibrntious, Commt. cd. sc.petro. 8 (1728), 1732,. 13; Oer omni, Lusnne et Geneve 1742, t. III,. 198] nd BROOK TAYLOR [( ), Methodus incrementorum direct et invers, Londini 1715,. 26 ] re in exct greement, is the following: 20. Let the weight holding the string =, the weight of the string = q, nd the length of the string =, from which three given quntities the number of vibrtions to be found in given time cn be roosed. I find the number of oscilltions set forth in second to be : , 7 q where should be mesured in scrules. [Recll tht the eriod of the endulum mesuring single seconds f for single swing from left to right of vice vers is given by T = π, where f is 3166 scrules. Hence, g 3166 g the time of one second corresonds to T = π. If the tension in the string is s force, the density of the string is q/ s mss er unit length, then the seed of trnsverse wve in the string is given by q v =, while the time to trvel length is given by t = / v = / =. If the recirocl is the q q frequency of the ssocited sound wve, then the frequency is q. If is given s mss, then the ssocited force is g, nd hence the frequency is given by q g ist =. This eriod comred with the endulum is sound g q while the ssocited eriod T q / / q g 1 q sound T =. g π = g π g 3166 = The frequency of the sound is hence π π s Euler hs shown.] Since the number of vibrtions of sound is in roortion to this number, q sounds sent forth by different strings re mongst themselves in the rtio q, i. e. the frequencies of the sounds re in squre root rtio comosed in direct roortion to the weight of the tension, nd indirectly to the length nd weight of the string. I do not deduce more rticulr consequences from this, but I will

6 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 6. inquire into the nture of known sounds, nd the number of vibrtions nswering to these formule from the exeriment tht is set u by me in the end. 21. If I tke coer string with the density of tht kind of substnce, which is indicted in 18, of length 980 scrules, which hs weight of ounds, nd I hve stretched this with weight of 11 4 ounds, the sound ulses from which re to be tken in greement with tht instrument in the chorl mode, s they sy, to be fitted [i. e., by someone singing the sme tone], which to the musicl is herd s ds. Hence therefore, one is ermitted to count u, how often tht sound is herd, nd hence how my vibrtions of the other sound in the given time re herd from the mechnicl device being struck ; indeed in the given generl formul, if 980 is substituted in lce of, in lce of 11 4 ounds, in lce of q 49, then the sound ds is found to hve 559 vibrtions er second nd since ds shll be to c s 6 to 5 [ minor third], the sound c hs 466 vibrtions [er second] nd thus t the minimum C is 116 vibrtions [er second]. 22. To this method of sound roduction, reference is mde to the sounds roduced by the tongue s well s by elstic ltes [vlves] inserted in tube in strong irflow, though both these methods re relted lso to the third mnner of sound roduction ; for indeed they re ech relted to the other. Mechnisms of this kind re to be seen in the vrious comressed ir ies such s trumets nd bugles, s in the imittion of the humn voice in song, ll of these instruments must hve ir blown into them in order tht they cn roduce sound. The comressed ir, by itself seeking to ss out, uncovers the tongue from the oening, like vlve, but holds tht oening to n excessive extent, so tht the tongue cts s vlve nd closes gin by stretching to its former stte, which is thus oened new, in order tht the ir ssing through cn be given vibrtory motion. Indeed it is unvoidble, with the ir flow roduced by uniform brething, tht this kind of vlve finlly comes to rest nd the sound ceses; but this wrning lies only to the wind instruments themselves, for if the ir is exelled from bellows [s in n orgn], it strikes the oenings of the mechnism in non-uniform mnner, nd with the hel of n inserted vlve the wind in the tube is crried long nd the vibrtion continues. 23. Clerly the humn voice is roduced in the sme wy; indeed the eiglottis holds in lce the set of the tongue in the orgn of seech, the vibrtion of which is mintined by the ssge of the ir scending through the rough windie. Besides, the vibrtory motion of the ir escing from the end of the rough windie is chnged in the cvity of the mouth in number of wys, by which the low nd high-itched tones of the voice cn be roduced, nd different vocl effects re formed, which with the hel of the tongue, lis, nd the hrynx rovide sounds with consonnts So with the nose, for it is rent tht the ir in continuous vibrtion coming from the eiglottis cn leve by the nose too; but with regrd to the vrious high nd low itched notes which cn be roduced in this wy, they re unble to give rise to either cler seech nd neither cn consonnts be reserved, which re thus different from sounds roduced by the mouth. 24. But these sounds sent forth by the vibrtions of the tongue, unless they re strengthened by ies or tubes, re soon to become very wek, s is to be seen in the vibrtions of thin ltes [in contct with the mouth, one imgines], where hrdly nything is erceived by the er. But the mircle is the wy in which these sounds re mlified in tubes, nd lso the humn voice by the mouth cvity, not in the lest being the mnner sounds of low nd high notes sent into ie undergo gret chnge. Truly this is not the lce to sow the seeds [for further study] of how sound wves re intensified nd reflected within ies; there would need to be work with secil title, in which tht mteril could be set out more crefully, nd in which the mirculous mlifiction of sound by meghones could lso be estblished, s well s the rincile of the echo nd mny other henomen; but the tsk of crefully considering tht mteril more crefully is not yet comlete, nd concerning wht is held in other writings, lot of things tht I hve exmined in these sources re very confused nd in the greter rt flse. 25. By the second clss of sounds I refer to these sounds which rise either from the sudden relese of noteworthy quntity of comressed ir or which strike the ir with greter force. The ir in the ltter mode is lso comressed, since it ttemts not to leve the lce to the vibrting body, from where the ir left hs gin exnded. Thus the cuse of the sounds belonging to the second clss is the restitution of the ir to its

7 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 7. stte before being comressed. It is rent tht the sound rises from tht restorton, the ir which ws comressed itself exnds too nd hence gin contrcts nd so on, with which undultion of the ir shll be, s the smllest of the ir globules too, which of course mke u the mss of the ir, rticite in tht vibrtory motion nd in consequence led the sound forwrds; it is to be noted, tht if greter body of ir is comressed, deeer sound is roduced, nd if it is smller, the sound will be shrer. Truly sounds roduced in this wy re not ble to endure for long time, but from wht is left they must sto, becuse the vibrtory motion of the ir set u long wy off is lost t once. 26. Therefore ll the cuses revil for the roduction of suitble sound, whereby either the ir is comressed for the sound to be sent, or truly it is thus comressed gin in order tht the exnsion cn be stoed. Whereby ll the swifter motions of bodies in the ir ought to emit sound; indeed from the motions of bodies the ir on ccount of its own inerti is unble to go freely nd is comressed, nd gin by exnding itself the vibrtory motion of the ir is induced in the smllest of ir globules, to roduce suitble sound. Thus sounds flow from the stronger vibrtions of brnches, s from bodies moving more swiftly. Also the origin of the sounds of the breeze nd of the wind rise from this font; indeed the ir from the receeding is comressed s for tht following exmle, s from hrd body. 27. Esily the loudest of the sounds which rise from the relxtion of suddenly comressed ir re those ssocited with rtillery nd thunder. Moreover vrious exeriments set u with sulhur nd sltetre owder rove tht the cuse of these immense sounds is the restortion of comressed ir to its former stte; since it is found tht the ir there is in stte of mximum comression when the owder is set light nd wys re found for it to emerge, so tht it cn burst forth with the mximum force. Since moreover from the sltetre nd fiery dust, nd the mny constituent clouds nd vours tht re resent, is it not mircle tht fire rises from tht mteril, nd the stonishing sounds tht then reverberte. 28. The sounds of flutes constitute the third kind of sounds. The exlntion of the nture of these sounds hs been stonishingly twisted by scrutineers t ny time. Most hve thought tht with the infltion of the ies the smllest rticles of the inner surfce re struck nd the vibrtory motion is set u, thus the internl surfce of the ies restores the vibrtion by infltion nd enbles the oscilltions to be communicted with the ir; but how tht exlntion is consistnt with the lws of nture nd of motion, they themselves my exmine. I myself m certinly unble to conceive this, ll the sme it is ossible to exlin how ies of different heights cn give rise to differences of sounds of the sme mlitude ; for indeed if the internl rticles cn bring bout motion t ny time, then I m unble to see how, for tubes of different heights, it is ossible to hve different kinds of oscilltions. In next to no time I cn decide by setting u n exeriment even with single ies, how it is ossible to exlin the theory. 29. Moreover, since I will resolve this mtter, t first the structure of flutes is to be considered, nd then wht indeed is going on inside while ir is blowing through them hs to be considered with gret cre. For flutes re ies or tubes, for which beyond the junction of the tubes there is cvity for the mouth [the embrouchure or blow hole] in order to receive suitble ir blown in, which is lced towrds the end of the tube nd which chnges the ir flow by striking shr edge on the inner side of the hole on the side of the tube. This finlly llows ir from the mouth to be sent in through the hole in the tube, nd the ir is llowed to move slowly bck long the inner surfce of the tube [Euler thus seems to believe tht reflected ir moves bck long the inner surfce of the tube slowly, rther thn reflected sound]; if the ie is mde in this wy, sound is emitted on ir being blown in, tht is redily rent, for if some tube on its own is blown into by the mouth, so tht the ir in the tube cn move slowly long on the inner surfce, s it lso sends out the sme sound s the flute. The internl surfce of the tube should be hrd nd go to the left, so tht the ir cn neither intrude, nor be given lce in tht extended tube [to the right] where it is ossible to exnd [i. e., stnding wves re estblished in the left-hnd rt of the tube, the smll sce to the right cting s sort of buffer, evens out sounds of different frequencies], wherefore the ies should be rered from tubes with closed end nd smooth rigid sides. [Note however tht the flute cts lmost like tube oen t both ends, due to the resence of the blow hole; thus, if ll the note holes re closed, then the sound roduced will hve wvelength roximtely twice the length of the instrument. The closed end now hs stoer, nd its exct osition is imortnt for tuning.]

8 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce We my now see wht my eventute in the tube while ir is blown in, wht vibrtion of the ir it is ossible to retin, or by wht rtio the ir moving slowly in the tube in the sid mnner cn be retined by holding the vibrtion. It is cler, for the ir entering the flute, with the understnding tht the ir is going to be comressed long the length of the tube; where since it will exnd by itself gin, truly gretly, gin it will be comressed by the weight of the tmoshere, s the vibrtory motion in the tube is roduced, which vibrtion is the reson for the sound nerby. Thus truly the reson for the sound roduced by flutes hs been found, but the relity nd truth of this will become bundntly well-known; but to get to the hert of the mtter, the first considertion is the mnner by which the motion cn roduce tht vibrtion. [We my note tht the lyer hs lot to do with the resonnce, by feeding in ir t the correct seed vi the blow - hole to mintin the oscilltion; it is the ressure difference introduced by this ir tht is resonsible for the ressure vritions in the tube, nd there is ressure node t ech end; the tmosheric ressure is brought into the scheme of things s it determines the seed of sound, s Newton (lmost) demonstrted.] 31. An ir column in ie itself undultes following the mlitude of exnsions nd contrctions in the sme mnner s strings, nd thus I cn consider tht sme ir column s bundle of ir strings with the tension given by the weight of the tmoshere. But lthough the weights stretching the strings cn try to ull them rt, here indeed the direct oosite effect is obtined, with the ir in such column mde nrrower by the weight of the tmoshere, nevertheless the nlogy is legitimte one; indeed the weight of the tmoshere exerts the sme effect on n ir column, which weight does stretching the strings, if indeed we comre both sides, we hve here the weight of the tmoshere, nd there the weight of the strings being stretched, nd the gretly extended strings re gin drwn together. But in lce of ordinry strings, since they cn emit sound ulsting from one oint, for these ir strings with ulse mde from one oint, the whole ir column my be unble to vibrte on ccount of the discontinuity of the rt, for likewise the whole length ought to be ulsting, which shll be the cse in flutes, where the ir moving slowly through the whole ir length in the tube is content to be comressed. [Thus, if tuning fork is struck nd lced in n orgn ie, the ie my not vibrte s whole, deending on the resonnt frequencies of the fork nd the ie involved; on the other hnd, lucked string will lwys resonte ccording to its tension, length, line density, etc This my be the sort of sitution Euler hs in mind. In ny cse, the nlogy is rther loose one. ] 32. In order tht the oscilltions of the ir in flutes cn be found, or the number of wves tht cn be set in motion for some ie ccording to our revious rgument, to the ir in the ie from the weight of the tmoshere hs the sme weight s the tension in the chords in both cses ; nd this will be found by considering the oscilltions in 21. Let the length of some string, i. e. of some ie = ; (the weight holding the string) = to the weight of the tmoshere or column of mercury in brometer, i. e. from the minimum 2260 to the mximum 2460 scru.; q (the weight of the string) moreover will be = to the weight of the ir in the tube. Agin the secific grvity of mercury to ir will be = n : 1 nd k the height of the mercury in the brometer; to q is in the rtio comosited from n to 1 nd k to or to q is s nk to. 33. Thus by utting these roortions nk nd in exression 20 in lce of nd q the number of vibrtions er second of the sound emitted is found to hve the vlue nk Thus it is rent, since n nd k my chnge with the seson, tht the sound lso is to be chnged, it is rent tht with n increse in nk the itch will rise, nd with decrese the itch will fll. Therefore the sounds of flutes will be shrest itch with the mximum het, nd the ir the lest dense, but to be lowest itch with the mximum cold nd the most dense. This difference of sounds is esecilly observed by musicins nd orgnists. But since ll flutes hve the sme chnge in lce eqully, the melody is not chnged. [From 20, we hve the seed v P ρhg gk = = = ngk. The fundmentl frequency ρ ρ ir ngk f = v / λ = v / 2 = ; now, the endulum tht mesures the seconds is mens of determinig g, the 2 ccelertion due to grvity, nd in this cse, with the units used, this mounts to ngk π 3166nk T = = π ; or g = π Hence, f = = ; the frequency in Euler's formul is g 2 2 twice this mount, corresonding to the first hrmonic.]

9 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce In order tht the number of oscilltions of flutes cn be exressed by numbers, if the frequency of the sound is desired with the liquids in brometers ut in osition t the gretest height, for n ut nd for k ut 2460 scrules; then the number of oscilltions sent forth in one second from flute scrules long is found to be = = Truly for the gretest cold of the most svge seson for the digits of liquid brometers nd thermometers, ut for n nd for k ut 2260 scrules, the number of oscilltions sent forth er second is found to be equl to = = Hence the rtio is therefore rent, whereby flutes sent out sounds in recirocl roortion with their lengths nd where the width mkes no difference, nd indeed lso where the mteril of flutes does not introduce different sounds. Though neither the width nor the mteril of the body of the flute results in ny chnge in the rising or lowering of the itch, nevertheless these contribute much ffection nd chrm to the flute ; moreover the width of the tube is the bsis of the strength of the sound, for wider tube gives stronger sound lso ; the width of flutes is well-known to be nlogous to the density of strings. And s not every string is suitble for the roduction of ny sound, truly s lower notes re required so the density of the strings becomes greter, thus lso it revils for flutes tht the tller these re, so lso greter width is requires. 36. For the whole number rtio of the notes which vry between themselves s 8 to 9, under different conditions of blowing the ir, the sme flute cn give rise to both mximum nd minimum tones. Let the flute be 4 feet long, which is used to roduce the note C s in the singing mnner [i. e. middle C]; nd there will be vibrtions er second of this which re t most 240 nd t lest 210; which grees well enough with these, which we found before, when we considered the ction of strings; there indeed the number of oscilltions er second for the note C ws found to be 116, thus it is rent tht the note C of flutes is nerly n octve higher thn tht note C of the string. Becuse they stnd whole octve rt, they re commonly tken s being equl, which is not to be wondered t, since concerning notes, the most difficult to judge re those which re heterogenous or discordnt, or else they re in unison; now truly it is sufficient to hve n octve or in short with two or more octves set rt, since in one octve le I cn find these dissonnt sounds or discrencies, which confirms my theory well enough. 37. U to this stge we hve been concerned with the sounds roduced by ies, but the oening of cylindricl ie should be understood, where the exit for the ir blown in from the to of the tube lies oen. For since the tube oening bove is covered [i. e. the end of the tube beyond the blow hole], the ir blown in from bove is unble to esce, nd thus it is necessry to go bck, in order tht it cn emerge from the lower oening. Thus it shll be the cse tht so much more of the length is mde free with the ir reflected from the uer stoered end of the tube to the other oening before reching the oen exit. And by following the ir in the tube string of twice the length s it were hs to be considered, obviously strings tht bend t the joint hve to be considered Thus the sound is gthered by the closed end of the tube to be sent forth in the sme wy s for tube twice s long [one oen t both ends], or it emits sound n octve lower thn if it were oen. [There seems to be mistke here, s both the string nd the oen-ended ie suort fundmentl wvelength equl to twice the length of the string or ie; this error is due to the fctor of 1 / 2 missing in the revious formul, which results in n octve shift.] Whereby such ies do not sent out sounds everywhere of the sme wholeness, i. e. the sounds my be converging or diverging, likewise even from the rt of the cover of the bove ie. I ut this work forwrd to be exmined with the distinguished cndittes. [Some of the ltter mteril resented by Euler on flutes is suitble for someone with n interest in the history of the flute: good strting oint for such erson to look is in Vol. 23 of the 8th Edition (1878) of the Encyloedi Britnnic,.519 onwrds. We my note tht in Euler's dy, the flute ws in eriod of trnsition to one or other of its modern forms, nd hd smll U shed ie with the mouth iece t the to end, s well s hving joints to insert tubes of differing lengths to ccomodte severl octves s required. The end cover ws cork which could be removed to dischrge condenstion, nd its osition ws criticl for tuning. Those considered t n erlier dte by Mersenne in his Hrmonie were stright, nd resembled recorders in construction, I think.] The modern theory of the flute nd other musicl

10 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 10. instruments cn be found on the web t the site of the Physics Dertment of the University of N.S.W. in Austrli, which is well worth reding. Attchments. 1. The systems of the body nd the reformed hrmonies of the mind, from which the ctions of the mind nd body in turn ssert themselves, is not consistent with the truth. [In other words, even with sne mind in sound body, tht is not enough to sto you mking fool of yourself!] 2. The ttrctive force of NEWTON is the most suitble wy for ll henomin of celestil bodies to be exlined. And I believe beyond doubt the ide ut forwrd tht ll bodies by their own mutul ttrction drw together. 3. From thw osition of the centre of the erth (which moreover is truly fr wy nd lien) ny bodies re to be ttrcted by the inverse squre of their distnce, nd hole is to be drilled through the centre of the erth. It is sked, for stone sent down the hole, wht will hen, when it reches the centre, whether it will either remin there ermnently, or rogress wy from the centre without using nd return to us gin soon from the centre of the erth. I confirm tht the ltter is the cse.. 4. The strengths [i.e. kinetic energies] of moving bodies re comosed in the rtio of their msses to the first ower nd of their seeds to the second ower. 5. A shere is descending rotting on n inclined lne, in the bsence of ll resistnce, the seed tht it cn cquire from flling erendiculrly through the sme height, is found to be much less. Indeed it will be in the rtio to tht cquired when body flls normlly in the rtio 5 to Msts on shis should not be exceedingly high, so tht the force of the wind does not csize the shi. Moreover we cn ut the mst equied with sils exceedingly high, s surely the shi is knocked over by the given wind. I sy, tht if wider sils re ttched, in order tht the force driving the shi forwrds, cn be stronger, to be less lible to uset the shi. And lwys, whtever shll be the height of the mst, the width of the sils of these cn be incresed, in order tht even in the strongest wind, it will not be ossible to reort the loss of the shi. Tht's Enough! Cut I. De Ntur et Progtione Soni. 1. Obscur dmodum tque confus fuit vererum Philosohorum soni exlictio, quntum ex scritis eorum nobis relictis intelligi otest. Alii, cum Eicuro sonum, istr fluminis ex cororibus sonoris ulstis emnre sttuerunt. Alii utem & rærimis interretes ARISTOTLIS ltini cum illo nturm soni osuerunt in frctione æris, quæ oritur ex collisione vehementior cororum. Inter recentiores HONORATUS FABRI tque CARTESIUS invenerunt sonum consistere in æris tremore, de isto utem tremore riter confuse sentiebnt. Acutissimus NEUTONUS, hnc rem ccurtius exendere tque exonere ggressus est, ræciue soni rogtionem exlicndo, verum rum feliciori successu. Ardum ergo hnc de sone mterim, istc in disserttione trctre, tque ro viribus dilucidre constitui, duobus citibus em comrehenendo. Priore hoc cite scilicet erendetur, qure sonus consistt, & quomodo b uno loco d lium rogetur. In osteriore utem, tres sonum roducendi : considerbuntur. 2. Antequm utem isius soni trcttionem ggredir, quædm de ëre, utote soni subiecto ræmittend sunt. Aërem conciio constntem ex globulis in rvis, comressis b incumbente ondere tmoshærico & tnto gudentibus elterio, ut semot vi comrimente sese quent in sttum nturlem restituere. Cum

11 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 11. itque ondus ëris suerioris inferiorem comrimt, rohibetque ne globuli ërei extendntur, vis globulorum ëreorum elstic æqutur onderi tmoshæræ; quocirc em exerimentis definire licet. æqulis neme est, mximo existente ondere tmoshæræ ; columnæ Mercurili ltæ 2460 scruul seu millesims edis Rhenni [1 es Rhennus est mm], qum mensurm in osterum semer dhibebo; sin utem tmoshær minimo ondero gvis fuerit, æquivlens derehenditur vis ëris elstic columnæ Mercurili ltitudinis 2260 scruulorum. Quin etim ondus ëris oe ntliæ neumticæ determintum est ; grvits enim secific rgenti vivi se hbere observt est d grvittem secificm ëris, mximo clore, ut 1200 d 1, & summo frigore, ut d 1 circiter. 3. Si conciimus in serie globulorum ëreorum unum reliquis mgis comressum, ille sui iuris fctus diltbitur, globulos circumiectos ququ versus imellendo d comressionem in illos effundendo, qui ulterius lios imellent, ut globuli rocul dissiti liquntillum comressionem sentint. Atque hc rtione sonus in li loc trnsfertur. Cum utem motus, quo globulus ille se exndit, ostqum in æqulem cum ceteris sttum redierit subito cohiberi nequet, nimium is extendetur; unde reliquis rursus comrimetur, denuo tmen nimis ; ut itque motu tremulo unusquisque b illo rimo non nimis dissitus globulus modo se diltet, modo se rursus contrht. Iste utem tremor globulorum ëris in instnte cessre debet ob globulorum infinite exigum mgnitudinem, & inde deendens infinite breve unius oscilltionis temus, edendæ igitur essent b huiusmodi globulo temore finito oscilltiones seu undultiones innumeræ, quod vero ob motus cuiusvis globuli continum diminutionem fieri nequit. Quum utem d sensum in nobis excitndum temus requirtur finitum, in isto ëris motu tremulo sonus consistere nequit. 4. Tum demum oritur sonus, cum idem globulus vi lien, intervllis interositis finitis, crebriores titur comressiones; requiritur scilicet d sonum excitndum, ut idem globulus lterntim contrhtur tque relxetur, verum temor hrum oscilltionum non infinite rv esse debent, sed finit, ut numeros vibrtionum seu oscilltionum illrum dt temore determinri quet; numerus scilicet ulsuum in uris orgnum dto temore finito illidentium tntus esse debet, ut numeris exrimi ossit. 5. Cognito im temore, in quo sonus consistit, fcile erit exlicre sonorum diversittes, hic nonnisi rimris dducm. Distribuitur vulgo sonus in mgnum & rvum. Mgnus est vel vehemens, cum comressiones globulorum æreorum sunt vlidiores, sonus vero debilis vel rvus est, cum comressiones illæ debiliores sunt. Quum sonus globulo tremulo fcto rogetur communictione comressionis cum globulis undequque circumositis, horum utem numerus cresct in rtione dulict distntirum loco originis, decrescet soni vehementi in rtione distnrium dulict invers, ni forte sonus liunde ugment cciit. 6. Mximi momenti soni distinctio est in grvem tque cutum. Grvis est, cum vibrtiones globulorum ëriorum trdius se invicem insequuntur, seu cum dto temore rriores eduntur undultiones. Acutus utem est sonus cuius vibrtiones breviores interolits hbent moruls, ut deo lures eodem temore erguntur oscilltiones. Et hinc soni, resectu grvis & cuti, sunt inter se in rtione numeri oscilltionum dto temore fctrum. 7. Sonus etim est vel simlex vel comositus, Simles sonus est, cuius vibrtiones æquliter inter se sunt distnces æqules fortes. Comositus constt luribus sonis simlicibus simul sonntibus, hic constituit vel consonntim, vel dissonntim. Consonnti erciitur sonis simlicibus comonentibus rtionem servntibus simliciorem v. g. dulm ut in dison, vel sesquilterm ut in diente &c. Dissonntiæ utem sunt, cum rtio sonorum comonentium mgis est bstrus v. g. suerbirtiens setims, quemdmodum in tritono. 8. Contemlemur im soni rogtionem liquntum ttentius, id quod non incongrue fiet. id ex Theori sur jct comutetur stium, quod sonus temore dto ervdere otest, v. g. minuto horæ secundo, observtum enim est sonos omnes, sive mgnos sive rvos, sive grves sive cutos, eodem temore er dtum stium ferri, nec non eos eretuo edem velocitte romoveri. Ut illud ræstetur, quærendum est, qunto temore globulus æreus comressus comressionem d dtm distntim rotrudere quet. Id quod ex regulis communictionis motus & contemltione nturæ æris hud difficulter erui otest; isum quidem inveniendi modum, ut evitem iconismos; omitto, quod utem inde resultt, ono.

12 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce Sit (ut rem generliter comlectr) grvits merurii secific d æris grvittem ut n d 1, ltitudo mercurii in brometro = k, longitudino enduli = f, secundum cuius oscilltiones temus, quo sonus er intervllum trnsmittitur, dimetiri lubet. Hisce fctis denomintionibus, ego invenio, quod temus unius oscilltionis enduli, f, se hbet d temus rogtionis soni er intervllum ut 1 d. 10. Si et k determinentur in scruulis, loco f utem ontur 3166, indigitbit hic vlor 4 nkf nk minutis sucundis sonun er intervllum rogri debet. Est enim longitudo enduli singulis minutis secundis oscillntis scru Cum itque distnti bsolvtur temore, erit distnti, d qum sonus uno minuto secundo diffunditur scru nk nk, quot 11. Unde hec fluunt consectri. Mnente nk eodem celerits soni edem quoque erit deoque, si fuerint densittes ëris elsticittibus roortionles, soni edem celeritte rovehentur, scilicet in ëre qum mxime comresso sonus d sensum non celerius qum in ëre mxime rrefcto romovetur. Et hinc sonun in summis montibus edem velocitte rogredidebet, qu in imis vllibus, nisi lie cuse ccesserint mox exonende. 12. Crescente fcto nk soni celerits ugeri debet. Densitte ergo ëris mnente vel minut, elterio utem ucto soni celerits mior erit; sin vero e contrrio ëris densits cresct, elterio mnente vel minuto, sonus retrdbitur. Atque hinc colligitur, cum ëris tellurem cingentis et ondus seu densits et vis elstic vriis obnoxi sit muttionibus, soni velocittem subinde quoque vriri. Mxim ergo soni celerits erit mximo clore coeloque sudo seu ccurtius liquoribus in brometro et thermometro d summm ltitudinem elevtis. Acerbissimo vero frigore et sevissim temestte celerits soni minim esse debet, id quod evenit liquoribus in brometris et thermometris in infimis locis existentibus. 13. Mxim ergo soni celerits reerietur, si ontur loco n et loco k 2460 scru., ut deo stium uno secundo sono ercursum reeritur scru = , i. e. sonus mxim celeritte ervdere debet secundum istm mem Theorim intervllo minuti secundi 1222 edes Rhennos. Minim vero soni celerits hbebitur onendo ro n et ro k 2260, ut deo stium secundo emensum sit scruulorum = , seu 1069 edum. Distnti ergo, d qum sonus secundo disergi debet, continetur inter hos limites 1222 et 1069 ed. 14. Si ist cum exerienti conferntur, egregie cum e consentire reerientur, id quod mem methodum confirmbit. Observrunt enim FLAMSTEDIUS [J. Flmsteed ( ) in observtorio b e in oido Greenwich instituto un cum Ed. Hlley ( ) soni celertem in ëre determinvert.] et DERHAMIUS [W. Derhm ( ), Exeriment et observtiones de soni motu liisque d id ttinentibus, Philosohicl Trnsctions (London) 26, 1708, no. 313,. 1] ccurtissime institutis exerimentis sonum temore minuti secundi ercurrere 1108 edes, qui numerus fere medium tenet inter limites inventos. Si im condideremus, que NEUTONUS [I. Newton, Philosohie nturlis rincii mthemtic lib. II ro. 50, scholium; editio secund, Cntbrigie 1713, ] hc de re hbet Phil. lib. II sectione VIII, invenit ille ro distnti, qum sonus minuto secundo ercurrit (d nostrum loquendi modum eius rtiocinio reducto) scru. Rhenni 3166nk denotnte d rtionem dimetri d eriherim, i. e. qum roxime 7 : 22. Est itque eius exressio nostr minor, si quidem NEUTONUS 3166 nk duct in 3 1, ego utem loco huius numeri dhibem Hinc ergo mirum non est, quod cutissimus NEUTONUS nimis exigum invenit distntim, d qum sonus secundo minutio ertingit; miorem em non determint qum 947 ed., que sne ingens est discrenti b ill distnti, que exerimentis ert invent; quod utem d confirmtionem methodi ffert, tribuendo istm discrentim imuritti ëris, mer est tergiverstio. Utcumque enim ër voribus sit d

13 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 13. infectus, vis eius elstic equlis semer est onderi tmosherico ondusque ëris inde d sensum quoque non muttor. His vero obtinentibus soni celerits muttionem ullm ereti non otest. Nec mgnitudo moleculrum ërerum quicqum d rem fcit. CAPUT II DE PRODUCTIONE SONI 16. Ad roducendum sonum requiritur, ut ër eo, quem cite recedente exosui, modo tremulus reddtur, scilicet ut globuli ëris hbent contrctiones tque exnsiones finitio temore se invicem serts. Huiusmodi tremulum motum ëris trilici modo diverso, imrimis ex trilici sonorum genere concludere otui. Quocirc isto in cite de tribus diversis sonum roducendi modis verb erunt fciend. Refero utem d genus sonorum rimum sonus chordrum, tymnorum, cmnrum, instrumentorum lingulis instructorum, etc., omnes scilicet sonos, qui originem sum debent corori solido contremiscenti. Ad secundum genus referendi sunt soni tonitrus, bombrdrum tque virgrum et quorumvis cororum vehementius commotorum, omnes nimirum soni orti subitne restitutione ëris comressi, ut et vlidiore ercussione ëris. Tertio generi utem nnumero sonos tibirum, quorum nturm, cum nemo hctenus quicqum solidi hc de re dederit, diligentius exendm. 17. Ad rimum sonorum genus hctenus omnes, quntum scio, cunctos lne sonos referebnt rbitrbnturque nullum sonum nisi corore solido contremiscente exoriri osse; flsits utem huius sentientie mox ob oculos onetur, cum duos reliquos sonum roducendi modos exlicturus ero. Nunc utem modus, quo soni excitntur, rimus ccurtius erendendus est. Verum in resentim nonnisi, cum reliqu fcile eo reduci quent chords, quomodo et qules ednt sonos, contemlbor. Ad quod exctius obtinendum chords ut ondere tenss considero, cum lis circumvolutione circ columnm extenduntur, ut ccurte vim chordm extendentem metiri licet. 18. Ante omni observndum est chords esdem equles rtione grvis et cuti edere sonos, qucunque vi ulsentur, licet ingens esse ossit discrenti rtione vehementie et debilittis; soni enim vehementi est ut celerits, qu chord ërem ercutit sonique eque fortes sunt, si ër edem vi imellitur. Quocirc, cum soni musici tm grves qum cuti equliter fortes esse debent, ut dulcis hrmoni hbetur, in fbrictione instrumentorum musicorum robe in id incumbendum est, ut soni rtione fortitudinis seu roboris equles edntur, d quod obtinendum sequentes regule, que quidem recentioribus rtificibus ex multilici rxi crsse im erute sunt, qurum vero verits ex sequentibus erfecte tibit, diligenter observnde sunt. I. Chordrum longitudines sint in reciroc rtione sonorum, i. e. numeri vibrtionum dto temore edendrum. II. Chordrum crssitudines seu sectones trnsverse sint quoque in rtione reciroc sonorum, si scilicet eiusdem mterie chorde in usum vocentur; sin vero minus, tum cum rtione crssitudinis densittis rtio invers coniungend est. Ad instrument tibiis instruct regule iste quoque licri ossunt, sumendo ibi loco longitudinis chordrum longitudinem seu ltitudinem tibirum et loco crssitudinis chordrum mlitudinem tibirum internm. 19. Qundo chord oscilltur, ëreos globulos ferit, qui, cum in instnti cedere nequent, comrimuntur;durnte utem oscilltorio motu globuli ërei continuo novs tiuntur comressiones, unde sonus oritur. Aër itque toties ferit urem seu tymnum uris, quoties chord redierit. Adeoque reeriri oterit numerus ercussionum uniuscuiusque soni dto temore urem invectrum, investigndo scilicet numerum oscilltionum chorde sonum illum edentis edem temore. Me solutio utem, que cum solutionibus Cl. Cl. D. D. JOH. BERNOULLI [JOH. BERNOULLI ( ), Medittiones de chordis vibrntious, Commt. cd. sc.petro. 8 (1728), 1732,. 13; Oer omni, Lusnne et Geneve 1742, t. III,. 198] tque BROOK TAYLOR [( ), Methodus incrementorum direct et invers, Londini 1715,. 26 ] excte consirt, est hec.

14 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce Sit ondus chordm tendens =, ondus chorde = q et longitudo chorde =, ex quibus tribus dtis numerus vibrtionum dto temore inveniendus roonitur. Invenio ego ro numero oscilltionum uno minuto secundo editrum 22 7, q 3166 ubi determinri debet in scruulis. Huic numero cum sonus roortionlis sit, soni diversis chordis editi erunt inter se ut q, i. e. soni sunt in rtione subdulict comosit ex onderis tendentis direct et reciroc longitudinis et onderis chorde. Plur hinc mgis rticulri consectri non deduco, sed inquirm in nturm sonorum cognitorum tque numeros vibrtionum illis resondentes ex exerimento me hunc in finem instituto. 21. Sumsi chordm ërem ex eius crssitiei genere, quod No. 18 indigittur, longitudinis 980 scru., que onderbt libr., emque tetendi ondere 11 4 libr., qu uls derehendi sonum convenisse cum eo, in instrumento chorli modo, ut iunt, dtoto,qui musicis udit ds. Hinc ergo licebt suutre, quoties iste sonus et roin quivis lius dto temore uditus orgnum ferit; in dt enim formul generli, si substitutur loco 980, loco 11 4 libr., loco q 49, sonus ds minuto secundo hbere invenietur vibrtiones 559 et cum sit ds d c ut 6 d 5, hbebit sonus c 466 et roinde infimum C 116 vibrtiones. 22. Ad hunc modum roductionis soni quoque referendi sunt soni lingulis seu lminis elsticis tubo insertis infltione venti editi, qunqum quoque ex rte d tertium modum ertinent; d utrumque enim ertinent. Huiusmodi mchins videre est in vriis orgnis neumticis tubrum, buccinrum, ut et hominum cntus imitntibus, que instrument omni inflri debent d id, ut sonum ednt. Ventus, trnsitum sibi querendo, erit lingulm instr vlvule, nimium utem em eriendo tendit, ut rursus vlvul retrocedt, in riorem sttum tendendo que roinde denuo eritur, ut it motu tremulo ërem trnseuntem inficit. Necesse quidem esset, ut vento equbiliter flnte vlvul tndem quiesceret sonusque cessret; d hoc utem cvendum ventus ise, reterqum quod er se, dum e follibus roellitur, non equbiliter orifici mchinrum imett, oe vlvule tubo ventum deferenti inserte tremulus redditur. 23. Eodem lne modo vox humn genertur; lingule enim locum in orgno loquele obtinet eiglottis, que tremul redditur b ëre er rterim serm scendente. Tremulus iste motus ëris egredientis cum in cite rterie sere tum in cvitte oris vriis modis immuttur, ex qu vox grvis tque cut inflectitur vriique vocles formntur, qui soni oe lbiorum, lingue tque fucium consonntibus exornntur. Quin et nso, cum ëri b eiglottide tremulo reddito exitus er nsum quoque tet, vrii soni resectu grvis et cuti edi ossunt, qui utem sonis oris in eo differunt, quod nec voclibus distincte interstingui nec consonntibus condiri ossint. 24. Isti utem soni lingulis tremulis editi, nisi in tubis confirmentur, dmodum debiles essent, ut ercii vix ossent, quemdmodum observre est in lmin contremiscente, ubi nil fere uribus erciitur. Mirum utem in modum soni isti intenduntur in tubis tque vox humn in ore, nec non quod grvittem tque ciem ingens muttio huiusmodi sonis infertur in tubo. Verum de hisce soni intensionibus c inflexionibus hic non est locus fusius disserere; eculiri ous esset cite, quo ist mteri ccurtius exenderetur, ubi exlicnd quoque veniret mirific soni in tubis stenterohonicis mlifictio, ut et doctrin de Echo lurque li; sed istm mterim ccurtius erendere nondum vcvit, et que in liorum scritis continentur, quntum ex iis ersexi dmodum confus sunt et mximm rtem fls. 25. Ad secundum sonorum clssem retuli eos sonos, qui oriuntur vel notbili ëris quntitte comress subito dimiss vel vlidiore ëris ercussione. Posteriori modo ër quoque comrimitur, cum corori verbernti loci cessionem denegre conetur, unde ër iterum sibi relictus sese exndit. Cus itque sonorum d secundm clssem erinentium est restitutio ëris nte comressi. Istm vero restitutionem sonum generre debere exinde tet, quos ër comressus sese diltndo nimium exndt et roinde iterum contrhtur et it orro, qu undtione ëris sit, ut quoque minimi ërei globuli, quie qui ëris mssm comonunt, motum istum tremulum rticient tque er consequens sonum roducnt; ubi

15 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce 15. notndum, ut, quo mior ëris coi sit comress, eo grviorem edi sonum, c quo minor e sit, eo cutiorem. Huiusmodi vero soni diu durre nequeunt, sed e vestigio cessre debent, qui ër, motum in longe dissit loc diffundendo, motum tremulum sttim mittit. 26. Omnes ergo cuse, que ërem vel im comressum dimittere vel vero comrimere, it tmen, ut se sttim relxre ossit, vlent, d sonum roducendum te sunt. Quocirc omnes velociores cororum motiones in ëre sonum edere debent; motis emim cororibus ër ob rorim inertim liberrime cedere nescius comrimitur rursusque se diltndo motum tremulum globulis ëreis minimis inducit, d sonum roducendum tum. Hinc fluunt soni vehementius vibrtrum virgrum, ut et omnium velocius motorum cororum. Soni quoque fltuum tque ventorum ex hoc fonte originem ducunt; ër enim recedens b insequente etim comrimitur, quemdmodum corore duro. 27. Soni, qui oriuntur ëre im comresso subito relxto, fcile vlidissimi sunt tormentorum tque tonitrui. Horum utem immensorum sonorum cusm esse restitutionem ëris comressi comrobnt vri exeriment ulvere yrio tque nitro institut, qundoquidem reertum sit ërem inibi qum mxime esse condenstum, cui inflmmtione nitri vie exitum ei rebentes deriuntur, ut mximo imetu erumere quet. Cum utem ex mteri nitros et ulvis yrius, et multi vores nubes constituentes constent, mirum non est, ignem ist mteri conciiente, tm stuendos inde resultre sonos. 28. Tertium sonorum genus constituunt soni tibirum. Horum sonorum exlictio quovis temore nture scruttores mirum in modum torsit. Plerique existimvere infltione tibirum minims interne suerficiei rticuls imelli tque d motum tremulum sollicitri, ut it intern tibirum suerficies infltione tremens reddtur fcitque oscilltiones cum ëre communicnds; sed quomodo ist exlictio cum legibus nture et motus consistnt, isi inquirnt. Ego sne conciere nequeo, quomodo duntxt differenti sonorum tibirum diverse ltitudinis non mutt erum mlitudine exinde exoni ossit; qure enim rticule interne, si unqum motum conciint, ro divers ltitudine tuborum diversimode oscillri debent, videre non ossum, brevi vix rbitror vel unicum tibiis institutum exerimentum ex ist theori exlicri osse. 29. Ut utem verm huius rei exlictionem nnciscr, rimum tibirum structur, et quenm in illis, dum inflntur, evenint, ccurtius erendend sunt. Sunt tibie seu fistule tubi, quibus infr iunctum est eristomium cvum ëri reciiendo tum, quod versus tubum in crenm desinit directe oositm lteri cuidm interne tubi suerficiei, eum in finem, ut ër eristomio infltus er fissurm in tubum secundum eius longitudinem irrut, reendo suer suerficie tubi intern; si fistul hoc modo construct sit, sonum inflt edit, uti fcile tet, si quivis tubus eristomo destitutus it quoque infletur, ut ër in tubum suer suerficie intern ret, tum enim sonum etim sicut tibii edit. Intern utem tubi suerficies dur levisque esse debet, ne ëri irruenti cedere, nec illi i tubo contento locum sese exndendi dre ossit, quocirc fistule ex tubis d lter clusis tque rigidis interneque non scbris rri debent. 30. Videmus im, quid in fistul, dum infltur, evenit, quod ërem tremulum reddere ossit, seu qu rtione ër dicto modo in tubum retns ërem in tubo contentum tremulum reddere quet. Mnifestum est, ëre in tibim ingrediente, comrehensum in ill ërem secundum longitudinem comressum iri; quo cum sese rursus exndt, verum nimis, comrimetur rursus ondere remente tmosherico, ut it motus tremulus in tubo roductur, qui tremor cus soni roxim est. Atque sic detect est cus sone tibirum ver, cuius utem relits et verits bundius innotescet; enitius utem rius considerndus est modus, quo motus ille tremulus rodicitur. 31. Column ëre in fistul sese secundum mlitudinem exndendo c contrhendo more chordrum undt tque idcirco istm columnm considerbo ut fsciculum chordrum ërerum tensrum ondere tmosherico. Licet utem onder chords tendenti es divellere conentur, hic vero directe contrrium obtinet, cum column ill ëres ondere tmosherico corctetur, nihilo tmen minus nlogi legitim est; eundem enim ondus tmoshere exerit in columnm ërem effectum, quem onder tendenti in chords, si quidem utrinque, ibi ondus tmoshere, hic onder tendenti, chords nimium extens rursus comrimunt. Loco utem, quod chorde ordinrie unico in uncto ulste sonum ednt, chorde ille ëree, cum ulsu unico in uncto fcto tote ob discontinutionem rtium contremiscere nequent, simul er integrm longitudinem ulsri debent, id quod sit in tibiis, ubi ër irreens er totm longitudinem ërem in tubo contentum comrimit.

16 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce Ad inveniendum itque oscilltiones ëris in tibiis seu d determinndum numerum undultionum cuiusvis fistule res eo redit, ut ër in tubo hbetur ro chord tens utrinque ondere tmosherico; tque hoc intuitu oscilltiones reerientur ex 21. Sit scilicet longitudo chorde, i. e. tubi, = ; erit (ondus chordm tendens) = onderi tmosherico seu columne mercurii in brometro, i. e. d minimum 2260 d mximum 2460 scru.; q (ondus chorde) utem erit = onderi ëris in tubo. Sit rursus rtio grvittum secificrum mercurii et ëris = n : 1 et k ltitudo mercurii in brometro. Erit d q in rtione comosit n d 1 et k d seu erit d q ut nk d. 33. Posit im in exressione 20 loco et q eorum roortionlibus nk et reerietur ro numero vibrtionum uno minuto secundo editrum iste vlor nk Unde tet, cum n et k ro divers temestte mutentur, sonum quoque mutri, scilicet crescente nk ille sit cutior, decrescente utem nk sit grvior. Erunt ergo soni tibirum cutissimi mximo clore et ëre onderosissimo, grvissimi utem mximo frigore ëreque levissimo. Que differenti sonorum egregie quoque observtur musicis tque orgnriis. Qui utem ist muttio in omnibus tibiis equbiliter locum hbet, hrmoni non muttur. 34. Ut numeris exrimtur numerus oscilltionum tibirum, onntur, si desideretur sonus liquoribus in brometris et thermometris d mximm ltitudinem consistentibus, ro n tque ro k 2460 scru.; reerietur numerus oscilltionum minuto secundo editrum in tibi scru. long = = Mximum vero frigus sevissimmque temesttem indigitntibus liquoribus brometrorum c thermometrorum, onendo ro n et ro k 2260 scru., reerietur numerus oscilltionum minuto secundo editrum equlis = = Hinc ergo rtio tet, qure tibie ednt sonos longitudinibus reciroce roortionles et qure mlitudino d rem nihil fcit, immo etim, qure mteri tibirum nullm sono diversittem infert. Qunqum utem nec mlitudo nec mteri tuborum quicqum d soni grvittem seu ciem immutndm confert, tmen hec d ffectionem tque suvittem multum contribuit; ill utem mlitudo tubi fundmentum est vis soni, ut, quo mlior sit tubus, eo fortior quoque sit sonus; mlitido scilicet in tibiis nlogi est crssitiei in chordis. Et quemdmodum non quevis chord d quemvis sonum edendum t est, verum d grviorem crssior requirtur, it etim in tibiis istud locum obtinet, ut, quo ltiores ee sint, eo mior mlitudo requirtur. 36. Cum rtio sonorum tonum integrum se invicem distntium sit ut 8 d 9, edem tibi ro divers ëris conditione sonos d summum lus qum tono discrentes edere otest. Sit tibi 4 edes long, que dhibetur d sonum C in chorli modo edendum; et erunt eius vibrtiones secundo minuto edite d summum 240 tque d minimum 210; id quos stis convenit cum iis, que nte invenimus, ubi de chordis ctum fuit; ibi enim numerus oscilltionum soni C reertus fuit 116, unde tet sonum tibirum C esse qum roxime octv sueriorem eo sono C in chordis. Quod utem tot octv distent, vulgo tmen ro equlibus hbentur, mirndum non est, cum de sonis heterogeneis difficillimum sit iudicre, n unisoni sint; num vero octv vel rorsus dubus ut luribus octvis distent, sufficit, quod unic sltem octv eos sonos discrentes reererim, id quod mem theorim stis confirmt. 37. Que hucusque de sono fistulrum llt sunt, intelligi debent de fistulis cylindricis ertis, ubi ëri inflto sur in suremo tubo exitus tet. Cum utem tubus sur tectus fuerit, er infltus sur egredi nequit, ideoque eum retrogredi necesse est, ut d orificium inferius emergt. Unde fit, ut qusi d oerculum sur tubum reflectt lterumque tntum stii bsolvt, ntequm exitus ei tet. Et er consequens ër in tubo tnqum chord dulo longior est considernd, quie chorde ut comlicte concii debent. Unde colligitur fistulm tectm sonum edere eundem cum ert dulo longiore, seu edet sonum octv grviorem ert. Qules utem ednt sonos fistule non ubivis eiusdem mlitudinis, i. e. vel convergentes vel divergentes, item fistule sur ex rte sltem tecte, Clr. Cometitoribus exminndum roono. ANNEXA.

17 Euler's Disserttion De Sono : E002. Trnslted & Annotted by In Bruce System cororis et nime hrmonie restbilite, quo ctiones cororis et nime minime se invicem deendere sseruntur, veritti non consentneum est. 2. Vis ttrctiv NEUTONI tissimus cunct cororum coelestium henomen exlicndi modus est. Et ide extr dubium ositum esse credo coror omni ex su ntur se mutuo trhere. 3. Posito, centrum telluris (quod utem vero longe est lienum) quevis coror ttrhere i reciroc rtione dulict distntirum, terrmque er centrum esse erfortm. Queritur,lide er formen demisso, quid eveniret, cum d centrum erveniret, utrum ibi vel quiescens ermnsurus vel rotinus ultr centrum rogressurus, n vero vestigio d nos ex centro reversurus esset. Postremum ego ffirmo. 4. Vires cororum motorum sunt in rtione comosit ex simlici mssrum et dulict celerittum. 5. Globus suer lno inclinto rotndo descendens, bstrhendo b omni resistenti e celeritte qum ex edem ltitudine erendiculriter cdendo cquireret, multo minorem nnciscetur. Erit enim ill d istm normliter cdendo cquisitm ut 5 d Mli in nvibus nimis lti esse non debent, ne venti vis nvem subvertt. Ponmus utem mlum velis instructum nimis ltum, ut scilicet nvis vento dt certo rosterneretur. Dico, ltior vel si enderentur, ut vis nvem roulsns fortior esset, minus fore nvem subversioni obnoxim. Et semer, quntumvis mlus ltus sit, ltitudo velorum eiusque ugeri oterit, ut nequidem vehementissimus ventus nvi dmnum fferre ossit. TANTUM.