Harmonic generation in organ pipes, recorders, and flutes N.H. Fletcher and Lorna M. Douglas Departmentof Physics,University of NewEngland, Armidale New South Wales 2351, Australia (Received25 September1979; accepted for publication6 March 1980)

A simplifiedtreatment is givenof the mechanismof soundproduction in musicalinstruments driven by air jets, whichis sufficientlyexplicit that semiquantitativepredictions can be madeabout the effectsof certainvariables upon the harmonicstructure of the soundproduced. In particularit is foundthat the amplitudeof the evenharmonics, generally, and of the secondharmonic, particularly, is quitecritically dependentupon the offsetof the pipelip from the symmetryplane of thejet. A completelysymmetrical relationship(zero offset)reduces the generatedamplitude of the secondharmonic by a large factor. Experimentalresults with an adjustableorgan pipe are found to confirm thesepredictions. The implicationsof theseresults for the voicingof organpipes and recordersand for subtletonal variation in flute playingare briefly discussed.

PACS numbers:43,75.Np

INTRODUCTION and is the openingin the player's lips in the case of the flute), travels some distance across the mouth of the The mechanism of sound production by flue pipes, pipe, and then interacts with the pipe lip. The acoustic under which heading are included organ flue pipes, re- flow out of the pipe mouth associated with the vibration corders (German: Blockfltte), and flutes, is now fairly of the pipe air column induces waves on the jet which well understood,x-? not only in its general principles grow in amplitude as they propagate so that the trans- but also in relation to generation of the harmonic spec- verse motion of the jet at the pipe lip is generally com- trum of the soundproduced 4'6 and the characteristic parable with the jet width when the pipe is sounding attacktransient. ? The full theory of harmonic genera- normally. The jet blowing alternately into and outside tion is complicated6'a and involves solution of a large the pipe at its lip maintains the pipe oscillation so that number of couplednordinear differential equations,• or steady sound results. their related integral equations• so that the essential features of the mechanism are somewhat obscured. It Careful consideration s'• shows that the driving force turns out to be possible, however, to deduce a simpli- exerted on the pipe modes by the deflecting jet has fied versionof thetheory which accounts well for the terms proportional, respectively, to the volume flow behavior of the pipe and the effects of several voicing of the jet into the pipe and to the square of this quan- or playing adjustments and which is free from all the tity, but under most conditionsthe simple volumeflow mathematical elaboration of the detailed treatment. term is dominant. Phase relationships are a little It is the purpose of this paper to outline and justify this complex6'•ø but this neednot concernus here. If the simplified approach, to set out its predictions, and to jet velocityis suchthat there is just half a wavelength compare these with experimental measurements for of the transverse disturbance between the flue and the several cases of interest. In this way we arrive at a lip, then the pipe will soundexactly at its resonance straightforward and semiquantitative understanding of frequency. If the blowingpressure is increased to several important features of pipe voicing adjustments shorten the jet travel time then the soundingfrequency andtheir effect on tonequality. will rise slightly to introduce a compensatingphase shift, while if the blowingpressure is lowered the I. THEORETICAL PRINCIPLES sounding frequency will fall slightly.

The sounding mechanism of flue pipes has been dis- Harmonic generation occurs primarily because the cussed in detail elsewhere • and a recent review •ø sets jet velocityhas a bell-shapedprofile so that, evenif out the principles in a concise way. Briefly, and re- the jet is being deflectedsinusoidally, its flow into the ferring to Fig. 1, an air jet emerges from the flue pipe will be a distorted sinusoid containinghigher har- (which is a fixed structure in an organ pipe or recorder monic componentswhose frequencies approximate those of higher resonant modes of the pipe. There is a small additional upper harmonic componentgenerated by a flue• ear•l flip Bernoulli term in the flow interaction s'6 but this is of small amplitudein a normal pipe configuration.6 In our simplified development of the theory we con- sider only the upper harmonics generated by the jet profile, thus computingwhat might be called a "source spectrum" which is then modified by interaction with languid •cut the array of not-exactly harmonic resonancesof the FIG, 1, The geometry of a typical org an flue pipe, ¾or our pipe, treated as a "filter function." This is the ap- experiments the pipe wa• cut along the plane shown and the proachoriginally advocated by Sundberg,• thoughhe body moved relative to the foot and flue, was forced to assume rather than to derive an ap-

767 J. Acoust. Soc. Am. 68(3), Sept. 1980 0001-4966/80/090767-05500.80 ¸ 1980 AcousticalSociety of America 767 ,propriate source spectrum. The method cannot be ler. •3'•4 The mean velocity profile of a turbulent jet justified for a general case, since in principle each does not have exactly the form (1), but this function is harmonic interacts back upon the jet to modify the a sufficiently close approximation to serve our purpose source spectrum, but in the present case it represents and has the advantage of algebraic simplicity. a good approximation. There are several reasons for this: (1) for many flue pipes of practical interest, the Referring to Fig. 2, where the function (1) is plotted, fundamental is the strongest component of the radiated suppose that the pipe lip cuts the jet at the position Yo sound; (2) the internal velocity spectrum of the pipe is when in equilibrium, and that the jet then oscillates further weighted towards low frequencies at 6 dB per in the y direction with amplitude a and angular fre- quency w. Then the jet flow entering the pipe has the octave relative to the radiated spectrum because of the relation between source strength and radiated sound form, pressure; (3) the internal acoustic displacements (which are what really concern us in calculating jet U(t)=f_•ø+asinWtv(y) dy displacement) have in addition the further low-frequency emphasis of 6 dB per octave arising from the fact that = Vo(1+ tanh[(Yo +a sinwt)/b]}. (2) displacement is the integral of velocity. The exceptions Clearly, U(t) is not sinusoidal but contains harmonies to these generalizations occur only for the lowest notes of the Boehm flute and for organ flue pipes of "string" of all orders, that with frequency no) having amplitude quality, where the fundamental is relatively weak. Thus, to a good approximation, the displacement of the jet at the pipe lip is simply sinusoidal, with frequency equal to and amplitude proportional to those of the pipe where the sine is used for odd n and the cosine for even fundamental, and we can proceed to calculate the source n. U, clearly depends both on the lip offset Yo and on the spectrum by examining the waveform of the resulting amplitude a. jet flow into the pipe. It is possible also to see the qualitative form of the II. JET SOURCE FUNCTION variation of the U, with y by expanding U as a function of Yoin a Taylor's series about y =Yo. Recalling that For a symmetrical planar jet in the laminar flow re- sin%t can be written •s as a sum of terms in either gime at low Reynolds' numbers the velocity profile has sin(n - 2m)•t or cos(n- 2m)•ot for m = 0, 1, 2, ..., and the form •2 keeping just leading terms, which is appropriate for V (y) = Vo sech2(y /b ) , (1) a/b <<1, then, where y is the distancefrom the axial plane, Vo is the U•o:(a/b)"[dn/dy•,:,occ (a/b)'•ld"-XV/dy"-•[•=•, ø. (4) central velocity, and b is a scale factor determining The immediately interesting observation is that the the width of the profile (see Fig. 2). As the jet spreads even harmonics are all identically zero when the lip is downstream from the flue V o decreases and b increases. symmetrically placed in relation to the jet (Yo: 0) while The jets in real musical flue pipes are rarely in the the odd harmonics are maximal in this configuration. laminarflow regimebut are usuallyhomogeneously This effect was already noted in the earlier complete turbulent. The nicking on the languids of some organ treatment 4 but now emerges, as we see later, as one pipes perhaps serves to establish this homogeneity. of the most important practical consequences of voicing The behavior of a turbulent jet is, however, quite simi- or playing adjustments. The other result worthy of lar to that of a laminar jet as far as its function in a comment is that, from (4), the amplitude of the nth flue pipe is concerned, and indeed the jet divergence harmonic varies as the nth power of the amplitude of and wave propagation behavior is considerably simp- the fundamental, a result derived for a somewhat dif- ferent case by Worman?

As we have already remarked, the amplitude a of jet displacement at the lip is not generally small compared with the jet scale width b at this point, so that we need to calculate the full expression (3) for the Fourier com- ponents of the excitation spectrum for a/b ~ I in order to compare the predictions of the theory realistically with experiment. These calculations are shown in Fig. 3 from which it can be seen that, while the qualitative conclusions about even harmonics remain unaltered, the variation of the odd harmonics now shows a plateau region around the symmetrical configuration yo= 0. We conclude that lip adjustment is not critical to the -2b -b 0 Yob 2b Y low-numbered odd harmonics but may be critical to the even harmonics. FIG. 2. Axial velocity distribution across a lainfriar jet as given by (1). The mean axial distribution across a turbulent Before turning from theory to experiment we should jet is very similar. note several additional points. Firs fly what we have

768 J.Acoust. Sec. Am., Vol. 68, No. 3, September1980 N.H. Fletcherand L. M. Douglas:Harmonic generation inorgan pipes 768 languid as suggestedby Nolle•? so that the foot and flue could be moved backwards and forwards relative to the lip and pipe body, thus varying the offset Yo- This mo- tion was effected by a screw connected to a multi-turn potentiometer to provide displacement along one axis of an XY plot, the other coordinate being provided by the appropriately filtered output of a microphone mounted on the axis of the pipe, about 0.5 m from the open end, in an artechole room. By plotting the sound pressure logarithmically, the experimental results can thus be compared directly with the theoretical re- suits of Fig. 3.

•:-60 We should note that some of the results detailed be- low have already been measuredby Nolle, •? thoughin a less systematic fashion since he did not have the pre- dictions of an explicit theory to guide him in deter- -80 mining what parameters are important.

This experimental arrangement behaved consistently -3 -2 -1 0 I 2 3 for a cousiderable range of blowing pressures, flue Lip Offset Yo/b widths and lip cut-up distances. Figure 4 shows the results of one of the better measurements which is, FIG. 3. Calculated relative pressure levels of the first four harmonics in the jet source function as functions of the rela- however, quite typical of all those obtained. There is tive lip displacementyo/b, as givenby (2). The relative jet clearly quite good semiquantitative agreement with displax•ementamplitude is ,•/b = 2. For clarity the curvesfor the theoretical curves of Fig. 3, if a/b is assumedto successive harmonics are displaced downwards by 10, 20, and have a value near 2, which is quite reasonable. More 30 dB relative to the scale for the first harmonic. importanfiy, however, the experimental curves ex- hibit very clearly the miniran in the even harmonics, calculatedis the internal sourcespectrum of the jet and and specially the second harmonic, expected from the this must be modified both by the filter function of the theory. This feature is important not only as a test pipe resonator andby the radiation functionof its open of the theory but also because variation in the intensity ends in order to derive the radiated spectrum. These of the second harmonic of a complex tone is a very together, determine the absolute level of each har- effective means of achieving a variation in tone color, monicbut do not dependon the lip offset voicingad- such variations supplying a large measure of the dif- justmentYo- We shouldthus be able to compare mea- ference between open and stopped organ pipes or, sured sound output curves with the calculated curves of Fig. 3 if absolutelevels are ignored. Secondly,we have left out of consideration the small amount of secondharmonic (and thenceof other even harmonics), o I generated by the Bernoulli nonlinearity of the jet-drive mechanism.a Inclusionof this effectwould simply mean a small componentproportional to a 2 and inde- pendentof Yoadded to U2, and similarly for the higher even harmonics. This would change the zeros shown in Fig. 3 into simple minima. Finally, we shouldpoint out that the lip offset Yois measuredfrom the center • -20 plane of the undisturbedjet at the lip position. As dis- _> cussedelsewhere 6 there is generally a small static displacement of the jet caused by static pressure in- • -30 crease within the pipe, so that experimental determina- tion of Yo may not be simple.

-40 III. ORGAN PIPE EXPERIMENT

To examine experimentally the predictions of this theory we took an adjustableflute-type organ pipe used in a previous study6 and having a square cross section -2-0 -1-0 0 1'0 about 40x 40 mm and length 460 mm giving a funda- Lip Offset Yo Imm) mental near 330 Hz. Pipe material was Perspex 6 mm FiG. 4. Measured relative sound pressure levels of the har- thick and the languid was of the type knownto organ monics in the radiated sound oœthe experimental pipe for a builders as "inverted" with a 60 ø bevel inside the foot cut up of 10 ram, a flue width of ] ram, and blowing pressure of the pipe. For the present measurements "ears" of 50 Pa, as functions of lip offset :Y0from the nominally sym- were added and the pipe was cut through just above the metric position. (]00 Pa= ! cm water gauge.)

769 J. Acoust.Soc. Am., Vol. 68, No. 3, September1980 N.H. Fletcherand L. M. Douglas:Harmonic generation in organpipes 769 among reed-driven instruments, between low-register notes on the oboe and the clarinet. As expected, the even harmonics do not go strictly to zero, partly be- cause of the omitted nonlinear term in the jet drive and partly, no doubt, because of incomplete symmetry in -20 / i the actual jet flow and lip shape. This also probably accounts for the slight asymmetry of the whole set of curves.

By examinationof acousticwaveforms on an oscil- loscope we readily see that the phase of the second har- monic changesby 180ø across the minimumat Yo=0. This is just what is expected from the theoretical re- sult. (4) if the algebraic rather than the absolute value -co 100 200 300 100 200 300 of the derivative is considered. Blowing Pressure•Pa) Finally, we examined the effects of the sharpness of FIG. 5. Measured relative sound pressure levels of the har- the upper lip of the pipe by beginning with a sharp wedge monicsfor the notesC5 (523Hz) andD5 (587Hz), as a f•nc- of about30 ø angle and then squaringor roundingit off lion of blowing pressure, played on an alto recorder by to a thickness of up to about 3 ram. These changes Schreiber. Both noteshave a pleasantlyrich sound. made no difference to the absolute sound pressure levels of the harmonics or to their behavior with lip eluded angle about 80 ø) of the embouchurehole. The offset, though more subtle characteristics like back- jet geometry is under the continuous control of the ground air noise or onset transient might well have player, as also is the blowingpressure. 2x Suchsubtle been influenced. adjustment possibilities make any objective study of tone quality in relation to jet geometry ex•emely dif- ficult, but it is noteworthy that good flute players are IV. THE RECORDER able to produce significant changes in tone color by changing minor aspects of jet geometry and blowing Members of the recorder or BlockflUte family are pressure. In the light of our discussion above it seems essentially built like organ flue pipes except that the likely that one of the most important quantities ac- windway below the flue is long and narrow (typically cessible to the player may be the jet direction, by ad- 10 mm x 2 mm in cross section and 60 mm in length justment of which he can change the relative intensity for an alto recorder) and the tapered body is provided of the second harmonic. A qualitative experiment sug- ßwith finger holes. The lip cut-upis small (3 to 4 ram) gests that this is indeed so, but measurement is made andthe lip is a•rangednearly centr•ly in relationto difficult by the normal presence of ribtaro caused by the flue opening(so yo= 0). From our previous discus- rhythmically varying blowing pressure? • sion we should therefore expect the even harmonics in recorder sound to be weak relative tothe odd harmonics and indeed this effect, which gives to recorders of Vl. CONCLUSIONS good quality their characteristic "hollow" sound, has been previously noted by Lupkexs and by Herman/9 The simple theory presented above enables us to though attributed by Herman for some reason to the understand, in a semiquantitative way, the mechanism slight conicily of the bore. of harmonic production in flue pipes. We have essen- tially dealt only with the source function for the inter- In a set of measurements2ø taken some time ago in action of the jet with the pipe lip. Upon this must be this laboratory we followed the development of in- superposed the highly selective filtering action of the dividual harmonics in the sound of a typical recorder pipe resonator as discussedby Sundberg• and by of moderate quality as a function of blowing pressure. Benade? Figure 5, extracted from these results, is fairly typi- c• •nd shows the ne•r!y independent beh•_:•or of the odd A result of major import•nce to understn_n•ng the and even harmonics, as well as the low relative level effect of voicing adjustments in organ pipes and re- of the even harmonics. This behavior is rather what corders and of tone variation in flute playing is the we should expect from our simple theory, while the theoretical prediction, confirmed experimentally, that independent coupling of odd and even harmonics follows the amplitudes of the even harmonics generally and of in greater detail as a consequence of a nonlinearity in the second harmonic specifically depend quite critical- which the cubic term is large compared with the quad- ly upon the offset of the pipe lip relative to the center ratic term. 6 plane of the jet.

V. THEFLUTE ACKNOWLEDGMENT

In the normal transverse orchestral flute, the This studyis part of a programin musical'acoustics player's lips take the place of the fixed flue of the organ supported by the Australian Research Grants Commit- pipe and direct a jet of air against the sharp edge (in- tee.

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771 J.Acoust. Soc.Am., VoL 68, No. 3, September 1980N.H. Fletcher andL.M. Douglas: Harmonic generation inorgan pipes 771