Harmonic generation in organ pipes, recorders, and flutes
N.H. Fletcher and Lorna M. Douglas
Department of Physics, University of New England, Armidale New South Wales 2351, Australia
(Received 25 September 1979; accepted for publication 6 March 1980)
A simplified treatment is given of the mechanism of sound production in musical instruments driven by
air jets, which is sufficiently explicit that semiquantitative predictions can be made about the effects of
certain variables upon the harmonic structure of the sound produced. In particular it is found that the
amplitude of the even harmonics, generally, and of the second harmonic, particularly, is quite critically
dependent upon the offset of the pipe lip from the symmetry plane of the jet. A completely symmetrical
relationship (zero offset) reduces the generated amplitude of the second harmonic by a large factor.
Experimental results with an adjustable organ pipe are found to confirm these predictions. The
implications of these results for the voicing of organ pipes and recorders and for subtle tonal variation in
flute playing are briefly discussed.
PACS numbers: 43,75.Np
INTRODUCTION
The mechanism of sound production by flue pipes,
under which heading are included organ flue pipes, re-
corders (German: Blockfltte), and flutes, is now fairly
well understood, x-? not only in its general principles
but also in relation to generation of the harmonic spec-
trum of the sound produced 4'6 and the characteristic
attack transient. ? The full theory of harmonic genera-
tion is complicated 6'a and involves solution of a large
number of coupled nordinear differential equations, • or
their related integral equations • so that the essential
features of the mechanism are somewhat obscured. It
turns out to be possible, however, to deduce a simpli-
fied version of the theory which accounts well for the
behavior of the pipe and the effects of several voicing
or playing adjustments and which is free from all the
mathematical elaboration of the detailed treatment.
It is the purpose of this paper to outline and justify this
simplified approach, to set out its predictions, and to
compare these with experimental measurements for
several cases of interest. In this way we arrive at a
straightforward and semiquantitative understanding of
several important features of pipe voicing adjustments
and their effect on tone quality.
I. THEORETICAL PRINCIPLES
and is the opening in the player's lips in the case of the
flute), travels some distance across the mouth of the
pipe, and then interacts with the pipe lip. The acoustic
flow out of the pipe mouth associated with the vibration
of the pipe air column induces waves on the jet which
grow in amplitude as they propagate so that the trans-
verse motion of the jet at the pipe lip is generally com-
parable with the jet width when the pipe is sounding
normally. The jet blowing alternately into and outside
the pipe at its lip maintains the pipe oscillation so that
steady sound results.
Careful consideration s'• shows that the driving force
exerted on the pipe modes by the deflecting jet has
terms proportional, respectively, to the volume flow
of the jet into the pipe and to the square of this quan-
tity, but under most conditions the simple volume flow
term is dominant. Phase relationships are a little
complex 6'•ø but this need not concern us here. If the
jet velocity is such that there is just half a wavelength
of the transverse disturbance between the flue and the
lip, then the pipe will sound exactly at its resonance
frequency. If the blowing pressure is increased to
shorten the jet travel time then the sounding frequency
will rise slightly to introduce a compensating phase
shift, while if the blowing pressure is lowered the
sounding frequency will fall slightly.
The sounding mechanism of flue pipes has been dis-
cussed in detail elsewhere • and a recent review •ø sets
out the principles in a concise way. Briefly, and re-
ferring to Fig. 1, an air jet emerges from the flue
(which is a fixed structure in an organ pipe or recorder
flue • ear•l flip
languid •cut
FIG, 1, The geometry of a typical org an flue pipe, ¾or our
experiments the pipe wa• cut along the plane shown and the
body moved relative to the foot and flue,
Harmonic generation occurs primarily because the
jet velocity has a bell-shaped profile so that, even if
the jet is being deflected sinusoidally, its flow into the
pipe will be a distorted sinusoid containing higher har-
monic components whose frequencies approximate those
of higher resonant modes of the pipe. There is a small
additional upper harmonic component generated by a
Bernoulli term in the flow interaction s'6 but this is of
small amplitude in 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 computing what might be called a "source
spectrum" which is then modified by interaction with
the array of not-exactly harmonic resonances of the
pipe, treated as a "filter function." This is the ap-
proach originally advocated by Sundberg, • though he
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 Acoustical Society of America 767
,propriate source spectrum. The method cannot be
justified for a general case, since in principle each
harmonic interacts back upon the jet to modify the
source spectrum, but in the present case it represents
a good approximation. There are several reasons for
this: (1) for many flue pipes of practical interest, the
fundamental is the strongest component of the radiated
sound; (2) the internal velocity spectrum of the pipe is
further weighted towards low frequencies at 6 dB per
octave relative to the radiated spectrum because of the
relation between source strength and radiated sound
pressure; (3) the internal acoustic displacements
(which are what really concern us in calculating jet
displacement) have in addition the further low-frequency
emphasis of 6 dB per octave arising from the fact that
displacement is the integral of velocity. The exceptions
to these generalizations occur only for the lowest notes
of the Boehm flute and for organ flue pipes of "string"
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
fundamental, and we can proceed to calculate the source
spectrum by examining the waveform of the resulting
jet flow into the pipe.
II. JET SOURCE FUNCTION
For a symmetrical planar jet in the laminar flow re-
gime at low Reynolds' numbers the velocity profile has
the form •2
V (y) = V o sech2(y /b ) , (1)
where y is the distance from the axial plane, V o is the
central velocity, and b is a scale factor determining
the width of the profile (see Fig. 2). As the jet spreads
downstream from the flue V o decreases and b increases.
The jets in real musical flue pipes are rarely in the
laminar flow regime but are usually homogeneously
turbulent. The nicking on the languids of some organ
pipes perhaps serves to establish this homogeneity.
The behavior of a turbulent jet is, however, quite simi-
lar to that of a laminar jet as far as its function in a
flue pipe is concerned, and indeed the jet divergence
and wave propagation behavior is considerably simp-
-2b -b 0 Yo b 2b
Y
FIG. 2. Axial velocity distribution across a lainfriar jet as
given by (1). The mean axial distribution across a turbulent
jet is very similar.
ler. •3'•4 The mean velocity profile of a turbulent jet
does not have exactly the form (1), but this function is
a sufficiently close approximation to serve our purpose
and has the advantage of algebraic simplicity.
Referring to Fig. 2, where the function (1) is plotted,
suppose that the pipe lip cuts the jet at the position Yo
when in equilibrium, and that the jet then oscillates
in the y direction with amplitude a and angular fre-
quency w. Then the jet flow entering the pipe has the
form,
U(t)= f_•ø+asinWtv(y) dy
= Vo(1 + tanh [(Yo + a sinwt)/b ]}. (2)
Clearly, U(t) is not sinusoidal but contains harmonies
of all orders, that with frequency no) having amplitude
where the sine is used for odd n and the cosine for even
n. U, clearly depends both on the lip offset Yo and on the
amplitude a.
It is possible also to see the qualitative form of the
variation of the U, with y by expanding U as a function
of Yo in a Taylor's series about y =Yo. Recalling that
sin%t can be written •s as a sum of terms in either
sin(n - 2m)•t or cos(n - 2m)•ot for m = 0, 1, 2, ..., and
keeping just leading terms, which is appropriate for
a/b << 1, then,
U•o: (a/b)"[dn/dy•,:,occ (a/b)'•ld"-XV/dy"-•[•=•, ø . (4)
The immediately interesting observation is that the
even harmonics are all identically zero when the lip is
symmetrically placed in relation to the jet (Yo: 0) while
the odd harmonics are maximal in this configuration.
This effect was already noted in the earlier complete
treatment 4 but now emerges, as we see later, as one
of the most important practical consequences of voicing
or playing adjustments. The other result worthy of
comment is that, from (4), the amplitude of the nth
harmonic varies as the nth power of the amplitude of
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
low-numbered odd harmonics but may be critical to
the even harmonics.
Before turning from theory to experiment we should
note several additional points. Firs fly what we have
768 J. Acoust. Sec. Am., Vol. 68, No. 3, September 1980 N.H. Fletcher and L. M. Douglas: Harmonic generation in organ pipes 768
•:-60
-80
-3 -2 -1 0 I 2 3
Lip Offset Yo/b
FIG. 3. Calculated relative pressure levels of the first four
harmonics in the jet source function as functions of the rela-
tive lip displacement yo/b, as given by (2). The relative jet
displax•ement amplitude is ,•/b = 2. For clarity the curves for
successive harmonics are displaced downwards by 10, 20, and
30 dB relative to the scale for the first harmonic.
calculated is the internal source spectrum of the jet and
this must be modified both by the filter function of the
pipe resonator and by the radiation function of its open
ends in order to derive the radiated spectrum. These
together, determine the absolute level of each har-
monic but do not depend on the lip offset voicing ad-
justment Yo- We should thus be able to compare mea-
sured sound output curves with the calculated curves
of Fig. 3 if absolute levels are ignored. Secondly, we
have left out of consideration the small amount of
second harmonic (and thence of other even harmonics),
generated by the Bernoulli nonlinearity of the jet-drive
mechanism. a Inclusion of this effect would simply
mean a small component proportional to a 2 and inde-
pendent of Yo added to U2, and similarly for the higher
even harmonics. This would change the zeros shown
in Fig. 3 into simple minima. Finally, we should point
out that the lip offset Yo is measured from the center
plane of the undisturbed jet at the lip position. As dis-
cussed elsewhere 6 there is generally a small static
displacement of the jet caused by static pressure in-
crease within the pipe, so that experimental determina-
tion of Yo may not be simple.
III. ORGAN PIPE EXPERIMENT
To examine experimentally the predictions of this
theory we took an adjustable flute-type organ pipe used
in a previous study 6 and having a square cross section
about 40x 40 mm and length 460 mm giving a funda-
mental near 330 Hz. Pipe material was Perspex 6 mm
thick and the languid was of the type known to organ
builders as "inverted" with a 60 ø bevel inside the foot
of the pipe. For the present measurements "ears"
were added and the pipe was cut through just above the
languid as suggested by 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.
We should note that some of the results detailed be-
low have already been measured by Nolle, •? though in a
less systematic fashion since he did not have the pre-
dictions of an explicit theory to guide him in deter-
mining what parameters are important.
This experimental arrangement behaved consistently
for a cousiderable range of blowing pressures, flue
widths and lip cut-up distances. Figure 4 shows the
results of one of the better measurements which is,
however, quite typical of all those obtained. There is
clearly quite good semiquantitative agreement with
the theoretical curves of Fig. 3, if a/b is assumed to
have a value near 2, which is quite reasonable. More
importanfiy, however, the experimental curves ex-
hibit very clearly the miniran in the even harmonics,
and specially the second harmonic, expected from the
theory. This feature is important not only as a test
of the theory but also because variation in the intensity
of the second harmonic of a complex tone is a very
effective means of achieving a variation in tone color,
such variations supplying a large measure of the dif-
ference between open and stopped organ pipes or,
o I
• -20
_>
• -30
-40
-2-0 -1-0 0 1'0
Lip Offset Yo Imm)
FiG. 4. Measured relative sound pressure levels of the har-
monics in the radiated sound oœ the experimental pipe for a
cut up of 10 ram, a flue width of ] ram, and blowing pressure
of 50 Pa, as functions of lip offset :Y0 from the nominally sym-
metric position. (]00 Pa= ! cm water gauge.)
769 J. Acoust. Soc. Am., Vol. 68, No. 3, September 1980 N.H. Fletcher and L. M. Douglas: Harmonic generation in organ pipes 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
the actual jet flow and lip shape. This also probably
accounts for the slight asymmetry of the whole set of
curves.
By examination of acoustic waveforms on an oscil-
loscope we readily see that the phase of the second har-
monic changes by 180 ø across the minimum at Yo=0.
This is just what is expected from the theoretical re-
sult. (4) if the algebraic rather than the absolute value
of the derivative is considered.
Finally, we examined the effects of the sharpness of
the upper lip of the pipe by beginning with a sharp wedge
of about 30 ø angle and then squaring or rounding it off
to a thickness of up to about 3 ram. These changes
made no difference to the absolute sound pressure
levels of the harmonics or to their behavior with lip
offset, though more subtle characteristics like back-
ground air noise or onset transient might well have
been influenced.
IV. THE RECORDER
Members of the recorder or BlockflUte family are
essentially built like organ flue pipes except that the
windway below the flue is long and narrow (typically
10 mm x 2 mm in cross section and 60 mm in length
for an alto recorder) and the tapered body is provided
ß with finger holes. The lip cut-up is small (3 to 4 ram)
and the lip is a•ranged nearly centr•ly in relation to
the flue opening (so yo= 0). From our previous discus-
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
good quality their characteristic "hollow" sound, has
been previously noted by Lupke xs and by Herman/9
though attributed by Herman for some reason to the
slight conicily of the bore.
In a set of measurements 2ø taken some time ago in
this laboratory we followed the development of in-
dividual harmonics in the sound of a typical recorder
of moderate quality as a function of blowing pressure.
Figure 5, extracted from these results, is fairly typi-
c• •nd shows the ne•r!y independent beh•_:•or of the odd
and even harmonics, as well as the low relative level
of the even harmonics. This behavior is rather what
we should expect from our simple theory, while the
independent coupling of odd and even harmonics follows
in greater detail as a consequence of a nonlinearity in
which the cubic term is large compared with the quad-
ratic term. 6
i
-20 /
-co
100 200 300
Blowing
100 200 300
Pressure •Pa)
FIG. 5. Measured relative sound pressure levels of the har-
monics for the notes C5 (523 Hz) and D5 (587 Hz), as a f•nc-
lion of blowing pressure, played on an alto recorder by
Schreiber. Both notes have a pleasantly rich sound.
eluded angle about 80 ø) of the embouchure hole. The
jet geometry is under the continuous control of the
player, as also is the blowing pressure. 2x Such subtle
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
able to produce significant changes in tone color by
changing minor aspects of jet geometry and blowing
pressure. In the light of our discussion above it seems
likely that one of the most important quantities ac-
cessible to the player may be the jet direction, by ad-
justment of which he can change the relative intensity
of the second harmonic. A qualitative experiment sug-
gests that this is indeed so, but measurement is made
difficult by the normal presence of ribtaro caused by
rhythmically varying blowing pressure? •
Vl. CONCLUSIONS
The simple theory presented above enables us to
understand, in a semiquantitative way, the mechanism
of harmonic production in flue pipes. We have essen-
tially dealt only with the source function for the inter-
action of the jet with the pipe lip. Upon this must be
superposed the highly selective filtering action of the
pipe resonator as discussed by Sundberg • and by
Benade?
A result of major import•nce to understn_n•ng the
effect of voicing adjustments in organ pipes and re-
corders and of tone variation in flute playing is the
theoretical prediction, confirmed experimentally, that
the amplitudes of the even harmonics generally and of
the second harmonic specifically depend quite critical-
ly upon the offset of the pipe lip relative to the center
plane of the jet.
V. THEFLUTE
In the normal transverse orchestral flute, the
player's lips take the place of the fixed flue of the organ
pipe and direct a jet of air against the sharp edge (in-
ACKNOWLEDGMENT
This study is part of a program in musical' acoustics
supported by the Australian Research Grants Commit-
tee.
770 J. Acoust. Soc. Am., VoL 68, No. 3, September 1980 N.H. Fletcher and L. M. Douglas: Harmonic generation in organ pipes 770
iL. Cremer and H. Ising, "Die selbsterregten Sehwingungen
yon Orgelpfeifen," Acustica 19, 143-153 (1967).
2J. W. Coltman, "Sounding mechanism of the flute and organ
pipe," J. Acoust. Sec. Am. 44, 983-992 (1968).
3S. A. Eider, "On the mechanism of sound production in organ
pipes," J. Acoust. Sec. Am. 54, 1554-1564 (1973).
½N. H. Fletcher, "Non-linear interactions in organ flue pipes,"
J. Acoust. Soc. Am. õ6, 645-652 (1974).
'•N. H. Fletcher, "Jet-drive mechanism in organ pipes," J.
Acoust. Soc. Am. 60, 481-483 (1976).
6N. H. Fletcher, "Sound production by organ flue pipes," J.
Acoust. Soc. Am. 60, 926-936 (1976).
IN. H. Fletcher, "Transients in the speech of organ flue
pipes--a theoretical study," Acustica 34, 224-233 (1976).
aN. H. Fletcher, 'q•Iode locking in non-linearly excited inhar-
monic musical oscillators," J. Acoust. Soc. Am. 64, 1566-
1569 (1978).
•R. T. Schumacher, "Self-sustained oscillations of organ flue
pipes: An integral equation solution," Acustica 39, 225-238
(1978).
1øN. H. Fletcher, "A•r flow and sound generation in musical
wind instruments," Ann. Rev. Fluid Mech. 11, 123-146
(1979).
tlj. Sundberg, '•Viensurens betydelse i $ppna labialpipor"
(with English summary), Acta Univ. Ups., Stud. musicol.
Ups. (N.S.)3, 1-224 (1966).
tiW. G. Bickley, "The plane jet," Philos. Mag. 28, 727-731
(1937).
13N. H. Fletcher and S. Thwaltes, "Wave propagation on an
acoustically perturbed jet," Acustica 42, 323-334 (1979).
t4S. Thwaites and N.H. Fletcher, "Wave propagation on turbu-
lent jets," Acustica (in press).
l•I. S. Gradshteyn and I. W. Ryzhik, Tables of i•tecrals, se•'ies
andpzoducts (Academic, New York, 1965), pp. 25-26.
i•. E. Worman, "Self-sustained non-linear oscillations of
medium amplitude in clarinet-like systems," Ph.D. thesis,
Case Western Reserve University, Cleveland, OH (1971)
[Uni. Microfilms, Ann Arbor (Her. 71-22869)], pp. 1-154.
t?A. W. Nolle, "Some voicing adjustments of flue organ pipes,"
J. Acoust. Soc. Am. 66, 1612-1626 (1979).
18A. v. L6pke, "Utersuchungen an Blockfi•ten," Akust. Z. 5,
39-46 (1940)•
19R. Herman, '•3bservations on the acoustical characteristics
of the English flute," Am. J. Phys. 27, 22-29 (1959).
2øWe are grateful to Richard Ward for taking these measure-
ments.
2tN. H. Fletcher, "Acoustical correlates of flute performance
technique," J. Acoust. Soc. Am. 57, 233-237 (1977).
22A. H. Benade, 'qtelation of air-column resonances to sound
spectra produced by wind instruments," J. Acoust. SOc. Am.
40, 247-249 (1966).
771 J. Acoust. Soc. Am., VoL 68, No. 3, September 1980 N.H. Fletcher and L. M. Douglas: Harmonic generation in organ pipes 771