We turn to the final family of wind instruments, the air-jet instruments like the flute, recorder and flue organ pipe. Looking back over previous sections of this chapter, the logic behind the organisation should now become clear: we started with the best-understood instruments, and we have progressively moved to the ones where physical understanding is more challenging, and therefore more rudimentary.
For reed instruments like the clarinet, we were able to construct a fairly simple physical model with quantitative experimental support for all aspects. The pipe acoustics was anchored in measured input admittance which could be interpreted via simple theory, while the nonlinear characteristics of the reed mouthpiece were based on uncontroversial theory with some direct measurements in support. The result was a simulation model with some claims to realism. This could be used to explore questions about how the player might control the instrument to maximise the range of musical options, and also how the instrument maker might help by “playability” improvements.
Moving on to the brass instruments, our understanding of tube acoustics was still supported by direct measurements of input impedance, but the model for the lip dynamics was less convincing. We started with essentially the same model as for the reeds, except for a reversal of sign to reflect the opening-reed behaviour of lip vibration as opposed to the closing-reed character of the reed woodwinds. This super-simple model, with just a single degree of freedom, can only be expected to give a rather crude representation of the vibration of the squashy flesh of lips. We then explored a rather ad hoc extension of the model in order to incorporate at least some aspects of the vibration of real lips. The resulting simulation models gave qualitatively plausible results, correctly reflecting many aspects of the behaviour of this family of instruments.
When we moved to the free reed instruments, things got more complicated. This time, the behaviour of the reeds themselves was reasonably uncontroversial, but it was far from clear what additional physics needed to be included in order to model the excitation mechanism with any claim to realism. For some problems, with strong acoustical feedback from a well-characterised system, it was easy to complete a model in the same style as the original clarinet model and obtain quite convincing agreement with experiments.
But in other cases we met snags of two different kinds. First, no impedance measurements appear to have been made on instruments like the concertina, let alone on the harmonica for which the player’s vocal tract is a significant part of the system. Second, we saw strong hints that acoustical feedback is not always the predominant excitation mechanism, suggesting that something more complicated is needed. Initial modelling efforts have been made, based on idealised analysis of the fluid flow in the immediate vicinity of a vibrating reed, but the story is by no means complete.
Now we turn to the air-jet instruments, and we are squarely faced with problems involving non-trivial fluid dynamics. These instruments have no moving parts analogous to reeds or brass-player’s lips. The behaviour of the instrument is determined by the interaction of the internal acoustics of the instrument tube with air-flow from a mouthpiece slot or from a flute-player’s lips. The tube acoustics can be characterised by measured input impedance (or its inverse, input admittance: as we saw back in section 11.1, an air-jet instrument is expected to play notes determined by peaks of admittance, or antiresonances of impedance). But the fluid dynamics governing how an air jet interacts with the solid part of a mouthpiece and with the internal acoustical field is far more complicated than anything we have grappled with in previous sections.
A. Introducing the fluid dynamics of air-jet instruments
We can get a first idea of how flute-like instruments work, and simultaneously gain some insight into the complications, by looking at examples of flow visualisation. Figure 1 shows a kind of rectangular organ pipe, or one-note recorder. It has a transparent section in the pipe wall to allow schlieren imaging (look back at section 10.6 for an account of how the schlieren system works). Air is injected from the bottom, passes through a slot, and the emerging jet then interacts with a sharp edge. It is this interaction between the air jet and the edge (known as the “labium” in the terminology of recorders) which is the key to understanding how such instruments work. For the purposes of the flow visualisation, some carbon dioxide is also injected to give a density contrast with the air and allow the imaging process to work.
Figure 2 shows a sequence of images spread through a single cycle of periodic oscillation, when the “recorder” is playing a steady note. Note the orientation: the resonating pipe lies below the labium, in the lower part of each figure — exactly as in the normal configuration for playing a recorder. To interpret these images, we will need to invoke some of the fluid-dynamical concepts outlined in section 11.2: we can see both laminar and turbulent flow, we can see flow separation at the sharp edges of the labium and of the slot, we can see vortices being formed.
For this particular geometry and blowing pressure, the jet emerging from the slot (at the left of each image) remains laminar long enough to reach the sharp edge. But just a little further downstream, you can see that the jet becomes turbulent. Looking more carefully at the sequence of images, during this cycle of oscillation the jet “switches” from one side of the labium to the other. Look particularly at the fourth image in the sequence (right-hand column, second from the top). The jet is still lying predominantly above the labium (in other words, outside the pipe), but in the gap between the slot and the sharp edge (the “mouth” of the recorder) you can see that the jet is bending downwards. By the next picture in the sequence, it has switched to the lower side (but you can still see the residual turbulence left behind on the upper side). The final image in the sequence shows the opposite transition: the jet has bent upwards, and is in the process of switching back to the upper side.
This switching process involves two physical phenomena. The one we can see some evidence of in the images is instability of the jet: when the jet is perturbed, a disturbance travels along the jet and it grows as it travels. What is causing the perturbation is the second phenomenon, not directly visible in these images. The “recorder” is playing a note with a frequency close to the lowest resonance of the tube, and the tube is open at the mouth, so there is an acoustic air flow in and out through the mouth. This acoustical flow interacts with the jet, triggering a disturbance in the jet near the slot. The disturbance grows as it is carried along by the jet, and ultimately causes switching from one side of the labium to the other.
Each time the jet switches past the sharp edge, it sheds a vortex. We can see vortices even more clearly in Fig. 3, which shows a similar sequence of schlieren images, but this time they show the very early stages of a transient after the air supply has been abruptly turned on. The first two images in this sequence show a very clear mushroom-shaped vortex being generated from the sharp edges of the slot on the left-hand side. When this vortex reaches the labium, it interacts to generate more complicated vortex structures.
The images in Fig. 2 illustrate a regime of vibration which is generally accepted to form the basis for “normal” playing on a recorder or a transverse flute. However, this is not true for all instruments, or for all playing techniques. In Fig. 2 we see a relatively short, relatively thin jet which is laminar throughout the region of most interest. This thin jet tends to move as a whole, in a sinuous motion. But sometimes we might have a laminar jet which is wider, in which case a different-looking fluid-mechanical phenomenon occurs, illustrated in Fig. 4 (reproduced from section 11.2). The two separate shear layers on each side of the wide jet show a classic instability pattern in which they generate vortices on alternate sides before the jet reaches the labium.
If the Reynolds number of the air-jet is higher, either because the flow is faster or the jet is longer, the jet becomes turbulent before it reaches the labium. Flue organ pipes are commonly designed like this, for example. The jet may still be able to interact in a way that gives a periodic movement of the jet centre-line, but the sound of this periodic oscillation will be accompanied by a significant component of “wind noise” generated by the interaction of the turbulence with the edge. The result is a more “breathy” sound quality, desirable in some instruments (such as the Japanese shakuhachi).
Jets that show different combinations of short or long, narrow or wide, laminar or turbulent would require different theoretical modelling. Only for a few of these combinations have credible models been developed that can allow simulations of the kind we have shown for the other wind instruments. We will confine our attention to just one of these cases, the one illustrated in Fig. 2. For this type of problem, a procedure known as the “jet-drive model” has been found to give reasonably good results. The approach was pioneered by John Coltman back in the 1960s [1,2]. This model has three main ingredients: the time delay and exponential growth associated with a travelling, growing perturbation on the jet; a simplified form of how the acoustical air-flow from the resonant tube perturbs the jet; and a dipole sound source from the interaction of the switching jet with the labium, which generates acoustical pressure that completes the feedback loop. Some details of this model are given in the next link, based on the descriptions given by Auvray, Ernoult, Fabre and Lagrée [3], and by Fabre in chapter 10 of Chaigne and Kergomard [4].
There is only one other case for which a simple model suitable for simulation has been developed and validated against experiments. This is the case illustrated in Fig. 4, for which a “discrete vortex model” has been presented in the same two references [3,4], building on pioneering work by Holger, Wilson and Beavers in the 1970s [5]. This oscillation regime has some relevance to musical performance, but it seems to be rather marginal compared with the jet-drive model, which relates to the most common playing regime on the recorder or transverse flute. So we will concentrate on the jet-drive model, keeping in mind that it involves major simplifications to the messy fluid mechanics seen in Fig. 2 — and it doesn’t even attempt to describe transient behaviour like Fig. 3, since the model assumes that the jet is “already there” when the note starts. As a result, we can hope to predict some qualitative patterns of behaviour but we should not expect quantitative transient predictions relating to the subtle things players do to start notes in particular ways.
B. The influence of cork position in a transverse flute
But before we plunge into simulation results, we need to say a few words about the easy part of the problem, the linear acoustics of the tube. For all these instruments, whether it is a flute, recorder or flue organ pipe, we have a more-or-less cylindrical tube, open at both ends (or stopped at the far end, in the case of a stopped organ pipe). For the recorder or the organ pipe, we already know more or less how the input admittance will behave: it will resemble the example shown back in section 11.1, in Fig. 14 there.
But the transverse flute brings in another ingredient, which has a consequence similar in some ways to the mouthpiece of a brass instrument, discussed in section 11.6. As shown schematically in Fig. 5, the embouchure hole of a flute is not right at the end of the tube, like the mouth of a recorder. Instead, there is a short length of closed tube extending in the opposite direction to the main playing length, with the fingerholes. The effective acoustical length of this closed tube is determined by the position of a tight-fitting cork, which can be adjusted.
The influence of this closed tube is significant. The black dashed curve in Fig. 6 shows the measured input admittance of a modern Boehm flute for the note $C_4$, 262 Hz, with all but the last fingerhole closed. The red curve shows the predicted input admittance for an ideal double-open tube, with a suitable length corresponding to the distance from embouchure hole to tube end. We will see in a moment that when we correct for the extra closed tube, this length of plain tube will produce more or less the same set of resonance peaks as the measurement, although at this stage they are all a little too high. But the most striking disparity between the two curves concerns the peak heights rather than their frequencies. It can be seen that the first few peak heights are rather similar to the measurement, but the measured admittance dies away rapidly in amplitude while the theoretical curve carries on with strong peaks. (The measurement only extended up to 4 kHz.)
Figure 7 shows the effect of adding the extra closed tube to the calculation, as explained in the next link. The red curve from Fig. 6 is included as a dotted line, for comparison. Using a carefully-selected cork position (in other words, a length for the closed tube), the result is an excellent match to the measurement. The peak frequencies and amplitudes are well captured for the first nine or so resonances. Above that, we start to see small deviations. The inclusion of the closed tube has clearly made a crucial difference to the frequency response, but the simple model has captured this difference very well.
Figure 8 shows what the simple model predicts if we move the cork, to change the length of the closed tube. The red curve is the same as in Fig. 7, while the blue and green curves (displaced vertically in the plot for clarity) show the effect of a shorter and a longer tube, respectively. The differences here are far larger than a player or flute-maker would normally make, to show the effect clearly. The admittance peaks fade away to zero at a frequency that depends on the cork position: making the closed tube shorter raises the frequency, making it longer reduces the frequency.
The effect was discussed by Benade [6], and we can illustrate his explanation with some computed examples of the pressure distribution inside the tube of the flute. Figure 9 shows the theoretical admittance from Figs. 7 and 8, and a selection of the resonance peaks have been marked. At each of these frequencies, making use of the analysis described in the previous link, the pressure distribution along the tube has been plotted in Fig. 10. The main portion of the flute, from embouchure hole to the open end, is indicated by blue curves, while the short portion inside the closed tube is plotted in red. All the curves have been scaled to the same peak amplitude.
The first four curves, at the top in Fig. 10, correspond to the four lowest resonances of the tube. The pressure distributions show more or less what we expect: successively one, two, three and four half-wavelengths fitted in to the length of the tube. At the open end (at the right) the simple model imposes a nodal point of pressure. But this is not quite true at the embouchure hole. The hole is smaller than the bore of the tube, and there is a small but non-zero mass of air in the short “chimney” sketched in Fig. 5. The combined effects of these two things is that the pressure inside the tube does not fall to zero at the hole.
The red portions of the curves, inside the closed tube, show different behaviour. At the closed end the pressure variation must have a horizontal tangent, corresponding to the fact that the volume flow rate must be zero there. The pressure rises towards the closed end, but for these modes at low frequency the rise is very slight, barely visible in the figure. This is as we expect: at frequencies such that the length of the closed tube is very short compared to the wavelength of sound, the pressure must be more or less constant. (We used the same approximation when we talked about the Helmholtz resonator back in section 4.2.1, and we used it again when discussing the effect of the mouthpiece volume in conical woodwind instrument like the saxophone, in section 11.3, part F.)
The next group of three plots correspond to three resonance peaks in Fig. 9 which are approaching the “dead zone” where the admittance shows no peaks. The final plot in Fig. 10 corresponds to a peak beyond this dead zone. For all four of these frequencies, Figure 10 shows rather similar behaviour. The red portion of each curve joins rather smoothly on to the blue portion, and the combined pressure variation looks very much like what we would see if the embouchure hole was blocked, so that we had a closed-open tube with no side-hole. The shapes look sinusoidal, with an antinode at the closed end and a node at the open end.
Now look closely at the pattern of the joins between the red and blue curves — you may find it helpful to look at Fig. 11, which shows a zoomed view of Fig. 10 concentrating near the region of the join at the embouchure hole. For the lower four plots, in the vicinity of the dead zone, we see a systematic pattern of behaviour. If you look at the position of the first nodal point of pressure, you will see that it moves closer and closer to the red-blue join as the dead zone is approached. Then in the lowest plot, above the dead zone, the nodal point has moved into the red portion of the curve. Now we can see that the centre of the dead zone is determined by the condition that there is exactly a quarter-wavelength fitted into the closed tube. The embouchure hole then falls exactly at a nodal point of pressure, so at that frequency it is not possible to excite the internal tube resonance from the embouchure hole. That is the reason the peaks heights in the admittance fade away to zero at this frequency, then grow again once the wavelength is shorter.
C. Simulation results
We are ready to combine the jet-drive model set out in section 11.8.1 with measured input admittance, to run some simulations. We need to choose an input admittance: the three cases to be used here are shown in Fig. 12. The lowest plot, shown in a black dashed line, is the flute measurement we have already seen in Figs. 6, 7 and 8. But we will not start with that, for reasons that will emerge shortly. The top curve, shown in red, is a three-mode approximation to the measured admittance of a recorder, as used in reference [1]. In order to check the simulation code, we will start with a run matching that reference — more details of this comparison, including the parameter values used in the simulations, are given in the first link, section 11.8.1. After that we will look at a comparable note on a modern transverse flute, shown in the blue curve in Fig. 12.
In reference [3], the results for playing frequency and amplitude were extracted from single long simulations in which the jet speed was slowly ramped up or down. Instead, the results shown here use a separate transient for each of a set of steady values of the jet speed, in steps of 0.1 m/s. We do not expect this model to give accurate results for initial transients, for reasons explained above, so we are mainly interested in the possible pattern of steady notes that can be produced on this “recorder”. To focus on this question, each transient was “seeded” with a small pressure fluctuation in the lowest mode, as was done by Auvray et al. [1], and also as we sometimes did in earlier sections on reed and brass instruments.
Figure 13 shows a first set of results. The playing frequency has been normalised by the nominal fundamental frequency to give what we earlier called the “correlation harmonic number”. For in-tune notes, this should take values close to 1, 2, 3 etc. This quantity is plotted against the jet speed, and the resulting plot is surprisingly complicated. At least as predicted by this crude model, the frequency jumps all over the place, especially with very slow jet speeds. The “normal” playing regime for this note corresponds to the flattish portion of the plot for jet speed in the range 16—32 m/s. The line can be seen to slope slightly uphill: the frequency of the note is close to the nominal fundamental, but it increases slightly as the jet speed increases. At higher speed we see another flattish portion with the normalised frequency roughly equal to 2: the “recorder” has overblown to the octave. This line also slopes gently upwards.
The curious cascades of frequencies at low jet speeds are related to an advanced flute-playing technique known as “Aeolian tones”. To see what is going on in the model to produce these tones, it is useful to plot the data in a different way. Instead of using the jet speed on the horizontal axis, we can use the delay time for disturbances on the jet, as they travel the distance from the slot to the labium. This delay time is simply determined by the jet speed and the distance to the labium: in recorder terminology, this is the width of the mouth. The disturbances do not travel exactly at the jet speed, but within this simplified model they travel at 40% of that speed.
The resulting plot is shown in Fig. 14. It is important to note that by using delay rather than jet speed on the axis, we have reversed the order of the points: the left-most clusters of points here correspond to results on the right-hand side of Fig. 13, and vice versa. The delay time has been expressed as a fraction of the period length of each note, and it can immediately be seen that the plotted points cluster around particular values of this normalised delay: there are columns of clustered points near the values $\frac{1}{4}$, $1 \frac{1}{4}$, $2 \frac{1}{4}$etc.
The explanation of this pattern is given in the earlier side link, section 11.8.1. For any note that can be played at a frequency close to a peak of admittance, the normalised delay needs to be in the vicinity of $n+\frac{1}{4}$, where $n$ could take the values 0, 1, 2,… The normal playing regime on a flute or recorder corresponds to $n=0$, while the Aeolian tones correspond to higher values of $n$. What is happening is illustrated in sketch form in Fig. 15: the disturbance on the jet needs to be in a particular phase when it reaches the labium, and this can only be achieved by fitting certain numbers of quarter-wavelengths into the width of the mouth. With a fast jet speed, only the longest wavelength option can be achieved, but with a slow jet it is possible to fit extra wavelengths in. It is then clear that the four columns of points evident in Fig. 14 correspond to the values $n=0,1,2,3$. The “normal” fundamental and octave notes both appear in the $n=0$ column.
Aeolian tones like this can really occur on flutes or recorders, but the details of them predicted by this model should be taken with a generous pinch of salt. In reality a jet with these extra wiggles will tend to break up into separate vortices, which is the case addressed by the discrete-vortex model, which we will not go into here. According to this web site (addressing non-standard flute performance techniques) “Aeolian sounds are colored air sounds with no normal flute tone. The air stream across the embouchure produces an airy pitch resonance in the first octave only, low B- middle D#”. We can perhaps attribute the “airy” sound to the fact that the jet is more likely to become turbulent under these conditions, generating “wind noise” when it interacts with the labium. The statement that the effect is only found in the lowest octave agrees with the experimental findings of Auvray et al. [3]: they were able to obtain an Aeolian tone with their artificially-blown model recorder corresponding to the fundamental pitch with $n=1$, but not for higher overtones or higher values of $n$. So it appears that the jet-drive model is considerably over-predicting the occurrence of Aeolian tones.
The next step is to see what changes if we use input admittance measured from a transverse flute. The blue curve in Fig. 12 shows a measurement from Joe Wolfe’s website, for a modern Boehm flute with B foot, fingered in a manner that he says flautists use for both $C_5$ and $C_6$, adjusting the speed, length and shape of the air jet to choose between these two notes. On a recorder, the only one of these available for a player to vary is the jet speed. The transverse flute allows extra degrees of freedom, as the player can shape the lip opening and vary the distance from the edge as well as adjusting flow speed.
Figures 16 and 17 show equivalent plots to Figs. 13 and 14, keeping the same parameter values as those runs so that this could be thought of as a transverse flute played using a recorder mouthpiece. It is immediately clear that the results are very similar. In this case, the inclusion of extra modes in the admittance, and the small differences of peak frequencies, heights and bandwidths, all seem to make rather little difference to the outcome.
The circles in these two plots pick out a case playing the fundamental (i.e. $C_5$) and a case playing the octave ($C_6$), The detailed waveforms for these two cases are shown in Figs. 18 and 19. In each group, the top plot shows the acoustic pressure inside the mouthpiece and the middle plot shows the acoustic volume flow rate through the mouth (or embouchure hole). The bottom plot shows an indication of the jet position: a value approaching $-1$ indicates that the jet is fully above the labium and outside the tube, while a value approaching $+1$ indicates the converse with jet fully below the labium. (Specifically, in terms of the model set out in section 11.8.1, this plot shows the combination $\tanh \left( \dfrac{\eta(w,t)-y_0}{b} \right)$ that appears in equation (6).)
Apart from the octave difference in frequency, the waveforms are recognisably similar. The jet is switching in and out the tube once every cycle, and the acoustic volume flow rate is broadly following the jet position. The pressure waveforms show a pair of main peaks in every cycle, one up and the other down. To estimate the sound radiated from the embouchure hole by these waveforms, we can note that the hole is very small compared to the wavelength, so (from the analysis in section 11.7.1) the radiated sound pressure will be proportional to the rate of change of the volume flow rate through the hole. The resulting sound of the two simulations from which Figs. 18 and 19 were extracted can be heard in Sound 1. Do these sound recognisably like a flute? Not really, it must be admitted. There are two main reasons. First, the main body of each sound is accurately periodic, with none of the “wind noise” that accompanies a real flute note, and none of the subtle modulation (such as vibrato) that a human player would add. Second, although each note here has a starting transient, it is not one that we have any great confidence in because of the limitations of the model. Nevertheless, perhaps these sounds capture something a bit flute-like?
Now, finally, we can look at some results based on the input admittance for the Boehm flute fingered for the note $C_4$, the black dashed curve in Fig. 12 and the one we used earlier when we were looking at the influence of the cork position. Using the same parameter values as the previous simulations, this admittance leads to the behaviour shown in Figs. 20 and 21 in the same format as earlier plots. The general pattern is recognisable based on the earlier discussion, but the model is predicting that the fundamental tone is never produced. The clusters of points for Aeolian tones and for “normal” tones show normalised frequencies close to 2, 3, 4 up to 7, but never 1. As is well known, beginners on the flute often struggle to achieve good tone on the lowest notes on the instrument, and to avoid overblowing to the octave, but it is not supposed to be completely impossible!
We can perhaps guess what the problem is. We can see whereabouts in Fig. 20 we would expect the “normal” playing regime at the fundamental to sit, but in this range of jet speed the model is choosing instead to play Aeolian tones at high frequency. But we already know that this jet-drive model seriously over-estimates the prevalence of Aeolian tones. We can test this idea with a simple numerical experiment. We can take the same input admittance, but only use the first 4 resonances and ignore the higher ones. Without those higher resonances, it should not be possible for the associated Aeolian tones to be produced.
Figures 22 and 23 show the result, keeping all other details the same as in Figs. 20 and 21. By suppressing the possibility of Aeolian tones higher than the 4th harmonic, we have opened a gap in the range of jet speed previously filled with Aeolian tones. Figure 22 shows that this gap is populated in part by extending the range when the octave is played (normalised frequency around 2), but it also now includes a short interval in which the fundamental is played (normalised frequency around 1). Figure 23 confirms that this short interval is indeed associated with values of the delay around a quarter-period, in other words with $n=0$. This model is, of course, artificial: the higher resonances really are there in the admittance. But the experiment is enough to confirm that by suppressing some Aeolian tones that are never in practice heard, it becomes possible for the model to produce this fundamental note.
We have seen the crucial role of the delay, the time taken for disturbances to travel along the jet from slot (or lips) to labium. In a recorder, the geometry is fixed by the instrument maker, and the only way the player can influence this delay is by changing the jet speed. But in a transverse flute, the player has extra degrees of freedom. For the low note we were considering in the last example, rather a long delay is needed for normal tone. In the simulations, the distance has been kept fixed, and the appropriate delay was achieved in a narrow range of rather slow jet speeds.
On a transverse flute, the same delay could be achieved with a faster jet and a correspondingly longer distance: the player can roll the embouchure hole away from the lips, while blowing harder. But this combination of changes will also increase the Reynolds Number, and make it more likely that the jet will become turbulent. Possibly the player can counteract this tendency, to an extent, by careful shaping of the lips to tailor the details of the flow leaving the mouth. The act of rolling the tube so the embouchure hole is less covered will also affect the intonation of the note, by changing the end correction at the hole. All this gives a hint of the subtleties that a flautist must learn to master. But to represent such effects in a simulation lies beyond the reach of the very simple model we are using here. Indeed, it is far from clear that any simple model could be constructed that would capture such effects accurately. This challenging task is a topic for future research.
[1] John W. Coltman, “Sounding mechanism of the flute and organ pipe,” Journal of the Acoustical Society of America 44, 983–992 (1968).
[2] John W. Coltman, “Jet drive mechanisms in edge tones and organ pipes,” Journal of the Acoustical Society of America 60, 725–733 (1976).
[3] Roman Auvray, Augustin Ernoult, Benoît Fabre and Pierre-Yves Lagrée, “Time-domain simulation of flute-like instruments: Comparison of jet-drive and discrete-vortex models”, Journal of the Acoustical Society of America 136, 389—400 (2014)
[4] Antoine Chaigne and Jean Kergomard; “Acoustics of musical instruments”, Springer/ASA press (2013)
[5] D.K. Holger, T. A. Wilson and G. S. Beavers: “Fluid mechanics of the edgetone”, Journal of the Acoustical Society of America 62, 1116—1128 (1977).
[6] Arthur H. Benade; “Fundamentals of Musical Acoustics”, Oxford University Press (1976), reprinted by Dover (1990). See chapter 22.