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Wind Instruments: Variable Cross Section and Toneholes

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Acoustics of Musical Instruments

Part of the book series: Modern Acoustics and Signal Processing ((MASP))

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Abstract

Wind instruments are studied in two steps. In the first step, one-dimensional models are presented. The horn equation is equivalent to the heterogeneous string studied in Chap. 3 It can either be used for tubes with discontinuities, such as chimney pipes, flutes, or trumpet mouthpieces, or for tubes with a continuous change in cross section, such as conical instruments or bells of brass instruments. The natural modes are calculated with some approximations for several basic shapes of wind instruments. The geometry has an important and complex effect on eigenfrequencies and on amplitudes of input impedance peaks. The role of dissipation is more simple, because it can be averaged over the length of the instrument. However, it depends on the radius, and this yields non-proportional damping. In the second step, it is investigated how to reduce three-dimensional geometric elements, such as toneholes or bends, to lumped elements. For this purpose, the definition of duct modes is given, and the mode-matching method is presented. The basis is a general formulation of the junction of several waveguides at low frequencies. Useful formulas are given for many elements of this kind. The chapter ends with the use of the theory of periodic media in order to analyze the tonehole lattice, with the explanation of important features that distinguish baroque from modern instruments. Attempt is made to give the most recent formulas of use for designing the resonators of wind instruments.

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Notes

  1. 1.

    This simple result follows the fact that, mathematically, the Fourier transform of Eq. (7.6) is a Sturm-Liouville equation, i.e., an ordinary differential second-order equation with variable coefficients. Making the change of variable p⟶pR to obtain the transform of (7.8) is equivalent to make canonical the Sturm–Liouville equation by eliminating the term containing the derivative of order 1 with respect to x. This is a general result.

  2. 2.

    An alternative method is the optimization in the frequency domain [13, 39, 40]: it is sought to minimize the error between the impedance of a target shape and an estimate thereof, based on a number of geometrical parameters.

  3. 3.

    For the sake of simplicity, radiation will be ignored, thus results are limited to not too high frequencies. Dispersion is ignored as well. One difficulty is that the modes of the problem with losses are not quite orthogonal, because damping is varying with the radius . This can be avoided by applying the residue calculus (see Chap. 4, Sect. 4.6.2) to the closed-form expression, which exists, but this leads to heavier calculations.

  4. 4.

    We will assume that there is no influence coming from boundary layers (\(Z_{v} = j\omega \rho /S(x)\) on both sides), although this is not strictly true.

  5. 5.

    Remember that strictly speaking, as the peaks are finite, all modes have to be taken into account at a given resonance frequency. However, we will consider that dissipation is low enough in order to consider in the infinite series only the mode whose eigenfrequency is the considered resonance frequency.

  6. 6.

    More accurately, only \(\varLambda _{n}\) occurs far from the resonances, and \(\varLambda _{n}^{{\prime}}\) occurs near the resonances.

  7. 7.

    The extrema of these two curves are different, as those of the input impedance of Cylinder 1 depend on dissipation, while the modulation factor \(\varLambda _{n}\) is “reactive,” independent of dissipation. The result cannot be the product of the input impedances of two cylinders: we can never have the product of two damping factors . The simple sentence “the first cylinder impedance modulates that of the second” is therefore improper.

  8. 8.

    An interest of the above-discussed problem is that it can be treated as a limiting case of the mass-type or compliance-type termination, while maintaining simple boundary conditions (zero or infinite impedance). The idea has already been touched on when considering the radiation impedance of a pipe as a length correction (see Chap. 4). This can be advantageous for the sake of simple calculations.

  9. 9.

    We can presume that enhancing the impedance peaks also enhances the harmonics of lower notes whose frequency is close to the mouthpiece resonance: this is a delicate issue, because resonance inharmonicity explains why the frequencies of the harmonics rarely coincide with those of the peaks, and consequently why their impedance can be small (with a large imaginary part compared to the real part).

  10. 10.

    Notice that if the cup is cylindrical, Eq. (7.32) yields a pressure uniform within the cup if k n ℓ 1 < < 1. Moreover notice that we choose to consider here the pressure as dimensionless; therefore the flow rate has the dimension of an admittance.

  11. 11.

    Remember that the dimension of an acoustic mass is kg m−4.

  12. 12.

    If Pipe 2 had a large section, the ratio M 2∕M 1 could be neglected with respect to unity, and this would lead to V∕2V 0. In this ratio, the value of the numerator comes from the fact that at zero frequency, there is only potential energy in the entire instrument, since the flow is zero at the input, and therefore everywhere! Otherwise the value of the denominator arises from the fact that at the mouthpiece resonance, we have equipartition of energy (hence the factor 2) in the mouthpiece, and very low energy in the main part of the instrument.

  13. 13.

    Considering this source as the one that produces the sound is adopting a linear point of view, which is wrong. Actually, it is not the player who sets this source because we are dealing with a closed loop system, which will require knowledge of another equation. The important thing for reed instrument is to know the ratio P∕U s , in particular at the entrance.

  14. 14.

    Of course the flow oscillates and thus goes in and out of the pipe, but we define it as the volume flow rate through the pipe surface, the vector \(d\overrightarrow{S}\) being here oriented to the right.

  15. 15.

    The discontinuity between the mouthpiece hole and the pipe requires adding a series impedance of acoustic-mass type, denoted by \(Z_{d} = j\omega \rho \varDelta \ell_{dm}/S_{m}\) at the input impedance of the pipe Z e . This will be explained in Sect. 7.6.3.2. The order of magnitude of the length Δ ℓ dm is similar to that of Δ ℓ m . Since in the mouthpiece hole there is flow rate continuity (the fluid being considered to be incompressible), we can simply add this acoustic mass to the radiation acoustic mass, and replace in the radiation impedance Z m the length Δ ℓ m by the equivalent total length of the mouthpiece hole, ℓ m , whose order of magnitude is 1. 5 times the radius R m .

  16. 16.

    Why the − sign? In order to find the phase difference of the radiated pressures, we must consider the flow rate “outgoing” from the instruments (see fourth Part).

  17. 17.

    Actually it can be shown that the ratio increases with frequency, because, if \(jk_{n} = s_{n}/c\):

    $$\displaystyle{ - \frac{U(\ell)} {U(0)} \simeq \frac{1 + k_{n}^{2}\ell_{m}^{2}(S/S_{m})^{2}} {1 + k_{n}^{2}\varDelta \ell^{2}}. }$$

    We have seen that the mouthpiece-hole length correction is significantly larger than that of the end ℓ.

  18. 18.

    If the frequency tends to 0 in (7.83), we find, (after some calculations): \(A = D = 1\), B = jω L a , C = jωC a , where L a and C a are the mass and the acoustic compliance found in (7.16), respectively. We find that the cross section of the cylinder equivalent to a truncated cone of length ℓ is: for the acoustic mass \(S =\pi R_{1}R_{2}\); and for the acoustic compliance: \(S =\pi (R_{1}^{2} + R_{1}R_{2} + R_{2}^{2})/3.\) This type of result is useful, e.g., for sizing a conical tonehole , by calculating the equivalent cylindrical hole.

  19. 19.

    In the case of a cylinder or a cone, these functions can be expressed using trigonometric functions.

  20. 20.

    Moreover let us not forget that is only an approximation, because Eq. (7.114) does not depend on the termination of the bell, and the geometrical shape is not an exact power law.

  21. 21.

    One can say that everything is for the best: the mouthpiece, used to improve the emission of quite high notes, offsets the effect of the bell. A more human point of view is that makers have succeeded after years of trials and errors to find well-adapted shapes for the mouthpiece and the bell. However, Bouasse [15, p. 308] noted that in any case, at higher frequencies, the ratio between successive harmonics for the two basic series (odd or complete) tends towards the same values!

  22. 22.

    We can improve the convergence by dividing the instrument into small cone portions, but then, because of the variation of boundary layers thicknesses with the radius, analytical solutions exist only when calculating the visco-thermal effects for a fixed equivalent radius [see Eq. (7.106)]. This method has been used for the calculation in Fig. 7.12.

  23. 23.

    Changing U i in \(\widetilde{U}_{i}\) is equivalent to consider the flow rate at the output along the axis opposite to that of the input, and thus to consider the two ends of a two-port in a symmetrical way. Conversely the orientation chosen for a transfer matrix distinguishes input and output, and is obviously essential in order to juxtapose a sequence of cascaded two-ports.

  24. 24.

    Note: for a segment without losses, it is easy to show that the coefficients A and D are real numbers, and B and C purely imaginary. Furthermore the coefficients of the admittance matrix are all purely imaginary numbers.

  25. 25.

    To take visco-thermal effects (losses and dispersion) into account, we replace the delays δ(t − T) by digital filters, which represent these effects in an approximate way.

  26. 26.

    We should pay attention to notations: we use the symbol K for the reflection coefficient, while Smith [78] uses k (he uses R for the characteristic impedance , inversely proportional to the cross section area).

  27. 27.

    One solution is the hot wire anemometer, which, however, implies that there is a mean flow, or better the Microflown probe that was developed to measure low particle velocity. More complex methods, such as the Laser Doppler Anemometry can also be considered.

  28. 28.

    On the one hand, at lower frequencies, the gradient is small and, on the other hand, at higher frequencies, the gradient can vanish when the distance is equal to one half wavelength. In both cases, accuracy is poor.

  29. 29.

    Physically, the modes can only decrease on both sides of a discontinuity or of a source, the existence of growing modes is thus linked to a choice of orientation of the x axis. Such modes are typically found near the open end of a tube.

  30. 30.

    The superposition of increasing and decreasing modes, which occurs, for example, between two close discontinuities, can transport energy if they are not in phase, which is equivalent to the tunneling effect in quantum mechanics.

  31. 31.

    This mass is added to that due to the planar mode, which is ρ ℓ∕S for a small tube of length ℓ and section S, and we will see that its order of magnitude is the mass of air in a pipe whose cross section is that of the diaphragm and the length is that of the radius.

  32. 32.

    If the guides have a common symmetry, the double infinity can be reduced to a simple infinite: in the circular case, if both guides are concentric, we have a radial symmetry, and the azimuth \(\theta\) is not relevant. Similarly for the case of a 2D rectangular geometry, if the two guides have a common transverse dimension, the discontinuity cannot create any higher order modes in the common dimension.

  33. 33.

    The approximation of the plane piston can be generalized. We write in (7.159):

    $$\displaystyle{ \mathbb{Y}_{2}^{{\prime}} + ^{t}\mathbb{F}^{{\prime}}\mathbb{Y}_{ 1}^{{\prime}}\mathbb{F}^{{\prime}} = \mathbb{Y}_{ 2}^{{\prime}}\boldsymbol{(}\mathbb{I}\boldsymbol{ +} \mathbb{Q}\boldsymbol{)}\text{ where }\mathbb{Q} = \mathbb{Z}_{ 2}{}^{{\prime}}{}^{t}\mathbb{F}^{{\prime}}\mathbb{Y}_{ 1}^{{\prime}}\mathbb{F}^{{\prime}}\text{,} }$$

    and we use the Neumann series expansion, valid if the matrix norm of \(\mathbb{Q}\) is less than unity \(\boldsymbol{(}\mathbb{I}\boldsymbol{ +} \mathbb{Q}\boldsymbol{)}^{-1} = \mathbb{I}\boldsymbol{ -} \mathbb{Q}\boldsymbol{ +}\ \mathbb{Q}^{2}\boldsymbol{ -} \mathbb{Q}^{3}\boldsymbol{+\ldots }\) The characteristic impedances are inversely proportional to the sections, so \(\mathbb{Q}\) is proportional to S 1∕S 2: intuitively, we therefore understand that the convergence is even better when this ratio is small. The plane piston approximation is the zeroth order of the expansion.

  34. 34.

    The value of 0. 25 for a zero-thickness diaphragm instead of 0. 26164 for a discontinuity reveals the (weak) interaction effect between the input and output of the diaphragm. This interaction is that of decreasing and increasing evanescent modes in the diaphragm, when the latter has a very small thickness.

  35. 35.

    When the radius R 2 tends to infinity, the discontinuity mass is the same as that of the low frequency radiation, provided that the pipe radiates in an infinite screen. The case of tube radiation is covered in Chap. 12.

  36. 36.

    We can also mention that we could have chosen to find a length correction for guide 2, assuming that the input impedance of guide 1 is not too large. If we label it Δ ℓ (2), where obviously \(\varDelta \ell^{(2)} =\varDelta \ell S_{2}/S_{1},\) we see that it has a more complicated relationship with the radii, particularly for large discontinuities, where Δ ℓ is simply proportional to the radius of the small guide.

  37. 37.

    A simple way to summarize the discussion is as follows: a short length of pipe corresponds to the effects of a mass and a spring. Assimilating a mass added to a length correction therefore yields the addition of a spring that does not exist.

  38. 38.

    We can refine the effect of the compliance and mass using continued fraction expansions (Sect. 7.5.2.1). Let us take the example of a cylindrical cavity of length ℓ and of section S. Its resonance frequency is given by \(\rho cS^{-1}\cot k\ell =\omega M_{a}\). We can always write \(M_{a} =\rho L_{a}/S_{\mathrm{int}}\), where L a is a length. Expanding the function \(\cot k\ell\) to the third order, the approximate solution is written:

    $$\displaystyle{ k\ell = 1/\sqrt{\frac{SL_{a } } {S_{\mathrm{int}}\ell} + \frac{1} {3}}. }$$
  39. 39.

    In the case where we can use a Green’s function expanded in the modes of the cavity volume V, we can get an idea of the method. The shape of this function is given by (4.52), replacing the simple sum by a triple sum. The expansion can be done on the increasing frequencies: the term of lowest order is in brackets, corresponding to the uniform mode of the cavity, which explains the simplicity of the first term of Eq. (7.169).

  40. 40.

    This simplification can be applied to a cavity of zero volume, but also to an incompressible fluid. The various elements of the new mass matrix can be calculated using the incompressible fluid approximation simply verifying the Laplace equation . In particular for two-dimensional geometry, this allows the conformal mapping technique to be used in different termination cases to obtain the various terms of the matrix [17, 49].

  41. 41.

    The calculation is similar to that of a closed pipe [See. Eq. (7.184)]. Here, we focus on the resonance frequencies, and therefore ignore the real part of the radiation impedance.

  42. 42.

    Similar phenomena are encountered in connection with speech production, with the effect of the nasal tract: for producing the nasal vowels, the velum is lowered to connect the nasal and vocal tracts, thereby producing rejections in the sound radiated from the mouth.

  43. 43.

    They are made by not only closing the holes upstream of the first open hole, but also by closing one or two holes downstream. The case of the recorder is typical.

  44. 44.

    Theobald Boehm (1794–1881) redesigned the flute , now often called the Boehm flute, and his invention had repercussions for many “woodwind instruments,” including the saxophone , invented by Adolphe Sax in 1846.

  45. 45.

    This reasoning is based on the first approximation of the model of a hole, which assumes equivalence with the effect of cutting the pipe where it is located, and ignores the effect of closed holes.

  46. 46.

    One might just as well choose either another symmetrical cell, of length 2ℓ, ended by two half-holes or an asymmetrical cell.

  47. 47.

    To multiply the transfer matrices, we use the same orientation for both ends of the two-port, u n and u n+1; we must be careful when using the results for a tonehole, obtained with the other orientation convention (see Fig. 7.25).

  48. 48.

    However, these notes are not used because they are too isolated. Raising the pitch by a tone, for example, requires some complicated movements, and furthermore a register hole would still be required to completely remove the fundamental frequencies from the sound. We will discuss this issue in the third Part.

  49. 49.

    In the latter case the attenuation is called “reactive” attenuation, which does not correspond to dissipation. This is the type used in automotive mufflers, where the aim is essentially to prevent the source from emitting power by presenting to it a purely imaginary impedance, rather than to dissipate energy.

  50. 50.

    In addition there are bore deviations from perfectly cylindrical or conical shapes, and the effect of the closed holes.

  51. 51.

    A general theoretical result is known for random lattices, that can be obtained by randomizing, for example, the diameters of the holes or their spacing. It has been shown that when the number of cells (i.e., of holes) increases to infinity, there are no longer any pass bands, all frequencies being exponentially attenuated. But in practice this result does not apply for wind instruments, because the deviation from periodicity (the “disorder”) is too low.

  52. 52.

    Some horn shapes, which are duals of the previous ones, also give rise to an analytical solution [71], such as \((1/R)^{{\prime\prime}}/(1/R) =\) constant, for which the flow rate is easily calculated (see Sect. 7.2.2).

  53. 53.

    Strictly speaking, the existence of a cutoff frequency between propagating waves and evanescent waves is a global limit property of a medium with uniform characteristics, like an exponential, infinite, and lossless horn. The only thing that is easy to show in the case of a horn of arbitrary shape, is that for the horn alone, there will be no resonance below the lowest local cutoff, corresponding to the horn entrance [45]. But this leads to nothing more, for a Bessel horn, than Eq. (7.117)!

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    Jean Kergomard

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Kergomard, J. (2016). Wind Instruments: Variable Cross Section and Toneholes. In: Acoustics of Musical Instruments. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3679-3_7

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