OPTIMIZATION AND SIMULATION ALGORITHMS FOR THE SOUND DESIGN OF LABIAL ORGAN PIPES

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THE SOUND DESIGN OF LABIAL ORGAN PIPES

Péter Rucz, Fülöp Augusztinovicz, Péter Fiala

Budapest University of Technology and Economics, Dept. of Telecommunications, H-1117 Budapest, Hungary, e-mail: rucz@hit.bme.hu

Judit Angster, Andr´as Mikl´os and Thomas Trommer

Fraunhofer Intistute for Building Physics, Group of Musical Acoustics and Photoacoustics, 70569 Stuttgart, Germany

In the present contribution optimization and simulation methods are investigated in order to attain optimal scaling of labial organ pipes. The goal is to determine the optimal dimensions of the pipe, by which a desired steady state sound spectrum can be achieved. By modifying the geometry of the pipe body, the eigenfrequencies of the acoustic resonator can be tuned in order to amplify or repress given harmonic partials in the pipe sound. Since the dependence of the eigenfrequencies on the pipe dimensions is quite complex, obtaining the optimal scaling parameters is not trivial. To overcome this difficulty two alternative approaches are suggested and examined.

In case of simple pipe forms, such as chimney pipes, the transfer (input admittance) function and the eigenfrequencies of the pipe are calculated by means of a one-dimensional model.

However, when the pipe geometry is irregular (e.g. pipes with tuning slot) constructing a simple pipe model is not trivial. Therefore, numerical (finite/boundary element) methods are applied in order to predict the transfer function. These modeling techniques serve as guidelines in the development of an optimization algorithm. The usefulness and applicability of the methodology are proven by validation measurements performed on pipes built with optimized dimensions.

1. Introduction

Scaling rules of thumb are applied in organ pipe mensuration practice long since. These traditional methods generally lead to satisfactory sounding characteristics. However, in case of special pipe forms they give suboptimal dimensions and leave some potentials of the resonator unexploited.

Chimney and tuning slot organ pipes serve as good examples, since the design of their pipe body influences the resulting sound character to a great extent. The goal of the research presented herein, is to find optimal scaling dimensions of organ pipe resonators for the desired sound spectrum. Despite the fact that properties of the air jet excitation of labial organ pipes also play an important part in the sound character of the pipe, our discussion is limited to pure acoustic phenomena, and thus, effects of the flow and pipe voicing are not considered herein.

In section 2 optimization methods for the sound design of chimney organ pipes are discussed.

The one-dimensional waveguide model of the resonator is also described therein. The input admittance function and the eigenfrequencies of the pipe are calculated by means of this model later

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Main resonator

Pipe foot Chimney

Z_M ZP0

L_P

ZC0

L_C Z_R

Z_S Z_P Z_C

Figure 1.Schematic and one-dimensional waveguide model of a chimney pipe

on. Section 3 presents the validation of the novel scaling technique by comparing simulation results against measurements carried out on chimney pipes built with optimized resonators. Section 4 de- scribes a numerical simulation technique that is capable of calculating transfer properties of tuning slots in the frequency domain. Finally, conclusions and outlook on future work are given in Section 5.

2. Optimization of chimney pipe resonators

Chimney organ pipes are a special family of labial organ pipes, with a pipe body consisting of a main resonator and a chimney part. The chimney provides a small opening at the end of the main resonator, causing the complete pipe body to act as a coupled acoustic resonator, as depicted in Fig. 1.

Chimney pipes can provide a special sound character with certain harmonics being reinforced by the coupled resonator. In the following a novel scaling methodology is presented, which provides optimal parameters for chimney pipe resonators in order to amplify specified harmonic partials.

To describe the behavior of the resonator in the frequency domain, the acoustic input admittance functionY_S is used, which is defined as the ratio of volume velocity and sound pressure at the pipe mouth. As the resonator is an acoustically open system, its eigenfrequencies are coincident with the local maxima of the input admittance function. The first eigenfrequency (or fundamental) is also the musical pitch of the pipe in most cases. The input admittance also influences the strength of harmonic partials: if a partial overlaps with an acoustic resonant or anti-resonant frequency it will be amplified or repressed, respectively. Hence, by the appropriate selection of pipe dimensions the eigenfrequencies can be tuned and the spectrum of the pipe reshaped.

To be able to calculate the input admittance function and eigenfrequencies of the resonator, the one-dimensional waveguide model is used, as shown in Fig. 1. This model consists of distributed (waveguide) and concentrated parameter (radiation impedance) elements and is valid under the cut-on frequency of transversal modes.

The radiation impedance at the open endZ_R can be calculated using the formula deduced by Levine and Schwinger [1]. The radiation impedance at the mouth opening in the low frequency range can be modeled by a mass of air with the volume of ∆L_MW_MH_M, withW_M and H_M denoting the width and height of the mouth and ∆L_M representing the end correction term. The latter can be approximated by the formula given by Fletcher [2] as

∆L_M = 2.3·R²_P

√W_MH_M and Z_M = iωρ₀∆L_M

S_P , (1)

whereiis the imaginary unit,ωis the angular frequncy,ρ₀represents the average density of air, and R_PandS_Pdenote the radius and cross-sectional area of the main resonator, respectively.

The input impedance of the complete system Z_S can be written with expressing the input impedance of the chimneyZ_C first. The chimney is terminated by its radiation impedance, whereas the main resonator is terminated by the input impedance of the chimney

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Z_C =Z_C0Z_R+ iZ_C0tan(kL_C)

Z_C0+ iZ_Rtan(kL_C) and Z_S=Z_M+Z_P0Z_C+ iZ_P0tan(kL_P)

Z_P0+ iZ_Ctan(kL_P), (2) withk representing the acoustic wave number,L_CandL_P,Z_C0andZ_P0denoting the physical lengths and acoustic plane wave impedances of the chimney and the main resonator, respectively. Viscous and thermal losses can also be taken into account, using complex valued wave impedances wave numbers.

In the calculations presented herein losses were incorporated using the theory described by Zwikker in [3].

The input admittance functionY_S(f) = 1/Z_S(f)depends on six scaling parameters (the length and diameter of the main resonator and chimney parts and the width and height of the mouth). Since the speed of sound and viscothermal losses are dependent on the air temperature, the optimization is always performed with the tuning temperature given.

In the following this set of parameters is denoted byP = P^∗ ∪ P0 with P^∗ representing the unknown variables (optimization targets) andP₀ denoting the known parameters (fixed values). In the next paragraphs a specialized and a more general approach is discussed for differentP^∗sets.

2.1 Optimizing the lengths

In this section one of the simplest cases is examined, namely when the length of the chimney and the main resonator are unknown, while other variables are kept constant, i.e. P^∗ = {L_P, L_C}.

The goal of the optimization is to tune the first eigenfrequency to the desired fundamental frequency f₁and to tune an other eigenfrequency to be coincident with then^th partial,f_n =nf₁.

Making use of the fact that the one-dimensional homogeneous Helmholtz equation holds inside the pipe, the reflection coefficients at the pipe mouth, the joint point of the chimney and the open end can be expressed by impedance terms. By means of these coefficients phase constraints of acoustic resonance can be written. This condition is written at the fundamental frequency (k = k₁) for the main resonator length as

2k₁L_P= arg

Z_M−Z_P0

Z_M+Z_P0 · Z_C−Z_P0 Z_C+Z_P0

(3) and similarly at the frequency of the n^th harmonic (k =k_n) for the chimney length

2k_nL_C= arg

Z_P⁰ −Z_C0

Z_P⁰ +Z_C0 ·Z_R−Z_C0 Z_R+Z_C0

, (4)

withZ_P⁰ denoting the input impedance of the main resonator terminated by the radiation impedance of the mouth at one side.

Since the length of the chimney must be known to solve equation (3) for the main resonator length and vice-versa, a simple iteration can be constructed by making an initial guess on the chimney length (L⁽⁰⁾_C ), calculating L_P from (3) and then updatingL_C andL_P using (4) and (3) subsequently, in an alternating manner. To find the initial chimney length, the chimney can be taken as a half- wavelength resonator radiating into free space at both sides on the frequencyf_n.

This simple iteration proves to be efficient in determining the lengths of the main resonator and the chimney. An example result of the iteration process is displayed in Table 1. The fixed parameters were set as D_P = 79.0 mm, D_C = 28.7 mm, W_M = 60.0 mm, H_M = 25.7 mmand T = 293 K.

The goal frequency wasf₁ = 140.0 Hzandnwas set to5. Fig. 2 displays the original and optimized input admittance curves and the normalized modal waveforms of the example pipe. In the optimized case, the 4^th eigenfrequency was tuned to overlap with the fifth harmonic, and hence its amplification can be expected. The modal waveforms clearly show the effect of the chimney, especially at the3^rd and4^th modes, which are relatively close in frequency.

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Iteration LC[mm] |∆L_C| [mm] LP[mm] |∆L_P| [mm]

0. 213.11 — 600.41 —

1. 245.18 32.07 588.27 12.14

2. 249.66 4.48 586.77 1.50

3. 250.36 0.70 586.54 0.23

4. 250.47 0.11 586.51 0.03

5. 250.49 0.02 586.51 <0.005

Table 1.Results of the simple iteration process for an example chimney pipe. After5iterations the change of the lengths was under0.01 mm.

Figure 2. Effect of the optimization on the input admittance (left) and the resulting eigenmodes (right) 2.2 Optimization using a cost function

Besides optimizing the chimney and main resonator lengths, theoretically, optimizing all geometry parameters is possible. However, the practically most relevant case is when the diameter of the chimney is to be determined together with the lengths. When optimizing more than two parameters, an iterative process can not be defined straightforwardly, hence an alternative approach is needed.

Since the dependence of eigenfrequencies on the pipe dimensions is quite complex, applying a general optimization technique is more feasible than developing a heuristic method. As the optimization goals are well-defined, a cost function can be constructed that measures the distance of the actual configuration from the ideal one in a special metric. Once the cost function is set up, a global minimization technique can be applied to find the optimal parameters. A general method for function minimization is the simplex algorithm first published by Nelder and Mead [4]. This process shows rapid convergence for a wide family of functions, as it is discussed in [5].

To construct the cost function, the deviations of the desired and real eigenfrequencies are evaluated. For the fundamental the first eigenfrequency, and for then^th harmonic, the nearest eigenfrequency is taken into account. For example, a simple quadratic cost function reads as

C(P) = w₁(f₁−f₁^∗(P))²+w_n(f_n−f_n^∗(P))², (5) wheref₁^∗(P)is the first eigenfrequency dependent on the scaling parameters, andf_n^∗(P)is the eigenfrequency closest to f_n, whereas w₁ and w_n are positive weights. Trivially, the cost function (5) gives zero result when both frequency criteria are satisfied. This simple quadratic function has shown good performance and stability in the test process. The optimal ratio of weights was found to be w₁/w_n≈10.

Compared to the simple iteration process, optimizing by means of the Nelder-Mead technique requires remarkably more computational effort, since the input admittance function needs to be calculated for the whole frequency domain of interest at each time when the cost is to be evaluated. To find a good starting point for the algorithm, which speeds up the convergence, the iteration process described above can be applied using a fixed chimney diameter.

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Table 2.Dimensions of experimental chimney pipes

Parameter Value [mm] Common parameters Value [mm]

Reference A2 B4 C5 Resonator diameter 79.0

Selected partial — 3^rd 4^th 5^th Mouth width 60.0

Chimney length 180.0 193.3 99.1 56.0 Mouth height 19.0

Chimney diameter 19.0 10.7 29.4 29.3 Fundamental freq. 131 Hz Resonator length 600.0 564.0 742.9 852.5 Tuning temperature 23^◦C

3. Results of chimney pipe optimization

To validate the novel scaling method, chimney pipes were built with the dimensions given by cost function-based optimization method. A reference pipe with traditional scaling were also manufactured. Sound spectra of the pipes were measured using the same pressure in the pipe foot in an anechoic room. All pipes were tuned to the same fundamental frequency of131 Hz to facilitate subjective comparison of sounds. The dimensions of the pipes are displayed in Table 2. It is important to note, that the pipes were not voiced, only tuned to the given pitch.

Fig. 3 shows the comparison of measured sound spectra and calculated input admittance for the four test pipes. As it can be seen, the eigenfrequencies of the reference pipe are not overlapping with any of the partials, which implies weak harmoncis and a strongly fundamental pipe sound.

Pipe A2, which was optimized to enhance the 3^rd partial, provides14 dBamplification for the third. All other harmonics are amplified slightly at the same time, making the pipe sound louder by a total of3 dBcompared to the reference pipe. Pipe B4, optimized for the 4^th partial, amplifies it by as much as30 dB, while repressing the 3^rd and 5^th by more than10 dB. Pipe C5, which was optimized for the 5^th, provides17 dBamplification, also enhancing the 2^ndnearly as much, while repressing the 3^rdmore than20 dB. Interestingly, optimal amplification of the 3^rdwas achieved using a very thin and slightly longer chimney than the original one, while the 4^th and 5^th was amplified the most by wider and shorter chimneys.

For all pipes very good match of the calculated input admittance and the base line of the measured sound spectra is observed. Locations of eigenfrequencies can clearly be seen in the spectra as well. Peaks of the calculated input admittance function are usually sharper than the eigenfrequency peaks in the measured spectra, which means that losses are underestimated by the model.

As it is seen, the desired amplifications were achieved by the optimized design, nevertheless, the amount of amplification were remarkably different in the three cases. Also, the optimiziation has significatntly affected the amplitudes of other partials, mostly for pipe C5. These effects are not yet taken into account in the optimization process, but can be predicted using the original input admittance curve as a reference. Optimization for more than one partials is also possible by a simple extension of the cost function. At the same time, pipe voicing (intonation) also has a great effect on the strength of the partials and can reinforce the amplifications. To model these effects a coupled fluid flow and acoustic model would be neeeded, which is out of the scope of this paper.

4. Simulation of tuning slots

Tuning slots are tuning devices most often used on open straight metal pipes. By opening the resonator with a slot, the geometry becomes irregular and the one-dimensional pipe model has to be extended to treat this discontinuity. Several models were developed and applied to model the similar case of woodwind instrument tone holes with success. However, tuning slots differ from them by their non-circular shape and the thin pipe walls. Due to these dissimilarities tone hole models can only be applied for tuning slots as a rough approximation. Thin walls prevents the separate analysis of the interior and exterior sound fields near the slot, which makes the analytic treatment of the tuning slot cumbersome.

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0 200 400 600 800 1000 1200 0

10 20 30 40 50 60 70 80 90 100 110

Frequency [Hz]

Sound pressure measured at mouth [dBSPL] 107 70 77 60 69 56 41

0 200 400 600 800 1000 1200

−40

−20 0 20 40 60

Input admittance [dB re. 1/ZP0]

SPL measured at mouth Calculated admittance

(a) Reference pipe (original scaling)

0 200 400 600 800 1000 1200

0 10 20 30 40 50 60 70 80 90 100 110

Frequency [Hz]

Sound pressure measured at mouth [dBSPL] +2 +3 +14 +6 +7 +4 +6

0 200 400 600 800 1000 1200

−40

−20 0 20 40 60

(b) Pipe A2 (optimized for the 3^rd)

0 200 400 600 800 1000 1200

0 10 20 30 40 50 60 70 80 90 100 110

Frequency [Hz]

Sound pressure measured at mouth [dBSPL] +1 +7 −14 +30 −13 +9 −6

0 200 400 600 800 1000 1200

−40

−20 0 20 40 60

(c) Pipe B4 (optimized for the 4^th)

0 200 400 600 800 1000 1200

0 10 20 30 40 50 60 70 80 90 100 110

Frequency [Hz]

Sound pressure measured at mouth [dBSPL] −0 +16 −21 +12 +17 −4 +14

0 200 400 600 800 1000 1200

−40

−20 0 20 40 60

(d) Pipe C5 (optimized for the 5^th)

Figure 3.Measured spectra and calculated input admittance of experimental chimney pipes. Absolute amplitudes (reference pipe) and amplifications (optimized pipes) indBSPLof partials are displayed.

Recently, Lefebvre & Scavone [6] proposed a simulation method to investigate properties of woodwind tone holes with improved accuracy. Their technique can be interpreted to tuning slot simulation, as shown in Fig. 4(a). To emulate free field radiation conditions near the tuning slot opening, the Perfectly Matched Layer (PML) technique was used, implemented in the unsplit frequency domain formulation as proposed by Berm´udez et al. in [7]. Making use of the symmetricity of the arrangement, only one quarterth of the geometry has to be modeled, which allows better spatial reso- lution with a lower number of degrees of freedom.

To be able to identify the transfer matrix (TM) of the tuning slot separate from the pipe, the slot is placed into a symmetrical cylindrical section with2L_cyllength. The transfer matrixTis defined as

p_in Z₀v_in

=

T₁₁ T₁₂ T₂₁ T₂₂

p_out Z₀v_out

, (6)

withp_in, v_in andp_out, v_out denoting the sound pressure and particle velocity sampled at the input and output planes, respectively. The TM of the objectT_objand the slotT_slotreads as

T_obj =T_cylT_slotT_cyl and T_slot =T⁻¹_cylT_objT⁻¹_cyl, (7) where T_cyl denotes the TM of the cylindrical section of length L_cyl, which can easily be evaluated analytically.L_cyl has to be large enough to make the irregularities of the sound field introduced by the tuning slot negligible at the output and input planes. To be able to determine the four values inT_slot the simulation is performed using two different sets of boundary conditions, as it is described in [6].

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Tuning slot

Input plane

Domain extension (PML on surface) Symmetry plane A (x= 0) Symmetry plane B (z=L_cyl)

x z y

(a) Tuning slot simulation setup

(b) Simulation result,ka= 0.75

(c) Simulation result,ka= 0.88

Figure 4. Simulation of the tuning slot, arrangement and example results

Once the TM is calculated, an equivalent symmetric T-circuit of the slot can be constructed. The elements of the T-circuit are usally reinterpreted as inner (shunt) and outer (series) length corrections to the tube.

To validate the method, a woodwind tonehole with the same dimensions that Keefe analyzed [8]

was simulated. The only difference was that a rectangular slot with the same effective diameter was taken instead of a circular one. Fig. 5(a) compares the evaluated shunt length correction compared to the analysis of Keefe [8] and simulation of Lefebvre & Scavone [6]. As it is seen, the results show very good match on low frequencies, slight deviations are experienced in the higher frequency range.

The tuning slot model was inserted into a one-dimensional pipe model similar to the one shown for chimney pipes in Fig. 1. As it can be seen in Fig. 5(b) the calculated admittance and the measured spectrum show a good fit, which means that the presented method is capable of calculating the effect of the tuning slot accurately.

These results can serve as a basis for the development of an optimal scaling method for tuning slot organ pipes, and also can be used as guidelines for setting up an accurate analytical model for calculating tuning slots.

5. Summary and outlook

The first part of this paper presented an optimization method for the sound design of chimney organ pipes. The basic idea of the optimization was to tune the eigenfrequencies of the resonator to overlap with given harmonic partials, amplifying them in the pipe sound. A simple iteration method and a global optimization technique based on the minimization of a cost function was described. The effectiveness of the novel scaling methodology was shown by measurements performed on experimental chimney pipes built with optimized dimensions. The method could be further developed by better approximation of viscothermal losses and taking into account new terms in the cost function.

In the second part, numerical simulation of tuning slots was discussed. Using the PML extension of the finite element technique, the transfer matrix of the slot is determined. From the TM,

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10⁻² 10⁻¹ 10⁰ 15

20 25 30 35 40 45 50 55

ka [−]

Open shunt length correction ts [mm]

Analytic (Dalmont2002a) Our FEM simulation Lefebvre 2010a FEM

(a) Comparison result for the reference tonehole

0 500 1000 1500

0 10 20 30 40 50 60 70 80 90 100 110

Frequency [Hz]

Sound pressure measured at mouth [dBSPL]

0 500 1000 1500

−30

−20

−10 0 10 20 30 40 50 60 70

SPL measured at mouth Simulated admittance

(b) Input admittance calculation using the tuning slot model

Figure 5.Results of tuning slot modeling for the reference tone hole and a real pipe

an equivalent circuit is created, which can be inserted into the one-dimensional model of the pipe.

Validation simulations have shown good agreement, proving that the presented technique can model the tuning slot accurately. Finally, an optimization method can be implemented later, based on the presented simulation results.

Acknowledgments

Contributions of Societ´e de Construction d’Orgues Muhleisen (experimental pipes) and Thomas Barthold organ builder (participation in laboratory experiments) and the financial support of the research by the European Commission (Grant Agreement Ref# 222104) is gratefully acknowledged.

P. Rucz acknowledges the support of the Hungarian research grant T ´AMOP - 4.2.2.B-10/1–2010- 0009.

REFERENCES

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2 N. H. Fletcher and T. D. Rossing. The physics of musical instruments, page 475. Springer, 1991.

3 C. Zwikker and C. W. Kosten. Sound Abosrbing Materials, pages 25–40. Elsevier, 1949.

4 J. A. Nelder and R. Mead. A simplex method for function minimalization.The Computer Journal, 7:308–313, 1965.

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6 A. Lefebvre and G. P. Scavone. Refinements to the model of a single woodwind instrument tonehole. InProceedings of 20th International Symposium on Music Acoustics (Associated Meeting of the International Congress on Acoustics), Sydney and Katoomba, Australia, August 2010.

7 A. Berm´udez, L. Hervella-Nieto, A. Prieto, and R. Rodr´ıguez. An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. Journal of Computational Physics, 223:469–488, 2007.

8 D. H. Keefe. Theory of the single woodwind tone hole. Journal of the Acoustical Society of America, 72(3):676–687, 1982.