comparison between the radiated sound produced ...

Bagpipe model vs. experiments

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COMPARISON BETWEEN THE RADIATED SOUND PRODUCED BY A SCOTTISH BORDER BAGPIPE CHANTER, AND THE SOLUTION OF A SIMPLE PHYSICAL MODEL USING HARMBAL Sandra Carral Institute of Musical Acoustics University of Music and Performing Arts, Vienna [email protected] Abstract In a previous study, the reed of a Scottish Border bagpipe chanter was stored in different relative humidity conditions. Its physical parameters were measured, as well as the radiated sound produced by the chanter and reed system. It was found that the relative humidity at which the reed was stored influences the reed parameters, as well as the pitch and spectral centroid of the radiated sound produced by the instrument. In this paper, the reed parameters that were measured earlier are introduced into a simple physical model of the chanter and reed system, and solved with the Harmonic Balance Method. The reed is modelled as a simple harmonic oscillator, the impedance curve of the bore is calculated using a transmission line model, and the pressure difference across the mouthpiece is related to the volume flow following Bernoulli’s equation. By solving this model, a program called harmbal calculates the playing frequency and the internal spectrum of the sound inside the mouthpiece. The former is compared to the pitch and the latter is convolved with the transfer function of the chanter to obtain the radiated spectrum, from which the spectral centroid is calculated, and compared to what was obtained experimentally.

INTRODUCTION [Carral and Campbell, 2005] found that the relative humidity at which the bagpipe chanter reed is stored affects its physical parameters, as well as the sound the instrument produces while being played with an artificial blowing machine. This paper is aimed at investigating to what extent a simple physical model of the chanter and reed system is able to mimic the results obtained in [Carral and Campbell, 2005], and raises questions on where the limitations of the model might be. Ultimately, having a more realistic physical model that behaves in the same way as a real instrument does, might bring us closer to a better understanding of the sound production process in woodwinds, as well as which and to what extent the variations on the reed parameters are responsible for the change in the produced sound of the instrument. The Harmonic Balance Method was first applied to find the periodic solutions of self-sustained musical instruments by [Gilbert et al., 1989]. [Farner et al., 2006] have developed a computer program called harmbal, that applies the Harmonic Balance Method to solve problems where a linear exciter is non linearly coupled to a linear resonator. This program has been so far used to solve clarinet models: [Fritz et al., 2003] used harmbal to study how the vocal tract of the musician affects the playing frequency of the clarinet. [Fritz et al., 2004] have used harmbal to study the influence of the reed mass and damping in the spectrum of the clarinet. [Vergez and


ISMA 2007 mouthpiece lip p

−h

U bore

pm 0 lip y

Figure 1: Schematic of a clarinet mouthpiece Lizée, 2005] presented a method to find the stability in the solutions determined by harmbal. In this paper, a simple physical model of a Scottish Border bagpipe chanter and reed is proposed, where the reed parameters are those measured in [Carral and Campbell, 2005]. PHYSICAL MODEL The chanter and reed system are modelled as a self-sustained oscillator with a linear exciter (the reed) that is coupled non linearly to a linear resonator (the air column). This section presents the equations that were used to model these three components. Reed [Almeida et al., 2002] have provided evidence that the two blades of a double reed have symmetric displacement. This means that the motion of only one blade needs to be modelled as a simple harmonic oscillator: d2 y dy 1 2 + g + ω y = − ∆P r r dt2 dt µr

(1)

where y is the displacement of the reed, gr its damping factor, ωr its resonance frequency, µr its mass per unit area, ∆P = pm − p, pm is the pressure inside the mouth or wind cap and p is the pressure inside the reed. The stiffness k of the reed is k = µr ωr2

(2)

This linear approximation only holds for non-beating reeds [Fritz et al., 2004], [Gilbert et al., 1989]. The maximum negative value that y can take is −h (see Figure 1), at which point the reed gap is closed and the air flow into the mouthpiece is completely blocked. This occurs when the mouth pressure pm is equal or greater to the closing pressure pM : pM = µr ωr2 h = kh

(3)

Air column The air column is usually characterised by its input impedance Zin , which describes the interaction between the volume flow and the pressure inside the mouthpiece. in the frequency domain:


10

10

2

0

in

|Z | (Ohms)

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10

−2

0

2

4

6

8

4

6

8

in

∠ Z (rad)

2

0

−2 0

2

Frequency (KHz)

Figure 2: Calculated input impedance of the chanter

P (ω) = Zin (ω)U(ω)

(4)

Zin was calculated from the bore profile of the chanter provided by the maker, using a transmission line model as described in [Plitnik and Strong, 1979]. It is shown in Figure 2. Nonlinear coupling The volume flow is related to the pressure across the mouthpiece by following Bernoulli’s equation as follows [Kergomard, 1996]: U = w(y + h)

s

2∆P sign(∆P ) ρ

(5)

HARMONIC BALANCE METHOD Equations 1 (reed), 4 (air column) and 5 (nonlinear coupling) can be solved by the Harmonic Balance Method [Fritz et al., 2004]. To keep these equations as general as possible, these equations are converted into dimensionless quantities by substituting: y h p p˜ = pM ˜ t = tωp pm γ= pM y˜ =

(6) (7) (8) (9)

where ωp is the angular frequency of the first resonance peak of the air column. Similarly, equation 1 using dimensionless quantities becomes: d˜ y d2 y˜ M 2 + R + K y˜ = p˜ − γ ˜ dt dt˜

(10)


ISMA 2007 Salt KC2 H3 O2 MgCl2 Mg(NO3 )2 NaCl KCl K2 SO4

Relative Humidity 26% 46% 59% 82% 94% 99%

h (mm) 0.35 0.3 0.3 0.3 0.225 0.25

k MPa m 70 65 60 55 55 45

ωr krad s 30.16 28.9 28.27 28.27 27.96 26.39

gr krad s 2.51 2.01 2.26 1.88 2.42 2.2

R ×10−3 1.6693 1.4541 1.7094 1.4245 1.8694 1.9078

M ×10−2 1.5842 1.7250 1.8025 1.8025 1.8432 20.692

ζ ×10−1 7.2309 6.9472 7.2309 7.5524 6.5406 7.8896

Table 1: Approximate values of height of opening at rest h, stiffness k, resonance frequency ωr and damping factor gr of the reed (taken from [Carral and Campbell, 2005]), as well as the dimensionless reed parameters R, M and ζ that were input to the program harmbal The parameters: ωp 2 M= ωr ωp gr R= 2 ωr

(11) (12)

are the the dimensionless mass and damping respectively, and since the reed closes when pm = pM , K = 1 (dimensionless stiffness) [Fritz et al., 2004]. Similarly, equation 5 becomes: q

U˜ (˜ p, y˜) = ζ(1 + y˜) |γ − p˜|sign(γ − p˜)

(13)

as long as y˜ > −1, otherwise U˜ (˜ p, y˜) = 0 [Fritz et al., 2004]. The “embouchure” parameter s

ζ = Z0 wh

2 ρpM

(14)

is a parameter that characterises the mouthpiece [Fritz et al., 2004]. Finally, the dimensionless form of the input impedance is obtained by: Zin Z˜in = Z0

(15)

where ρc (16) S S being the crosssectional area of the air column (cylindrical section) at the reed input. The dimensionless quantities in the frequency domain are: Z0 =

˜ P˜ (ω) = Z˜in (ω)U(ω)

(17)

The program harmbal 1 requires the parameters R, M and ζ, which are specified in a parameter file. In the version of harmbal used in this study (v1.27b), the input impedance of the air column can be specified in a file. For a detailed discussion on the Harmonic Balance Method, as well as how the program harmbal solves this 1

Written by Snorre Farner: http://www.pvv.ntnu.no/ farner/pub/harmbal/


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b) 4 26% 82% 99%

60

Spectral Centroid (kHz)

Pitch variation (Cents)

80

40 20 0 −20 −40 1

26% 82% 99% 3.5

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Figure 3: a) Pitch and b) spectral centroid variation vs pressure. The threshold pressure is indicated by a circle model, the reader is referred to [Gilbert et al., 1989], [Fritz et al., 2004] and [Farner et al., 2006]. Physical parameters of the reed In [Carral and Campbell, 2005] the reed of a bagpipe chanter was stored in containers with controlled relative humidity conditions by using various aqueous salt solutions, following the method described in [ASTM, 1998]. After having stored the reed in a particular container for several days, it was taken out of the container and the following parameters were measured: the opening height at rest h, stiffness k, resonance frequency ωr and damping factor gr . The results obtained in that study are summarised in Table 1. Reed parameter calculation The dimensionless parameters R, M and ζ are calculated with equations 11, 12 and 14, using the parameters obtained experimentally The speed of sound was taken kg to be c = 343.37 m s , and the density of air ρ = 1.2 m3 . The resulting parameters, which were passed to harmbal, are presented in Table 1. RESULTS Pitch and spectral centroid as measured experimentally The pitch and spectral centroid variation for three cases, where the mean relative humidity was approximately 26%, 82% and 99% are presented in Figure 3. In the pitch variation curves presented throughout this paper, 0 cents corresponds to 586.67 Hz, which is the frequency at which the studied note of the chanter was tuned (D5 in a just intonation scale 2 ). Figure 3 shows that the threshold pressure, as well as the pitch and spectral 2 This information was provided by the manufacturer Nigel Richards from Garvie Bagpipes: http://www.borderpipes.co.uk/

Bagpipe model vs. experiments −20 −40 −60 −80

Phase (rad)

Magnitude (dB)

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2

1

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2 0 −2

Frequency (kHz)

Figure 4: Transfer function of chanter centroid vs. pressure curves tend to drop gradually as the storage relative humidity increases. Pitch and spectral centroid obtained by harmbal The program harmbal calculates the playing frequency and the spectrum inside the mouthpiece. The pitch variation can be calculated directly from the playing frequency, and is shown in Figure 5 a). To calculate the spectral centroid of the radiated sound, the spectrum inside the mouthpiece was first convolved with the transfer function of the chanter, which was obtained experimentally, and is shown in Figure 4. The spectral centroid obtained is shown in Figure 5 b). The plots shown in Figure 5 stop at the point where the reed started beating. The reduction of threshold pressure as the relative humidity increases was predicted by harmbal, as was found experimentally. Discussion A comparison between the results obtained experimentally and the results obtained by solving the model using harmbal shows that the model was able to predict the reduction in threshold pressure with increasing relative humidity, as was found experimentally. However, there are also large discrepancies: • The pitch of the chanter in the experiment was mostly below the first air column resonance (located at around 50 cents in Figures 3a) and 5 a) ), whereas in the model, the playing frequency is below the air column resonance only for pressures close to the threshold pressure • The experiment shows a decrease in pitch and spectral centroid as the relative humidity increases. In the model the pitch seems to be independent on relative humidity. It is possible that the pitch drops once the reed starts beating, and that a model that can be solved for pressures above the beating threshold could show that behaviour as well. In the model, the rate at which the spectral centroid grows for increasing pressures drops for higher relative humidity.


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b) 2

26% 46%

300

Spectral Centroid (kHz)

Pitch variation (cents)

350

59% 82%

250

94% 99%

200

First air column resonance

150 100 50 0 0

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Pressure (kPa)

10

12

1.5

26% 46% 59% 82% 94% 99%

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0.5

0 0

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Pressure (kPa)

10

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Figure 5: a) Pitch and b) spectral centroid variation vs pressure calculated with harmbal • For low pressures, the spectral centroid calculated from the model is close to the frequency of the first harmonic. A close inspection at the evolution of the first few harmonics as the pressure increases (see a detailed analysis in [Carral, 2005], Chapter 7) shows that for low pressures the first harmonic dominates, and as the pressure increases, higher harmonics start to build up, as the effect of the non linearity in equation 13 increases. The spectral centroid increase shown in in Figure 5 b) is a direct consequence of this. Contrary to this, in the experiment, for low pressures the spectral centroid is high, which means that even at pressures close to threshold, the spectrum is rich in harmonics. • The threshold pressures observed in the experiment range between 3 and 5 kPa, whereas in the model range of pressures observed experimentally was between 2 and 8 kPa. This can be due to the fact that the stiffness measurements presented in [Carral and Campbell, 2005] had large uncertainties. • The chanter exhibits hysteresis when it is played: it is possible to play below the minimum pressure required to start the vibrations. This behaviour was not observed in the model. CONCLUSIONS AND FUTURE WORK The objective of this paper was to compare the experimental results obtained in [Carral and Campbell, 2005] with the solution of a simple model of the instrument obtained by harmbal. The model was able to predict a reduction of threshold pressure as the storage relative humidity increases. The reduction in pitch and spectral centroid with increasing relative humidity observed in [Carral and Campbell, 2005], as well as the hysteresis effect (the fact that it is possible to play the chanter at pressures lower than the threshold pressure, once the chanter is playing) were not reproduced by the model. Suggested modifications to the model include: • Calculating the reed stiffness parameter from other experimental data, for example, from the threshold pressure


ISMA 2007

• Increasing the number of harmonics used to be able to get solutions over the beating threshold • Implementing the flow characteristics curve that [Almeida et al., 2007] found for oboe reeds • Modifying the model to include hysteresis effects References Almeida, A., Vergez, C., and Caussé, R. (2007). Quasistatic nonlinear characteristics of double-reed instruments. Journal of the Acoustical Society of America, 121(1):536– 546. Almeida, A., Vergez, C., Caussé, R., and Rodet, X. (2002). Physical study of double-reed instruments for application to sound-synthesis. In Proceedings of the International Symposium on Musical Acoustics, pages 215–220, Mexico City, Mexico. ASTM, editor (1998). Anual book of ASTM standards, volume 11.03, chapter Standard practice for maintaining constant relative humidity by means of aqueous solutions E104 − 85, pages 781–783. ASTM International. Carral, S. (2005). Relationship between the physical parameters of musical wind instruments and the psychoacoustic attributes of the produced sound. PhD thesis, University of Edinburgh. Carral, S. and Campbell, D. M. (2005). The influence of relative humidity on the physical characteristics of the reed, and on psychoacoustic attributes of the sound of a bellows blown scottish border bagpipe chanter and reed. In Proceedings of the Forum Acusticum Conference, Budapest, Hungary. Novotel Budapest Congress Centre. Farner, S., Vergez, C., Kergomard, J., and Lizée, A. (2006). Contribution to harmonic balance calculations of self-sustained periodic oscillations with focus on single-reed instruments. Journal of the Acoustical Society of America, 119(3):1794–1804. Fritz, C., Farner, S., and Kergomard, J. (2004). Some aspects of the harmonic balance method applied to the clarinet. Applied Acoustics, 65:1155–1180. Fritz, C., Wolfe, J., Kergomard, J., and Caussé, R. (2003). Playing frequency shift due to the interaction between the vocal tract of the musician and the clarinet. In Proceedings of the Stockholm Music Acoustics Conference, volume 1, pages 263–266, Stockholm, Sweden. KTH Speech, Music and Hearing. Gilbert, J., Kergomard, J., and Ngoya, E. (1989). Calculation of the steady-state oscillations of a clarinet using the harmonic balance technique. Journal of the Acoustical Society of America, 86(1):35–41. Kergomard, J. (1996). Elementary considerations on reed-instrument oscillations. In Hirschberg, A., Kergomard, J., and Weinreich, G., editors, Mechanics of musical instruments, Lecture notes CISM, chapter 6, pages 229–290. Springer, Vienna. Plitnik, G. R. and Strong, W. J. (1979). Numerical method for calculating input impedances of the oboe. Journal of the Acoustical Society of America, 65(3):816– 825. Vergez, C. and Lizée, A. (2005). A frequency-domain approach of harmonic balance solutions stability. In Proceedings of the Forum Acusticum Conference, pages 539– 543, Budapest, Hungary. Novotel Budapest Congress Centre.