COMPARISON BETWEEN THE RADIATED SOUND PRODUCED BY A SCOTTISH BORDER BAGPIPE CHANTER, AND THE SOLUTION OF A SIMPLE PHYSICAL MODEL USING HARMBAL

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COMPARISON BETWEEN THE RADIATED SOUNDPRODUCED BY A SCOTTISH BORDER BAGPIPE

CHANTER, AND THE SOLUTION OF A SIMPLEPHYSICAL MODEL USING HARMBAL

Sandra Carral

Institute of Musical Acoustics

University of Music and Performing Arts, [email protected]

Abstract

In a previous study, the reed of a Scottish Border bagpipe chanter wasstored in different relative humidity conditions. Its physical parameters weremeasured, as well as the radiated sound produced by the chanter and reedsystem. It was found that the relative humidity at which the reed was storedinfluences the reed parameters, as well as the pitch and spectral centroid of theradiated sound produced by the instrument. In this paper, the reed parame-ters 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 differ-ence across the mouthpiece is related to the volume flow following Bernoulli’sequation. By solving this model, a program called harmbal 1 calculates theplaying frequency and the internal spectrum of the sound inside the mouth-piece. The former is compared to the pitch and the latter is convolved withthe transfer function of the chanter to obtain the radiated spectrum, fromwhich the spectral centroid is calculated, and compared to what was obtainedexperimentally.

INTRODUCTION

[Carral and Campbell, 2005] found that the relative humidity at which the bag-pipe chanter reed is stored affects its physical parameters, as well as the sound theinstrument 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 andCampbell, 2005], and raises questions on where the limitations of the model mightbe. Ultimately, having a more realistic physical model that behaves in the same wayas a real instrument does, might bring us closer to a better understanding of thesound production process in woodwinds, as well as which and to what extent thevariations on the reed parameters are responsible for the change in the producedsound of the instrument.

The Harmonic Balance Method was first applied to find the periodic solutionsof self-sustained musical instruments by [Gilbert et al., 1989]. [Farner et al., 2006]have developed a computer program called harmbal, that applies the HarmonicBalance Method to solve problems where a linear exciter is non linearly coupled toa linear resonator. This program has been so far used to solve clarinet models: [Fritzet al., 2003] used harmbal to study how the vocal tract of the musician affects the

1Written by Snorre Farner: http://www.pvv.ntnu.no/∼farner/pub/harmbal/



.

pm

lip

lip

mouthpiece

bore−h

0

y

pU

Figure 1: Schematic of a clarinet mouthpiece

playing frequency of the clarinet. [Fritz et al., 2004] have used harmbal to study theinfluence of the reed mass and damping in the spectrum of the clarinet. [Vergez andLizee, 2005] presented a method to find the stability in the solutions determined byharmbal.

In this paper, a simple physical model of a Scottish Border bagpipe chanterand reed is proposed, where the reed parameters are those measured in [Carral andCampbell, 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 aircolumn). This section presents the equations that were used to model these threecomponents.

Reed

[Almeida et al., 2002] have provided evidence that the two blades of a doublereed have symmetric displacement. This means that the motion of only one bladeneeds to be modelled as a simple harmonic oscillator:

d2y

dt2

+ grdy

dt

+ ω2

ry = −1

µr∆P (1)

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

k = µrω2

r (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 pointthe 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 = µrω2

rh = kh (3)



.

0 2 4 6 810

−2

100

102

| Z i n

| ( O h m s )

0 2 4 6 8−2

0

2

∠ Z i n

( r a d )

Frequency (KHz)

Figure 2: Calculated input impedance of the chanter

Air column

The air column is usually characterised by its input impedance Z in, which de-

scribes the interaction between the volume flow and the pressure inside the mouth-piece. in the frequency domain:

P (ω) = Z in(ω)U (ω) (4)

Z in 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 isshown in Figure 2.

Nonlinear coupling

The volume flow is related to the pressure across the mouthpiece by followingBernoulli’s equation as follows [Kergomard, 1996]:

U = w(y + h)

2∆P

ρsign(∆P ) (5)

HARMONIC BALANCE METHOD

Equations 1 (reed), 4 (air column) and 5 (nonlinear coupling) can be solvedby the Harmonic Balance Method [Fritz et al., 2004]. To keep these equations asgeneral as possible, these equations are converted into dimensionless quantities bysubstituting:

y =y

h(6)

˜ p =p

pM

(7)

t = tω p (8)

γ =pm

pM

(9)





.

a)

1 2 3 4 5 6 7 8−40

−20

0

20

40

60

80

P i t c h

v a r i a t i o n ( C e n t s )

Pressure (kPa)

26%

82%

99%

b)

1 2 3 4 5 6 7 8

2.5

3

3.5

4

S p e c t

r a l C e n

t r o i d ( k H z

)

Pressure (kPa)

26%

82%

99%

Figure 3: a) Pitch and b) spectral centroid variation vs pressure. The threshold pressure is indicated by a circle

P (ω) = Z in(ω)U (ω) (17)

The program harmbal requires the parameters R, M and ζ , which are specifiedin a parameter file. In the version of harmbal used in this study (v1.27b), the inputimpedance of the air column can be specified in a file. For a detailed discussionon the Harmonic Balance Method, as well as how the program harmbal solves thismodel, the reader is referred to [Gilbert et al., 1989], [Fritz et al., 2004] and [Farneret al., 2006].

Physical parameters of the reed

In [Carral and Campbell, 2005] the reed of a bagpipe chanter was stored incontainers with controlled relative humidity conditions by using various aqueous

salt solutions, following the method described in [ASTM, 1998]. After having storedthe reed in a particular container for several days, it was taken out of the containerand the following parameters were measured: the opening height at rest h, stiffnessk, resonance frequency ωr and damping factor gr. The results obtained in that studyare summarised in Table 1.

Reed parameter calculation

The dimensionless parameters R, M and ζ are calculated with equations 11, 12and 14, using the parameters obtained experimentally. The speed of sound was taken

to be c = 343.37 m

s

, and the density of air ρ = 1.2 kg

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 relativehumidity was approximately 26%, 82% and 99% are presented in Figure 3. In the



.

1 2 3 4 5 6

−80

−60

−40

−20

M a g n i t u d e ( d B )

1 2 3 4 5 6

−2

0

2

P h a s e

( r a d )

Frequency (kHz)

Figure 4: Transfer function of chanter

pitch variation curves presented throughout this paper, 0 cents corresponds to 586.67Hz, 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

centroid vs. pressure curves tend to drop gradually as the storage relative humidityincreases.

Pitch and spectral centroid obtained by harmbal

The program harmbal calculates the playing frequency and the spectrum insidethe mouthpiece. The pitch variation can be calculated directly from the playingfrequency, and is shown in Figure 5 a).

To calculate the spectral centroid of the radiated sound, the spectrum inside themouthpiece was first convolved with the transfer function of the chanter, which was

obtained experimentally, and is shown in Figure 4. The spectral centroid obtainedis 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 predictedby harmbal, as was found experimentally.

Discussion

A comparison between the results obtained experimentally and the results ob-tained by solving the model using harmbal shows that the model was able to predictthe 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 aircolumn resonance (located at around 50 cents in Figures 3a) and 5 a) ), whereasin the model, the playing frequency is below the air column resonance only forpressures close to the threshold pressure

2This information was provided by the manufacturer Nigel Richards from Garvie Bagpipes:http://www.borderpipes.co.uk/



.

a)

0 2 4 6 8 10 120

50

100

150

200

250

300

350

Pressure (kPa)

P i t c h v a r i a t i o n ( c e n t s )

26%

46%

59%

82%

94%

99%

First air column resonance

b)

0 2 4 6 8 10 120

0.5

1

1.5

2

Pressure (kPa)

S p e c t r a l C e n t r o i d ( k H z )

26%

46%

59%

82%

94%

99%

Figure 5: a) Pitch and b) spectral centroid variation vs pressure calculated with harmbal

• The experiment shows a decrease in pitch and spectral centroid as the relativehumidity 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, andthat a model that can be solved for pressures above the beating thresholdcould show that behaviour as well. In the model, the rate at which the spectralcentroid grows for increasing pressures drops for higher relative humidity.

• For low pressures, the spectral centroid calculated from the model is close tothe frequency of the first harmonic. A close inspection at the evolution of thefirst few harmonics as the pressure increases (for a detailed analysis, see[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 effectof 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, inthe experiment, for low pressures the spectral centroid is high, which meansthat 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 between2 and 8 kPa. This can be due to the fact that the stiffness measurementspresented in [Carral and Campbell, 2005] had large uncertainties.

• The chanter exhibits hysteresis when it is played: it is possible to play belowthe minimum pressure required to start the vibrations. This behaviour wasnot 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 instrumentobtained by harmbal. The model was able to predict a reduction of threshold pres-sure as the storage relative humidity increases. The reduction in pitch and spectralcentroid 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 atpressures lower than the threshold pressure, once the chanter is playing) were notreproduced by the model.

Suggested modifications to the model include:

• Calculating the reed stiffness parameter from other experimental data, forexample, from the threshold pressure

• Increasing the number of harmonics used to be able to get solutions over thebeating threshold

• Implementing the flow characteristics curve that [Almeida et al., 2007] foundfor oboe reeds

• Modifying the model to include hysteresis effects

References

Almeida, A., Vergez, C., and Causse, R. (2007). Quasistatic nonlinear characteristics of double-reedinstruments. Journal of the Acoustical Society of America , 121(1):536–546.

Almeida, A., Vergez, C., Causse, R., and Rodet, X. (2002). Physical study of double-reed instru-

ments 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 practicefor maintaining constant relative humidity by means of aqueous solutions E 104 − 85, pages781–783. 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 physics andpsychoacoustics of a Scottish bellows blown Border bagpipe chanter and reed. In Proceedingsof the Forum Acusticum Conference, pages 379–384, Budapest, Hungary. Novotel BudapestCongress Centre.

Farner, S., Vergez, C., Kergomard, J., and Lizee, A. (2006). Contribution to harmonic balance cal-culations 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 methodapplied to the clarinet. Applied Acoustics, 65:1155–1180.

Fritz, C., Wolfe, J., Kergomard, J., and Causse, R. (2003). Playing frequency shift due to theinteraction between the vocal tract of the musician and the clarinet. In Proceedings of theStockholm Music Acoustics Conference, volume 1, pages 263–266, Stockholm, Sweden. KTHSpeech, Music and Hearing.

Gilbert, J., Kergomard, J., and Ngoya, E. (1989). Calculation of the steady-state oscillations of aclarinet 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, Lecturenotes 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 Lizee, A. (2005). A frequency-domain approach of harmonic balance solutionsstability. In Proceedings of the Forum Acusticum Conference, pages 539–543, Budapest,Hungary. Novotel Budapest Congress Centre.

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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