Operational transfer path analysis of a piano

Citation for published version (APA):Tan, J. J., Chaigne, A., & Acri, A. (2018). Operational transfer path analysis of a piano. Applied Acoustics, 140,39-47. https://doi.org/10.1016/j.apacoust.2018.05.008

DOI:10.1016/j.apacoust.2018.05.008

Document status and date:Published: 01/11/2018

Document Version:Accepted manuscript including changes made at the peer-review stage

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.

Download date: 02. Oct. 2021

Operational transfer path analysis of a piano1

Jin Jack Tana,b,1, Antoine Chaignea,2, Antonio Acric,d,32

aIWK, University of Music and Performing Arts Vienna, Austria3

bIMSIA, ENSTA-ParisTech-CNRS-EDF-CEA, France4

cPolitecnico di Milano, Italy5

dVirtual Vehicle, Austria6

Abstract7

The piano sound is made audible by the vibration of its soundboard. A pianistpushes the key to release a hammer that strikes the strings, which transfer the en-ergy to the soundboard, set it into vibration and the piano sound is heard due to thecompression of air surrounding the soundboard. However, as piano is being played,other components such as the rims, cast-iron frame and the lid are also vibrating.This raises a question of how much of their vibrations are contributing to the soundas compared to the soundboard. To answer this question, operational transfer pathanalysis, a noise source identification technique used widely in automotive acoustics,is carried out on a Bosendorfer 280VC-9 grand piano. The ”noise” in a piano systemwould be the piano sound while the ”sources” are soundboard and the aforemen-tioned components. For this particular piano, it is found out that the soundboardis the dominant contributor. However, at high frequencies, the lid contributes themost to the piano sound.

Keywords: piano acoustics, structural vibration, source identification, operational8

transfer path analysis9

1. Introduction10

The study of piano acoustics has traditionally been focused on its piano action11

[1, 2], interaction between a piano hammer and string [3, 4], string vibration [5, 6, 7],12

soundboard vibration [8, 9, 10] and its radiation [11, 12, 13]. As computational13

power becomes cheaper, it is possible to model the piano as a coupled system that14

involves the string, soundboard and surrounding air [14, 15]. However, this raises15

a question whether the considered system is complete enough to have a realistic16

reproduction of the piano sound. The importance and role of a soundboard in17

1Present affiliation: Faculty of Engineering and the Environment, University of SouthamptonMalaysia Campus, Iskandar Puteri, Malaysia

2Present address: 13 allee Francois Jacob, 78530 BUC, France3Present affiliation: R&D Department, Brugola OEB Industriale spa, Milan, Italy

Preprint submitted to Elsevier April 30, 2018

piano sound production have been extensively studied [16, 17, 18, 19, 20, 21] but18

not for other components. Is there any other components that are contributing to19

the piano sound production that has not been accounted for? Anecdotally, when a20

piano is played, vibration can be felt not only on the soundboard but also on the21

rim, the frame, the lid etc. In a Bosendorfer piano, spruce, a wood usually used for22

the soundboard by other manufacturers, is used extensively in building the case of23

the piano. Bosendorfer claims that the use of spruce, especially on the rim of the24

piano, allows the whole instrument to vibrate and is the reason that gives the unique25

Bosendorfer sound [22]. Based on the fact that vibration is felt on other parts of the26

piano and how Bosendorfer uses spruce extensively, it necessitates an investigation27

if the vibrations of these parts contribute to the production of the sound.28

Current work takes inspiration from noise source identification techniques used29

commonly in automotive acoustics [23, 24, 25]. However, in the case of piano,30

the ”noise” is the resulting piano sound and the ”sources” to be identified are the31

piano components to be investigated. These ”sources” may emit different ”noise”32

contributions that characterise the resulting ”noise”, i.e. the piano sound. One33

technique that can be used to identify the contribution of the piano components34

to the final sound is the operational transfer path analysis (OTPA) [26]. OTPA35

computes a transfer function matrix to relate a set of input/source(s) measurements36

to output/response(s) measurements. In this case, the inputs are the vibration of the37

components of piano and the output is the resulting piano sound. Initial result of the38

work was first presented at the International Congress of Acoustics [27] but more39

thorough analysis has since been conducted with more convincing and confident40

results obtained. These results are being presented in this paper. In Section 2, the41

theory of OTPA is presented. The experiment designed for OTPA is then detailed42

in Section 3 before the results are shown and discussed in Section 4.43

2. The Operational Transfer Path Analysis (OTPA)44

Operational transfer path analysis (OTPA) is a signal processing technique that45

studies the noise source propagation pathways of a system based on its operational46

data [28]. This is in contrast to classical transfer path analysis (TPA) where the47

source propagation pathways are established by means of experimental investiga-48

tions with specific inputs. Indeed, OTPA was developed in wake of the need of49

a fast and robust alternative to the classical TPA. In both methods, the source50

propagation pathways are determined by studying the transfer functions between51

the sources (inputs) and the responses (outputs). In TPA, the relationship between52

sources (excitation signals) and responses are estimated by frequency response func-53

tions and these are meticulously determined by series of experimental excitation of54

known forces (e.g. by shaker or impact hammer). Where necessary, part of the55

system is also removed or isolated. In this way, TPA is able to trace the flow of56

vibro-acoustic energy from a source through a set of known structure- and air-borne57

pathways, to a given receiver location by studying the frequency response functions.58

2

On the other hand, OTPA measures directly the source and response signals when59

the system is operated and establishes the source propagation pathways based on60

the experimentally determined transfer functions. OTPA is a response-response61

model where measurement data are collected and analysed while the system, when62

operated, provides the excitation. Detailed comparisons between the classical TPA63

and OTPA can be found in [29, 30, 31].64

While OTPA may appear to be simpler to use, it is also prone to error if it is65

not designed and analysed properly. OTPA requires prior knowledge of the system66

as neglected pathways could not be easily detected. In a multi-component system,67

cross-coupling between the components could affect the accuracy of an OTPA model.68

Several techniques, which are detailed in the following Section 2.2, can be employed69

to mitigate the effects of cross-coupling [28].70

2.1. Theory of OTPA71

In a linear system, the input X and output Y can be related by:72

Y(jω) = X(jω)H(jω), (1)

where:73

Y(jω) is the output vector/matrix at the receivers;74

X(jω) is the input vector/matrix at the sources;75

H(jω) is the operational transfer function matrix, also known as the transmissibility76

matrix. The dependency of frequencies for all three matrix is as denoted by (jω)77

[28].78

The inputs and outputs signals can be the forces, displacements, velocities or79

pressures of the components in the system. Given that there are m inputs and n80

outputs with p set of measurements, Equation (1) can be written in the expanded81

form:82 y11 · · · y1n...

. . ....

yp1 · · · ypn

=

x11 · · · x1m...

. . ....

xp1 · · · xpm

H11 · · · H1n

.... . .

...Hm1 · · · Hmn

, (2)

where for clarity purposes, the frequency dependency jω is dropped. In order to83

quantify the contributions of the inputs to the outputs, the transfer function matrix84

needs to be solved. If the input matrix X is square and invertible, this can be solved85

by simply multiplying the inverse of X on both sides:86

H = X−1Y. (3)

However, in most cases, p 6= m. Thus, for the system to be solvable, it is required87

that the number of measurement sets is larger than or equal to the number of inputs,88

i.e.89

p ≥ m, (4)

3

thereby forming an overdetermined system with residual vector µ:90

XH + µ = Y. (5)

The transmissibility matrix H can then be obtained via the following equation [28]:91

H = (XTX)−1XTY = X+Y, (6)

where the + superscript denotes the Moore-Penrose pseudo-inverse [32].92

2.2. Enhanced OTPA with singular value decomposition and principal component93

analysis94

In essence, the basics of OTPA is analogous to the multiple-input multiple-95

output (MIMO) technique in experimental modal analysis [33]. However, solving the96

transfer function H directly is prone to error if the input signals are highly coherent97

between each other. High coherence is caused by unavoidable cross-talks between98

the measurement channels as they are sampled simultaneously. To mitigate this99

error, an enhanced version of OTPA can be employed. Singular value decomposition100

(SVD) and principal component analysis (PCA) can be carried out [28, 26]. There101

are two main reasons in using SVD. Firstly, it can be used to solve for X+, even102

though it is not the only way to solve for Moore-Penrose pseudo-inverse. Secondly,103

the singular value matrix can later be repurposed to carry out PCA to reduce the104

measurement noise.105

The input matrix X, as decomposed by economy size SVD, can be written as106

X = UΣVT, (7)

where107

U is a unitary column-orthogonal matrix;108

Σ is a square diagonal matrix with the singular values;109

VT is the transpose of a unitary column-orthogonal matrix, V.110

111

The singular values obtained along the diagonal of Σ are also the principal112

components (PC). The PC are defined such that the one with the largest variance113

within the data is the first PC (the first singular values), the next most varying is the114

second PC and so on. The smallest singular value thus corresponds to the weakest115

PC that has little to no variation. A matrix of PC scores can then be constructed116

as:117

Z = XV = UΣ. (8)

The contribution of each PC can be evaluated by dividing Z with the sum of all the118

PC scores. For each PC, this yields a value between 0 to 1. The larger the number,119

the more significant the PC is. In other words, a weakly contributing PC can be120

identified and thus be removed by setting it to zero as they are mainly caused by121

4

noise influences or external disturbances [28]. Then, the inverse of the singular value122

matrix Σ−1 can be recalculated by:123

Σ−1e =

{1/σq if σq ≥ θ,

0 otherwise,(9)

where σq indicates the q-th singular value along the diagonal Σ matrix and the124

overbar indicates normalised singular value against sum of all singular values [26].125

On the other hand, θ represents a threshold value (where 0 ≤ θ ≤ 1) while the126

subscript e indicates that the matrix has been enhanced by SVD and PCA.127

Subsequently, the modified pseudo-inverse of X can be written as [34]:128

X+e = VΣ−1

e UT. (10)

Introducing Equation (10) into Equation (6), the treated transmissibility matrix He129

can then be written as:130

He = X+e Y. (11)

Once the treated transmissibility matrix He is computed, it is possible to derive the131

synthesised responses Ys such that [28, 25]:132

Ys = XHe. (12)

However, to quantify the contribution of each input to the outputs, one could con-133

sider the following summations:134

cijk = xikhkj, (13)

where i = 1, . . . , p, j = 1, . . . , n and k = 1, . . . ,m while xik and hkj represent the135

element in the matrix X and He respectively. In other words, Equation (13) quan-136

tifies the contributions c from a transmission path (i.e. input) k to a measurement137

point (i.e. output) j for a given measurement i. Summing up all measurement sets,138

one can thus obtain the overall contributions, Cjk, from an input j to an output k:139

Cjk =

p∑i=1

cijk, (14)

where the individual k-th synthesised response could be recovered if one sums Cjk140

overs all the j inputs.141

3. Experimental setup142

The soundboard, inner rim, outer rim, the cast-iron frame and the lid are all143

considered to be potential transmission paths of the piano sound (see Figure 1a).144

The vibrations of these components are sampled by sets of accelerometers and can145

5

be treated to be the inputs of OTPA, X. To fully capture all the modes of interest,146

the accelerations are sampled extensively over the surfaces in grids of 20cm x 20cm.147

For the cast-iron frame, only the part highlighted in the green area in Figure 1b are148

sampled. The other part has a comparatively small surface area, and it is assumed149

that its contribution does not play a significant role in the final sound. In the same150

figure, the surface sampled for inner rim and outer rim are also highlighted (albeit151

partially) in blue and purple respectively. Of course, the outer rim measurement152

is also extended to the straight part of the rim which is not seen in the figure.153

The soundboard and lid measurement are sampled on the surface that is visible in154

Figure 1b (i.e. the soundboard surface that faces up and the lid surface that faces155

down) but are not highlighted so as to not obscure the presentation. The total156

number of measurements for each components is outlined in Table 1. Meanwhile,157

the output Y is the sound pressure at a point away from the piano, recorded by a158

microphone, marked M in Figure 2. The experiment is conducted in the anechoic159

chamber of the University of Music and Performing Arts Vienna so as to eliminate160

wall and floor reflections, and conveniently ignore radiation from the bottom surface161

of the soundboard. The piano used is a Bosendorfer 280VC-9 equipped with a CEUS162

Reproducing System. The CEUS Reproducing System is a computerised system that163

is able to record and reproduce the playing of the piano by means of optical sensors164

and solenoids.165

(a) Illustrated view of piano, showing thecomponents investigated. Modified from[35].

(b) A Bosendorfer 280VC-9. Highlightedgreen, blue and purple areas are the frame,inner rim and outer rim respectively.

Figure 1: Anatomy of a piano.

One of the advantages of OTPA is its flexibility to use operational data rather166

than controlled excitation from a shaker or impact hammer. Thus, measurement167

6

y

x

SBL

IR,OR

F M

Figure 2: Location of the microphone (marked M) with respect to the outline of the piano sound-board. Assuming that the origin of an (x,y) coordinate system as shown in the figure, the mi-crophone is at x=2.2m, y=1.0m and is the same height with the soundboard. The midpoints onthe x-y plane of each summed input are also shown where SB, L, F, IR and OR represent thesoundboard, lid, frame, inner and outer rim respectively. The image is to scale.

data are collected as the piano is being played. As vibrational data need to be taken168

in batches due to limited number of accelerometers, the CEUS Reproducing System169

is used where the piano playing can be reproduced with the same force and timing.170

The CEUS system can be controlled via pre-produced .boe files, which contains171

information on the keys to play, their pushing profiles and the hammer velocities172

[36]. For this experiment, two .boe files are prepared via in-house MATLAB/GNU173

Octave script, each corresponding to different excitation patterns.174

For an OTPA, it is desired that the inputs are as incoherent as possible to175

minimise cross-couplings between them. The two excitations are designed with that176

in mind. The first is simply playing each of the 88 notes sequentially, from the177

lowest A0 (27.5Hz) to the highest C8 (4186 Hz). Each note is played for 3 seconds178

with a rest interval of 0.6 seconds in between. Each 3-second sequence is a unique179

measurement and thus p = 88 as is defined in Equation (2). The second sets of180

measurement contains 12 playing sequences with each of them playing all the same181

notes, starting with all A (i.e. A0, A1, A2 ... A7) and followed by all of the A#s,182

all of the Bs, all of the Cs until finally all of the G#s as is illustrated in Figure 3.183

Each sequence is played for 5 seconds with a 1 second interval in between and184

yields p = 12. Both sequences are played at a moderate dynamic level that gives a185

subjective loudness of mezzoforte.186

As outlined in Table 1, there are a total of 154 vibration signals that can be187

treated as inputs. This would violate Equation (4) if m = 154. It is thus necessary188

to either increase the number of measurements or reduce the number of inputs. For189

this study, the latter is performed where the vibrational signals are summed via190

7

Figure 3: Illustration of the 12-sequence excitation. Vertical axis indicates the note number played.The dips indicate the keys being pressed.

Rayleigh integral so as to obtain an input variable with the dimension of a sound191

pressure. The net pressure p at point R as summed by Rayleigh integral can be192

defined as:193

p(R, ω) =ρ

2π

∑i

ai(ω)Si

rie−jkri , (15)

where ρ, ai, ri, k and Si represent density of the air, the acceleration, distance to the194

summed point, wavenumber and area covered by source point i respectively and the195

term ω indicates dependence on frequency. The acceleration data of each component196

are summed to the midpoint of the accelerometer locations of each component (see197

Figure 2), with an averaged uniform area assumed (i.e. total calculated area divided198

by number of measurements on each component). As a result, this yields a total of199

only 5 sources, i.e. m = 5, making them suitable to be used for OTPA.200

Table 1: Number of accelerometers for each components.

Components Number of accelerometersSoundboard 35

Inner rim 24Outer rim 24

Frame 25Lid 46

The use of Rayleigh integral is an approximation as it is defined for an infinitely201

large flat baffle [37]. A rapid calculation allows to estimate the frequency range for202

8

which the Rayleigh approximation may lead to an overestimation of the sound pres-203

sure. The soundboard and lid have a nominal length Ln of about 1.47m. With this204

order of magnitude for the acoustic wavelength, an acoustical short circuit might205

appear for wave modes below 230Hz in the reality, thus leading to a sound pressure206

lower than the one predicted by the Rayleigh integral. However, the overestimation207

may not be so prominent due to the complex shape of the piano and the presence208

of the semi-opened cavity defined by the rim and the lid. In practice, this overesti-209

mation should not have appreciable consequences in the analysis, since the Rayleigh210

integral is used here for the purpose of reducing the number of inputs, only, and211

also it is a comparative study. The only consequence is that the contributions of212

the smaller components (such as the frame) might be slightly overestimated in the213

low-frequency range, compared to the large components (soundboard, lid) through214

the use of the Rayleigh integral.215

4. Results and discussion216

A quality check of OTPA can be performed by comparing the experimentally217

measured output against the synthesised output as obtained from Equation (12).218

The error between the two outputs can be defined as:219

ε = |Y −XHe|. (16)

Using different threshold values θ as defined in Equation (9) will yield different220

synthesised output and as a result, different ε. To minimise the overall ε, different θ221

is used for each frequency point and is chosen for when ε is lowest at that particular222

frequency point. This is possible since the system is linear, and its linearity is further223

verified by varying the number of measurement blocks, i.e. p with no significant224

difference in the transmissibility matrix, He [28].225

The first 300Hz of the outputs of the first sequence of both 88-sequence and226

12-sequence excitations are as shown in Figure 4. For the 88-sequence excitation,227

the averaged threshold value θ is 1.7% and varies between 1% and 4% while for the228

12-sequence excitation, the averaged θ value is 1.2% and varies between 1% and 6%.229

The distribution of number of principal component (PC) kept after the principal230

component analysis is performed is also presented in Table 2. Comparing between231

the measured and synthesised data in Figure 4, there are very good agreements232

except at the frequency range between 10 Hz and 40Hz for the 88-sequence excita-233

tion. Additionally, the synthesised responses produce few extra peaks that are not234

observed in the measured responses, such as at 124Hz for both excitations. These235

peaks appear to persist even when a high value of θ (e.g. 25%) is used, or when p is236

varied, ruling them out being noises or effect due to nonlinearity. It must be high-237

lighted that the variable θ approach across the whole frequency range is important238

as it manages to remove several other peaks that would otherwise be present if a239

common θ value is used instead.240

9

The presences of these peaks might be due to the contribution of other input241

signals that are not recorded in the experiment. Indeed, there has been study242

that suggest that the presence of finger-key impact and the key-bottom impact243

are perceivable to listeners [36]. The former does not exist in the experiment as244

all keys are played by the CEUS system while the role of the latter might not be245

significant as the key is not played with excessive force and to a mezzoforte level246

only. Nonetheless, it remains a point to revisit in future study.

Table 2: Distribution of number of PC kept after principal component analysis is performed.

Number of PC 1 2 3 4 512-sequence 740 2276 2400 3728 5639288-sequence 1141 2823 2211 1832 57529

247

0 50 100 150 200 250 30040

60

80

100

120

140

Frequency (Hz)

Sou

nd

pre

ssu

re(d

B) measured

synthesised

(a) 88-sequence excitation, playing only A0 note.

0 50 100 150 200 250 30040

60

80

100

120

140

Frequency (Hz)

Sou

nd

pre

ssu

re(d

B) measured

synthesised

(b) 12-sequence excitation, playing all A note.

Figure 4: Comparison of the synthesised and measured output (sound pressure at M as shown inFigure 2). Good agreements between the data indicate that the models are representative of themeasurements.

The next part of the analysis is to study the contributions of each source in the248

output. The contribution level of each input can be obtained via Equation (14) over249

the whole frequency range. To represent the result in a clear and readable manner,250

five frequency groups, as divided based on the partition made by the frame of the251

Bosendorfer piano (see Figure 1b on how the stress bars terminate on top of the252

10

keyboard) are identified and all results obtained within each frequency group are253

averaged and presented in bar charts as shown in Figure 5. The frequency group is254

summarised in Table 3.

Table 3: Key number of the frequency groups and their corresponding frequency ranges.

Key number Frequency range (Hz)1 to 20 27.5 to 82.421 to 38 82.4 to 233.139 to 54 233.1 to 587.355 to 72 587.3 to 1661.273 to 88 1661.2 to 4186.0

255

Figure 5 shows the contribution level of each input (i.e. soundboard, inner rim,256

outer rim, frame and lid) and the corresponding output (labeled ”total”) in two sets257

of bar charts: the top graph shows the results from the 88-sequence excitation, while258

the bottom shows results from the 12-sequence excitation. On top of each individual259

bar, the sound pressure in dB (relative to 20µPa) is displayed. From both results,260

some main observations can be made:261

• From key 1 to 72, the soundboard is the main contributor to the sound.262

• From key 73 to 88, the lid is the main contributor.263

• From key 73 to 88, the rims and the frame have similar level of contributions264

compared to the soundboard.265

However, there are also some discrepancies between the two sets of results:266

• The differences in sound pressure between soundboard and the other compo-267

nents are noticeably smaller in key 1 to 20 and key 55 to 72 in the 88-sequence268

excitation (about 4dB) compared to the 12-sequence excitation (8 to 10 dB).269

• The frame is a clear secondary contributor from key 1 to 38 in the 12-sequence270

excitation, but less obvious in the 88-sequence excitation.271

• Overall, 12-sequence excitation has higher sound pressure level.272

The consistent results between the two types of excitation confirm the importance273

of soundboard in piano sound production, which is not surprising. However, the274

results also reveal some new findings, particularly at higher frequencies (key 73 and275

above), that the lid plays an important role in sound production. The result can be276

best illustrated by plotting the contribution of input and the output sound pressure277

as is shown in Figure 6 and 7.278

In Figure 6, the synthesised output of the 28th sequence in the 88-sequence279

excitation is shown together with the contribution of each component. The thick280

11

1-20 21-38 39-54 55-72 73-88

40

60

80

63

6764

55

45

58

50

56

47 46

56

50

55

50

46

5758

5350

46

59 57

5351

50

6668

66

58

54

Sou

nd

pre

ssu

re(d

B)

1-20 21-38 39-54 55-72 73-88

40

60

8070

7573

63

48

56 56

60

5249

53 54

63

53

49

62

66

58

53

48

5961

55

52 51

71

7674

65

56

Key number

Sou

nd

pre

ssu

re(d

B)

soundboard innner rim outer rim frame lid total

Figure 5: Average contribution level from the analysis for (on the top) 88-sequence excitation and(on the bottom) 12-sequence excitation. The frequency range is defined based on the piano keynumber.

black line (labeled ”Total”) represents the final sound and the other lines correspond281

to the inputs. The 28th sequence corresponds to playing the C3 note and its equal282

temperament frequency is shown as a dotted vertical line at 130.8Hz. The measured283

frequency of C3 of the piano investigated is 131.1Hz, but the difference of +4 cents284

is essentially negligible. From the figure, the spectral content of the played note,285

as represented as the highest peak, is made up almost entirely by the soundboard.286

At other peaks (about 10dB less than the C3 peak), soundboard, frame and lid are287

seen to be contributing as well. Similar spectra can be observed at some other notes288

of lower ranges, which explain the high contribution of soundboard.289

Next, a higher key belonging to the highest frequency group is investigated. The290

spectra in which the 80th key (E7) is played is as shown in Figure 7. The equal291

temperament frequency of E7 is 2,637Hz (shown as dotted line) but for this piano,292

it is tuned at 2,672Hz (+23 cents), which is in agreement with the Railsback curve293

[38]. One can see that while soundboard still contributes at the frequency of the294

played note (i.e. at 2,672Hz), for other spectral contents, the lid plays a dominant295

contribution, such as at around 2,549Hz and 2,651Hz. These additional components296

12

116 118 120 122 124 126 128 130 132 1340

20

40

60

Frequency (Hz)

Sou

nd

pre

ssure

(dB

)

C3 Soundboard Outer rim LidTotal Inner rim Frame

Figure 6: Contribution of the five inspected components for the synthesised output (labeled ”To-tal”) for the 28th sequence of the 88-sequence excitation. The dotted vertical line corresponds tothe equal temperament frequency of the played note (C3).

might be due to eigenmodes of the lid or other elements of the piano structure which297

are transmitted to the lid through prop and hinges. However, such an assumption298

could only be confirmed after a thorough theoretical and experimental analysis of299

the transmission of vibrations through the various components of the piano.300

Attempt was made to determine whether the contribution of the lid to the total301

sound field is only due to transmission of vibrations through prop and hinges (i.e.302

structure-borne transmission), or reflection of the pressure radiated by the sound-303

board (i.e. air-borne transmission), especially in the treble range. For this purpose,304

Figure 8 shows the measured acceleration map of the lid for the two played notes305

A6 (Nr 73, 1771 Hz) and A7 (Nr 85, 3587 Hz). Both maps show higher acceleration306

level in the vicinity of the prop (red circle), and of some hinges (2nd hinge from top307

for A6, first hinge from bottom for A7). In contrast, both pressure maps computed308

in the lid plane (in the absence of the lid) due to the radiation of the soundboard309

for these two notes show a much smoother pattern, with a regular decrease of the310

pressure from the left bottom to the outer edge (see Figure 9). These results sug-311

gest that the acceleration of the lid is not entirely due to the pressure field radiated312

by the soundboard, and that transmission of vibrations from other piano parts to313

the lid also exists. More experimental work is needed for clearly separating the314

transmission-induced from the radiation-induced vibrations on the lid.315

Bosendorfer claims that the rims play a role in the sound production as part316

of their ”resonance case principle”. The main idea is to construct the rims using317

spruce, the wood used widely in the construction of soundboard. This will result in318

a complete resonating body consisting of the soundboard and rim that is responsible319

for the sound production [22]. However, from the analysis performed in this study,320

their contributions appear to only be influential from 1661Hz (key 73) onwards.321

13

2,540 2,560 2,580 2,600 2,620 2,640 2,660 2,68010

20

30

40

Frequency (Hz)

Sou

nd

pre

ssure

(dB

)

E7 Soundboard Outer rim LidTotal Inner rim Frame

Figure 7: Contribution of the five inspected components for the synthesised output (labeled ”To-tal”) for the 80th sequence of the 88-sequence excitation. The dotted vertical line corresponds tothe equal temperament frequency of the played note (E7).

Finally, the discrepancies between the two types of excitation need to be ad-322

dressed. Physically, the excitations differ by the number of key pushed simultane-323

ously and the frequency content of each excitation and the corresponding output.324

Pushing more keys at the same time introduce more energy to the piano, which325

explains the higher sound pressure level in the 12-sequence excitation. Since the326

frequency contents of the signals of two excitation actually differ, it is thus probable327

that the same processing would yield slightly different results on the sound pressure328

level of secondary contributors.329

5. Conclusion330

An operational transfer path analysis (OTPA) has been conducted for a Bosendorfer331

piano to identify any additional vibrating components (other than the soundboard)332

that could contribute to the sound production. A variable threshold approach is333

implemented during the principal component analysis to enhance the computation.334

Across the frequency range inspected, the soundboard has been the major contrib-335

utor except for the highest frequency range (≥ 1661.2Hz). In that range, the lid336

contributes the most to the piano sound.337

Current study represents a rare interdisciplinary example of technique originally338

developed for automotive analysis being applied to musical instruments. However,339

it is indeed possible to apply other source identification techniques to investigate340

current problem and obtain a better understanding on the transfer pathways from341

the strings to the listener. For recommendation, one can conduct a classical transfer342

path analysis. Components like lid can be removed and boundary conditions between343

the soundboard and the rims can be artificially modified to isolate the components.344

14

0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x (m)

y(m

)

−40

−30

−20

(a) Note A6

0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x (m)

y(m

)

−40

−30

−20

(b) Note A7

Figure 8: Lid acceleration map in dB (re 1 m/s2) for the two notes A6 (1771 Hz) and A7 (3587Hz). Red circle: position of the prop. Red rectangles: positions of the hinges. The circle andrectangle are not to scale with actual prop and hinges.

6. Acknowledgement345

The research work presented is funded by the European Commission (EC) within346

the BATWOMAN Initial Training Network (ITN) of Marie Sk lodowska-Curie action,347

under the seventh framework program (EC grant agreement no. 605867). The348

authors thank the Network for the generous funding and support. The research has349

been further supported by the Lise-Meitner Fellowship M1653-N30 and the stand-350

alone project P29386 of the Austrian Science Fund (FWF) attributed to Antoine351

Chaigne.352

The authors also thank Werner Goebl for his guidance on using the CEUS Re-353

producing System and Alexander Mayer for his contribution in setting up the ex-354

periment.355

References356

[1] A. Thorin, X. Boutillon, J. Lozada, X. Merlhiot, Non-smooth dynamics for an efficient simu-357

lation of the grand piano action, Meccanica (2017) 1–18 doi:10.1007/s11012-017-0641-1.358

[2] R. Masoudi, S. Birkett, Experimental validation of a mechanistic multibody model of a ver-359

tical piano action, Journal of Computational and Nonlinear Dynamics 10 (6) (2015) 061004–360

061004–11.361

15

0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x (m)

y(m

)

50

60

70

(a) Note A6

0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x (m)

y(m

)

50

60

70

(b) Note A7

Figure 9: Pressure map in the lid plane in dB (re 2.0 10−5 Pa) for the two notes A6 (1771 Hz) andA7 (3587 Hz).

[3] A. Chaigne, Reconstruction of piano hammer force from string velocity, The Journal of the362

Acoustical Society of America 140 (5) (2016) 3504–3517.363

[4] S. Birkett, Experimental investigation of the piano hammer-string interaction, The Journal364

of the Acoustical Society of America 133 (4) (2013) 2467–2478.365

[5] A. Chaigne, A. Askenfelt, Numerical simulations of piano strings. I. a physical model for a366

struck string using finite difference methods, The Journal of the Acoustical Society of America367

95 (2) (1994) 1112–1118.368

[6] J. Chabassier, S. Imperiale, Stability and dispersion analysis of improved time discretization369

for simply supported prestressed Timoshenko systems. Application to the stiff piano string,370

Wave Motion 50 (3) (2013) 456–480.371

[7] A. Stulov, D. Kartofelev, Vibration of strings with nonlinear supports, Applied Acoustics 76372

(2014) 223–229.373

[8] A. Chaigne, B. Cotte, R. Viggiano, Dynamical properties of piano soundboards, The Journal374

of the Acoustical Society of America 133 (4) (2013) 2456–2466.375

[9] J. Berthaut, M. Ichchou, L. Jezequel, Piano soundboard: structural behavior, numerical and376

experimental study in the modal range, Applied Acoustics 64 (11) (2003) 1113–1136.377

[10] R. Corradi, S. Miccoli, G. Squicciarini, P. Fazioli, Modal analysis of a grand piano soundboard378

at successive manufacturing stages, Applied Acoustics 125 (2017) 113–127.379

[11] H. Suzuki, Vibration and sound radiation of a piano soundboard, The Journal of the Acoustical380

Society of America 80 (6) (1986) 1573–1582.381

[12] H. Suzuki, I. Nakamura, Acoustics of pianos, Applied Acoustics 30 (1990) 147–205.382

doi:10.1016/0003-682X(90)90043-T.383

16

[13] P. Derogis, R. Causse, Computation and modelisation of the sound radiation of an upright384

piano using modal formalism and integral equations, in: Proceedings of the 11th ICA, Vol. 3,385

Trondheim, Norway, 1995, pp. 409–412.386

[14] J. Chabassier, A. Chaigne, P. Joly, Modeling and simulation of a grand piano, The Journal387

of the Acoustical Society of America 134 (1) (2013) 648–665.388

[15] N. Giordano, M. Jiang, Physical modeling of the piano, EURASIP J. Appl. Signal Process.389

2004 (2004) 926–933. doi:10.1155/S111086570440105X.390

URL http://dx.doi.org/10.1155/S111086570440105X391

[16] K. Wogram, Acoustical research on pianos, Vibrational characteristics of the soundboard, Das392

Musikinstrument 24 (1980) 694–702, 776–782, 872–880.393

[17] I. Nakamura, The vibrational character of the piano soundboard, in: Proceedings of the 11th394

ICA, Vol. 4, 1983, pp. 385–388.395

[18] A. Mamou-Mani, J. Frelat, C. Besnainou, Numerical simulation of a piano soundboard under396

downbearing, The Journal of the Acoustical Society of America 123 (4) (2008) 2401–2406.397

[19] K. Ege, X. Boutillon, M. Rebillat, Vibroacoustics of the piano soundboard:(non) linearity398

and modal properties in the low-and mid-frequency ranges, Journal of Sound and Vibration399

332 (5) (2013) 1288–1305.400

[20] X. Boutillon, K. Ege, Vibroacoustics of the piano soundboard: Reduced models, mobility401

synthesis, and acoustical radiation regime, Journal of Sound and Vibration 332 (18) (2013)402

4261–4279.403

[21] B. Trevisan, K. Ege, B. Laulagnet, A modal approach to piano soundboard vibroacoustic404

behavior, The Journal of the Acoustical Society of America 141 (2) (2017) 690–709.405

[22] F. Brau, The Touching Sound - Part 1, The Resonance Case Principle, Bosendorfer, the406

magazine by Bosendorfer Austria (2008) 14–15.407

[23] N. B. Roozen, Q. Leclere, C. Sandier, Operational transfer path analysis applied to a small408

gearbox test set-up, in: Proceedings of the Acoustics 2012 Nantes Conference, 2012, pp.409

3467–3473.410

[24] D. De Klerk, M. Lohrmann, M. Quickert, W. Foken, Application of operational transfer411

path analysis on a classic car, in: Proceeding of the International Conference on Acoustics412

NAG/DAGA 2009, 2009, pp. 776–779.413

[25] J. Putner, H. Fastl, M. Lohrmann, A. Kaltenhauser, F. Ullrich, Operational transfer path414

analysis predicting contributions to the vehicle interior noise for different excitations from the415

same sound source, in: Proc. Internoise, 2012, pp. 2336–2347.416

[26] M. Toome, Operational transfer path analysis: A study of source contribution predictions at417

low frequency, Master’s thesis, Chalmers University of Technology, Sweden (2012).418

URL http://publications.lib.chalmers.se/records/fulltext/162546.pdf419

[27] J. Tan, A. Chaigne, A. Acri, Contribution of the vibration of various piano components in420

the resulting piano sound, in: Proceedings of the 22nd International Congress on Acoustics,421

2016, pp. ICA2016–171.422

[28] D. de Klerk, A. Ossipov, Operational transfer path analysis: Theory, guidelines and tire noise423

application, Mechanical Systems and Signal Processing 24 (7) (2010) 1950–1962.424

[29] A. Diez-Ibarbia, M. Battarra, J. Palenzuela, G. Cervantes, S. Walsh, M. De-la Cruz, S. Theo-425

dossiades, L. Gagliardini, Comparison between transfer path analysis methods on an electric426

vehicle, Applied Acoustics 118 (2017) 83–101.427

[30] P. Gajdatsy, K. Janssens, W. Desmet, H. Van Der Auweraer, Application of the transmis-428

sibility concept in transfer path analysis, Mechanical Systems and Signal Processing 24 (7)429

(2010) 1963–1976.430

[31] C. Sandier, Q. Leclere, N. B. Roozen, Operational transfer path analysis: theoretical aspects431

and experimental validation, in: Proceedings of the Acoustics 2012 Nantes Conference, 2012,432

pp. 3461–3466.433

[32] R. Penrose, A generalized inverse for matrices, Mathematical Proceedings of the Cambridge434

Philosophical Society 51 (3) (1955) 406–413.435

17

[33] N. Maia, J. Silva, A. Ribeiro, The transmissibility concept in multi-degree-of-freedom systems,436

Mechanical Systems and Signal Processing 15 (1) (2001) 129–137.437

[34] G. Golub, C. Van Loan, Matrix computations, Vol. 3, JHU Press, 2012.438

[35] E. Blackham, The physics of the piano, Scientific american 213 (6) (1965) 88–95.439

[36] W. Goebl, R. Bresin, I. Fujinaga, Perception of touch quality in piano tones, The Journal of440

the Acoustical Society of America 136 (5) (2014) 2839–2850.441

[37] A. Chaigne, J. Kergomard, Acoustics of Musical Instruments, Springer-Verlag, New York,442

2016.443

[38] D. W. Martin, W. Ward, Subjective evaluation of musical scale temperament in pianos, The444

Journal of the Acoustical Society of America 33 (5) (1961) 582–585.445

18