operational transfer path analysis of a piano
TRANSCRIPT
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
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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
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