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Acoustics of the Sagrada Familia Church and the Gaudi's bells

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A Catalan architect, Antoni Gaudí (1852-1926) seems to have envisaged the Sagrada Família as a gigantic musical instrument. Each tower seems to be designed as a belfry, in which many tubular bells will be installed. Its upper section of the Nativity Façade has numerous windows with characteristic louvers. The lower section is almost closed and connected to the nave through muffler-like structures. Therefore, Gaudí's bell music is to be propagated to the navethrough the internal passage as well as to be radiated to the exterior space. Acoustic radiation from the windows and propagation through the lower structures are simulated by two-dimensional FDTD (Finite-DifferenceTime-Domain) method. The window yields a broad directivity pattern in lower frequencies (63 Hz –250 Hz) and a beam pattern in higher frequencies (500 Hz – 2 kHz). A beam pattern consisting of several sharp lobes is similar to that formed by a linear array consisting of several point sources. In Gaudí's windows, both edges of the louvers and the gaps between the louvers yield secondary sources for the beam-like radiation. Six receiving points along a 1/25-scaled model of the lower structure indicate interactions between the direct sound from the top of the lower section and the reflection from the floor. Also, an impulse-response measurement on this model suggests that an attenuation of 40 dB is estimated above 100 Hz and that the frequencies below 110 Hz are strongly cut off. This cutoff frequency can be estimated from the baffled-piston radiation. In addtion, numerical simulations exibit that the curved ceiling in cylindrical rooms can serve to reduce the reverberation time but cannot yield a stronger attenuation. The vibrational and radiational characteristics of a miniture bell similar to a Gaudí's tubular bell are investigated by the FE (Finite-Element) method and the acoustic measurement in an anechoic room. The measued radiational pattern indicates a good agreement with the simulated result.
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ISMA 2007 Acoustics of the Sagrada Familia Church
ACOUSTICS OF THE SAGRADA FAMÍILIA CHURCH AND THE GAUDÍ’S
BELLS: JOINT RESEARCH BETWEEN POLYTECHNICAL UNIVERSITY
OF CATALUNYA AND KYUSHU UNIVERSITY
Shigeru Yoshikawa
Department of Acoustic Design, Faculty of Design, Kyushu University
4-9-1 Shiobaru, Minami-ku, Fukuoka 815-8540 Japan
shig@design.kyushu-u.ac.jp
Abstract
A Catalan architect, Antoni Gaudí (1852-1926) seems to have envisaged
the Sagrada Família as a gigantic musical instrument. Each tower seems to
be designed as a belfry, in which many tubular bells will be installed. Its
upper section of the Nativity Façade has numerous windows with
characteristic louvers. The lower section is almost closed and connected to
the nave through muffler-like structures. Therefore, Gaudí’s bell music is
to be propagated to the nave through the internal passage as well as to be
radiated to the exterior space. Acoustic radiation from the windows and
propagation through the lower structures are simulated by two-
dimensional FDTD (Finite-Difference Time-Domain) method. The win-
dow yields a broad directivity pattern in lower frequencies (63 Hz –250
Hz) and a beam pattern in higher frequencies (500 Hz – 2 kHz). A beam
pattern consisting of several sharp lobes is similar to that formed by a
linear array consisting of several point sources. In Gaudí’s windows, both
edges of the louvers and the gaps between the louvers yield secondary
sources for the beam-like radiation. Six receiving points along a 1/25-
scaled model of the lower structure indicate interactions between the direct
sound from the top of the lower section and the reflection from the floor.
Also, an impulse-response measurement on this model suggests that an
attenuation of 40 dB is estimated above 100 Hz and that the frequencies
below 110 Hz are strongly cut off. This cutoff frequency can be estimated
from the baffled-piston radiation. In addtion, numerical simulations exibit
that the curved ceiling in cylindrical rooms can serve to reduce the rever-
beration time but cannot yield a stronger attenuation. The vibrational and
radiational characteristics of a miniture bell similar to a Gaudí’s tubular
bell are investigated by the FE (Finite-Element) method and the acoustic
measurement in an anechoic room. The measued radiational pattern
indicates a good agreement with the simulated result.
INTRODUCTION
The construction of the Sagrada Família Church began in 1882. Gaudí was
commissioned with the direction of the work in 1883 shortly after the resignation of the
architect Francesc de P. Villar. However, Gaudí died by traffic accident in 1926 at the
first stage of the huge Sagrada Família Project. In 1936 the Civil War destroyed the
Gaidí’s studio and workshop, from which Gaudí’s original materials were lost.
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Nevertheless, the construction is still going on. It is thus important for the present and
future generations to look for Gaudí’s original intention based on the surviving things
and matters as well as to create new ideas and incorporate them into the construction.
According to the ground plan of the Sagrada Família [Bonet, 2000; Campos,
2002], three facades (the Nativity, the Passion, and the Glory Façade) are to be
constructed facing to the east, west, and south, respectively. Each Façade consists of the
four bell towers. The height of the longest tower exceeds 100 meters. The Nativity
Façade was almost completed by Gaudí himself, and the Passion Façade was completed
by the next-generation architects. The Glory façade has not been completed yet. The
altar faces the north. Furthermore, six main towers will rise over the central nave, which
is surrounded by the side aisles. The 12 towers rising from the three facades are the
belfries. It is said that Gaudí planed to install many bells and transmit bell music to the
city of Barcelona. In fact, Gaudí carried out the propagation experiment using a model
bell in 1914. His acoustical image and concept can be inferred from Figs. 1 and 2
[Campos, 2002]. Also, it is designed to lead this bell music to the central nave in the
Sagrada família. Although the model bell that was used at the propagation experiemnt
still remains, it is doubtful if the shape of this tubular bell (an almost straight tube plus a
triangular-like enlarging frange at the end is the final image of Gaudí’s tubular bell.
Since there seemed to be a lot of interesting topics on the acoustical aspects of
the Sagrada Família, a framework agreement for the research collaboration was
concluded in June of 2003 between the three: The Polytechnical University of Cata-
lunya (Depts. of Architectural Presentation-I, Architectural Technology-I, and Mecha-
nical Engineering), the Kyushu University (Faculty of Design, formerly the Kyushu
Institute of Design), and the Gaudí Club Cultural Association (Barcelona). However,
the research object is very huge and complicated, in addition Gaudí’s original material
has been lost. Nevertheless, our collaboration has been managed with mutual research
interests. Although this paper describes most results yielded during the first stage at the
Kyushu University, various informative supports provided from the UPC and the Gaudí
Club as well as mutual discussions are involved.
Fig. 1. Image of sound radiation
from the Sagrada Família belfries
[Campos, 2002].
Fig. 2. Image of bell sound transmis-
sion through the louvered windows
[Campos, 2002].
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BASIC STRUCTURE OF THE NATIVITY FAÇADE
The Façade of Nativity, which was constructed by Gaudí, is divided into two
both horizontally and vertically. Figure 3 depicts the cross-sectional view. The Façade
is vertically divided at the central surface passing through the choir loft. That is, the
Façade has twin-tower structure. We thus consider the left half. Also, the Façade may
be horizontally divided into the upper and the lower structure. The upper structure,
which is enclosed by a blue rectangle in Fig. 3, has many windows with characteristic
louvers. The bell sound is radiated from these windows. The lower structure, which is
enclosed by a red rectangle, consists of many cylindrical rooms through which the bell
sound is conveyed to the nave. The cylindrical rooms, which have very few windows,
are connected vertically and horizontally. Particularly, the vertical connection is done
with a short cylindrical neck. This kind of passage made of constrictions and enlarge-
ments might function as an acoustic filter or an acoustic attenuator (silencer) [Kinsler et
al., 2000].
Also, the choir loft is con-
structed at the height of about 30
meters. Therefore, we may
assume that Gaudí envisaged the
music of bells and chant in the
central nave. In other word, the
Sagrada Família was considered
as a kind of huge musical inst-
ruments, whose music was to be
heard both outside and inside.
The objective of this paper is to
examine the acoustical properties
of the belfries and bells based on
numerical simulations and model
experiments on the radiation and
propagation.
Specifically speaking,
acoustic radiation from the
windows in the upper section
enclosed by a blue rectangle (see
Fig. 3) and acoustic propagation
through the uppermost cylinders
in the lower section enclosed by
a red rectangle will be con-
sidered by introducing many
simplifications. The geometries
of the interested sections are
determined from a reference
book [Tanaka, 1987], where the
detailed structures and configura-
tions of the windows and cylin-
drical passages are drawn. How-
ever, since it is difficult to assume
the installation of the bells, only a
model bell is considered.
Fig. 3. Cross section of the Nativity Facade.
The sections enclosed by a blue rectangle and a
red rectangle are the present reserach objects.
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A
B
A
B
ACOUSTIC RADIATION FROM THE LOUVERED WINDOWS
Basic structure of the belfry windows
The belfry windows are shown in Fig. 4 (a) and the window section enclosed by
a blue rectangle in Fig. 3 is illustrated in Fig. 4 (b) [Tanaka, 1987]. Basically, this
window section consists of two types. Type A is made of four to six smaller windows
with louvers and is arranged along the inner spiral stairs. Type B is made of three or
four larger windows with louvers. Such an arrangement of Types A and B is rotating up
along the parabolic tower in the interval of about 30 degrees. Also, the louver has a
slant angle of about 45 degrees, and we may expect the downward radiation of the bell
sound as depicted in Fig. 2. Moreover, the louver surface is not flat but corrugated,
showing Gaudí’s characteristic design. This corrugation might bring any acoustical
effect to the radiation.
Numerical simulations using FDTD method
The 2-D (two-dimensional) FDTD (Finite-Difference Time-Domain) method
[Suzuki et al., 2007; Bottledooren, 1994] is applied to our numerical calculation. This
FDTD method is more suitable for the visualization than the FE (Finite- Element) and
BE (Boundary-Element) methods. The acoustic pressure and the particle velocities (in x
and y directions) are discretized by placing a Staggered-Grid mesh. The equations of
motion and the equation of continuity are expressed by the finite-difference scheme.
Fig. 4. The bell tower windows. (a): A photo showing the windows with slant
louvers; (b): Cross-sectional view of the window section in Fig. 3 [Tanaka, 1987].
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0.70
0.80
0.85
0.84
0.52
1.22
1.50
1.60
0.83
5.13
3.71
Type-A Type-B
(m)
(m)
7.0 7.0
3.0
14.0
5.0
10.0
3.03.0
O1
O3
O2
(m) (m)
Specifically, the Matlab source code in 2-D FD equations is created but it is omitted
here for saving the space (cf. [Suzuki et al., 2007]).
Computational assumptions and conditions
Since it is difficult to analyse a series of windows shown in Fig. 4, only Type A
or Type B is treated separately as depicted in Fig. 5. For example, Type A is put in the
region for computation as indicated in Fig. 6, which has the area of 14 m × 14 m, and
the radiated sound pressure level is computed in the meshed area (10 m × 5 m) in steps
of 0.5 m × 0.5 m. The edges of the right-half area are guarded by a quasi-reflection-free
layer [Yokota et al., 2002]. The grid widths in the x and y directions are respectively
given as x = 0.01 m and y = 0.01 m. The sampling time is given by t =x /1.5c =
19.6 µs, where c denotes the sound speed. This t satisfies the stability condition.
The sound source pIN is simplified as the following Gaussian pulse: pIN = exp{ -
[(t – 30t)/(8t)]2}. The frequency characteristic of this pulse is almost flat up to 3 kHz.
The symbols O1, O2, and O3 in Fig. 6 denote the assumed source positions. Also, the
wall is assumed upside and downside the window as shown in Figs. 5 and 6. Acoustic
impedance of the normal incidence to the wall surface is assumed to be 50 times the air
impedance. This wall impedance corresponds to the reflection coefficient of about 0.96
when defined by the acoustic pressure. The magnitude of the corrugation (凹凸) made
on the actual louver surface is about 0.3 to 0.5 meters.
Simulations of sound propagation to the exterior
The acoustic propagation is visualized in Figs. 7 (a) and (b) after the above
Gaussian pulse is generated at point O2 and O3 (cf. Fig. 6), respectively. The window is
Type A with the surface-corrugated louvers. At t = 10 ms after the pulse generation, we
can clearly see the wave front passing through the space between the louvers in Fig. 7
(a), however, the pulse has just arrived at the lowest louver in Fig. 7 (b). Also, the
circular wave fronts of the reflecting waves are very clear, and the sources of these
waves are the left-side edges of the louvers. At t = 15 ms the situation of sound
propagation is already considerably complicated. Observing the wave fronts in detail,
we may understand that the passages between the louvers and the right-side edges of
the louvers are working as the secondary sources. Since the phase difference between
these secondary sources is ambiguous, it may be assumed that several point sources
Fig. 6. Point-source position (O1, O2,
or O3) and calculation area (meshed).
Fig. 5. Geometry of the separated windows
used for simulations [Tanaka, 1987].
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with the opposite phases are formed at the window in an average sense. It might be
considered that a kind of linear array is formed along the louvered window. As a result,
the sound propagates toward almost all directions as shown in (a) at the frame of t = 20
ms. On the other hand, when the source lies at O3, sound propagation is roughly divided
into two directions as depicted in (b). Particularly, we have to notice that a relatively
large energy is propagated into the upward direction. Also, a strong reflection toward
the interior space is observed in (b) compared with (a). We can thus recognize that both
edges of the louvers form the secondary sources for the outward radiation and for the
inward reflection, respectively. The upward and downward propagations are a little
retarded than the frontal propagation in (a), and thus the reflections between the
louvers are suggested (see Fig. 8). When the pulse source is located at O1, the image
illustrated in Fig. 2 is applicable.
Fig. 7. Sound propagation through the louvered window (Type A with the surface-
corrugated louvers). (a): pulse source at O2; (b) pulse source at O3.
Fig. 8. Schematic on the multiple reflections between the louvers.
Such multiple reflections cause the upward radiation.
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63Hz
500Hz
250Hz
125Hz
1kHz
2kHz
(dB)100-20 -10-30
(a)
63Hz
250Hz
125Hz
500Hz 1kHz
2kHz
(dB)100-20 -10-30
(d)
63Hz 125 Hz
250Hz
500Hz 1kHz
2kHz
(dB)100-20 -10
-30
(b)
63Hz
250Hz
125Hz
500Hz 1kHz
2kHz
(dB)100-20 -10-30
(c)
Frequency dependence of the radiation directivity
The frequency dependence of the acoustic radiation is illustrated in Fig. 9 when
the Gaussian pulse is generated at point O2 or O3 and the window type is A or B. The
responses in 1/3-octave bands whose central frequencies are 63, 125, 250, 500, 1000,
Fig. 9. Frequency dependence of acoustic radiation field. (a): window type A with
louver corrugations, source O2; (b): type A, O3; (c) type B, O2; (d): type B, O3.
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and 2000 Hz are demonstrated. When the sound source lies at O2, relatively similar
radiation patterns are shown regardless of the window type [cf. Figs. 9 (a) and (c)]. On
the other hand, when the source lies at O3, the radiation pattern depends on the window
type as well known from the 125-Hz frames in (b) and (d). These radiation directivity
patterns can be inferred from the impulse response patterns illustrated in Fig. 7.
Particularly, it should be noted that the upward radiation is stressed in (b) except the
250-Hz frame. Also, the pressure level is very low in (d) except the 63-Hz frame.
Summing up, the lower source position O3 relative to the window center is inappropri-
ate to propagate the bell sounds to the people on the ground (cf. Fig. 8).
Roughly speaking, broad directivity patterns are indicated in 63, 125, and 250
Hz. The wavelength of the 125-Hz component is 2.7 m, and the window lengths are 3.7
m and 5.1 m. Thus, the radiation directivity is not sharp in these lower frequencies.
However, at 500 Hz a beam-like directivity pattern is indicated. There seem to be five
beams. Also, at 2 kHz many speckles due to the interference are recognized, particular-
ly in (a). These radiation characteristics in higher frequencies (0.5 – 2.0 kHz) may be
interpreted as the results of so-called “line array” formed along the window. The
number of (secondary) sources arranged along this linear array can be varied according
to the frequency and the louver interval.
Comparing the results on the louvers with and without the surface corrugation,
we know that there is no appreciable difference in the directivity pattern below 2 kHz,
although the sound pressure level in the case of the louvers with the corrugation is
reduced by a few dBs only in higher frequencies. See Ref [Nishimoto and Yoshikawa,
2006] for more detailed information and discussion.
SOUND PROPAGATION THROUGH THE BELFRY LOWER STRUCTURE
Measurement on a 1/25-scaled model of the lower structure
According to Gaudí’s plans that were actually measured by Dr. Hiroya Tanaka
of the Gaudí Club Cultural Association in Barcelona, a simplified 1/25-scaled model of
the lower structure was made of acrylic resin (Plexiglas). This model corresponds to the
section enclosed by a red rectangle shown in Fig. 3. The details in structure are simpli-
fied, but major geometries are correct. Figure 10 shows (a) a drawing by the computer
Fig. 10. A 1/25-scaled model of the lower structure. (a): computer graphics;
(b): experimental model. Size: 0.5 × 0.8 × 1.7 m. Actual height: about 45 m.
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graphics and (b) a photo of the acrylic resin model. It is well demonstrated that the
lower structure consists of five or six rooms, which are vertically connected by short
cylindrical necks. Moreover, each room has a few small openings. The openings in
lower rooms may collect the bell sound and radiate it toward the nave space. In this
section attenuation effects of the lower structure are examined experimentally. However,
we have no intention to carry out the scaled-model experiment in its rigorous meaning.
Since our experimental model was
scaled in 1/25, acoustical measurement was
carried out in low- and high-frequency
ranges using different speaker systems.
According to the frequency characteristics
of the measuring systems, the measurement
was feasible in a lower frequency range of
500 Hz to 17.5 kHz (20 Hz to 700 Hz in
actual scale) and in a higher frequency range
of 15 kHz to 50 kHz (600 Hz to 2 kHz in
actual scale). The result measured in the
lower frequency range is illustrated in Fig.
11. The speaker is set up on the top of the
lower tower (the outward tower when the
belfries are seen from the outside, cf. Fig. 3).
A TSP (Time-Stretched Pulse) signal is used
as the input signal. The sampling frequency
is 48 kHz and the DFT points are 216.
Alphabets “a” to “k” shown in Fig. 11
denote the measurement points in the model.
The measurement result of Fig. 11
shows relatively flat frequency character-
istic when the frequency higher than about
100 Hz is considered. If an octave band of
the centre frequency 1 kHz is examined, the
acoustic pressure received at point “f” is
weaker than that at point “a” by 37 dB. The
response in the higher frequency range (600
Hz to 2 kHz in actual scale) indicates
similar flat attenuation, which can be
attributed to the diffusive-like field formed
in large rooms. Since the speaker is not
omnidirectional, this level difference of 37
dB should be corrected and we may possibly
expect the attenuation of over 40 dB
between 100 Hz and 2 kHz. The attenuation
level at each point in Fig. 11 is also calcul-
ated by considering the lower structure as
the coupling rooms on the hypothesis of the
diffusive field. The result (not shown here)
confirms an increase in the attenuation level.
Spectral attenuation of about 40 dB between
about 100 Hz and about 2 kHz seems to be a
reasonable result.
Fig. 11. Sound propagation through the
lower structure model (0 – 700 Hz).
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Low-frequency cutoff
On the other hand, the frequency response is strongly cut off below about 100
Hz as clearly shown in the data of points “a”, “b”, and “c” in Fig. 11. A series of the
enlargement and constriction in cross section can be considered as the acoustic low-
pass filter [Kinsler et al., 2000], but it is correct only if the change in cross section is
small. In Fig. 11 the left, top room consists of an opening neck of diameter 1.5 m and a
cylinder of diameter 4 m. We may thus assume the situation that the sound is radiated
from a vibrating piston surrounded by an infinite baffle. Low-frequency radiation is
very weak in this configuration as estimated from the radiation impedance. As a result
the cutoff frequency is estimated by “wavelength = 2 × diameter” and is then appro-
ximately given as fcutoff = 110 Hz. This theoretical result seems to support the validity
of the measured result.
However, it should be noted there are sharp responses below 50 Hz as shown in
“a” to “c” in Fig. 11. Although these responses might be due to the Helmholtz reson-
ance, the calculated result does not well agree with the measured result. Anyway, the
frequency components below about 100 Hz are strongly attenuated (cf. “a” and “f” in
Fig. 11). Therefore, it may be concluded that bell sounds below 100 Hz will not be well
heard in the central nave of the Sagrada Familia Church.
Numerical simulations by 2-D FDTD method
Since the height of the upper structure is about 40 m, the bell sound can be
propagated to the opening of room “a” as the plane wave in the first approximation.
Although the data are not shown here, the following results are obtained:
(1) The impulse responses suggest the significant influence of the reflection at the floor.
(2) The low-frequency cutoff is seen near 200 Hz. The discrepancy with the experi-
mental value (100 Hz) may be due to the 2-D assumption for the simulation.
(3) The attenuation between 200 Hz and 2 kHz is about 20 dB in average. This very
low attenuation may be also due to the 2-D assumption.
Furthermore, the effects of the curved ceiling seen in actual structure are
simulated by comparing with the flat ceiling. The reverberation time of a room with a
curved ceiling is 0.35 s and that of a room with flat ceiling is 0.42 s. This is possibly
because the curved ceiling can diverge the wave direction. Such a reduction of the
reverberation time might be desirable to hear bell music in the nave. However,
simulations on the curved ceiling in the configuration of Fig. 11 do not indicate any
appreciable difference in the attenuation characteristics. See Refs. [Narita and
Yoshikawa , 2005] and [Yoshikawa and Narita, 2004] for more detailed discussions.
ACOUSTIC RADIATION CHARACTERISTICS OF GAUDÍ’S BELLS
It is not exactly known what bells Gaudí envisaged. The shape of the tubular bell
used for the propagation experiment in 1914 does not seem to be his final design. Gaudí
would adopt a parabolic or hyperbolic envelope as his tubular bells. For example, one of
the Catalan musical instruments, the gralla (double-reed woodwind) has the Gaudí-like
geometry as shown in Fig. 12 [Tanaka, 2004]. Probably Gaudí liked such a musical
instrument. Also, miniature bells with the geometry similar to the gralla were designed
by Tanaka and made by the Koizumi Factory Inc. a few years ago [Koizumi, 2006]. One
of them, which is indicated in Fig. 13, was used for our experiment on the radiation
characteristics.
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Fig. 12. The gralla. Fig. 13. Miniature Gaudí bell.
Experimental setup
Two kinds of microphone array are constructed: a linear array parallel to the bell
axis and a semi-circular array surrounding the bell axis at the center. Our experimental
setup is shown in Fig. 14, where a miniature Gaudí bell (287 mm long, 434 gram) and a
linear array (13 microphones are located with an interval of 30 mm and the top one
corresponds to the bell top) are arranged in an anechoic room of Kyushu University.
The positions of this vertical array, that is, the measurement points are 60, 90, 120, 180,
240, 300, 420, 540, 720, 900, and 1080 mm distant from the bell top edge. A semi-
circular array has its radius of 970 mm and 13 microphones are arranged. The measure-
ment is carried out at the height of 127 mm and 237 mm from the bell top and also 37
mm and 200 mm below the bell bottom. The microphone outputs are connected to the
lab. PC via the LMS Test Lab Mobile, which amplifies the microphone signals and
simultaneously digitize the amplified signals. The bell is struck by an experimenter. The
waveforms of the impact magnitude and its frequency characteristic measured by a load
cell put on the hammer head are observed to confirm almost the same impact. The
impact positions are 77, 142, 162, and 275 mm from the bell top
Experimental results
Only the result on the linear array is demonstrated. The analysis of bell tones
gives the peak responses at 483, 1320, 1672, and 2412 Hz. These frequencies corres-
pond to the first to the fourth mode frequencies as estimated below. The radiation
patterns measured at these frequencies are shown in Fig. 15, where the SPL is plotted
and the contour map of equal SPL is drawn. Each frame is interpreted as follows:
(a) 483 Hz, (2, 0) mode with two nodes along the free-free rod: The radiated pressure is
very low near 6 and 21 cm along the bell, and these two points correspond to the
nodal points. Three loops are confirmed near both ends and 12 cm from the top.
(b) 1320 Hz, (3, 0) mode: There seem to be three nodes near 3, 15, and 24 cm, although
the separation (resolution) is poor near both edges because of small numbers of
microphones. The frequency ratio of 1320 Hz to 483 Hz is 2.733. This value is very
close to 2.758 for the free-free rod vibration.
Fig. 14. Experimental setup to measure
the radiation field of the bell.
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(c) 1672 Hz, (4, 0) mode: the node positions might be estimated near 3, 9, 15, and 27
cm. The radiation of this (4, 0) mode is significantly weaker than that of (3, 0) mode.
This may be because the node near 15 cm in (3, 0) mode suppresses the loops
around 15 cm in (4, 0) mode. This explanation may be applicable to (2, 0) mode.
The frequency ratio of 3.462 is much smaller than 5.406 for the free-free rod.
(d) 2412 Hz, (5, 0) mode: Two nodes near 9 and 18 cm are clear, but the resolution is
too low to separate five nodes from each other.
The lateral directivity pattern (measured by a vertical line array) and the radial
directivity pattern (measured by a semi-circular array) show a fairly good agreement
with those simulated by the FE and BE methods [Horimoto et al., 2006].
The pitch sensation of the strike note for our Gaudí bell seems to be the lowest
frequency of (2, 0) mode. This is identified as the “spectral pitch” [Terhardt and
Seewann, 1984]. However, simple tubular bells (the wind chime made of aluminum)
tuned G5 = 781 Hz give the pitch as the spectral pitch determined by the (4, 1) mode
[Yoshikawa, 2005]. The pitch of the strike note is a complicated problem.
CONCLUSIONS
Some suggestions may be derived from our joint research. (1) The relative
position between the louvered window and the acoustic center of the bell is important to
the outward radiation toward the people on the ground. (2) The attenuation of the bell
sound is roughly estimated as 40 dB when heard in the nave. (3) Very-low-pitched bells
are probably useless due to the cutoff frequency (at around 100 Hz) of the lower
structure. (4) The reverberation in the nave might be reduced by scattering sounds
caused by the floor reflection. The acoustical research on the Sagrada Família Church
and Gaudí’s bells should be much developed in the future.
Fig. 15. Lateral radiation field of a miniature Gaudí bell. (a): 483 Hz;
(b): 1320 Hz; (c): 1672 Hz; (d): 2412 Hz.
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ACKNOWLEDGMENTS
The author would like to express his thanks to Prof. Joaquim Agulló, Prof.
Joaquim Lloveras, and Prof. Francesc Daumal of the Polytechnical University of Cata-
lunya and Dr. Hiroya Tanaka of the Gaudí Club in Barcelona for their sincere interest
and help to the joint research on the Sagrada Família Church since 2003. It is grateful
that this research was financially supported by the Japan Society for the Promotion of
Science from 2003 to 2004 fiscal year. Furthermore, the author is grateful to Prof.
Takeshi Toi and his students of the Chuo University in Tokyo for their support and
advice to the simulations and experiment on the Gaudí bell. Finally, it is noted that this
research was not developed without the lasting enthusiasm and endeavor of Takafumi
Narita, Yasuko Nishimoto, and Kazusa Horimoto in author’s laboratory.
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ResearchGate has not been able to resolve any citations for this publication.
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In this paper, we propose a simple method that considers boundary conditions in a finite difference time domain (FDTD) scheme by varying density, sound speed and flow resistance. A method based on a Rayleigh model is also proposed, and by these methods, we can design the frequency characteristics of normal incident absorption coefficient arbitrarily. These methods have three advantages: 1. easy coding, 2. easy designing of a frequency characteristic of normal incident absorption coefficient and 3. easy configuration of material thickness. For example, by our method, we can simulate the sound field in a reverberation chamber with a thick material such as glass wool. To confirm the accuracy of the model used, we compare the normal incident absorption coefficient with a one-dimensional exact solution. Results show that the model is sufficiently accurate. Although our method requires a high cost for calculation power and memory, a practical increase in elapsed time can be ignored. This method provides an easy way of analyzing the inner region of a material.
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The Sagrada Familia Cathedral in Barcelona, Spain was constructed in 1882. According to Antoni Gaudi, who worked over its grand plan, the Cathedral was supposed to be a huge musical instrument as a whole in the event of completion. As as result, the music of bells was expected to echo through the air of Barcelona from the belfries. However, Gaudi's true intention cannot be exactly known because the materials prepared by him were destroyed by war fire. If his idea of the Sagrada Familia as an architechtural music instrument is true, an acoustical balance should be considered between the roles of the Cathedral: bell music from the belfries and quiet service in the chapel. Basic structure of the Sagrada Familia seems to be an ensemble of twin towers. Following such speculation, we made a simplified acrylic 1/25-scale model of the lower structure of a twin tower located at the left side of the Birth Gate. The higher structure of this twin tower corresponds to the pinnacle where the bells should be arranged. The lower structure (about 43 m in actual height) has five passages connecting two towers. One of two towers includes five or six tandem columns whose ends are both squeezed to about 1.5 m in diameter. These columns seem to function as a kind of muffler. The location and shape of the roof over the nave is indefinite and tentatively supposed at the top of the lower structure. Based on our scale model, acoustical characteristics of the lower twin-tower structure as a muffler and acoustical differences between the exterior field and nave field will be reported and discussed.
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The finite-difference time-domain (FDTD) approximation can be used to solve acoustical field problems numerically. Mainly because it is a time-domain method, it has some specific advantages. The basic formulation of the FDTD method uses an analytical grid for the discretization of an unknown field. This is a major disadvantage. In this paper, FDTD equations that allow us to use a nonuniform grid are derived. With this grid, tilted and curved boundaries can be described more easily. This gives a better accuracy to CPU-resource ratio in a number of circumstances. The paper focuses on the new formulation and its accuracy. The problem of automatically generating the mesh in a general situation is not addressed. Simulations using quasi-Cartesian grids are compared to Cartesian grid results.
On the acoustic radiation characteristics of the bell that Gaudi envisaged in the Sagrada Familia Cathedral -Using a miniature bell
  • K Horimoto
  • M Shinozuka
  • S Yoshikawa
  • Y Shiozawa
  • S Nishida
Horimoto, K., Shinozuka, M., Yoshikawa, S., Shiozawa, Y., Nishida, S., and Toi, T. (2006), "On the acoustic radiation characteristics of the bell that Gaudi envisaged in the Sagrada Familia Cathedral -Using a miniature bell," Tech. Rep. Musical Acoust. Group 25, MA2006-84 (in Japanese).
Acoustical study on the belfries of the Sagrada Familia Cathedral
  • T Narita
  • S Yoshikawa
Narita, T. and Yoshikawa, S. (2005), "Acoustical study on the belfries of the Sagrada Familia Cathedral," Tech. Rep. Musical Acoust. Group 24, MA2005-68 (in Japanese).
Acoustical radiation characteristics from the Sagrada Familia Cathedral belfries
  • Y Nishimoto
  • S Yoshikawa
Nishimoto, Y. and Yoshikawa, S. (2006), "Acoustical radiation characteristics from the Sagrada Familia Cathedral belfries," Tech. Rep. Musical Acoust. Group 25, MA2006-83 (in Japanese).
Gaudi: Architectural works – Drawing of actual measurement
  • H Tanaka
Tanaka H. (1987), Gaudi: Architectural works – Drawing of actual measurement, Shohkoku-sha, Tokyo (in Japanese).
Aural and algorithmic determination of the strike note of historical church bells
  • E Terhardt
  • M Seewann
Terhardt, E. And Seewann, M. (1984), "Aural and algorithmic determination of the strike note of historical church bells," Acustica 54, 129-144.
Vibro-acoustical analysis of tubular bells and the identification of their pitches
  • S Yoshikawa
Yoshikawa, S. (2005), "Vibro-acoustical analysis of tubular bells and the identification of their pitches," Tech. Rep. Musical Acoust. Group 24, MA2005-3 (in Japanese).