Abstract
Replicating ancient musical instruments is a method to protect fragile originals from extensive playing. In the case of stringed instruments, replicas are generally realized by luthiers using identical wood species and geometry, according to dimensional surveys. Although this procedure yields a highly similar visual aspect, the intrinsic variability of wood properties does not ensure an identical sound. Therefore, acoustic surveys are a fundamental step in reproducing the sounds of original instruments. In this work, we report the acoustical survey of a late baroque mandolin preserved at Museo degli Strumenti Musicali del Castello Sforzesco di Milano. The survey was conducted using portable equipment and included measurements of the radiated sound spectrum, admittance, monopole mobility, and mode shape assignment. Finite Element Analyses (FEA) enabled the assignment of mode shapes and quantification of the effect of a crack on the structural integrity and acoustics of the instrument. This study has laid the foundation for the creation of a replica that, beyond the visual aspect, would resemble the original instrument in terms of sound to the extent feasible.
1 Introduction
1.1 Replicas of Musical Instruments as a Means of Preservation
Ancient musical instruments are an important part of cultural heritage and history. Presently, owing to the rediscovery of ancient music by many musicians and the role of historically informed music performances, iconic eighteenth- century instruments are being sought again. However, few masterpieces have survived to date, and most of these are not in good condition or playable. Being rare and fragile objects, these instruments should be preserved rather than played intensively by musicians. Therefore, the actual trend is to create replicas intended to be played (rather than the original) (Schwarz & Chinnery 2002). Such replicas are generally created by skilled and experienced luthiers and resemble the originals with high accuracy. Nevertheless, the design and structure are occasionally modified according to current constructive and musical criteria and musicians’ specifications. This type of hybridization affects the musical properties of the new instruments. It can also significantly affect their sound, which can differ from that of the original instruments. Moreover, wood property variability implies that an identical acoustic response is not assured even when the geometry is reproduced accurately (French & Brubaker 2007; Zygmuntowicz 2021; Quintavalla et al. 2022b). Therefore, it is almost infeasible to accurately reproduce historical music unless replicas that are accurate from an acoustic perspective are realized.
1.2 Modern Neapolitan Mandolin
Among the many stringed instruments, one of the most significant and successful is the Neapolitan mandolin. This instrument was developed in the first half of the XVIIIth century. It partly replicates the older types of mandolins developed in the XVII century. The first mandolins featured gut strings and a flat soundboard. The first Neapolitan mandolins evolved to address the sound requirements of musicians and composers who initially required mixed gut (treble side) — metal (bass side) strings and subsequently imposed the use of completely metal strings (McDonald 2016). A larger string tension results in key modifications such as the use of a third reinforcing brace and introduction of a bent soundboard (Woll 2021). A definitive configuration was finally established at the beginning of the 19th century. This redesigned model was established as the modern mandolin and has remained almost unaltered. From a musical perspective, several well-known musicians and composers consider the mandolin worthy of the violin, and several compositions feature mandolin solos and orchestras (Bone 2013). Moreover, the mandolin is unique in that it is one of the few instruments considered convenient for women to play in public. Therefore, it represents a means for women’s artistic expression and emancipation (Desiata 2022). Over the past two centuries, mandolins have undergone development and dispersal worldwide. The Italian mandolin tradition continues. At present, thousands of individuals play this instrument and its descendants, for both classical and more contemporary repertoires such as bluegrass or folk (Tyler & Sparks 1996; McDonald 2016).
1.3 Aim of the Study
In this article we report the results of an acoustical survey of a late Baroque mandolin constructed by Neapolitan luthier Joseph Filano in 1784 and preserved at Museo degli Strumenti Musicali del Castello Sforzesco di Milano. This is a well-preserved example of a late Baroque mandolin. It features mixed gut–metal strings tuned by wooden tuning pegs and a bent soundboard. Because of these features and the fabrication date, it is a noteworthy instrument that manifests the evolution from gut to metal strings. This survey was aimed at fabricating a replica of this mandolin in the context of the research project Neapolitan wood craftsmanship of the 18th century: discovering ancient technologies in lutherie. In Section 2, the materials and methods are described. Moreover, the research project that financed this work is introduced briefly. In Section 3, we report the measurements of the acoustical and vibrational properties of the mandolin. These were performed at the museum and validated by Finite Element Analyses (FEA). We also comment on these from the perspective of fabricating an accurate replica of the instrument.
2 Materials and Methods
2.1 World Wood Day Foundation Research Project Background
This work was funded in 2021 by the World Wood Day Research Grant project Neapolitan wood craftsmanship of the 18th century: Discovering ancient technologies in lutherie. This project has three main goals: (1) to investigate the techniques and technologies used in the 18th century and their implications in the design and manufacture of wooden musical instruments, (2) to analyze a few of the original instruments preserved at the Museo degli Strumenti Musicali del Castello Sforzesco di Milano, and (3) to realize a philological replica of one of these instruments. In this article, we present and discuss acoustical characterization techniques deployed to analyze a particular instrument at a museum. The other phases of the research project would be discussed in future studies.
2.2 Mandolin Description
The instrument considered in this study is a late Baroque Neapolitan mandolin constructed by luthier Joseph Filano in Naples in 1784. It is curated at the Museo degli Strumenti Musical del Castello Sforzesco di Milano (inventory number 249) and is a well-preserved example of a late Baroque mandolin. The instrument features eight metal strings and a highly decorated spruce (Picea abies) soundboard. The soundboard is bent by approximately 8°. The soundhole rosette and false inlays (Rovetta et al. 2015) are made of mother of pearl held in place by a resin filler. Meanwhile, the pickguard and the back part of the neck are made of thin strips of tortoise shells. The back of the bowl is made of 18 maple (Acer sp.) ribs, two larger ribs, and a carved end clasp. A photograph of the mandolin is shown in Fig. 1. A detailed list of the materials is presented in Table 1.
Three views of the mandolin built by luthier Joseph Filano in Naples in 1784 and preserved at Museo degli Strumenti Musical del Castello Sforzesco di Milano
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
List of materials used to construct the mandolins
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
During an in situ acoustical survey performed at the Museo degli Strumenti Musicali del Castello Sforzesco di Milano, we identified a crack in the soundboard. An image of the soundboard with an internal light source, shown in Fig. 2, highlights the geometry of the crack running along the grain across the entire thickness of the soundboard. It has a maximum width of approximately 1 mm and length of 120 mm. Although the crack was never reported in previous surveys, the museum curator remembers that the mandolin had cracked since its nomination in 1997 and hypothesizes that the crack was present when the instrument was acquired in 1957. From a close visual inspection, its oxidized edges indicate that it occurred several years ago. This supports the hypothesis. Cracking along the wood grain is typical of old instruments that have been subjected to significant variations in the relative humidity of air, which causes the wood to expand or contract significantly more across the grain than along the grain (Forest Products Laboratory — USDA 2010). Although the relative humidity and temperature in the museum displays were not controlled, the humidity is unlikely to have decreased significantly. The museum is situated in a castle from the 1400s with thick walls in the city of Milan, where the average relative humidity is above 75%. Cracking could be a consequence of significant variations in climatic condition before the acquisition in 1957. However, no information is available regarding the preservation of the instrument before this date. The effect of this crack on the acoustic properties of the instrument is analyzed in detail in Section 3.5.
2.3 Acoustic and Accelerometric Measurements
The acoustic measurements were performed by exciting the soundboard with a small hammer, measuring the sound pressure level (SPL) with a microphone, and measuring the admittance Y with an accelerometer. These measurements quantify how the soundboard radiates sound in the air when compelled to vibrate by strings. Admittance is a measure of the velocity per unit force as a function of frequency. Combined with SPL, it allows for the measurement of the frequency response of an instrument and the frequencies of the modes of vibration of the soundboard. To measure the SPL, an electret microphone attached to a small magnet was placed inside the instrument box. A second small magnet was placed outside the box. This enabled the microphone to be held in place and moved conveniently inside the box. The soundboard was excited by hitting it with a hammer, following the scheme shown in Fig. 3. The instrument sound box functions as a high-pass filter (it attenuates external sounds of frequencies below approximately 2 kHz). Therefore, this technique allows for the acquisition of relatively low-noise measurements of the signature modes (i.e., the low-frequency modes of vibration that are unique to individual instruments) even in the absence of an anechoic chamber. Therefore, although it does not provide a calibrated and completely repeatable measurement, this method is effective for performing acoustical surveys in-situ. This is particularly so for museums, where the available space, available time, and ambient noise are typical limiting factors. Y was measured by hitting the instrument bridge with a calibrated impact hammer (Dytran 5800SL) and measuring the acceleration with an accelerometer (Dytran 3225F). The accelerometer was installed on a soundboard with wax at point 1 (Fig. 3) close to the position of the G string (bass side). It was excited by hitting the bridge bone with an impact hammer significantly close to the accelerometer. This method enables the measurement of the driving-point admittance, namely, the ratio between the velocity measured at a point and the force applied to the same point as a function of the frequency. The measurement is representative of the response of the entire mandolin soundboard (Taguti & Yamanaka 2006).
Picture showing a crack in the mandolin’s soundboard. The crack is illuminated from the inside of the mandolin with a lamp. Arrows show the crack extent
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
Tap testing routine scheme used for the acoustic measurements on the mandolin
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
2.4 Measurement of the Mandolin Soundboard Stiffness and Monopole Mobility
The monopole mobility M is related to the admittance of the first mode of vibration of the soundboard. It indicates the effectiveness with which the soundboard can be compelled to vibrate by strings (Gore & Gilet 2011). M is calculated as
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
where k is the mechanical elastic constant of the soundboard and m* is the effective mass. k is measured experimentally using the apparatus shown in Fig. 4. Four 100 g weights are added subsequently to exert downward pressure on the bridge, and a digital machinist gauge is used to measure the displacement with respect to the sides. The stiffness k is calculated by the linear regression of the force versus displacement relationship. m* is calculated as follows by assuming that the soundboard behaves as a simple harmonic oscillator:
Apparatus for the measurement of the mandolin soundboard mechanical stiffness
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
where ft is the vibration frequency of the soundboard monopole measured by closing the sound hole with a rigid polystyrene foam plug to decouple the soundboard vibration from the instrument sound box Helmholtz resonance (i.e., the resonance of the air enclosed in the instrument box across the soundhole).
2.5 Finite Element Analyses
2.5.1 FEA Software
FEA was performed using COMSOL Multiphysics software (COMSOL, 2023) and its built-in Solid Mechanics Modulus with Stationary study and Eigenfrequency analysis tool. Linearized equations of motion were solved using the built-in MUltifrontal Massively Parallel Sparse direct solver (MUMPS).
2.5.2 FEA Geometry
To perform the simulations, we considered only the soundboard. This was because it is the most important part for the sound radiation of mandolins (Quintavalla et al. 2022b). The soundboard geometry was replicated from a dimensional survey of the mandolin performed in 2016 by luthier Arianna Colombo. An excerpt of the survey showing the soundboard thickness and reinforcement brace geometry is presented in Fig. 5. The geometry was drawn in 3D with a constant thickness (2,38 mm), which was determined by averaging the available measurements (indicated within circles in the survey). It is also illustrated in Fig. 5. A second 3D drawing was performed to understand the effects of the cracks. The details are presented in Section 3.5.
(Left) Excerpt of the dimensional survey of the mandolin reporting the soundboard thickness and the bracing geometry. (Right) 3D reconstruction of the mandolin soundboard: front and back view. Frames of reference are included to highlight wood grain orientation in the FEA analyses
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
2.5.3 Meshing and Constraints
To perform the simulations we created a nonstructured tetrahedral mesh with adaptive refinement using the built-in COMSOL mesh tool. We ran the simulation refining the mesh size unless convergence was achieved (difference of less than 1% in the eigenmode frequencies within successive refinements). A regular mesh size was demonstrated to be sufficiently accurate for determining the mode frequencies and soundboard deformation even in a geometry with a crack. In this case, the mesh comprised elements with minimum and maximum sizes of 0.298 and 23.9 mm, respectively, for a total of 257385 degrees of freedom. An image of the mesh is shown in Fig. 6. In all the simulations, a fixed constraint was applied to the external rim of the soundboard.
Mesh elements used to run the FEA simulations on the mandolin soundboard
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
2.5.4 Materials Definition
We generated two orthotropic materials in COMSOL to represent wood with different grain orientations used for soundboards and braces. Referring to the frame of reference in Fig. 5 and considering the main directions of wood growth (namely longitudinal (L), radial (R), and tangential (T); Forest Products Laboratory — USDA 2010), the material orientation used in the simulations can be summarized as presented in Table 2.
Wood orientation and correspondence between the frame of reference of the simulation and the main direction of wood growth
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
Defining the orthotropic materials for this type of study requires knowledge of their densities and the nine elastic constants. The density and three elastic constants (namely, the longitudinal elastic modulus EL, the radial elastic modulus ER, and a shear modulus GLR, which are highly correlated with the vibrational behavior) were obtained from our previous study on spruce wood (Quintavalla et al., 2022). These are presented in Table 3.
Density and mechanical properties of the wood used for the soundboard and bracing of mandolin 1.
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
E and G are the elastic modulus (Young modulus) and shear modulus, respectively. The subscripts L and R indicate the longitudinal and radial directions, respectively.
The other elastic constants that displayed a marginal correlation with the acoustic properties of the soundboard (i.e., Poisson’s coefficients and the properties related to the tangential direction T) were retrieved from the literature (Forest Products Laboratory — USDA 2010). The soundhole rosette, pickguard, and mother of pearl inlays were considered as a 1.2 mm-thick homogeneous layer of isotropic material with a density of 1500 kg/m3 and elastic modulus of 2 GPa.
3 Results and Discussion
3.1 Soundboard Admittance and Acoustic Radiation
To study the acoustics of the mandolins, we measured Y and the acoustic radiation from the soundboard (details in Section 2). From the collected signals, we measured Y and the SPL, as shown in Fig. 7. The admittance measurements were performed according to the standard techniques reported in Section 2.3. The SPL measurements were performed according to a tap-testing routine developed using a simple and portable device, as described in Section 2.3. This technique enables portability and facilitates museum measurements. Therefore, SPL measurements are reproducible in terms of peak frequencies. However, notwithstanding the adoption of an identical pattern, a good reproducibility of peak intensity cannot be ensured. However, a good similarity between different measurements was observed. Peak frequencies can provide important information regarding the eigenmodes. Similar to other instruments such as violins and guitars it is feasible to distinguish a low-frequency response (up to approximately 1.5 kHz), where the peaks corresponding to the eigenmodes are separated significantly. This range corresponds to the so-called signature modes. The frequencies of all the peaks are reported in Table 4. Because of the crack at the top, it is reasonable to assume that the top stiffness was reduced with respect to the original pristine stiffness. Therefore, the mode frequencies could be lower. This aspect is investigated in detail in Section 3.5. It is also noteworthy that the mode frequencies acquired by admittance measurements were marginally lower than those acquired by SPL. This can be ascribed to the mass of the accelerometer (0.6 g), which was small but not negligible compared with that of the soundboard.
Relative Sound Pressure Level (SPL, top) and bridge admittance Y (bottom) of the mandolin. SPL signal with soundhole plugged is offset by –30 dB for clarity
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
Comparison of frequency peaks of SPL, SPL with soundhole plugged, admittance measurements, and FEA simulations
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
3.2 Soundboard Stiffness and Monopole Mobility
Two important parameters that can be considered while generating an accurate replica of an instrument are the soundboard mechanical stiffness (k) and M. These quantities are described in detail in Section 2.4. These complement the SPL and admittance measurements and can be used as reference during the construction of the instrument replica. k was determined by linear regression of the force vs. displacement relationships (as illustrated in Fig. 8) to be 4.2 × 104 N/m. The calculated effective mass was 7.01 g, and the monopole mobility was 58.4 × 10-3 s/kg.
Plot of force vs. displacement and linear regression used to measure the soundboard stiffness k.
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
3.3 Finite Element Analysis (FEA) of Modal Shapes and Modal Frequencies
The sound radiation strongly depends on the eigenmode shapes (Fletcher & Rossing 1998). To understand the acoustic response of the mandolin, it is important to determine which eigenmode shape corresponds to each signature mode peak (Cohen & Rossing 2003). The mode shapes are typically determined using techniques such as holographic interferometry, laser vibrometry, Chladni patterns, and accelerometric modal analyses (Rossing 2007). However, these techniques are not effective for in-situ measurements in museums because these require complex and non-portable setups, are noisy, or require a substantial length of time. Therefore, we performed an FEA. It is a numerical method used to solve partial differential equations that govern the motion and vibration of objects, structures, and musical instruments (Zienkiewicz et al. 2013). This technique enables the determination of modal shapes and eigenfrequencies as well as deformations under load, among many other analyses (Kaselouris et al. 2022). Details of the simulations are reported in Section 2.5. The results are shown in Fig. 9. The frequencies of all the signature modes gathered by the SPL, admittance measurement, and FEA are listed in Table 4.
The first 12 eigenmodes of the mandolin soundboard simulated by Finite Element Analysis. Simulations were run on a geometry that includes the crack in the soundboard
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
3.4 Mode Assignment
By observing the spectra and FEA simulation results, we assigned the peak modes. For consistency, we adopted the nomenclature of Gore and Gilet (2011). Herein, a letter (in this case, T, indicating the top or soundboard) is followed by two numbers between parentheses indicating the number of antinodes along the transverse (y) and longitudinal (x) directions. The subscript indicates the order of appearance of modes that share an equal number of antinodes. The first peak, which does not appear in the FEA analyses and disappears when the sound hole is plugged, is evidently mode T(1,1)1. That is, it is a monopole mode resulting from the coupling of the Helmholtz resonance of the sound box and the first eigenmode of the soundboard. The mode has a low intensity in the admittance spectrum. This indicates that the coupling between the Helmholtz resonance of the box and the soundboard is relatively weak with respect to guitars and violins (Hess 2013) or other types of mandolins such as bluegrass instruments (Cohen & Rossing 2003). Nevertheless, it is higher than that of more contemporary mandolins, wherein a significantly smaller frequency shift was measured after the sound hole was plugged (Quintavalla et al. 2022b). The second and third peaks are presumably related to T(1,1)2, i.e., the monopole mode relative to the first soundboard eigenmode coupled to the Helmholtz resonance of the sound box. The frequency of the first of these peaks was not predicted by the FEA, did not vary after the sound hole was plugged, and was presumably related to a body-neck vibration or a high-order air mode. The third and fourth peaks are related to the longitudinal T(1,2) and transverse T(2,1) dipoles of the soundboard, respectively. Their order is typical of lute- and mandolin-type stringed instruments with transverse bracing (Fletcher & Rossing 1998, Quintavalla et al. 2022b). From the modal shape of mode T(1,2), it is noteworthy that one antinode (i.e., that located toward the instrument neck) is located across the soundhole. This implies that its radiating area is smaller than that of the other antinode. Based on this consideration, mode T(1,2) is likely to be an efficient sound radiator, unlike what occurs in more contemporary mandolins and guitars. The subsequent modes assume more complex shapes but display relatively good correspondence between the simulations and measurements. A few modes predicted by FEA are not visible in the spectra. This is either because these are not good radiators of sound or because the bridge (where the accelerometer is attached) is on a nodal line and, therefore, does not move. Nevertheless, the generally good agreement between the measurements and simulations reveals that FEA can be used as a reliable tool to determine mandolin eigenmode shapes when other techniques are not suitable.
3.5 Influence of the Crack on the Mechanical and Acoustical Behavior of the Mandolin
As highlighted in Section 3.1, a crack exists along the soundboard of the instrument. Although relatively thin, the crack spans an important part of the soundboard. In addition to affecting the structural integrity of the instrument, it is likely to affect its acoustic behavior. It may also render the acoustic survey inconsequential. To understand the effect of the crack on the measurements, we performed FEA analyses on the pristine soundboard to compare the mechanical deformation under the string load and the eigenmode shapes and frequencies. To achieve this, we developed a 3D model of a soundboard without cracks. This geometry, in conjunction with the pristine geometry, was first used to perform a deformation analysis. Herein, a force of 1 N exerted a downward pressure on the bridge. The soundboard stiffness k is calculated as follows:
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
where dmax is the maximum displacement at the center of the bridge. The results are reported in Fig. 10. These reveal stiffness values of 9.1 × 104 N/m and 9.0 × 104 N/m for the pristine and cracked soundboards, respectively. The values differ from the experimentally measured stiffness values. This difference can be ascribed to several factors including (i) the difference in the boundary conditions of the soundboard on the instrument from those of the simulations, (ii) the different material properties, and (iii) the elasticity of the bridge on the original instrument. Because the instrument is preserved with nontensioned strings, the bridge may not adhere perfectly to the soundboard surface and may function as a spring with a low elastic constant. Nevertheless, the difference in the FEA simulations without and with the crack is -0,9 %. This indicates that the crack causes the soundboard to be only marginally less rigid. Although one would expect the crack to produce a significantly higher impact on the structural integrity, this small difference is reasonable for two main reasons (Quintavalla et al. 2022b): (i) the soundboard stiffness is mainly related to the stiffness of the transverse braces (which are in good condition) in conjunction with the longitudinal stiffness of the soundboard wood, and (ii) the transverse stiffness of the soundboard is significantly lower than the longitudinal stiffness and plays only a secondary role. An identical geometry was used to simulate the eigenmodes. Their shapes and frequencies are shown in Fig. 11 and Table 5, respectively, and were compared with those of the cracked soundboard. The results reveal that notwithstanding the crack, the eigenmode shapes are almost unperturbed. The exceptions are modes 11 and 12, which are exchanged. The frequencies show a consistent difference only for the modes whose stiffness is dominated by the transversal stiffness and whose antinodes are located close to the crack, such as modes 5, 6 and 7. Mode 1 (i.e., the soundboard monopole T(1,1)2), whose stiffness is primarily related to the brace stiffness and longitudinal soundboard stiffness, is almost unperturbed. Meanwhile, mode 2, whose antinode is located across the crack, is perturbed only marginally. These results agree with those of (Quintavalla et al, 2022). Therein, a sensitivity analysis revealed that the transverse elastic modulus (ER) of the mandolin soundboard plays a secondary role with respect to the longitudinal modulus (EL). This is because the stiffness of the braces dominates the transversal stiffness of the soundboard.
FEA simulation of static bridge loading with a force of 1 N. (Left) Pristine soundboard. (Right) Soundboard with crack
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
The first 12 eigenmodes of the mandolin pristine soundboard simulated by Finite Element Analysis
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
Comparison of eigenmode frequencies of the pristine soundboard and those of the cracked soundboard
Citation: International Journal of Wood Culture 4, 1 (2024) ; 10.1163/27723194-bja10030
Considering these results, notwithstanding the presence of cracks, the mode frequencies retrieved by the SPL and admittance measurements are highly accurate for the most important radiating modes such as monopole T(1,1)2 and longitudinal dipole T(1,2). Hence, the frequencies of these modes can be considered as a reference for realizing an accurate replica of the instrument.
4 Conclusions
In this study, we investigated the acoustic properties of a late Baroque mandolin preserved at Museo degli Strumenti Musicali del Castello Sforzesco di Milano. It is an example of a well-preserved late Baroque mandolin. The acoustic survey enabled the measurement of certain characteristic features of this instrument in terms of the radiated sound pressure level, bridge admittance, soundboard stiffness, and monopole mobility using simple measurements performed with a portable setup. These measurements were complemented by finite-element analyses performed on the soundboard geometry retrieved by a dimensional survey. The mandolin displayed a characteristic mode order owing to transversal bracing where modes with similar shapes (i.e., dipoles and tripoles) occurred at a lower frequency in a longitudinal configuration and at higher frequencies in a transversal configuration. FEAs allowed for the reliable assignment of mode shapes to the admittance and SPL curve peaks. The simulation of the pristine, non-cracked soundboard enabled the determination of the mode shapes and the frequency variations that occurred after damage. It highlighted that the most important radiating modes were almost unperturbed, whereas other modes underwent variations in frequency. These analyses verified that the data retrieved in this survey are a good starting point for constructing an accurate replica of the instrument.
Acknowledgements
This work was funded by a World Wood Day Foundation Research Grant 2019: Choice of wood in musical instruments: Italian Red Spruce and traditional mandolins. The authors gratefully acknowledge the World Wood Day Foundation and the International Wood Culture Society for the research grant for aiding public awareness of wood as an eco-friendly material and its significant role in world history. They also thank Museo degli Strumenti Musicali del Castello Sforzesco di Milano (particularly its curator Valentina Ricetti) for their collaboration with the project and consideration. They also thank George Stoppani (http://www.stoppani.co.uk/index.htm) for providing the software used during the data acquisition. Finally, they are grateful to Associate Editor Lee Newsom and the reviewers for their help in improving the manuscript quality.
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