The acoustics of the violin: a review

A Statistical Approach to Violin Evaluation

Comparing violins requires competence and involves both subjective and objective evaluations. In this manuscript, vibration tests were performed on a set of 25 violins, both historical and new. The resulting bridge admittances were modeled in the low and mid-frequency ranges through a set of objective features. Once projected into the new representation, the bridge admittances of three historical violins made by Stradivari and a famous reproduction revealed high similarity. PCA highlighted the importance of signature mode frequencies, bridge hill behavior, and signature mode amplitudes in distinguishing different violins.

A comparative analysis of the directional sound radiation of historical violins

The directivity pattern of a musical instrument describes the sound energy radiation as a function of frequency and direction of emission. Violins exhibit a rather complex directivity pattern, which is known to show rapid variations across frequencies, and whose behavior cannot be easily predicted except in the lowest frequency range. The acoustic behavior of the violin is a fascinating research topic that has prompted numerous published works, but a thorough, comprehensive, and comparative analysis of violin directivity patterns is long overdue. In this article, we propose a set of metrics for characterizing the radiative behavior of musical instruments and, in particular, for comparing their directivity patterns. We apply such metrics for a comparative analysis of the directivity patterns of some of the most prestigious historical violins ever made, including grand masters such as Antonio Stradivari, Giuseppe Guarneri "del Gesú" and members of the Amati family. The instruments are preserved in the Violin Museum of Cremona, Italy, where our lab is located. The analysis methodology introduced in this work allowed us to quantitatively evaluate the similarity of directivity patterns of such extraordinary instruments and draw some interesting conclusions.

Helmholtz vibrations in bowed strings

For almost 160 years, it has been known that Helmholtz oscillations, unique to vibrating strings in bowed instruments (violin, cello, etc.), have two distinct regimes: "slip" and "stick." During the slip regime, the force at the bow-string interaction is attributed to friction between the sliding bow hair and the vibrating string, with a friction coefficient that decreases with increasing relative velocity. Yet the hair-string interaction during the stick regime is less understood. We propose that the interaction force during the stick regime is proportional to the product of the longitudinal acoustic impedance of the bow hair to the relative bow-string velocity. We validate this hypothesis by solving the string's differential equation of motion, including an enhanced formulation to avoid parasitic high-frequency oscillations. This physical model enables us to analyze, in real time, the characteristics of the Helmholtz oscillations, including the string shape, excitation of harmonics, Schelleng ripples, and string energy, showing that the bowed string gains energy during the stick regime and loses energy during the slip regime.

A Review of Finite Element Studies in String Musical Instruments

String instruments are complex mechanical vibrating systems, in terms of both structure and fluid–structure interaction. Here, a review study of the modeling and simulation of stringed musical instruments via the finite element method (FEM) is presented. The paper is focused on the methods capable of simulating (I) the soundboard behavior in bowed, plucked and hammered string musical instruments; (II) the assembled musical instrument box behavior in bowed and plucked instruments; (III) the fluid–structure interaction of assembled musical instruments; and (IV) the interaction of a musical instrument’s resonance box with the surrounding air. Due to the complexity and the high computational demands, a numerical model including all the parts and the full geometry of the instrument resonance box, the fluid–structure interaction and the interaction with the surrounding air has not yet been simulated.

Effect of Bow Camber and Mass Distribution on Violinists' Preferences and Performance

Little is known about how bow mechanical characteristics objectively and quantitatively influence violinists' preferences and performance. Hypothesizing that the bow shape (i.e., camber) and mass distribution modifications would alter both violinists' appreciations of a bow and objective assessments of their performance, we recruited 10 professional violinists to play their own violin using 18 versions of a single bow, modified by combining three cambers and six mass distributions, in random order. A musical phrase, composed for this study, was played legato and spiccato at three octaves and two tempi. Each violinist scored all 18 bows. Then, experts assessed the recorded performances according to criteria inspired by basic musical analysis. Finally, 12 audio-descriptors were calculated on the same note from each trial, to objectivise potential acoustic differences. Statistical analysis (ANOVA) reveals that bow camber impacted the violinists' appreciations (p < 0.05), and that heavier bow tips gave lower scores for spiccato playing (p < 0.05). The expert evaluations reveal that playing with a lighter bow (tip or frog), or with a bow whose camber's maximum curvature is close to the frog, had a positive impact on some violinists' performance (NS to p < 0.001). The "camber-participant" interaction had significant effects on the violinists' appreciations (p < 0.01 to p < 0.001), on the expert's evaluation and on almost all the audio-descriptors (NS to p < 0.001). While trends were identified, multiple camber-participant interactions suggest that bow makers should provide a variety of cambers to satisfy different violinists.

Coupled numerical simulations of the structure and acoustics of a violin body

This study is aimed at predicting the characteristics of vibration and sound radiation of violins and understanding the relationships among the properties of its wood, vibration, and sound radiation. Numerical simulations of the vibration mode of a violin body are performed, and the sound radiated by it are analyzed using the finite element method. The geometry of a real violin is scanned using a micro-computed tomography scanner, and the orthotropic properties of spruce and maple, such as Young's modulus, rigidity modulus, and Poisson's ratio, are set as the parameters of the numerical simulation. The main vibration modes, such as A0 and center bout rotation, and the acoustic pressure level around the violin body are calculated. This paper describes the influence of the density and longitudinal stiffness on the eigenfrequencies and sound radiation.

Status and future of modeling of musical instruments: Introduction to the JASA special issue

Over the last decades, physics-based modeling of musical instruments has seen increased attention. In 2020 and 2021, the Journal of the Acoustical Society of America accepted submissions for a special issue on the modeling of musical instruments. This article is intended as an introduction to the special issue. Our purpose is to discuss the role that modeling plays in the study of musical instruments, the kinds of things one hopes to learn from modeling studies, and how that work informs traditional experimental and theoretical studies of specific instruments. We also describe recent trends in modeling and make some observations about where we think the field is heading. Overall, our goal is to place the articles in the special issue into a context that helps the reader to better understand and appreciate the field.

Materials Engineering of Violin Soundboards by Stradivari and Guarneri

We investigated the material properties of Cremonese soundboards using a wide range of spectroscopic, microscopic, and chemical techniques. We found similar types of spruce in Cremonese soundboards as in modern instruments, but Cremonese spruces exhibit unnatural elemental compositions and oxidation patterns that suggest artificial manipulation. Combining analytical data and historical information, we may deduce the minerals being added and their potential functions—borax and metal sulfates for fungal suppression, table salt for moisture control, alum for molecular crosslinking, and potash or quicklime for alkaline treatment. The overall purpose may have been wood preservation or acoustic tuning. Hemicellulose fragmentation and altered cellulose nanostructures are observed in heavily treated Stradivari specimens, which show diminished second‐harmonic generation signals. Guarneri's practice of crosslinking wood fibers via aluminum coordination may also affect mechanical and acoustic properties. Our data suggest that old masters undertook materials engineering experiments to produce soundboards with unique properties.

A data-driven approach to violin making

Of all the characteristics of a violin, those that concern its shape are probably the most important ones, as the violin maker has complete control over them. Contemporary violin making, however, is still based more on tradition than understanding, and a definitive scientific study of the specific relations that exist between shape and vibrational properties is yet to come and sorely missed. In this article, using standard statistical learning tools, we show that the modal frequencies of violin tops can, in fact, be predicted from geometric parameters, and that artificial intelligence can be successfully applied to traditional violin making. We also study how modal frequencies vary with the thicknesses of the plate (a process often referred to as plate tuning) and discuss the complexity of this dependency. Finally, we propose a predictive tool for plate tuning, which takes into account material and geometric parameters.

Eigenfrequency optimisation of free violin plates

We discuss how the modal response of violin plates changes as their shape varies. Starting with an accurate 3D scan of the top plate of a historic violin, we develop a parametric model that controls a smooth shaping of the interior of the plate, while guaranteeing that the boundary is the same as the original violin. This allows us to generate a family of violin tops whose shape can be smoothly controlled through various parameters that are meaningful to a violin maker: from the thickness in different areas of the top to the location, angle, and dimensions of the bass bar. We show that the interplay between the different parameters affects the eigenmodes of the plate frequencies in a nonlinear fashion. We also show that, depending on the parameters, the ratio between the fifth and the second eigenfrequencies can be set to match that used by celebrated violin makers of the Cremonese school. As the parameterisation that we define can be readily understood by violin makers, we believe that our findings can have a relevant impact on the violin making community, as they show how to steer geometric modifications of the violin to balance the eigenfrequencies of the free plates.

Vibro-acoustic modeling, numerical and experimental study of the resonator and its contribution to the timbre of Sarasvati veena, a South Indian stringed instrument

This paper is part of a special issue on Modelling of Musical Instruments. The Sarasvati veena is a South Indian plucked string wooden lute, whose unique timbre is characterized by the presence of nearly all harmonics, in the range from 0 to 2800 Hz, the augmentation of the second and third harmonic and the revival of higher harmonics with time. This can be attributed to the shape of its extended bridge and its wooden resonator. In this work, the vibrational modes of the resonator are studied and the corresponding frequencies related to the musical tones generated while playing the veena. Numerical modal analysis is performed by the finite element method on a computer-aided design model of the resonator top plate as well as the air cavity within the dome-shaped structure. The results are compared to the experiment for validation. The resonator modes provide support to the harmonics of the string vibrations and give rise to the typical timbre of the instrument. We also find that other mode frequencies of the resonator support microtones that are used in Indian raga music. This gives scientific basis to the role of the timbre of the veena in supporting the development and sustenance of South Indian classical music.

Multiple two-step oscillation regimes produced by the alto saxophone

A saxophone mouthpiece fitted with sensors is used to observe the oscillation of a saxophone reed, as well as the internal acoustic pressure, allowing to identify qualitatively different oscillating regimes. In addition to the standard two-step regime, where the reed channel successively opens and closes once during an oscillation cycle, the experimental results show regimes featuring two closures of the reed channel per cycle, as well as inverted regimes, where the reed closure episode is longer than the open episode. These regimes are well-known on bowed string instruments and some were already described on the Uilleann pipes. A simple saxophone model using measured input impedance is studied with the harmonic balance method, and is shown to reproduce the same two-step regimes. The experiment shows qualitative agreement with the simulation: in both cases, the various regimes appear in the same order as the blowing pressure is increased. Similar results are obtained with other values of the reed opening control parameter, as well as another fingering.

Bark Features for Identifying Resonance Spruce Standing Timber

Measuring the acoustic properties of wood is not feasible for most luthiers, so identifying simple, valid criteria for diagnosis remains an exciting challenge when selecting materials for manufacturing musical instruments. This article aims to verify whether the bark qualities as a marker of resonance wood are indeed useful. The morphometric and colour traits (in CIELab space) of the bark scales were compared with the structural (width and regularity of the growth rings and of the latewood) and acoustic features (transverse sound velocity, radiation ratio, impedance, and wood basic density) of the wood from 145 standing and 10 felled spruce trees, which are considered a resource of the resonance wood in the Romanian Carpathians. It has been emphasized that the spruce trees with acoustic and structural features that match the requirements for the manufacture of violins have a bark phenotype distinguishable by colour (higher redness, lower yellowness and brightness)—as well as by scale shape (higher slenderness and width). The south-facing side of the trunk and the external side of the scale are best for identifying resonance trees by their bark. Additionally, the mature bark phenotypes denote topoclinal variations and do not depend on tree age. Moreover, the differences among bark phenotypes are noticeable to the naked eye.

Reconstruction of an early viola da gamba informed by physical modeling

The reconstruction of an early viola da gamba is considered, using virtual prototyping by means of the finite element method. Based on iconographic sources, previous research has postulated an instrument design lacking a soundpost and a bass bar. This led to the hypothesis of a top plate with variable thickness. In order to investigate the acoustic efficiency of such a design, a finite element model of the instrument is formulated. The structural accuracy of the model is qualitatively verified by comparing calculated modal shapes with those of a reconstructed instrument, visualized with the aid of Chladni patterns and electronic speckle pattern interferometry. Furthermore, simulating the interaction between the vibrating surfaces of the instrument and the surrounding air shows that the posited asymmetric design can radiate sound more efficiently than a design involving a symmetric top plate. However, the asymmetry introduced by the gradually thickening top plate is weaker than that usually enforced by the presence of a soundpost and a bass bar. Therefore, low frequency structural modes of the instrument are less easily excited by a force acting parallel to the top plate.

Effect of back wood choice on the perceived quality of steel-string acoustic guitars

Some of the most prized woods used for the backs and sides of acoustic guitars are expensive, rare, and from unsustainable sources. It is unclear to what extent back woods contribute to the sound and playability qualities of acoustic guitars. Six steel-string acoustic guitars were built for this study to the same design and material specifications except for the back/side plates which were made of woods varying widely in availability and price (Brazilian rosewood, Indian rosewood, mahogany, maple, sapele, and walnut). Bridge-admittance measurements revealed small differences between the modal properties of the guitars which could be largely attributed to residual manufacturing variability rather than to the back/side plates. Overall sound quality ratings, given by 52 guitarists in a dimly lit room while wearing welder's goggles to prevent visual identification, were very similar between the six guitars. The results of a blinded ABX discrimination test, performed by another subset of 31 guitarists, indicate that guitarists could not easily distinguish the guitars by their sound or feel. Overall, the results suggest that the species of wood used for the back and sides of a steel-string acoustic guitar has only a marginal impact on its body mode properties and perceived sound.

Playability of the wolf note of bowed string instruments

Playability is an important aspect of the evaluation of bowed string instruments. The well-known "wolf note" of a cello is a particularly obvious playability issue, and it has been suggested that susceptibility to wolfiness might be deduced directly from a measurement of the Schelleng minimum bow force for the playing of a steady note. This prediction is explored by comparing physical measurements with the experience of players after making controlled mechanical changes to a cello. Experienced luthiers and musicians made subjective judgements of changes in the severity of the wolf note, under blinded conditions. The results strongly suggest a direct and intimate link between the measurable acoustical parameter and perceptual discrimination. This simple acoustical measurement can help instrument makers to identify problem notes, and to assess the effectiveness of different possible interventions.

Studying friction while playing the violin: exploring the stick-slip phenomenon

Controlling the stick-slip friction phenomenon is of major importance for many familiar situations. This effect originates from the periodic rupture of junctions created between two rubbing surfaces due to the increasing shear stress at the interface. It is ultimately responsible for the behavior of many braking systems, earthquakes, and unpleasant squeaky sounds caused by the scratching of two surfaces. In the case of a musical bow-stringed instrument, stick-slip is controlled in order to provide well-tuned notes at different intensities. A trained ear is able to distinguish slight sound variations caused by small friction differences. Hence, a violin can be regarded as a perfect benchmark to explore the stick-slip effect at the mesoscale. Two violin bow hairs were studied, a natural horse tail used in a professional philharmonic orchestra, and a synthetic one used with a violin for beginners. Atomic force microscopy characterization revealed clear differences when comparing the surfaces of both bow hairs, suggesting that a structure having peaks and a roughness similar to that of the string to which both bow hairs rubbed permits a better control of the stick-slip phenomenon.

Are there reliable constitutive laws for dynamic friction?

Structural vibration controlled by interfacial friction is widespread, ranging from friction dampers in gas turbines to the motion of violin strings. To predict, control or prevent such vibration, a constitutive description of frictional interactions is inevitably required. A variety of friction models are discussed to assess their scope and validity, in the light of constraints provided by different experimental observations. Three contrasting case studies are used to illustrate how predicted behaviour can be extremely sensitive to the choice of frictional constitutive model, and to explore possible experimental paths to discriminate between and calibrate dynamic friction models over the full parameter range needed for real applications.

Reliability of the input admittance of bowed-string instruments measured by the hammer method

The input admittance at the bridge, measured by hammer testing, is often regarded as the most useful and convenient measurement of the vibrational behavior of a bowed string instrument. However, this method has been questioned, due especially to differences between human bowing and hammer impact. The goal of the research presented here is to investigate the reliability and accuracy of this classic hammer method. Experimental studies were carried out on cellos, with three different driving conditions and three different boundary conditions. Results suggest that there is nothing fundamentally different about the hammer method, compared to other kinds of excitation. The third series of experiments offers an opportunity to explore the difference between the input admittance measuring from one bridge corner to another and that of single strings. The classic measurement is found to give a reasonable approximation to that of all four strings. Some possible differences between the hammer method and normal bowing and implications of the acoustical results are also discussed.