13 mins
X MARKS THE SPOT
For many years the design of the cello bridge has remained constant – but could it be improved? Sebastian Gonzalez presents the results of a comparison between the standard French bridge and a newly designed model, while on page 52 Gaian Amorim tracks the development of the bridge
A standard French-model cello bridge blank
The ‘Model X’ bridge designed by Luiz Amorim, in its final construction stages
FRENCH BRIDGE COURTESY MATTEO PONTIGGIA. MODEL X BRIDGE COURTESY AMORIM FINE VIOLINS
The cello is a beautiful and complex instrument, known for its rich, deep sound. The bridge is an essential component of its structure, responsible for transferring the vibrational energy of the strings into the body and contributing significantly to the overall sound of the instrument. To achieve the best possible sound, the cello bridge needs to be light enough to transmit the movement of the strings efficiently, yet rigid enough to support the static load of the strings.
Throughout history there have been many attempts at designing the ideal cello bridge. Over time, luthiers have arrived at two standard models: the French and the Belgian. These designs have been refined over the years, but the basic principles remain the same. However, in recent times the Amorim family of luthiers in Cremona, Italy, has developed a new cello bridge design, which has garnered some attention in the music world.
Inspired by Amorim’s fifth prototype of cello bridge, called the ‘Model X’ (see page 52), the research team at the Musical Acoustic Lab of the Polytechnic University of Milan conducted a study to investigate how much the shape of a cello bridge’s legs alters its static and vibrational behaviour. We created 3D simulations of both a standard French model bridge and the Model X for the experiment. To study different geometries of the bridge legs, we created simulations using the ‘Finite Element Method’. This reduces complex shapes such as the bridge to make smaller ‘elements’ – multiple triangles, albeit ‘finite’ in number – which make up the volume of the object. The laws of physics can then be applied to them in the computer.
The experiment also involved parametric modelling, i.e. a design that can be controlled with one parameter – in this case the legs of the bridge. In figure 1 the control point can be seen at point ‘P’.
We performed displacement and modal analysis for different boundary conditions, providing a detailed description of the mode shapes and their natural frequencies for different leg shapes. The full study was recently published in the Journal of Applied Sciences in a special issue dedicated to the mechanics, dynamics and acoustics of musical instruments and can be found here: bit.ly/3T4wScP
Our simulations showed that changing the shape of the bridge legs resulted in a change in the apparent stiffness of the bridge. Specifically, we found that the variation in apparent stiffness has a linear dependence with the parameter controlling the shape variation of the legs. To measure the stiffness of the bridge, we applied a constant force on the location of the strings and computed the displacement of each point of the bridge along each of the three dimensions. These displacements are called u, v and w, respectively for the axes x, y and z defined in figure 1.
Figure 2 shows the absolute value of the displacement along x: |u| for the French (left) and the Model X (right). As can be seen, the legs in the French model splay far more than those of the Model X. We take the absolute value – the magnitude, not the sign of the displacement – as the legs move in opposite directions. The displacement in the vertical direction, v, is far more similar. The results presented here are for a force of 1 Newton and should be multiplied by a factor of around 400 to obtain the displacement under real strings, but the linearity of the model ensures that the spatial distribution of the displacement is the same.
Stiffness is defined as the ratio between displacement and force. Since all the models are subjected to the same load, we can compute the variation in stiffness simply by looking at the variation of the average displacement for different models. This is precisely what can be seen in figure 3 (see page 53): we compute the average displacement along one direction for all the points on the bridge and divide that by the value obtained for the French model, <u_0>. The stiffer a model is, the less it will move under the same load.
Magnitude of the displacement along the x axis.
FIGURES 1 AND 2 COURTESY SEBASTIAN GONZALEZ
Displacement along the y axis
FIGURE 2 Displacement fields on each bridge under load.
‘P’ denotes the point at which we vary the control parameter of the legs.
FIGURE 1 Three examples of the 3D model and its evolution.
DEVELOPMENT OF THE ‘MODEL X’
A cello’s bridge is an essential component in the quality of its sound. Traditional cello bridges contain many excess parts, such as long, decorative arms, which make the sound’s transmission path longer than necessary. In contrast, the Model X features a shorter, more direct transmission path with less useless material. This allows for a quicker response and a more focused, crystal-clear sound.
The journey towards the Model X began in 2016 when a customer in the Amorim workshop showed Luiz Amorim an American research article suggesting that an X-shaped bridge format could give the best sound results for a cello. This starting point gave Amorim and his team the inspiration they needed to experiment.
The first model was not fully X-shaped, but it was less curved than traditional models, and it already showed some improvement in sound quality. Amorim then experimented with two half-moon cuts instead of a heart-shaped cut, in order to achieve a more direct transmission (figure 1). Although the central strings benefited from this change, there was still excess material present in the bridge.
In the second model, the team focused on reducing material at the waist and creating more dynamic movement in the central area (figure 2). They brought back the central heart, which improved the brightness of the sound but caused a slight loss of quality in the central strings.
The third model featured shortened arms and a reduced central mass (figure 3). This was achieved by digging out the legs internally, creating a concavity and decreasing material in that area. This resulted in much better quality and power in the sound transmission while still maintaining speed.
C ontinuing to refine the design, in the fourth model the team further shortened the arms and increased the size of the heart while reducing the width of the upper region (figure 4). This produced an even better result, with a more pronounced difference in sound quality compared to the previous models.
The final model (figure 5) features a higher and more crossshaped arch while maintaining shortened arms at the same height. The back and front parts of the legs gained an inclination, where the inner parts of the feet were thicker than the outer parts, creating a decrease in mass without compromising resistance. This has led to a significant improvement in sound quality, with the result being fully satisfactory.
The Model X has been used with many cellos, and the difference in sound quality compared to traditional models is significant. Customers experimenting with the Model X have reported that their instruments have more focus, power, and crystal-clear sound. The Model X is a testament to Luiz Amorim’s commitment to finding ways to improve instruments, and his dedication to the craft of lutherie.
1a
1b
FIGURE 1 Photo and drawing of the first model, with half-moons instead of a heart-shaped cut
FIGURE 2 A model with more dynamic movement
FIGURE 3 Shortened arms and reduced central mass
ALL IMAGES COURTESY AMORIM FINE VIOLINS
FIGURE 4 Reduced width in the upper region
FIGURE 5 The final Model X design
FIGURE 3
Left Spatial average of the displacement along the x axis, normalised for the French bridge, as a function of the model number. Right Spatial average of the displacement along the y axis, normalised for the French bridge, as a function of the model number.
FIGURE 3 COURTESY SEBASTIAN GONZALEZ
On both graphs, ‘R2 ’ refers to the first portion of the fit, as the constant value in the second portion is the mean.
THE SHAPE OF THE BRIDGE LEGS CAN BE USED TO CONTROL THE VIBRATIONAL AND STATIC RESPONSES OF THE CELLO, AND THER EFOR E TUNE ITS SOUND
The range of values of the stiffness reachable with this varying geometry corresponds to a 50 per cent increase in the vertical stiffness, and a 17 per cent increase in the horizontal stiffness between the traditional French and the Model X. This suggests that the shape of the bridge legs can be used to control the vibrational and static responses of the cello, and therefore tune its sound. In particular, thanks to the increase in the apparent stiffness of the bridge, it can be made lighter, helping the energy transmission from the strings to the corpus of the instrument. Furthermore, the stiffness in the x- and y-coordinates seems to be determined independently, from a certain angle of the legs.
Regarding the eigenmodes of the instrument, the results are a bit more complex to interpret. Depending on what boundary conditions we used (free, clamped on the feet, clamped with springs on the string location, or springs in the top and the feet of the bridge) the variation of frequencies was different. Some modes went up in frequency while others would go down, depending on the coupling between the springs and the bridge body. Stiffer legs don’t necessarily mean higher frequency modes, as they can couple better with the strings and show lower frequency modes instead. All the modes and videos of them can be found in the online version of the article.
Finally, we also computed and compared frequency response functions for the different geometries, but only under one boundary condition: fixed feet. Our results showed that the response in the mid-range can be continuously controlled by changing the geometric parameter and that distance between the third and fifth modes can also be controlled, as well as the amplitude of the admittance. This would allow an instrument maker to fine-tune the response of a cello in a particular frequency range without having to make structural changes to the instrument – only to the set-up.
However, an aspect that we have not studied in this article is the influence of design variations on the cello’s sound and playability. We know that changing the shape of the bridge will shift the position of the modes in the so-called ‘bridge hill’ region, so we would expect a change in the mid-range components of the sound. According to the musicians who have tried the Model X, the cello has a faster response and is ‘easier’ to play. This points towards a change in the minimum bow force, owing to the alteration in bridge resonances. Accurately studying the force needed to play an instrument is an exciting open area of research, and other groups around the world (in particular in Vienna and Paris) are developing the scientific methodologies needed to achieve this. We are eager to apply those methodologies to this new bridge model.
In conclusion, the study of the influence of the shape of the legs of the cello bridge on its static and vibrational behaviour is an exciting development in the world of lutherie. The research provides valuable insights into how the shape of the bridge can be used to tune the sound of the cello, and highlights the potential of computational modelling and simulation in studying musical instruments. The study is an excellent example of the intersection between science and art, and demonstrates how advances in one field can have a profound impact on another when their proponents are willing to work together.
Sibelius’s autographed photo to Maud Powell
G minor of Bruch. […] Not only is the solo part so wonderfully written for the instrument but the orchestration is superb. I hope to have a really sensational success with the Sibelius and am proud to be the first to play it in this country.’
Recognising the risk that Powell was taking, Sibelius responded: ‘I am so very pleased to hear that you take interest in my violin concerto and will produce it in America. If I am going to have a good reception in New York, I am convinced it is to you I will owe the great part of the success.’ He inscribed his photograph, ‘To the Violin Queen, Miss Maud Powell, with gratitude – Jean Sibelius.’ Powell had it framed for display in the window of G. Schirmer, New York’s largest music shop. After the performances, in a letter dated 14 December, she wrote that she had made as much ‘réclame as possible’, even playing the concerto in private, before critics, musicians and others, in order ‘to create interest and understanding’.
The concerts took place at New York’s Carnegie Hall. Bewildered New York critics commended Powell’s artistry and courage in presenting the concerto but were uncertain of its lasting value. While recognising its originality, Krehbiel despaired that ‘there are few opportunities for the violin to speak in tones of beauty, while the accompaniment is for the greater part either a mutter or a growl. Solo instrument and orchestra do not love each other in this new work’ (New ‐York Tribune). Richard Aldrich (New York Times) wrote that Powell was ‘working against odds too great’ due to the work’s ‘paucity of ideas, its great length and almost unrelieved sombreness of mood’. William J. Henderson (New York Sun) excoriated it: ‘This concerto is of the Finns, finny. It is of the North, rugged. It is of the Russ, rude. It is of the fiddle, technical. It is almost everything except beautiful […] It is bitter as gall and savage as wilderness.’ He acknowledged that Powell played it superbly, ‘But why did she put all that magnificent art into this sour and crabbed concerto?’
Nevertheless, the audience let loose with a great ovation, and Saint-Saëns, who had listened ‘most attentively’, applauded heartily. Safonov shook Powell by the hand, and on 2 December wrote in a note to her, ‘To conquer these almost insurmountable difficulties of technic and interpretation requires really an unusually artistic force.’
THE COMPOSER INSCRIBED HIS PHOTOGRAPH, ‘TO THE VIOLIN QUEEN, MISS MAUD POWELL, WITH GRATITUDE – JEAN SIBELIUS’
Maud Powell in 1919
ALL PHOTOS AND IMAGES COURTESY OF THE MAUD POWELL SOCIETY
Powell sent the reviews to Sibelius, remarking in her 14 December letter: ‘Alas, the work was not received as well as I could have wished. However, I have not lost courage and shall play it again with the splendid orchestra in Chicago, also in Cincinnati and I hope with the Boston Symphony under Dr Muck.’ Still full of enthusiasm, Powell reassured him: ‘The concerto went really very well and we had a most representative and distinguished audience. And I was proud to be “making history”.’
With daring spirit, she went on to Chicago, Cincinnati and Boston. For the Chicago performances on 25 and 26 January 1907 she collaborated with Frederick Stock, successor to Theodore Thomas as conductor of the Chicago Symphony Orchestra. In sharp contrast to New York critics, O.H. Hall (Chicago Daily Journal) marvelled:
The striking, original composition, racked by the vigour of the northland, has feverish moods fierce with the fire of the gypsy Czardas, weird, witching and involved, and is intensely violinistic. Miss Powell was at all times equal to the titanic task imposed, for the tournies of technic, remarkable as they were had an artistic alpha and omega in the soothing, savage and brilliant melodies that flow from her facile fingers, interpreting every phase of the composer’s strange and variegated imagery in music.
The orchestra members ‘followed with wonder at her amazing performance. She returned and repeated the last movement with a power and brilliancy that even surpassed her previous performance.’ The Musical Courier wired back to New York: ‘Great success for Maud Powell… Veritable ovation.’ On 1 February, Powell wrote to Sibelius:
I am happy to inform you of the triumph of the Concerto in Chicago. The conductor, Fred Stock, and the orchestra were all enthusiastic about it and played it wonderfully. Mr Stock belongs to our generation and worked in complete sympathy with the composition and with my interpretation. Wish you could have heard the performance. […] I am so overjoyed at the success – we really are to congratulate each other.