INTELLIGENT DESIGN

The science of violin acoustics has encompassed 3D scanning, CNC technology and good old-fashioned tap tones – so why not AI software? Sebastian Gonzalez presents the results of a project that could help predict an instrument’s tone qualities even before it’s made

The idea that the shape and thickness of a violin’s top and back plates can affect its sound is nothing new. Antonio Stradivari was undoubtedly aware of it 300 years ago, and the science behind it was scrutinised and written up at length by Carleen Hutchins in the 1950s. Since then, the phenomenon of violin ‘modes’ and resonances has been investigated by both violin makers and academics; indeed, for many luthiers, one of the first steps in making a new instrument will be to examine its ‘tap tones’, or speed of sound along the plates. But what if we could predict how an instrument will sound even before we build it?Could it be possible for modern technology to give an idea of the final plates’ vibrational behaviour, and even understand how a good violin could be made to sound ‘great’?

A team from the Politecnico di Milano’s musical acoustic laboratory has recently completed a project using artificial intelligence (AI) software to examine violin forms. The study demonstrates that a simple neural network can learn to predict the way a violin’s top plate will vibrate, just from parameters relating to its geometry and material. For this project,we took a fundamentally different approach from the standard method of investigation – which, in a nutshell, is ‘varying X while leaving everything else constant’.In other words, you start with a model of a violin, vary one parameter at a time, and see how the instrument’s eigenfrequencies change as a result of varying that one parameter. The eigenfrequency (or ‘modal frequency’) is the frequency at which a system tends to oscillate in the absence of any driving or damping force, and the shape this vibration has in space is called the ‘normal mode’. These modes can be understood by analogy to the vibrations of a string, where the fundamental is always accompanied by several harmonics: these are the normal modes of a string.

Conversely, in our approach we varied all the variables of our violin top plates at the same time, and used AI software to figure out the correlations between parameters and eigenfrequencies. So when people ask us, ‘What happens when we vary X?’ we have no answer, because it depends on how all the other variables of the violin are set. However, the AI will give us the answer for any particular configuration. Note here that the answer we get is in terms of eigenfrequencies, not in terms of violin tone quality (as in ‘good’ or ‘poor’). How the eigenfrequencies are correlated to what we hear as good tone is a completely different can of worms.

To train a neural network, one needs a generally large dataset. We had some violins scanned from the collection of Cremona’s Museo del Violino, but only perhaps a dozen or so, and not all of them usable. So the first task was to come up with a good parametrisation of the violin – that is, a set of numbers that can be used to describe any violin. We discovered the solution to this in the Museo’s collection, in the form of a drawing from the workshop of Cremonese maker Enrico Ceruti (figure 1). The drawing shows how the outline of a violin can be created from a series of circular sections. Thus, the coordinates, radii and angles that describe these various sections become the parameters of the outline. I simply uploaded the Ceruti drawing into my computer, and started to play with the numbers to obtain numerous reasonable outlines.

The idea of applying a neural network to musical acoustics still seemed quite far-fetched, and we focused mostly on having a good model of violin to start with. Among the Museo’s 3D scans of historical instruments, is Stradivari’s 1716 ‘Messiah’, which in 2016 was brought over from Oxford’s Ashmolean Museum on loan, to spend its 300th birthday in Cremona. On that occasion, the Politecnico di Milano together with the University of Pavia scanned the violin while performing other experiments. The advantage of having a scan of the instrument, besides its extreme accuracy, is that we could get a more complete idea of its arching than can be obtained from simple shop drawings.

FIGURE 1A c.1800 drawing from the Enrico Ceruti workshop (top), showing the radii and centres of the circles that make up a violin outline (above).These points formed the parameters that would be inputted into the AI software

THE IDEA OF APPLYING A NEURAL NETWORK TO MUSICAL ACOUSTICS SEEMED QUITE FAR-FETCHED

FIGURE 2 Examples of the thickness distribution of the violin top plates, with thin regions marked in red and thick regions in yellow. Notice how the random sampling of the outline parameters creates shapes that would be rather unusual for a violin that really existed

My colleague, the mandolin player Davide Salvi, did some optimisation and managed to find a set of parameters that approximated the outline of the ‘Messiah’ with an area difference less than one per cent. For the longitudinal arching we used a simple polynomial formula that gives the ‘Messiah’ arching as a function of the position from neck to tailpiece.

For the transverse arching of the top we decided to go with a mathematical formulation that can be easily implemented for any violin outline. The idea behind this is that by parametrising the arching as a polynomial function, in the future we will be able to vary them at will in a straightforward way. The plate thickness also has its own challenges, since to obtain a smoothly varying surface on the inside, one needs to be quite careful.

THE NEURAL NETWORK PERFORMED EXTREMELY WELL IN PREDICTING THE EIGENFREQUENCIES

As a compromise between historical accuracy and ease of implementation, we chose to have nine different thickness regions in the plate: three in the upper region, three in the centre, and three in the lower part (figure 2). The resulting violin top plates are clearly not realistic, but that is precisely the idea. By training the neural network for a number of possible configurations, we teach it to recognise the relationship between the different parameters and the eigenfrequencies.Besides, we can easily expand the number of regions later to an arbitrary high number. We have always seen this research as a first step in a long path towards the accurate simulation of violins with the help of AI.

Once we had made the parametric model of a violin top plate, it just remained to run the simulations. We used the Finite Element Method (FEM), the standard method of simulating mechanical structures in areas ranging from aerospace to construction. It has been used to study violins for at least 60 years already. To answer the question of why we did not use real wood in our experiments, let me explain why we decided to go virtual. First of all, it was the height of the Covid-19 pandemic. Secondly, we had no certainty or previous work that would tell us that our approach was going to work at all, so using simulations was less risky than going directly for wooden plates. Thirdly, we needed a proper 3D model of a violin to do CNC carving, so it made sense to publish something that related only to the model.

It took around a week of computational time for us to simulate plates in three different datasets. These depended on whether we varied just the outline (1,750 simulations), the thickness (1,000), the material (1,000) or everything together (1,500). We then felt that we had enough data to feed the neural network (figure 3). To our surprise, the neural network performed extremely well in predicting the eigenfrequencies from the geometrical parameters (figure 4).

FIGURE 3 The architecture of the neural network used for the prediction of the eigenfrequencies. It is based on a hidden dense layer of N neurons, connected to a linear output layer. The inputs were the 20 parameters that define the outline, while the outputs were the first ten eigenfrequencies of the resulting top plate

FIGURE 4 Predicted versus actual values for the first five eigenfrequencies in the test set for a network with N=7. The frequencies are scaled by the average actual values for each mode, so we could compare different frequency values in the same plot

We then tried with the thickness profile for a fixed outline, and the results were just as good. Finally, we also varied the material parameters of the wood, and the neural network kept performing extremely well.

Yet we didn’t want to finish there. One problem of neural networks is that they are ‘black boxes’: you may have an answer but the network doesn’t tell you why that is the answer. We therefore examined how the neural network was making its predictions. It was quickly apparent that only a few of our geometrical parameters were actually important for the vibrational response, in particular the width of the plate. We also learnt that the wood density and its longitudinal stiffness are by far the two most important material parameters. Finally, the thickness was the least important feature, and its correlation with the vibrational modes was sketchy at best. This last point completely contradicts the ‘plate tuning’ approach in violin making. We found out that two plates made from different materials can have a very similar vibrational response, insofar as their outlines and thicknesses compensate for the variation in density and stiffness. This result shows how, if your goal is to copy the way in which a violin from the Old Italians vibrates (and therefore sounds) with contemporary wood, you will need to alter the shape of the violin – or find an artificial way to age wood 300 years. On balance, varying the shape slightly would be easier.

We are well aware that these findings will not sit well with the traditional approach to violin making, where the design of the old masters seems to be regarded as the ne plus ultra of the craft, and where violin makers copy even the scratches of a famous violin during the antiquing process. Taste has seldom been guided by science. Yet our results point in a different direction.

THESE FINDINGS WILL NOT SIT WELL WITH THE TRADITIONAL APPROACH TO VIOLIN MAKING

One of the main criticisms we have received for our paper is the fact that sound doesn’t equate to vibrations. We are aware of this as well, yet the only way of producing sound is by having a vibrating material. As George Bissinger said in a panel discussion at the 2007 VSA convention: ‘If you influence the way [the violin] vibrates, you influence the sound. Everything else about the way it looks is window dressing.’ Studying the vibrational response with AI is a step forward in understanding the sound of the violin from a scientific point of view. And that was our objective in the lab: to understand violins.

Perhaps when people read about ‘violin making and AI’, they imagine robots doing the work of luthiers. Even though we plan to build CNC-carved violins in the near future to test our theories with actual instruments, that is not the aim of our research.We don’t expect to answer what is ‘the best’ shape for a violin, but rather to understand what makes a good violin sound great. And here probably lies the most interesting part of the research.Musicality is not an objective feature of the sound signal; it is something that happens to human beings with a certain cultural background listening to a given piece of music. So understanding what’s so special about Stradivari’s violins is not something that can be accomplished without studying the biological response music produces in human bodies. Having an AI that predicts how a given instrument will vibrate is just a small step in a much larger research path, and one we are eager to continue exploring.