STRADIVARI À LA MODE

George Stoppani presents the results of an investigation into the acoustics of the ‘Boissier, Sarasate’, focusing primarily on the ‘signature modes’

Experimental modal analysis (where a real object is measured rather than via a computer simulation) is helpful in lutherie in two main ways. Firstly, it provides a great deal of information about what instruments actually do when they vibrate, and how they function as sound-radiating machines. The vast majority of violins behave in a very similar manner below around 800– 900Hz but progressively diverge with rising frequency. Secondly, when a sufficient body of information has been accumulated for both similar and different instruments, we can characterise a particular instrument in terms of its physical properties and how it stands in comparison to others.

One such property is the stiffness – both static and dynamic. If we place a plank on bricks, supporting the ends, and then load the middle with more bricks, we can see how it bends under the load. From this we can estimate its stiffness, or in more technical terms, its ‘elastic modulus’. This is static stiffness.

Dynamic stiffness varies with frequency and is lower when driven near a resonance. If we place a toddler on a swing and give them a push, we can top up the swinging with an occasional gentle push (timed so that we push in the direction that the swing is already moving). So our toddler can be kept happily swinging with very little effort on our part. If we try to make the swing move faster or slower than it naturally wants to, we will have to expend a lot more effort. This is an example of a system with a single resonance, and with a period that is the time taken to pass from the rest position forwards, all the way back (passing through the rest position), then returning to rest position. This is one cycle, and will take around two to three seconds. We can actually see this happening for the swing, but for musical instruments the motion is not only very small but also too fast for the eye to detect, so is effectively invisible. Furthermore, musical instruments tend to have a large number of resonances. A violin typically has more than 100 in the important audible range.

MUSICAL INSTRUMENTS TEND TO HAVE A LARGE NUMBER OF RESONANCES. A VIOLIN TYPICALLY HAS MORE THAN 100

Roberto Jardón Rico (left) and George Stoppani examine the ‘Boissier, Sarasate’ Stradivari violin

ROBERTO JARDÓN RICO

In vibration engineering the word mode is used frequently but is difficult to define in a non-mathematical way. Essentially, it is a resonance but with added information. A mode is fully specified by a resonance at a particular frequency, an amplitude and a damping factor. For a tuning fork we would say that the pitch that it rings at is its natural frequency (e.g. 440Hz – Hz from the 19th-century physicist Heinrich Hertz), the amplitude is how far the prongs are displaced in space as it vibrates, and the damping is how much energy is lost during each cycle. Our tuning fork has low damping because it rings for a long time (losing only a little energy at each cycle) whereas a cardboard box has high damping – we just hear a dull thud.

If we can only have one structural acoustics measurement it would be the admittance. This is the velocity that the structure moves at when excited by 1 Newton of force at a certain point in the chosen direction. For a violin we would seek to find something representative of this response at the string notches on the bridge. This is of interest because it gives us a first indication of how loud it will be at any frequency, in the measured range, as well as an indication of how it will feel to a player. If we measure a large number of points around the surface (60 to 100 for a top) we can map the mode shape. For the first few modes of a ‘free plate’ (suspended in space with the absolute minimum of constraints) we can see their shapes from the Chladni patterns (see figure 1). This method does not work well for an assembled violin, nor does it provide reliable information about amplitude and damping.

FIGURE 1 Modes 1, 2 and 5 of the top plate of the ‘Boissier, Sarasate’ violin

ALL IMAGES GEORGE STOPPANI

Most luthiers are familiar with some of the modes of violin free plates, in particular #1, #2 and #5. When taken in conjunction with the plate weights they can serve as a predictor for the dynamic stiffness of the assembled instrument. While this is easy information to collect, the transition from free plates to closed box is not simple or readily predictable. This has been shown in the papers by Professor Colin Gough in which he models this transition using Finite Element Analysis.

There is a group of several modes, commonly called the signature modes (starting from the first significant radiator), which are more or less the same in all violins. The important difference between the modes of the free plates and those of the assembled violin is that the free plate modes are substantially different from those observed on the assembled instrument, and are not directly linked to the sound radiation. The modes of the assembled instrument, on the other hand, are directly responsible for the sound radiation and we therefore have a causal connection between the structure and the sound. At higher frequencies we may well find a one-to-one correspondence for many modes, or only for very few.

Eventually, it serves little purpose to try to identify individual modes because it becomes increasingly difficult (though not necessarily impossible) to do so reliably: the shapes can morph with quite small modifications to the structure by mixing with other shapes, or just because of a change in the weather. We may not notice any difference in sound or playing feel. Also, above 3–3.5kHz the shape is much less important for radiation efficiency. We care more that it does move than exactly how.

The chosen method for analysing the ‘Boissier, Sarasate’ was the ‘roving hammer and fixed accelerometer’ approach. The underlying principle is the frequency response function (FRF). Rather than a response from a single transducer, we look at the ratio of the excitation force (at the chosen driving point) and the response at each of a set of chosen points located on the structure being measured. The excitation is from a very small instrumented hammer and the response is provided by a very small accelerometer weighing around 0.2g. If we tap twice as hard with the hammer, the response is doubled. In this way the FRF is the same regardless of the strength of the tap. In addition, the FRF is the same if we swap the locations of the excitation and response. It is about the transfer of energy from one point to another. This is very convenient because instead of having to tap repeatedly at the excitation point, then glue, remove and reglue the accelerometer for each point, we can place the accelerometer at the ‘driving point’ and tap at the measurement points. If we were using a scanning Doppler laser system it would be the other way round: the excitation would be fixed and the laser would move to each point.

TABLE 1 Comparison of modes for five Stradivari violins

A0

B1-

C4

CBR-

B1+

ALL IMAGES GEORGE STOPPANI

FIGURE 2 Signature modes of the ‘Boissier, Sarasate’. A0, B1-, B1+ and C4 are shapes that have a net volume change component. Violins are too small to radiate frequencies as low as 270Hz effectively, but an ingenious system of coupling the net volume change of the corpus with the Helmholtz resonance of the enclosed air makes this possible. CBR is left/right symmetric, with the C-bout areas rotating in opposite directions. Sometimes this mode radiates some sound if it has a volume change component due to a degree of left/right asymmetry. In the case of the ‘Boissier, Sarasate’ there is no radiation peak for this mode. C4 does not have a large volume change component and is not readily excited by bridge rocking. However, it can make a small, but important, contribution to the radiation above B1+. If the radiation is very low in this region (620–850Hz, E5 to A5 ) the notes will lack fundamental and feel empty.

Suitable software is indispensable. We used a suite of applications developed by the author. Electrical signals from the transducers need a lot of post-processing. Some of the output is numerical but mode shapes need graphic representation (figure 3).

There are a number of things that we can say with confidence about this violin in the light of the measurements. It is structurally robust. We can see from the thickness measurements that the top is not excessively thin. The back, while thin in the upper and lower bouts, rises to 5mm in the C-bout area. We do not have density measurements but it would be a fair guess that the back is of at least medium- or higher-density wood. Additionally, the structured light scan shows minimal distortion – the gluing surfaces of the plates still lie on a plane. This suggests that the wood has never or rarely been stressed close to force levels that would result in creep. Along with the minimal wear on the edges and scroll, this violin is very attractive to copyists.

FIGURE 3 Sound radiation for the ‘Boissier, Sarasate’. The room was very reverberant. The red series shows the raw data with all room reflections. The blue series shows the same data but with some exponential smoothing. The frequency bands ‘Signature modes’, ‘Transition hill’, ‘Bridge/island hill’ and ‘Treble slope’ are terms used in the Oberlin Acoustics Workshop to help characterise the sound radiation spectra of violins.

GEORGE STOPPANI

The modal analysis and sound radiation measurements also point to an instrument that is not lacking in dynamic stiffness. The frequencies of the signature modes taken with the plate arching and graduation suggest a top plate of low density but by no means extreme. A best guess today would be around 0.36g/cm3 . From Table 1 (page 35) we see the frequencies for the ‘Boissier, Sarasate’ are at the high end of the range but still not extreme. The balance between being sufficiently stiff and sufficiently compliant is critical. If too stiff, the instrument is likely to be missing in depth of sound (low frequency response), perhaps over-brilliant to the point of being scratchy (excess high frequency). If too compliant, the G string may seem boomy or flabby while the articulation would be weak and the treble end too soft.

The sound radiation measurements are not calibrated, so cannot be used for direct comparison with other instruments. However, we can learn a lot about the balance of output in the various frequency bands. The disposition of A0, B1- and B1+ are typical for good violins – old or new. The transition hill is a little high in amplitude but not higher than for the signature mode region. The bridge/island hill is a little low, perhaps, for what we might try to sculpt on a new instrument or for setting up an old one. Since the set-up is essentially what it was in 1908 this spectral balance may be just what the luthier intended at that time. Undoubtedly, this spectrum could be modified by the usual set-up procedures but that would destroy a rare item of historical documentation.

A 3D rendering of the violin, looking from the top downwards

It was not possible to take this violin to an industrial CT scanner, which is now becoming a common practice for documentation and condition reports. Instead, portable structured light scanning equipment was brought into the museum by Francesco Piasentini. This method is unable to see below the surface but can capture the accessible surfaces with a high degree of accuracy. There is no physical contact with the instrument beyond the necessary support and therefore minimal risk of damage. Thickness measurements were obtained with a ‘MAG-ic probe’ – a device that uses the distance between two magnets to determine thickness. A CT scan would be able to provide both external shape and wood thicknesses but not necessarily more accurately than could be done in this case. The CT scan requires post-processing to determine where a solid body ends and air begins. Any errors (which are inevitable) in either the inner or outer surface will affect the thickness estimate. Structured light scanners project a series of known patterns on to the test object and process the reflections directly to CAD files such as ‘stl’ or ‘obj’. This technology is developing rapidly and becoming more affordable, accurate and easy to use. CT scanners are likely to remain expensive and non-portable for the foreseeable future.