ROOM TO BREATHE

In the first of two articles investigating how the arching of violin plates affects tone, Nigel Harris presents evidence, theoretical and experimental, regarding the ‘breathing action’ of the plates

CREDIT

The action of bowing a string applies forces to the bridge that cause it to rock in its own plane. When the rocking lifts the belly on the bass-bar side, it will depress a smaller area of the belly on the soundpost side, which in turn lifts an area in the back. Therefore, a slow in-plane rocking of the bridge alternately expands and contracts the volume of the body; a form of breathing action that is centred near the bass-bar side of the bridge. But that is not all. As the bridge lifts on the bass-bar side, the group of four strings increases in tension, and as the bridge drops there will be a corresponding relaxation of this tension. In this article, I will show theoretically and experimentally that this alternation in string tension applies forces within the body that cause displacements consistent with a breathing action centred in the upper and lower bouts.

FIGURE 1 The principal lines of force in a violin

Figure 1 shows a long section of a violin with the line of action of the principal forces superimposed. For simplicity, the four strings are represented by a single string acting at the centroid of the group. If the compression down the length of the violin applied by the string tension were to act in a line about mid-height of the ribs, the violin would not bend in its length. But the string leaves this line and diverts to the top of the bridge. The bending this causes is countered by a tension in the back that diverts to the bottom of the soundpost. The tension forces in the string and the back are resisted by a compression in the belly that runs to the base of the bridge. By assuming a string tension of 1 unit, the corresponding forces in the other lines of force can be found.

FIGURE 2 The relative magnitude of the principal forces in a violin

MAIN PHOTO GETTY. ALL FIGURES NIGEL HARRIS

Figure 2 shows that the strings apply a downward force on the bridge of about 37 per cent of the tension in the strings. The soundpost carries only about 20 per cent of the bridge force, the remaining 80 per cent being carried by the compressive force in the belly, which pushes up to the underside of the bridge. The tension in the back of the instrument is about a third of the compression in the belly.

Figure 3 (page 56) shows a violin belly placed on a table with a load P applied at the bridge position, where P is the downward force from the bridge minus the upward force from the soundpost. The force P can be carried either on the ‘long arch’ ABC, or on the ‘cross-arch’ DBE. The force in the cross-arch can get around the f-holes. In fact, the long arch and the cross-arch both contribute to the support; the proportion carried by each depends on the relative stiffness (or resistance to movement) of the two arches. In this article, the load carried on the long arch will be called L and the load on the centre cross-arch will be called C, such that P = C+L. The ratio of the cross-arch load to the long-arch load, called the C/L ratio, will be referred to frequently.

Figure 4 shows a section through the long arch of the belly, with the load L at the bridge. The load is resisted by horizontal reactions H at the ends provided by the compression in the belly. The broken line is traditionally known as the ‘line of thrust’ of the arch. For an arch in compression (such as the belly long arch), the offset between the line of thrust and the centre line of the wood causes the wood to bend and move further away from the line of thrust. Figure 4 shows that the wood at the bridge position moves downwards and the wood in the upper and lower bouts moves upwards. The stiffness of the long arch to a force L applied at the bridge increases if the centre line of the wood is closer to the line of thrust and also increases with the flexural stiffness of the wood (how resistant it is to bending).

We now look at the force C carried on the centre cross-arch. Figure 5 shows a violin belly with cuts made across the plate at the upper and lower bouts’ cross-arches. The force C at the bridge is carried on the centre cross-arch to the two side arches. The abutments of the side arches require a force that can be split into two components H and X. Force H is provided by the end arch at the neck and saddle, but not without requiring an additional horizontal force Y in the same direction as X. This requires the bouts to be cross-tied to provide the force X+Y necessary to hold the system together. The cross-tie force X+Y is provided by the bouts’ cross-arches. The tension in these cross-arches will widen the bouts and pull down the crown height of the bouts’ cross-arch.

Summarising, then, the load L that goes into the long arch lifts the bouts, and the load C that goes into the centre crossarch lowers them. Whether they go up or down depends on the height of the bouts’ cross-arches. If they were low, the upward buckling force in the long arch would be weakened. The force pulling the bouts’ cross-arch down would also be weakened, but not to the same extent, since the bouts would have to go completely flat to reduce the downward moving force to zero.

So, we reach the important conclusion that when the string tension increases, if the ratio of the bouts’ arch height to the centre arch height is high, the bouts will rise. If this ratio is low, the bouts will drop.

THE LONG ARCH AND THE CROSS-ARCH BOTH CONTRIBUTE TO THE SUPPORT

FIGURE 3 A belly showing the bridge load splitting into the long arch and the cross-arch

FIGURE 5 The forces in the cross-arch support of the bridge

FIGURE 4 The forces and displacements in a belly long arch

ALL FIGURES NIGEL HARRIS

FIGURE 6 An exaggerated and diagrammatic illustration of the difference between belly and back long-arch shapes characteristic of classic violin models

Now consider the back plate. Although the soundpost is off centre, in all other respects the plate is loaded in the same way as the belly, except that the direction of the load at the centre is reversed and the force down the length of the plate is a tension rather than a compression. So, the back behaves in the opposite way to the belly: when the string tension increases, if the ratio of the bouts’ arch height to the centre arch height is low, the bouts will rise. If this ratio is high, the bouts will drop. Classic violin models show, to varying degrees, a difference in shape between the front and back plate of the form shown in an exaggerated way in Figure 6. Because the ratio h1/H 1of the belly is high and the ratio h 2/H 2 of the back is low, an increase in the string tension would raise the bouts’ height on both the back and belly, expanding the volume of the body. A quasi-static alternating string tension would drive a breathing deformation of the body, centred in the upper and lower bouts; a very efficient radiator of sound.

THE BACK BEHAVES IN THE OPPOSITE WAY TO THE BELLY: WHEN THE STRING TENSION INCREASES, IF THE RATIO OF THE BOUTS’ ARCH HEIGHT TO THE CENTRE ARCH HEIGHT IS LOW, THE BOUTS WILL RISE. IF THIS RATIO IS HIGH, THE BOUTS WILL DROP

The ratios h1/H 1and h 2/H 2 will determine the forces on the upper bouts’ arches and the direction of their movement. The arching shape can largely be defined by two non-dimensional parameters that can be called ‘EAR’ and ‘deviation’. With reference to figure 6:

• The letters E.A.R. stand for ‘end arch ratio’

• The ‘EAR’ of the upper bouts is (h1/H 1+ h 2/H 2 )/2. It is therefore the average, for the back and belly, of the arch height at the upper bouts divided by the arch height at the centre. There is a corresponding EAR for the lower bouts.

• The ‘deviation’ of the upper bouts is (h1/H 1- h 2/H 2 ). It is therefore the difference between the belly and back, of the arch height at the upper bouts divided by the arch height at the centre. There is a corresponding deviation for the lower bouts.

EXPERIMENTAL MEASUREMENT OF DISPLACEMENTS OF THE BODY CAUSED BY STRING TENSION

Having explored the internal forces resulting from an increase of string tension theoretically, we will now look experimentally at the displacements. The displacement of a number of points on the body of a violin of optimum EAR was measured as the tension in the strings was taken from zero to normal tension. The displacements were measured to high precision at the National Physical Laboratory (UK), using a coordinate measuring machine (see bit.ly/3ZTSU2U). Figure 7 shows the displacement of the violin surface in mm at the black dots, relative to the average position in space of the 8 points marked O, which are located at the edges of the widest points of the bouts.

FIGURE 7 Static deformation caused by an increase in string tension, from zero to playing pitch (shown in millimetres). Positive deformations are away from the violin towards the reader.

COURTESY NIGEL HARRIS

The length of the back increases and the belly shortens. The bouts of both the back and the belly lift. The violin widens in the upper and lower bouts and at the centre of both the back and the belly. On the diagrams of the back and belly, the broken lines separate areas of outward movement from areas of inward movement in the plates. It is clear that there is an increase in the volume of the body. When combined with the breathing displacement caused by the bridge rocking, the body adopts a breathing displacement over a high proportion of the surface. The general form of these measured deformations is consistant with the conclusions reached by the theoretical analysis given above. This breathing displacement will only be realised fully at very low frequencies (quasi-static). As the frequency rises the breathing shape will break up into a combination of smaller modal shapes. The modes that will be best excited will be those that have resonance frequencies near that of the excitation frequency and modal displacements in the same direction as this underlying breathing shape.

RELATIONSHIP BETWEEN EAR, THE C/L RATIO AND THE SOUNDPOST SETTING

If the value of EAR is raised, the belly bouts will be further away from the line of thrust and so it will lift more. If the C/L is increased, the increase in C will strengthen the downwards force on the belly bouts, and the reduction in L will weaken the upwards force on the belly bouts. So, the bouts will lift less. The same applies to the back, but in reverse. The higher the C/L ratio is, the higher the EAR will need to be in order to achieve compatibility of movement of the back and belly. Therefore, the optimum value for EAR is not a fixed number; it depends on the C/L ratio and can only be found by making a series of violins with the same C/L ratio and varying the EAR until values are found for EAR (in upper and lower bouts) that optimise the tone.

A VIOLIN THAT IS TOO LOW IN EAR WOULD BENEFIT FROM A POST SET FURTHER AWAY FROM THE CENTRE LINE

Carving the arching of a back plate

GETTY

The variables that affect the C/L ratio are: the outline shape; the five templates used for the cross-arch shapes; the arching height at the centre of the front and back and the flexural stiffness of the wood; the position of the soundpost; and the stiffness of the bass-bar. My own method for inferring the flexural stiffness of the detached plates from their mass and resonance frequencies is fully described in the 2005 Journal of the Violin Society of America, Vol.1 No.1(bit.ly/3JpVqqD). Achieving the same plate flexural stiffness for all violins in a series, requires that they have different thicknesses, but the same pattern of thickness distribution should be maintained. I use that given in Sacconi’s The ‘Secrets’ of Stradivari.

If the soundpost is moved closer to the centre line of the instrument, the line of thrust of the centre cross-arch moves closer to the arch and raises its stiffness, which raises the C/L ratio and so the violin would require a higher EAR. The distance of the soundpost from the centre line is not particularly sensitive and may require a move of 2–3mm or more to have sufficient effect on the C/L ratio. Conversely, a violin that is too low in EAR would benefit from a post set further away from the centre line. The post position has the same effect on the C/L of both the upper and lower bouts. If the upper bouts’ EAR is too high and the lower bouts’ EAR is too low, moving the post cannot correct for both faults. The solution is to make violins with a well-judged EAR in both the upper and lower bouts.

THE EFFECT OF EAR ON TONE

The violin evolved with a difference in shape between the back and belly. This would not have happened if it did not have a beneficial effect on the tone of the instrument. In 2013 I made an objective test to see if players could discriminate between violins that differed only in EAR or deviation: three violins were made with values of EAR which in my opinion optimise the tone, but one was of low deviation, one medium and one high. All the variables affecting the C/L ratio were fixed. The distance of the soundpost from the centre line of the violin was set precisely the same in all the instruments tested. They were made from the same wood. When the violins were being assembled the fronts intended for the low- and high-deviation violins were interchanged, giving three violins, one of low EAR, one medium (optimum) EAR and one high EAR, but all had the same deviation. These violins were given to six professional players to play for up to an hour or so until they were quite sure of their preference. They all preferred the optimum EAR violin. Then the fronts of the high and low EAR violins were interchanged, making the originally intended three violins of optimum EAR, but of low, medium and high deviation. These were again tested by the same players. They all said there was so little difference between them they could not express a preference. This test showed that the tone was clearly dependent on the EAR and much less affected by the deviation. If the high EAR violin had been fitted with a post closer to the centre line, the C/L ratio would increase and become compatible with the high EAR. Similarly, a post further away from the centre line would make the C/L suit the low EAR violin. By keeping the post unchanged in these tests, the effect of EAR alone was isolated. (This paper was published in the Journal of the Violin Society of America, vol.24 no.2, Fall 2013.)

The second part of this article will show the effect of EAR on the tone of each string and details of workshop procedures that enable makers to optimise the tone by controlling the arching of the plates.