On Graduating the Thickness of Violin Plates to Achieve Tonal Repeatability

The various criteria used for graduating the thickness of violin plates are discussed. These are shown apparently to be conflicting. A case is made that the flexural stiffness of the plates should be given priority. A simple indirect method is given for finding the flexural stiffness of the plates without the need to flex them in the hands or to apply forces and measure deflections. The method requires the back and belly resonance frequencies in the “ring” and “X” modes (modes 2 and 5) to be combined and modified by the plate weight to find a number that has proportionality to the plate stiffness. A suggested value is given for this stiffness number. The same stiffness number can be used for violins, violas and cellos. The important tonal determinants in the violin Before discussing methods of plate graduation we need to have a clear understanding of what are the important tonal determinants in the construction of the violin. The bowed violin string assumes a waveform that approximates to a saw tooth shape. This can be represented as a series of simple harmonic vibrations of displacement amplitude n a that decline as 1/n. The transverse string vibration (referred to as TSV) puts a transverse force (TSV force) on the bridge, which tends to rock it in its own plane. The transverse displacement of the string increases its tension and causes a vibration of string tension (referred to as longitudinal string vibration, or LSV). The amount of LSV developed in this way is small, but a larger amount is developed by the modal action of the body, which alters the distance between the string supports. The ratio of the LSV force on the bridge to the TSV force on the bridge has been found experimentally [2] to be about 0.4 up to 5000Hz, and above 6000Hz about 0.1. The transfer of power from the string to the body, per unit force squared by LSV force, is however about ten times higher than that by TSV force. The LSV forces put a downward force on the bridge and an upward force on the saddle and nut, which tends to make the bridge bounce vertically in its own plane and the violin bend in its length. But internally, LSV induces forces in the body, which tend to drive the end bouts cross arches and radiate sound efficiently. The violin complete with its strings vibrates in patterns called modes. The modal displacements peak at resonance frequencies. At any harmonic frequency the forces applied to the body will excite those modes that have resonance frequencies that are close, and which have large displacements in the direction of the applied forces. Up to about 1500Hz, the body modes are fairly widely spaced and the spatial arrangement of the applied forces is not critical, the nearest mode will be excited. Above 1500Hz as the modal overlap increases the spatial arrangement of the applied forces becomes significant in determining which modes are excited. The modes that involve large displacements of the end bouts cross arches are driven by LSV. The magnitude and direction of the LSV forces that are induced in the end bouts cross arches is very dependant on the arching shape. For this reason the sound radiated above 1500Hz is very arching dependant. These higher order harmonics have a significant effect on the timbre of the tone and the projection. All these matters are fully discussed in a later paper that I will submit to this Journal [2]. Not only is the arching shape important, but so also is the flexural stiffness of the plates both along and across the grain. It is possible to reproduce the same arching shape from instrument to instrument but unless the flexural stiffnesses are the same the arch will not behave in the same way. Much study has been made of the lower order modes in the violin and methods of controlling their resonance frequencies and even the mode shapes have been suggested, but the higher order modes are too numerous and complicated to yield to the same analysis and be controlled by the same methods. However, close control of the plate flexural stiffnesses and certain relevant features of the arching shape will ensure that a consistent pattern of higher order modes is excited. This will give tonal repeatability. In this paper the concern is with repeating the same flexural stiffness. In later papers the important features of the arching shape are identified [2] and a method of control is given [3]. The difficulties of plate thickness graduation In making the wooden parts of a violin there are really only two things to get right, the shape and the thickness. We tend to think of the arching shape as being the shape of the outside surface. Acoustically, the shape of the arching of the plates is the shape of the centerline of the wood thickness. If we make the arching with the same outer surface shape in all our instruments but graduate the thicknesses differently, then the wood-centerline arching shape will be different each time. This difference is not as inconsequential as is often assumed, and is certainly sufficient to affect significantly how the body responds to LSV forces. To make progress in violinmaking it is very important to alter one thing only from one violin to another in order to test its effect on tone, or to change nothing if we want to repeat the same tone every time. A series of violins with differing wood but the same shapes and thicknesses, will not produce the same result. Ideally, we must find a way of adjusting the thicknesses of the wood in such a way that the wood-centerline arching shape, the plate flexural stiffness and the resonance frequencies are maintained constant. Suppose one makes a violin and then removes the belly and adjusts the thicknesses and reassembles the instrument until the tone is optimized. In doing this the maker will have altered the wood-centerline arching shape, the resonance frequency, the flexural stiffness of the plate to different degrees along and across the grain, and the mass. To attribute the resulting tonal change, to a change in resonance frequency alone (or any other single variable) is not justified, and the maker will have learned nothing from the exercise. A violin made as a copy of another cannot be the same unless the wood thicknesses are adjusted to give the same resonances and the same plate flexural stiffness, and the surface shape of the arching is adjusted to give the same wood-centerline shape with the different wood thicknesses. Criteria for graduating violin plate thickness There are at least four basic systems of plate thickness graduation that have been used, and are still used. They rely on different criteria, a case can be made in support of all of them, and they all have their advocates. Criterion 1. We should make the wood thickness the same every time. It is very important tonally that the same wood-centerline arching shape should be reproduced from violin to violin. This can most conveniently be achieved by making the plate thicknesses the same for every instrument. This method ignores the requirements of the valid criteria 2 and 3 below, thus detracting from its undoubted benefits. In a paper that I will submit to this Journal shortly [3], I show that by adjusting the surface arching shape, it is possible to make the plate thicknesses different from instrument to instrument and still maintain the same wood-centerline arching shape. Criterion 2. We should adjust the thickness of the wood to make the long-grain and crossgrain plate bending stiffness constant. The radiation of sound requires that the wood flex. The stiffness of the plate in flexure, both along and across the grain will need to be the same for all instruments made in order that the plates behave in the same way. There is no point in having identical arching shapes if the flexural stiffnesses are different. To ensure repeatable tonal results the wood thickness must be adjusted to give the same long-grain and cross-grain flexural stiffness. An instrument has a certain playing resistance that the player is sensitive to. Some violins speak too easily and others need really hard playing to respond. There are a large number of possible causes of this but the flexural stiffness of the plates must be involved here. Historically, many makers may have removed wood from the detached plates until some degree of flexibility was achieved. They probably assessed the flexural stiffness of a detached plate by pressing the thumbs into the center of the plate while at the same time pulling up on the edges with the fingers. Assessment of plate stiffness by feel may be reasonably accurate if a reference plate is available for comparison, but a less subjective method is presented in this paper. Criterion 3. We should adjust the natural resonance of the free plates to certain predetermined frequencies. The violin radiates sound from modes and it has been suggested that the frequency at which some of the major modes occur may be tonally significant. With this aim in view, Carleen Hutchins proposed her well-known method of plate tuning [5-7]. The thicknesses of the detached back and belly plates are graduated to bring the resonance frequencies of up to three of the free plate modes, the 1, 2 (or “X mode”) and 5 (or “ring mode”) to certain prescribed frequencies and certain relationships between the back and belly. The objective is that when these plates are assembled into an instrument it will produce a more even “loudness” across the instrument. My experience with tuning the detached plates to prescribed frequencies is that there is a tonal benefit. The balance between the upper and lower strings is affected. To that extent I find the method works. However, the problem with it is that plates of widely varying mass can all be tuned to the same resonance frequencies. These plates would all have different flexural stiffness and the requirement of criterion 2 would not be met. Joseph Curtin [8] pointed out his concern that thicknessing by resonance frequency alone, disregards other plate properties that seemed too important to neglect. My experience is that because of its failure to control the flexural stiffness of the plates, plate tuning to predetermined frequencies alone, does not give tonal consistency from one violin to another. Criterion 4. We should adjust the plate thicknesses of the assembled violin to match some of the principal modal frequencies and shapes to those of a recognized standard as revealed by the frequency response function (frf) of the radiated sound. This method was pioneered by Martin Schleske [9, 10] and is arguably a shift from plate tuning to body tuning. As such it can be criticized for the same reasons as plate tuning, there are many different masses and stiffnesses that would give the same modal frequencies, but not necessarily the same sound. The surface arching shape can be taken from the instrument being copied but the subsequent adjustment of the thicknesses will alter the arching shape. Advocates of this method might argue that the precise arching shape is unimportant because the mode shapes and frequencies are the ultimate concern. But these can only be controlled in a few of the low order harmonics. The high order harmonics are much more arching dependant. Many violinmakers may find this method frighteningly complicated. Personally, I find the method too holistic, in that too many variables are altered at once. I like to be able to control separately the wood-centerline arching shape, the flexural stiffness and the resonance frequencies, so that I can identify their effects on the tone. The criterion used in the method proposed in this paper While it would be good to satisfy all the above criteria, that is clearly not possible. Choices have to be made. Essentially what I do is, graduate the plate thickness to maintain the same long-grain and cross-grain stiffness in all my instruments (criterion 2). I make this my priority because the coupling of the LSV with the body is highly dependent on the arching shape and the long-grain and cross-grain wood stiffnesses. I satisfy the plate-tuning requirement (criterion 3) by careful choice of wood. I satisfy the requirement of criterion 1 by modifying the surface arching shape to maintain the same wood-centerline arching shape with the thicknesses used. I will now describe how I graduate the plate thickness to achieve prescribed plate flexural stiffness. Determination of the plate flexural stiffness The flexural stiffness of a detached violin plate is proportional to the product of its weight and the square of its resonance frequency. For those interested in the derivation of this basic relationship, it is given in an appendix to this paper. The relationship holds good for any size of plate, violin, viola or cello. The resonance frequency should be that of a mode that best samples the wood properties over a large area of the plate and in both directions of the grain. Of the various possible free-plate modes there are two that are particularly useful. These are traditionally described as the second and fifth modes, or as the ‘X mode’ and the ‘ring mode’. Many violinmakers are familiar with these modes and know how to find their resonance frequencies, or tap tones as they are often called. The methods need not be described here except to say that suspending the plate between the thumb and second finger, tapping it with a knuckle and comparing the modal frequency with a tuning fork or piano keyboard, achieves sufficient accuracy. The extra precision afforded by electronic methods is not needed but can be helpful to those with a less experienced ear. These procedures are well described by Hutchins [5]. If musical notation is used to define the tap tones, these must be converted to the frequency in Hertz (cycles per second). We calculate the plate stiffness factor using the formula: 2 2 mode x f mode ring f W K  