We now turn to something more workshop-based and close to the heart of many instrument makers: the selection, cutting and characterisation of wood. We will discuss which properties of wood might be most important and how they might be measured. We will also try to relate these properties to the cellular organisation of wood, which can only be made visible using the kind of high-tech measurements discussed in the previous section.
We will mostly be concerned with the specific agenda of wood for the bodies of stringed instruments, with a focus on properties that influence vibration and sound. But we should also recognise that there are many other issues surrounding wood for musical instruments, especially in connection with sustainability. Many instruments have traditionally made use of tropical hardwoods: rosewoods for guitar backs and sides, and for making marimba bars; African blackwood for wind instruments; ebony for fingerboards of many stringed instruments; pernambuco for making bows.
Most of these timbers are now threatened, and many are listed under the CITES convention (which is short for the Convention on International Trade in Endangered Species of Wild Fauna and Flora). As a result, there is considerable interest in finding or developing alternative materials. In some cases, synthetic materials can be considered: wind instruments made of plastic and violin bows made of carbon fibre composite can be good enough that they are finding favour with some musicians. In other cases, more sustainable timbers are being explored, such as European walnut for guitar bodies. In the case of fingerboards one possibility involves a return to the earlier practice of using a composite construction, with a core of lighter wood and just a veneer of ebony or some other hard-wearing wood for the working surface.
Each of these applications involves its own set of desiderata and constraints, and the scientific mindset can be applied to nearly all of them. I will give a very brief review of some of the issues, before returning to the primary agenda of properties affecting vibration and sound. There are two issues which, in one way or another, tend to influence all applications: how a wood looks, and how it behaves under a maker’s hands when they are trying to shape it. Visual appearance is outside our remit here, but we can say some things about behaviour on the woodworking bench.
If woodworking convenience was the only consideration, then instrument makers would probably choose the same types of timber that wood-carvers prefer: close-grained, not prone to splitting, not too hard so that controlled cutting is easy, and capable of holding crisp details. That recipe points towards timbers like boxwood, and fruit woods like apple, pear or plum. But in fact these are not very common in musical instruments. In the stringed instrument world they are mainly used for fixtures and fittings like pegs and violin tailpieces. Those components are indeed often decorative, and boxwood in particular is a common choice for up-market fittings. It is also a traditional wood to make chessmen, and perhaps it acquired its name because of use for making decorative snuffboxes.
There is an important exception to those rather dismissive comments, in the world of woodwind instruments. In earlier times, boxwood and some fruit woods were commonly used to make woodwind instruments like recorders and oboes, and they remain a popular choice for recorder making. They can make handsome instruments, with excellent musical qualities. However, they have a disadvantage. These woods are rather sensitive to changes in moisture content, which can arise from changes in the weather and also from the fact that a player blows warm, moist air into them.
Absorption of moisture by the wood can lead to distortion or cracking, and it can also impair the musical operation. The surface texture of the inside of the bore may change, as a result of “grain raising”. This can increase the damping of the internal resonances of the air in the instrument, which in extreme cases can lead to the instrument “choking”. The standard control measure for this problem is to oil the wood, traditionally by using a feather to apply oil to the inside of the bore.
However, some of the most familiar woodwind instruments, like the clarinet and the oboe, are black in colour, and are obviously made of a different wood. This is usually African blackwood (Dalbergia melanoxylon), also known as grenadilla. It is one of the very hardest and densest of all timbers, with little or no discernible grain. This makes it relatively immune to moisture-related problems, but at a price: the wood is very difficult to work. It is so hard that it can blunt tools very quickly. The tubes of clarinets and oboes have to be made using metal-working tools like tungsten carbide lathe cutters.
Fingerboards of stringed instruments are another application where hardness of the wood is at a premium. The problem this time is not to do with moisture, but the wear and tear resulting from repeated vigorous contact with the strings and the player’s fingers during normal playing. Hard woods like ebony are often used for fingerboards. These are not quite as hard as African blackwood, but they still present challenges in terms to keeping your tools sharp.
The favoured wood to make violin bows is called pernambuco (previously Caesalpinia echinata, renamed to Paubrasilia echinata in 2016). This choice goes back to Tourte, the originator of the modern bow (see section 9.7). The wood was originally imported to Europe in the 16th century, to make an important dye. But that is not what attracted Tourte: a bow requires a challenging combination of properties, and presumably he found that pernambuco combined them to perfection. The long, slender bow-stick must be strong enough, and be capable of being bent to the required cambered shape using heat. The cranked tip (see Fig. 2 of section 9.7) is carved from the same solid piece of wood, and it is crucial that this tip doesn’t break off easily, despite the fact that the grain of the wood runs across it at a very narrow point. Pernambuco has been over-harvested, and trade in the wood may be banned altogether before long. This is causing a serious problem for bow-makers, although of course it was not their relatively modest usage that led to the over-harvesting.
Percussion instruments like the marimba bring in a different aspect of wood properties and choice. Unlike violin bows or guitar fingerboards, the bars of a marimba are used directly to make sound. This means that the vibration properties of the wood are important, including the damping. Damping determines how well the sound rings on after a bar is struck, and this aspect of the sound quality is a major factor in the choice of particular woods. Again, the traditional choice has settled on some species of tropical hardwood.
All the examples just reviewed have chosen woods that are denser and harder than the wood-carver’s choice of boxwood or fruit woods. But when we turn to the soundboards of stringed instruments, we find the opposite choice being made: wood species that are somewhat difficult to work because they have very low density, rather than very high density. We already know the main reason for this choice. Back in section 5.2, with detail in section 5.2.1, we looked at a simple criterion for choosing soundboard material based on loudness. If you want to make a loud instrument, you need a material with low density but high stiffness. The material selection charts we showed there pointed towards softwoods. The most widespread choice for stringed instruments is Norway spruce, but other species such as Sitka spruce are also used, and in the guitar world it is common to use Western red cedar for soundboards.
Softwoods like Norway spruce give the instrument maker some problems. These timbers are very prone to cracking along the grain, compared to hardwoods which have a more “interlocking” cellular structure (compare Figs. 1 and 9 of section 10.2.1, for example). Also, the marked difference in hardness between the “spring wood” and the “summer wood” in the annual growth rings makes it more difficult to carve smooth curves. Your knife tends to move in jumps through the softer wood, stopping on the harder layers. Violin makers have to put up with this when carving details such as f-holes, but it would make this wood a difficult choice for carving chessmen.
The vibration of a soundboard, like any other vibration, is governed by mass, stiffness and damping. The material property governing the mass is the density of the wood, which is very easy to measure by weighing a block of wood of known volume. There is also a neat trick that violin makers sometimes use (if the wood dealer allows them to) when selecting spruce for top plates. You can deduce the density quickly by seeing how high the billet of wood floats when immersed in a bucket of water, as sketched in Fig. 1. The volume of water displaced must weigh the same as the block of wood, by Archimedes’ principle. Armed with a tape measure you now find the proportion of the length of the billet which is underwater. This tells you the density of the wood billet relative to the density of water — and we know that the density of water at normal temperatures is 1000 kg/m$^3$.
“Stiffness” is a far more complicated thing, for a material like wood that has different properties in different directions. Figure 2 reminds us of the three principal directions in a tree, conventionally labelled L (for longitudinal), R (for radial) and T (for tangential). For a straight-growing tree without knots or other blemishes, the wood structure is symmetrical in the LR plane, and also in the RT plane. If we disregard the curvature of the annual rings, it is also approximately symmetrical in the LT plane. So a small block of wood extracted from the tree has approximate symmetry in three mutually perpendicular planes. In the language of materials science, such a material is called “orthotropic”.
There is now a piece of standard textbook theory which tells us how many parameters are needed to give a complete description of the stiffness of such a material (see for example this Wikipedia page). The answer is 9: more complicated than we might have wished. We can visualise these parameters if we think about cutting thin rods from our piece of wood, aligned in the three directions L, R and T. We can imagine taking one of those rods, gripping the ends in a testing machine, and applying various forces to it.
If we apply tension and measure how much the rod stretches, the relevant stiffness is characterised by the Young’s modulus. (We met Young’s modulus back in section 3.2.1 when we looked at bending beams.) We can measure something else in this same test. As the rod is stretched lengthwise, it will usually get a little thinner at the same time. In our orthotropic material, the amount of thinning might be different in the two crosswise directions. The ratio of the lengthwise stretch to a crosswise contraction is called Poisson’s ratio. Finally, if instead of applying tension we twist the rod in our test machine, we can measure the amount of torsion created by a given torque. That is governed by a different stiffness called a shear modulus. We can repeat these tests with each of our rods. In total, we will find three Young’s moduli, three shear moduli, and six Poisson’s ratios. But there is a mathematical relation between some of these, so that we only need three of the Poisson’s ratios to complete our set of parameters.
Luckily, we don’t need to think of measuring all 9 parameters for every piece of wood we use. All soundboards of musical instruments are deliberately cut quite thin: either approximately flat as in a piano or guitar, or arched as in a violin or cello. Furthermore, the instrument maker will try quite hard to have the grain direction (the L axis) lying in the plane of the plate. Under those circumstances, we only need a reduced number of stiffness parameters. A flat plate cut in a suitable way for a guitar or piano soundboard has the 2D version of orthotropic symmetry: the structure is symmetrical in two mutually perpendicular axes lying in the plate. A maker would probably describe these axes as lying “along the grain” and “across the grain”.
The corresponding piece of textbook theory now tells us that we need 4 stiffness values — and it turns out that one of those is not very important (see McIntyre and Woodhouse [1]), so for a first approximation we only need three. These can be visualised as corresponding to things that a guitar maker would feel in their hands, by bending and flexing the soundboard wood. We can bend the plate along the grain or across the grain. We can also twist it: we will see a picture shortly of what that looks like.
Vibration resonance frequencies can give a very good way to measure stiffnesses. The approach relies on have a test specimen of a suitable shape that there is a simple theoretical expression for the frequencies in terms of the geometry and material properties. The geometry is easy to measure directly, and the density can then be determined by weighing. The only unknown quantity in the formula for the frequencies will then be a stiffness. You can turn the formula around: frequency is easy to measure quite accurately, using an FFT app in your computer or your phone, and so you can deduce the stiffness.
The simplest application of this approach is to beam-shaped samples, like the ones shown in Fig. 3. These particular examples are made from the offcuts of spruce after cutting violin top plates out of their billets. The lowest resonance of a beam like this is something we have already looked at, back in section 3.2. The mode shape is repeated here in Fig. 4. The resonance frequencies of a bending beam like this were discussed in section 3.2.1: the detailed formula is given in the next link. It can easily be used to determine the Young’s modulus from a measured frequency. If you want to do this measurement yourself, the link gives some suggestions for how to do the test in order to get the most accurate and reliable answers.
By cutting beam samples like this from your offcut wood, in the directions parallel and perpendicular to the grain, you can deduce two of the three stiffnesses that were mentioned above. However, it is not so easy to get the third constant this way: it relies on twisting motion rather than bending motion. If you are a guitar maker, though, you might have an easy way to measure all three stiffnesses. The wood for guitar soundboards is often supplied in the form of rectangular panels with essentially constant thickness. This makes them ideal for a measurement of all three important stiffnesses.
The reason is that a rectangular panel like this usually has three vibration modes looking more or less like the animations in Figs. 5, 6 and 7. (To find out why it is only “usually”, see the next link.) These three modes involve precisely the three types of motion described above: bending along the grain, bending across the grain, and twisting. The frequencies of these three modes can thus be used to deduce stiffnesses, as described in the link, which also explains the measurement procedure in detail.
The first thing you learn from doing beam or plate tests like these does not come as a surprise: the stiffness along the grain is always much higher than the stiffness across the grain. This is a direct result of the cellular structure of spruce. Remember the microscope images from section 10.2, and the analogy with a bundle of drinking straws? It is intuitively clear that a bundle of straws will feel very stiff if you press it lengthwise, but much more squashy if you press across the width. The reason is that to deform the bundle lengthwise, you need to compress every straw along its length. But to deform the bundle sideways, each straw simply needs to distort into an oval shape. The walls of the straws bend, rather than compressing or stretching, and because they are very thin they do not provide much stiffness against this bending. For more detail about this bending/stretching idea, see the book on foams and honeycombs by Gibson and Ashby [2].
The full story of the link between wood stiffnesses and cellular structure is, needless to say, more complicated than this simple picture. Several factors play a role. First, the annual ring structure introduces relative stiff planes (the dense summer wood) interleaved with the spring wood which is much less stiff. This layered structure has different effects on the different stiffnesses.
One counterintuitive effect is that it should make the stiffness in the T direction higher than the one in the R direction. The reason is that the layers with alternating stiffness are connected “in series” in the R direction but “in parallel” for the T direction. But the actual stiffnesses are the other way round: the R stiffness is slightly higher than the T stiffness. The explanation is that the effect of the layered structure is counteracted by another effect, which turns out to be stronger. The cells in the T direction are randomly staggered, but the cells in the R direction are quite strongly aligned. This is clear in the micrographs shown in section 10.2, one of which is repeated below as Fig. 10. The alignment is partly to do with how the cells divide in the growing tree, and partly to do with the constraining effect of the medullary rays which interpenetrate the tracheids. This alignment has the effect of increasing the R stiffness relative to the T stiffness. For details of all this, see Kahle and Woodhouse [3].
There is, of course, a link between the 9 stiffnesses that characterise the solid wood, and the 4 needed for a thin soundboard. Investigating that link reveals something interesting. There are a few published measurements of the full set of 9 stiffnesses for spruce of suitable quality for soundboards. Some of these results are surprisingly old [4]: they go back to the time when aircraft were made of wood, and naturally enough aircraft manufacturers wanted low density and high stiffness, just as instrument makers do. Using some standard results from the mathematical theory of elasticity (see [1] for details), we can calculate the 4 plate stiffnesses from the measured values of the 9 stiffnesses of the solid.
We can use that calculation to investigate a question of immediate significance to instrument makers. Suppliers of wood for musical instruments will usually take great care to cut the tree parallel to the grain. However, in the interests of getting the most out of each tree they do not necessarily cut every soundboard blank exactly in the LR plane, so called “quarter-cut” wood. Instead, the annual rings may go through the thickness of the plate at some angle which we can call the “ring angle”.
How much difference does this angle make to the behaviour of the soundboard? Figure 8 gives an example of the answer to that question. This shows the three “plate stiffnesses” as a function of ring angle, going from $0^\circ$ (perfectly quarter-cut) all the way round to $90^\circ$, when the annual rings are in the plane of plate. The bending stiffness along the grain, shown in red, doesn’t change very much. Neither does the twisting stiffness, shown in black. But the blue curve for the bending stiffness across the grain shows very dramatic variation (noting that the plot uses a logarithmic scale because the range of values on the vertical axis is very large). As the ring angle moves away from zero, the stiffness falls steeply — only $10^\circ$ is enough to reduce the stiffness by a factor of 2. It continues to fall until the ring angle reaches about $45^\circ$, reaching a value nearly 10 times lower than the quarter-cut case. After that the stiffness climbs again, ending up only about a factor 2 lower than where it started.
Figure 9 shows a spruce plate (sold for the purpose of making a guitar soundboard) in which the ring angle is close to $45^\circ$. The stiffnesses of this particular plate have been measured by the procedure described in the previous link (see [1] for the detailed results), and the values were in line with the predictions in Fig. 8. The ratio of long-grain stiffness to cross-grain stiffness was nearly 70:1, whereas for quarter-cut spruce that ratio would be more like 16:1. When you flex this plate in your hands, it is scarily bendy in the cross-grain direction.
We can understand why the cross-grain stiffness gets so low with this intermediate ring angle by again thinking about the cell structure of spruce. Figure 10 shows a repeat of Fig. 2 from section 10.2. Imagine a rod cut from this wood, aligned with the horizontal axis in this image. The annual rings would then cut across the rod with a ring angle in the vicinity of $45^\circ$.
Now think what would happen if you tried to stretch this rod. The dense portions of the annual rings would act like fairly rigid plates, but the very open spring wood could easily deform by shearing motion. At the cell level, the deformation would only involve bending the thin cell walls in the spring growth into a slight S-shape. The effective stiffness for that shearing is extremely low (see [3] for a careful analysis). The rod would stretch rather in the way a deck of playing cards can be spread, by sliding each card over its neighbour: each card corresponds to the dense layer in one annual ring. Another analogy of this effect of an intermediate ring angle is with woven fabric. Most fabric is quite stiff if you try to stretch it parallel to the threads of the weave, either warp or weft. But at $45^\circ$ the fabric stretches easily, by shearing motion: this is the basis of “bias cutting”.
A consequence of the dramatic variation shown in Fig. 8 is that, so far as cross-grain stiffness is concerned, the ring angle probably makes at least as big a difference as the variation between one piece of wood and another. Apart from anything else, the ring angle is not usually constant across the entire width of a soundboard. Look again at Fig. 3, and see the variation in ring angle within each of these 6 samples — and these particular beams were all cut from the same tree!
If you want to maximise cross-grain stiffness, you need to keep the ring angle close to zero. For a violin maker, this brings in another variable. The arched top plate of a violin is (usually) carved out of solid wood. An ingenious maker might try to choose wood where the variation of ring angle “fans out” in a way that follows the arch of the plate. Figure 11 shows that Stradivari might have done this, at least in the particular violin shown here. The image is a close-up of Fig. 5 from section 10.2: a CT scan of the violin, which shows the detailed configuration of each annual ring. In the section between the f-holes, the rings do indeed fan out so that they follow the arch to an extent.
Having dealt with the mass and stiffness properties of wood, the remaining material property relevant to vibration is the damping. Damping behaviour is tied very closely to the stiffnesses that we have just discussed. The details are a bit messy (see the next link), but we don’t really need to know more than the fact that each stiffness has a damping factor associated with it. They can be measured as part of the same beam or plate testing procedures that we have already described. As well as measuring the frequency of each beam or plate mode, you also measure its decay rate, or equivalently its Q factor.
If you are wanting to select wood for a marimba bar, then this is indeed the measure of damping that you will need to take into account. But if your interest is in the damping of modes of a violin body, things are more complicated. Damping is a measure of how rapidly energy is dissipated when something vibrates. What we have talked about so far is material damping: energy dissipation arising directly from the fact that the material, wood in our case, is stretching, bending or twisting. But there are other physical mechanisms for energy dissipation, and the total damping of a mode is the combined effect of all of them, added together.
Any kind of fixture for holding your instrument while you test it is likely to introduce extra damping. This was already mentioned in the previous side links on beam and plate measurement: for damping measurements based on beam or plate testing, it requires great care to support the beam or a plate in a way that doesn’t add a lot of extra damping. We will come back to the issue in section ?, when we talk about measuring frequency response functions. Furthermore, if you attach any kind of sensor to the test specimen to measure the vibration, that, too, will add damping. Energy may be dissipated by the sensor itself or in a layer of adhesive by which it is attached, and also energy may be lost along the the sensor’s signal cable.
If holding fixtures are so problematic, why have I suggested doing tests with free, unsupported beams and plates? Why not clamp a beam at one end, and measure the frequency and damping as it vibrates like one tine of a tuning fork? The answer is quite surprising: clamping a beam introduces a subtle but significant extra mechanism of energy dissipation. The sketch in Fig. 12 shows why. A beam (yellow) is held in a clamp (grey). Now think what happens if the beam bends downwards as indicated by the black lines. This bending involves stretching the top surface of the beam, and compressing the bottom surface. Near the edge of the clamp, the beam wants to move, slightly, as indicated by the red arrows. When this situation is analysed carefully, it turns out that for a sharp-cornered clamp like the one sketched here, the limit of friction will always be exceeded near the corners. The surfaces of the beam must slip a little against the clamp jaws, and this will result in energy dissipation by friction.
[1] M. E McIntyre and J. Woodhouse, “On measuring the elastic and damping constants of orthotropic sheet materials”, Acta Metallurgica 36, 1397—1416 (1988).
[2] L. J. Gibson and M. F. Ashby, “Cellular solids”, Pergamon Press (1988).
[3] E. Kahle and J. Woodhouse, “The influence of cell geometry on the elasticity of softwood”, Journal of Materials Science 29, 1250—1259 (1994).
[4] H. Carrington,”The elastic constants of spruce”, Philosophical Magazine 45, 1055—1057 (1923).