A. The family of hammered string instruments
There are some musical instruments that use hammers of one kind or another to hit strings. The obvious example is the piano, but there are others. There is a family of zither-like instruments known by a wide variety of names around the world: for example the hammered dulcimer, the cimbalom and the santoor. For these, the player uses a pair of hand-held hammers to strike the strings. Then there is the clavichord, a keyboard instrument in which a note is sounded by striking the string with a metal “tangent” which excites the vibration and then remains in contact to form one end of the string’s vibrating length.
The story of this family of instruments starts with the zither-like instruments. The earliest of these instruments was known as the psaltery. The strings were plucked with the fingers, but then someone had the bright idea of using small hammers instead. Figure 1 shows an example: this is a santoor, being played by the Kashmiri virtuoso Bhajan Sopori. Performance technique was entirely transformed by the shift to hammers: while it was no longer possible to play multi-note chords simultaneously, this was compensated by the possibility of rapid note repetitions and sequences, taking advantage of the fast bouncing of the lightweight hammers from the strings.
The possibilities for performance technique were transformed again when both the plucked and hammered versions of the psaltery family were “mechanised” into keyboard instruments. The earliest of these was the clavichord, thought to have been invented in the early 14th century. Plucked instruments came next: the harpsichord was probably invented in the late 14th century, although the precise origin is not known. Finally, the fortepiano was invented by Bartolomeo Cristofori around 1700. Over time this instrument developed into the piano as we know it today. We will look at the piano and the clavichord in this section: the harpsichord belongs with the plucked-string instruments we looked at earlier.
B. Bouncing from strings
Strings have a special property that affects how a hammer will bounce in an instrument like the santoor or the piano. We already noted this property in a different context back in section 9.2.1, when we were thinking about making a string vibrate by bowing it. If you could have a string that was infinitely long and you dropped a mass onto it, the mass would not bounce at all. Disturbances on the string would spread outwards in both directions from the point where the mass landed, but the mass would stay “stuck” to the string. However, a real string obviously has finite length, and after a short delay the outward-travelling waves will reflect from the ends of the string and return to the position of the mass. Eventually, the effect of these reflections will cause the mass to bounce off.
The best way to visualise this is by using our simulation model to make an animation. We will begin with an example with parameter values in the right kind of range for a santoor or hammered dulcimer — but we will consider a single string, although the real instruments normally have strings in groups of 2, 3 or 4. We will choose a steel string with diameter 0.5 mm and length 0.4 m, tuned to the note $A_4$ (440 Hz). All modes of this string are given the same Q-factor, with the value 1000. We will strike this string with a 1 g mass at a position near the centre, 40% of the way from one end. We can simulate the behaviour using the same model we developed in the previous section: we simply use the properties of the string (including the inharmonicity associated with its bending stiffness, as described in section 5.4.3) in place of the plate model we used previously.
Figure 2 shows the resulting motion of the string and mass, with a hugely exaggerated vertical scale. The “hammer” mass appears as a red star. It comes down and hits the string at the point marked by a green star. The animation continues just long enough to see the hammer being thrown off the string. Before that, the mass stays in contact with the string — but the simulation model includes a fairly stiff contact spring, so the red and green stars are sometimes slightly separated. The initial outgoing disturbances on the string can be clearly seen, followed by a complicated pattern of reflected waves.
The resulting waveform of force applied to the string by the bouncing hammer is shown in Fig. 3. It has a shape quite different from any of the examples seen in section 12.1. There is an initial rapid upward jump in force following first contact. The force then ramps downwards, before it jumps up again when the first reflected pulse arrives back from the nearer end of the string. Notice a curious “precursor wiggle” in the force just before the second upward jump. This is caused by the effect of bending stiffness in the string — waves at different frequencies do not all travel at the same speed, and the higher-frequency components arrive back at the mass a little sooner than the lower-frequency components. After the second jump the pattern grows more complicated as multiple reflections arrive, until eventually the mass is thrown clear and the force drops to zero.
Figure 4 shows (in red) the rather complicated frequency spectrum of this contact force. The plot also shows (in blue) a frequency spectrum relevant to the sound made by the instrument. The simulation model doesn’t include any representation of the soundbox of the santoor, which would be necessary if we really wanted to calculate the sound. But we can use a trick that we have used before to get an idea of the sound, by calculating the waveform of force that the string exerts on the bridge. If the soundbox were an ideal loudspeaker, this would translate directly into the sound. In reality the sound will be modified by the vibration resonances of the body, but this bridge force allows us to get an idea: you can listen to it in Sound 1 below.
The blue curve in Fig. 4 shows the frequency spectrum of this bridge force. The sharp peaks indicate the “harmonics” of the string — except that they are not truly harmonics, because we have allowed for the inharmonicity caused by bending stiffness. The pattern of peak heights is governed by two things: the complicated form of the force waveform shown in Fig. 3, and also the chosen striking position. We have struck the string at the point 2/5 of the way from the bridge, and this is the reason that every 5th “harmonic” is missing. (This was explained back in section 5.4 — see Fig. 2 in that section and its associated text.)
Sounds 2 and 3 give two other examples of bridge force waveforms, to compare with Sound 1. Sound 2 is the result of using a heavier hammer, with a mass of 3 g rather than 1 g. Sound 3 is the result of plucking the string at the same 2/5 position, with an ideal pluck — the kind of thing we called a “wire-break pluck” when we were studying plucked strings in earlier sections. All three sounds are recognisably different: hitting with a hammer produces a different sound from plucking, and the mass of the hammer makes a difference to the details of the sound.
However, these sounds are a bit misleading. In practice, a player of the santoor or hammered dulcimer normally strikes the strings much closer to the bridge: the striking position at 2/5 of the length was chosen to make the animation in Fig. 2 easier to follow. If instead we use our 1 g hammer to strike at a position 1/10 of the length from the bridge, Fig. 5 shows what happens to the force pulse. The new pulse is shown in blue, while the red curve is the same as in Fig. 3. The earliest part of the blue waveform is exactly the same as the red one, but the bridge is now closer so the first reflection arrives much sooner, and after that the two waveforms diverge. The resulting sound at the new striking point can be heard in Sound 4, and a corresponding pluck at the same position is in Sound 5.
For the benefit of any santoor or dulcimer players who may be reading this, I should say a few words about the masses used in these simulations. The hammers visible in Fig. 1 are surely far heavier than 1 g, so have I used unrepresentative values? There are two reasons why not. The masses I have mentioned, 1 g and 3 g, refer to an equivalent point mass, not the total mass of the hammer. In the drumstick model used in the previous section, based on a uniform rod hinged at the non-striking end, that equivalent mass was 1/3 of the total mass of the rod. So with rod-shaped hammers, the total mass would be of the order of 3 g and 9 g for the two cases.
But there is another factor. The simulations were based on a single string, but real instruments in this family have groups of 2, 3 or 4 strings for each note. To achieve the same effect on each string of the group, the hammer mass would be scaled up by the same factor of 2, 3 or 4. So an actual hammer with a mass of the order of 10 g would have a similar effect to my 1 g mass in the simulated cases. The 3 g case would correspond to an actual hammer weighing of the order of 30 g.
C. The piano, Part 1: the strings
The issues of hammer mass and striking position reappear in a slightly different form in the context of the piano. The key difference from the santoor or dulcimer is that the string properties vary over a much wider range, while on the other hand every note has its own hammer rather than the player using the same hammers for all notes. We need to start by looking at the typical pattern of stringing in a modern piano. As an example we will show some data for the same piano we met earlier, in sections 7.1 and 7.3: a Broadwood baby grand dating from early in the 20th century but refurbished more recently.
Figure 6 shows a general view of this piano, and Fig. 7 shows the pattern of strings in more detail. The lowest notes (17 of them in this particular piano) have single strings, with a steel core over-wound with copper. The next 13 notes each have a pair of strings, again steel-cored over-wound with copper. The remaining notes, 58 of them, all have a triplet of plain steel strings. The layout of these higher strings shows a harp-like pattern, with progressively shorter strings for the higher notes. Because the lengths can vary like this while still fitting inside the relatively small case, the wire gauges for these strings do not need to change very drastically to accommodate the range of tuned notes. But for the lower strings the constraint of the case length becomes severe, and these strings need to be progressively heavier to achieve the low frequencies required. Over-wound construction allows the string mass to be increased without too big a penalty in terms of inharmonicity caused by bending stiffness. But the very lowest notes on this small piano do not give a very satisfying sound — this is the reason that a concert grand piano needs to be so large, to allow longer and therefore relatively lighter strings to be used.
We can determine the mass per unit length of the strings by simple measurement. We can measure the diameter of the load-carrying steel core, and for the over-wound strings we can also measure the outer diameter of the wrapped portion. Using standard values for the densities of steel and copper, we can infer the mass. Some results are shown in Fig. 8. I have measured all the over-wound strings, but the plain steel strings are sufficiently similar that it was enough to measure all the C strings in different octaves. The values vary from just a few grams per metre for the plain strings, up to over 200 grams per metre for the heaviest bass strings. Some parameter values for the C strings are listed in the next link.
Now we can deduce the tensions of the strings, by combining the mass information with measurements of the vibrating length of each string, together with knowledge of the nominal tuned fundamental frequencies. (The formulae relating these quantities can be found back in section 7.2.1.) The results are shown in Fig. 9. The plain steel strings all have a tension in the vicinity of 650 N, while the over-wound strings have higher tensions in a kind of “sawtooth” pattern which may seem puzzling at first sight.
To explain this pattern, we need to calculate one more property, the impedance. Impedance determines how much force is needed to create vibration with a given velocity amplitude, so flimsy structures have low impedance and robust ones have high impedance. As we saw back in Chapter 5, the loudness of a given note on any stringed instrument is governed by the ratio of two impedances: the impedance of the string(s), and the impedance of the instrument body measured at the bridge. (The earlier discussion was phrased in terms of the admittance of the body, which is the inverse of its impedance.) The impedances of our set of piano strings are shown in Fig. 10 — but there is an important point to note here. The impedance of interest is of all the strings sounding a particular note. So for the paired strings, we need to double the value of the impedance of a single string; for the triplets we need to triple it.
Now we see a potential problem for the pianist. We might anticipate that there would be an undesirable jump in loudness between pairs of notes straddling the boundaries between 1, 2 and 3 strings per note. Figure 10 shows that the piano designer has tried quite hard to combat this problem. There is a jump in impedance between the single and double strings, but this jump is only from about 9 to about 11: much less than the factor of two we might have expected when the number of strings per note doubles.
This has been achieved by choosing strings so that both the mass per unit length and the tension jumped down when moving to the double strings, in such a way that the impedance of a single string, which is a combination of those two quantities, drops by nearly the factor of 2 that would be needed to cancel out the effect of doubling the number of strings. So the “sawtooth” in Fig. 9 is a means to an end: the important thing is that the total impedance plotted in Fig. 10 follows a fairly smooth trend across the entire range of the piano.
There is another important thing to note from Fig. 10. Back in section 5.1, Table 1 gave some values of impedance for strings of a violin, a guitar and a banjo. All the numbers were less than 1 Ns/m. Now look at the values on the vertical axis of Fig. 10: they are an order of magnitude larger! This tells us something important about how the modern piano has developed from its harpsichord-like ancestor in 1700. As we noted in section 5.2, the impedance ratio between strings and body govern two things: the loudness, and also the decay rate of the note once it has been started. The banjo was noted as an extreme example of an instrument that has a relatively large value of this impedance ratio: the banjo is loud, but the price of that loudness is that the notes die away very rapidly, giving the familiar “plunk” sound.
If you want to design a stringed instrument which is loud and also capable of legato playing, you need to increase the impedances of both the strings and body, keeping the ratio fairly small. In other words, you need heavy strings on a very robust body. That is, of course, a short description of the modern piano: we have just seen that piano string impedances are an order of magnitude bigger than those of stringed instruments played with fingers, plectrum or bow. The process of evolving the design of the piano towards greater loudness has taken this a long way: according to measurements by Conklin [1], the combined tension of all the strings on a concert grand piano is of the order of 200 kN: if this force was provided by hanging weights, they would add up to about 20 tonnes. To support this huge tension, modern pianos have a cast iron frame: you can see it around the edges in Figs. 6 and 7.
Each piano key is attached to a complicated mechanism. When the key is pressed, two things happen. Near the bottom of the key travel, the hammer is “fired” at the strings. This hammer is hinged at one end, and covered with a thick layer of felt at the other end. After being launched by the key mechanism, the hammer flies across a small gap and the felt-covered end hits the string(s). When it rebounds, it falls back to a resting position slightly away from the strings, then when the key is released it falls all the way back to its home position alongside all the other hammers. At the same time, all the time the key is depressed a damper is lifted off the relevant string(s) so that vibration is possible. Releasing the key drops this damper onto the string(s), stopping the vibration. Figure 11 shows some of this in action: the left-hand image shows the line of hammers in their home positions, the right-hand image shows the position of a hammer and its associated damper while its key is pressed, but after the impact and rebound from the string has occurred. However, as we will explore in some detail shortly, the hammers are not all the same: the bass hammers are heavier, with thicker felt. Conversely, at the treble end the hammers are lighter and the felt is much thinner.
D. The piano, Part 2: Simulations
Now we are almost ready to simulate some piano notes. The simulation model needs to include an important effect: a piano hammer exhibits significant nonlinearity. The thick layer of felt covering each hammer, visible in Fig. 11, behaves as a hardening spring (see section 8.2). This has consequences for the waveform of contact force, and causes a marked difference in tone quality when a note is played gently or loudly: we will illustrate this in a moment. There have been some direct measurements of this hardening-spring behaviour — at around about the same time in the late 1980s, several authors made useful contributions.
Conklin [2] used a commercial hardness tester called a “Durometer” on the felt of various hammers. You can see a picture of it in action in Fig. 8 here. By working from the side of the hammer, he was able to show that the felt properties varied through the thickness: softer on the surface, harder and stiffer deeper down. It is probably this gradation that is responsible for the nonlinear stiffness: as the string indents progressively deeper into the felt, it is influenced more and more by the deeper, stiffer layers. Conklin also reports that these details matter to musicians: in the search for the perfect sound, high-end piano technicians devote a lot of effort to subtle manipulations of the nonlinear stiffness by adjusting the felt properties.
Boutillon [3] looked at two different hammers from an upright piano, and showed force-compression plots during bouncing. As well as nonlinear behaviour, he also saw hysteresis: the curve during the unloading phase of a bounce was not identical to the loading curve. Hall and Askenfelt [4] used a different approach based on the duration of the force pulse when the hammer bounced off a rigid surface. They did not comment on hysteresis, but they reported that their measurements could be well fitted by a force-compression relation in the form of a power law:
$$F=K C^\alpha \tag{1}$$
where $F$ is the contact force, $C$ is the compression of the hammer felt, $\alpha$ is a non-dimensional exponent and $K$ is a magnitude scaling factor.
For a linear contact spring as we used in the simplest model of bouncing, the exponent $\alpha$ would be equal to 1, and $K$ would be the spring stiffness. We have already seen another power-law relation of this kind in section 12.1.1, when we discussed Hertzian contact between curved surfaces. For the Hertzian case, the exponent $\alpha = 3/2$. For the various piano hammers they tested, Hall and Askenfelt reported values for this exponent ranging between 1.5 and 3.5.
Some of their measured values were later used by Chaigne and Askenfelt [5] as input to a simulation model. The values to be used in the present study are taken from Table 1 in that reference. They list values of $K$ and $\alpha$ for three hammers from a grand piano, taken from the notes $C_2$ $(K= 4 \times 10^8$, $\alpha = 2.3$), $C_4$ $(K= 4.5 \times 10^9$, $\alpha = 2.3$) and $C_7$ $(K= 1 \times 10^{12}$, $\alpha = 3.0$). These numbers show the trend mentioned earlier: the $C_2$ hammer, towards the bass end, is softer (a much smaller value of $K$), and also the nonlinearity is relatively weak (the smallest value of $\alpha$). By contrast, the $C_7$ hammer at the treble end has a far higher stiffness, and the nonlinearity is more marked because of the thinner covering of felt.
These differences have a direct audible consequence, which we can illustrate with simulated examples. Some details of the simulation process can be found in the previous link. To get us started, Sound 6 gives simulated bridge force for notes $C_1 – C_6$. All 6 notes are “played” with the same hammer, with properties as listed above for $C_4$. They also all have the same hammer striking speed, 1 m/s. Boutillon [3] measured hammer striking speeds in the range from 0.11 m/s (pianissimo) to 6.83 m/s (fortissimo), so the speed used here is in middle range: it could perhaps be described as mezzopiano.
[1] H. A. Conklin, “Design and tone in the mechanoacoustic piano: Part II Piano structure”, Journal of the Acoustical Society of America 100, 695–708, (1996).
[2] H. A. Conklin, “Design and tone in the mechanoacoustic piano. Part I. Piano hammers and tonal effects”, Journal of the Acoustical Society of America 99, 3286–3296, (1996).
[3] X. Boutillon, “Model for piano hammers: experimental determination and digital simulation”, Journal of the Acoustical Society of America 83, 746—754 (1988).
[4] Donald E. Hall and Anders Askenfelt, “Piano string excitation V: Spectra for real hammers and strings”, Journal of the Acoustical Society of America 83, 1627—1638 (1988).
[5] Antoine Chaigne and Anders Askenfelt, “Numerical simulations of piano strings. II Comparisons with measurements and systematic exploration of some hammer-string parameters” Journal of the Acoustical Society of America 95, 1631–1640 (1994).