12.2 Hitting strings: the piano and its relatives


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.

Figure 1. A santoor, being played by Bhajan Sopori. Image Wolfgang Rieger, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

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.

Figure 2. Animation of the response of a string to being struck by a 1 g mass, indicated by the red star. The string in this example has diameter 0.5 mm and length 0.4 m, and is tuned to the note $A_4$ (440 Hz). The material of the string is steel, and the slight inharmonicity caused by bending stiffness is included in the simulation model. The model includes a linear contact spring with stiffness 20 kN/m.

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 3. The waveform of contact force from the hammer-string impact shown in Fig. 2.

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.

Figure 4. The frequency spectrum of the contact force shown in Fig. 3 (in red), and the corresponding frequency spectrum of the force exerted by the string at the bridge (in blue).

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.

Sound 1. Simulated bridge force from the santoor string struck by a 1 g hammer at a position 2/5 of the way from the bridge
Sound 2 Simulated bridge force from the santoor string struck by a 3g hammer at the same position as Sound 1
Sound 3. Simulated bridge force from the same string used in Sounds 1 and 2, following an ideal pluck at the same position as the hammer hits

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.

Figure 5. Red curve: the contact force from Fig.3, from hitting the string 2/5 of the length from one end. Blue curve: the corresponding contact force from hitting 1/10 of the length from one end.
Sound 4. Simulated bridge force similar to Sound 1, but striking the string at a position 1/10 of the way from the bridge
Sound 5. Simulated bridge force like Sound 3, from an ideal pluck at a position 1/10 of the way from the bridge

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.

Figure 6. A baby grand piano, an early 20th century model by Broadwood.
Figure 7. Detail of the string layout of the piano from Fig. 6. The cast iron frame can be seen round the edge. The vibrating length of each string is terminated at a bridge attached to the soundboard, which can be seen underneath the strings. The bridge for the triplets of unwound strings can be seen snaking up the picture about 3/4 of the way towards the right-hand side, while the separate bridge for the over-wound strings can be seen in the top right-hand corner. Each string zigzags around a pair of pins fixed in the bridge: these pins are clearly visible in the picture. At the keyboard end, each string is wrapped around a pin which the piano tuner uses to adjust the tuning. These tuning pins can be seen at the left-hand side.

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.


Figure 8. The mass per unit length of selected strings from the piano shown in Figs. 6–7. The over-wound strings were all measured and are shown as stars. Circles denote all the notes C in different octaves, and for the unwound strings only these Cs were measured. Notes 1–17 have single over-wound strings, notes 18–30 have pairs of over-wound strings, and the remaining notes 31–88 have triplets of unwound strings.

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.

Figure 9. String tensions deduced from the measurements in Fig. 8, calculated using the length and tuned frequency of each string.

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.

Figure 10. Impedance of the selected piano strings, deduced from the data shown in Figs. 8 and 9. This impedance includes the factors of 1, 2 and 3 for the multiple string groups, so that it represents the total impedance presented to the hammer and to the soundboard.

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 $C_2$ $(K= 4 \times 10^8$, $\alpha = 2.3$), $C_4$ $(K= 4.5 \times 10^9$, $\alpha = 2.5$) 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.

Sound 6. Simulated bridge force for the notes $C_1-C_6$, using the hammer properties appropriate to $C_4$ and a hammer striking speed 1 m/s.

To my ears, the low notes in Sound 6 are quite convincingly piano-like, but the higher notes become progressively less good. We will look for explanations of this behaviour later in this section, but first we can take advantage of the good-sounding low notes to demonstrate the effect of hammer nonlinearity. Sound 7 has 5 versions of the note $C_2$, played using the same hammer as in Sound 6 with a range of different striking speeds covering the full range reported by Boutillon [3]. The first note is so quiet that you may miss it. As the speed increases the sound naturally gets louder, but the sound quality also changes. By the final note, the sound has a “clanging” quality that a pianist might grumble about. But remember that the hammer we have used here is the one appropriate to $C_4$, not $C_2$. If we use the properties of the $C_2$ hammer instead, we get the result in Sound 8. There is still a progressive change in sound quality which is quite piano-like, but now the loudest note is more acceptable.

Sound 7. Simulated bridge force for the note $C_2$, played with the same hammer as in Sound 6 at 5 different striking speeds: 0.11 m/s, 0.5 m/s, 1 m/s, 2.5 m/s and 6.83 m/s.
Sound 8. The same notes as in Sound 7, but played with the hammer properties for $C_2$.

Finally, Sound 9 demonstrates what happens if we hit this string with the hammer designed for $C_7$. The sound is now decidedly harsh compared to a concert grand piano, but it is not entirely unfamiliar. With the growth in popularity of period instruments and authentic performance styles, the sound of an early piano (usually described nowadays as a “fortepiano”) has become familiar. Such pianos have a thin covering of leather over the hammers, rather than a thick layer of felt. The result is to give nonlinear stiffness characteristics even more extreme than the $C_7$ values used here. The sound of a fortepiano is indeed quite different — it is sometimes described as “harpsichord-like”.

Sound 9. The same notes as in Sound 7, but played with the hammer properties for $C_7$.

To understand the origin of the tonal difference as the striking speed is increased, it is illuminating to look at the waveforms of contact force. Figure 12 shows the 5 cases from Sound 8, with the nominal $C_2$ hammer. Not surprisingly, the values increase as the striking speed increases, but it is hard to compare the pulse shapes in this plot. Figure 13 shows the shape change much more clearly. What has been done here is to normalise each force waveform by dividing it by the striking speed. If the contact spring had been linear, the waveform of contact force would simply scale with the striking speed, so this normalisation would make all the curves coincide. But with the nonlinear hammer, this has not happened. It is now clear that the most gentle strike, plotted in green, has a very smooth shape, and there is a progressive sharpening-up of the shape as the speed increases. This is most clear at the left-hand edge of the force pulse: the growth in force becomes increasingly steep and abrupt as the striking speed increases.

Figure 12. The pulses of contact force associated with the 5 notes in Sound 8. The note $C_2$ has been played using hammer properties appropriate to that note, with striking speeds: 0.11 m/s (green), 0.5 m/s (blue), 1 m/s (magenta), 2.5 m/s (red) and 6.83 m/s (black).
Figure 13. The same contact force data as in Fig. 12, but each force has been divided by the corresponding striking speed to produce a normalised comparison.

Another way to see the effect is to look at the frequency spectra of the waveforms in Fig. 12: these are shown in Fig. 14, using matching line colours. At very low frequencies, the curves are all quite flat, with a level that goes up as the striking speed increases. At higher frequency, each curve reaches a frequency above which it falls, following an approximately straight line in this logarithmic plot. This turnover frequency, where the decline sets in, can be seen to rise systematically as the striking speed increases. For the quietest note (in green), the line only reaches about 100 Hz before it declines. By the loudest note (in black), the corresponding turnover is up around 1 kHz. These changes in force spectrum make it hardly surprising that the sound gets brighter as the striking speed is increased.

Figure 14. The spectra of the contact forces shown in Fig. 12, using corresponding line colours.

By the way, perhaps you thought that some of the simulated notes in these various sequences sounded a bit out of tune? This raises an interesting issue about our perception of pitch, and it is something piano tuners have to learn to cope with. All the simulated notes have fundamental frequencies exactly in accordance with equal temperament based on the pitch standard with $A_4$ at 440 Hz. The octaves are all based exactly on a factor of 2 in fundamental frequency.

As a final demonstration in this set, Sound 10 gives the same note sequence as Sound 6, with the same hammer and the same striking speed, but with the inharmonicity due to bending stiffness suppressed in the simulation. You might have expected that forcing all string overtones to be exactly harmonic would make the sounds more “musical”, but it is more complicated than that. The lowest note, in particular, sounds very peculiar in this new version. The frequency is so low (32.7 Hz) that our ear-brain system starts to perceive the individual repetitive pulses that make up the sound: this fundamental frequency is in the transition range where we gradually switch from hearing something as a frequency, to hearing it as a pattern in time. Inharmonicity allows this low note to “work”, presumably by blurring the temporal pattern. Without it, the sound is not like a piano at all!

Sound 10. The same note sequence as Sound 6, but with the inharmonicity due to bending stiffness suppressed so that all string overtones are exactly harmonic.

E. The piano part 3: designing the hammer set

So far we have simply used the measured properties of three piano hammers, but we can use the simulation program to make a more systematic study of the influence of hammer design. In a similar spirit to the “playability diagrams” shown in previous chapters, we can simulate the response of a string to being hit with a large set of hammers with different masses and contact stiffnesses. We can then plot results in a plane indexed by these two parameters, and highlight whatever aspect of the response we are interested to see.

Figure 15 shows a first example. The string is $C_4$, “middle C”. A set of hammer masses has been used, linearly spaced between a very small value and double the actual mass of the $C_4$ hammer. On the vertical axis the contact stiffness coefficient $K$ has been varied over a very wide range, distributed on a logarithmic scale centred on the measured value for a $C_4$ hammer. All cases use the same nonlinear exponent $\alpha$, set to the measured value for a $C_4$ hammer, 2.5. The hammer striking speed is 2.5 m/s in all cases, corresponding to forte playing. The result is a design chart for assessing the effects of any realistic modification of this hammer.

Figure 15. Simulation results for a grid of simulated piano impacts, based on the note $C_4$ with a hammer striking speed 2.5 m/s. The hammer stiffness model uses exponent $\alpha=2.5$, the measured value appropriate to $C_4$. The colour shading indicates total time of contact.

Figure 15 is colour-shaded to show the total time of contact between the hammer and the strings. The nominal hammer gives a time of the order of 3 ms. Choosing a lighter or stiffer hammer tends to make the time shorter; a heavier or softer hammer makes it longer. Figure 16 shows a different plot based on the same grid of simulations, this time colour-shaded to show the number of contacts. In the white region there is only a single contact, while the coloured pixels show multiple bounces. A stiffer hammer tends to make multiple bounces more likely, but the pattern is complicated.

Figure 16. Results from the same grid as Fig. 15, colour shaded to show the number of contacts made by the hammer with the string

Figure 17 shows yet another plot based on the same data, this time showing the spectral centroid of the waveform of contact force. This will correlate with the level of brightness of the resulting sound. A high spectral centroid (associated with a stiff or light hammer, indicated by whites and yellows in the plot) means that the contact force is capable of exciting a large number of string overtones to a significant level; conversely a low spectral centroid (with a heavy and soft hammer) means that the sound will be dominated by the fundamental, with relatively low levels of higher overtones.

Figure 17. Results from the same grid as Fig. 15, colour shaded to show the spectral centroid of the contact force waveform. The colour scale is truncated at 2 kHz, so all values higher than that appear white.

But, somewhat unexpectedly, this data can be used to tell a bigger story — at least approximately. If we are prepared to neglect the effects of damping and bending stiffness during the short time that the hammer and string are in contact, it turns out that there is a universal pattern to the response of any hammer hitting any string. The reasons, and the details of the calculation, are explained in the next link. We simply need to reinterpret the results of a plot like Fig. 15. Instead of plotting the actual mass on the horizontal axis, we plot a normalised version in which the hammer mass is divided by the total vibrating mass of the string(s). In a similar way, we normalise the stiffness coefficient $K$ by dividing by a stiffness associated with the string(s): specifically, the total tension divided by the vibrating length. Finally, we express time in multiples of the fundamental frequency of the string.


An example of the result is shown in Fig. 18. This is essentially the same data as in Fig. 15, except that we have chosen a different set of hammer masses, distributed on a logarithmic scale in order to make the results more clear. Apart from that change, Fig. 18 only differs from Fig. 15 in the three ways just described. The mass and stiffness have been normalised, and the total contact time is now expressed as a number of periods of the string vibration, rather than in milliseconds.

Figure 18. Results for the same hammer and string combination as Fig. 15, but with a somewhat different grid of values. The variation of mass, on the horizontal axis, is now shown on a logarithmic scale, and the values are normalised by the total vibrating mass of the $C_4$ string set. The values of $K$ on the vertical axis have been scaled by the ratio of tension to length for the string set. The total contact time is now expressed in units of the nominal period of the note $C_4$. The cyan squares mark the positions corresponding to all the notes C on the tested piano, assuming that they all used identical hammers.

But the interpretation is now much broader. It is no longer just a summary of computed results for the note $C_4$: it should apply to all 88 notes across the range of the piano. This interpretation is indicated by the line of cyan squares. These mark the positions, on these normalised axes, of the 8 notes C on the tested piano. (The parameter values for all these Cs were given in Table 1 of section 12.2.1.) The bass notes are on the left, starting with $C_1$; the treble notes are on the right, ending with $C_8$. The logarithmic spacing of musical frequencies is the underlying reason that we needed a logarithmic distribution of the hammer masses, so that these square markers are more or less equally spaced across the plot.

Before we try to unpack the implications of this plot, we need to do a little more work on it. I glossed over an important detail: the square markers show where all those Cs would fall if the actual hammers were all identical to the $C_4$ hammer. But, of course, they are not all identical. First, the hammers are heavier in the bass, lighter in the treble. When the actual masses are taken into account, the result is shown in Fig. 19: a new set of points marked by cyan stars shows how the squares move inwards a little from both ends of the line.

Figure 19. The same plot as Fig. 18, with the addition of the cyan stars marking the positions corresponding to all the notes C on the tested piano, with the hammer masses adjusted to the correct values but the hammer stiffnesses still identical.

The final step is a bit more speculative. So far, we have assumed that the nonlinear exponent $\alpha$ is the same for all hammers. But in fact we know that it isn’t: the thick felt of the bass hammers gives a lower value, the thinner felt of the treble hammers gives a higher value. The mathematical theory behind the universal nature of the behaviour for all strings can’t really cope with variations in the exponent. However, we can use the simulation results to give an empirical “fix”: the details are explained in the previous link. It turns out that, to a limited extent at least, we can compensate for variations in the exponent by changing the (scaled) value of the coefficient $K$.

The behaviour of the bass hammers, with their lower value of the exponent, can be captured approximately by scaling up the actual $K$ by a suitable factor. Conversely, the behaviour of the treble hammers, with a higher exponent, can be captured approximately by scaling the actual $K$ down by a suitable factor. The result of this semi-empirical “fudge” is shown in Fig. 20. We have measurements of hammer properties for $C_2$, $C_4$ and $C_7$. $C_4$ is our reference case, for which we have used the correct exponent. The green circles in Fig. 20 show the new (approximate) positions of $C_2$ and $C_7$ after applying the scale factors deduced from simulations. The scaling process has moved the effective values of $K$ much closer to the value for $C_4$, our reference case, but the circle for the $C_2$ hammer still lies below the corresponding star, and the one for the $C_7$ hammer lies above its star. The net effect is that the three circles mark out an approximately horizontal track across the plot: the empirical hammer design process seems to have resulted in rather similar values for the effective scaled stiffness coefficient across the entire range of the piano.

Figure 20. The same plot as Figs. 18 and 19, with the addition of three green circles marking the corrected positions of the notes $C_2$, $C_4$ and $C_7$ using the procedure described in the text to account, approximately, for the different nonlinear stiffness exponents.

Figures 21 and 22 show corresponding plots, colour-shaded to indicate number of contacts and spectral centroid respectively. In the scaled variables, the spectral centroid is expressed as a ratio with the fundamental frequency of the string — in other words, as an effective harmonic number. Taking the three plots together, we can summarise the behaviour of piano hammers across the range. Although in terms of grammes the bass hammers are a little heavier and the treble hammers a little lighter, the trend in the plots goes in the opposite direction. Relative to the vibrating mass of the strings, the bass hammers are very light, and the treble hammers very heavy.

Figure 21. A similar plot to Fig. 20, using the same data but colour-shaded to show the number of hammer-string contacts.
Figure 22. A similar plot to Figs. 20 and 21, colour-shaded to show the spectral centroid of the contact force waveform, expressed as a number of harmonics of the nominal fundamental frequency.

The pattern of behaviour stems mainly from this contrast. In the bass, the hammers are light compared to the strings, and the time of contact is a fraction of a period of the string’s vibration. In the treble, the hammers are heavy compared to the strings, and they remain in contact for several period-lengths of string vibration. The spectral centroid more or less follows the inverse of the contact time. The bass hammers have a high spectral centroid (i.e. many string overtones excited). The treble hammers have a low centroid, but because these strings have much higher fundamental frequencies they do not need very many overtones to reach the limit of human hearing: for $C_8$, the fifth overtone would be beyond that limit.

Some of this behaviour is illustrated directly in Figs. 23—25. These show animations, in a similar format to Fig. 2, of the hammer impact and string response for the $C_2$, $C_4$ and $C_7$ strings respectively. For $C_2$ in Fig. 23, the hammer (red star) is thrown clear of the string soon after the reflected wave returns from the nearer end of the string, and long before the reflection gets back from the far end. You will notice that the red star drops a little below the green star during the contact. The distance between the two stars shows the compression of the felt: in this case that compression is smaller than the maximum string displacement.

Figure 23. Animation in the same format as Fig. 2, showing the simulated motion of hammer and string for the note $C_2$ with striking speed 2.5 m/s. When the red star falls below the green star, the difference of their position indicates the compression of the felt.

Figure 24 shows the corresponding animation for the $C_4$ string and hammer. Now, the hammer bounces off after roughly one period of the string vibration, as the colour in Fig. 20 indicates. The compression of the felt is now comparable to the string displacement.

Figure 24. Animation in the same format as Fig. 23, showing the simulated motion of hammer and string for the note $C_4$ with striking speed 2.5 m/s.

By the time we reach $C_7$, shown in Fig. 25, this has changed. The red star disappears off the bottom of the plot for a while, before it reappears and then bounces off the string. The compression of the felt is far bigger than the motion of the string, and the hammer remains in contact for several period-lengths of the string vibration (as Fig. 20 indicates). You can see the vibration frequency of the free string at the end of the animation, after the hammer has bounced off. Because the hammer is significantly heavier than the string for this case, the lowest resonance frequency of the string is far lower than usual while the hammer is in contact. It might be this lower frequency of the mass-loaded string that governs the time-scale of bouncing.

Figure 25. Animation in the same format as Figs. 23 and 24, showing the simulated motion of hammer and string for the note $C_7$ with striking speed 2.5 m/s. This time, the compression of the felt is far greater than the displacement of the string, and for a while the red star disappears off the bottom of the plot.

It is worth listening to this simulated $C_7$ note: Sound 11 gives three repeats of the bridge force waveform. As we commented earlier, this high note does not sound as convincingly piano-like as the low notes, and we should explore the reasons for that. One factor may simply be the assumed value of Q-factor for the string vibration: our value (2000) was chosen somewhat arbitrarily. But there are several other factors that probably contribute. Figure 26 shows the early part of the bridge force signal for this note, together with the contact force pulse. One thing that is immediately apparent is a spike at the start of the waveform, much higher than the subsequent bridge force. This probably accounts for the audible “click” at the start of the note. The animation in Fig. 25 gives a clue about the origin of this peak. While the hammer mass is in contact with the string, its momentum carries the string downwards much further than it ever goes again in the subsequent vibration. This extra displacement automatically increases the bridge force during that initial phase.

Sound 11. Three strikes of the note $C_7$, using the measured hammer properties and a striking speed 2.5 m/s.
Figure 26. The contact force (red) and the bridge force (blue) for the early part of the $C_7$ note heard in Sound 11.

The fact that the sound is a little unrealistic does not mean that the waveform is incorrect — rather, it points to a disadvantage of using bridge force as a surrogate for the radiated sound of the piano. In a real piano the initial pulse of force applied to the bridge would excite a transient “thump” from the soundboard, as all its vibration modes respond. Perhaps if we replaced the force spike by a “soundboard thump”, the note would sound more realistic: Chaigne and Askenfelt [5] report that in their simulations they tried doing exactly this using a recorded “thump”, with encouraging results.

But this soundboard interaction is by no means the only thing missing from the simple simulations here. Two other effects have already been described in earlier chapters: the “double decay” envelope resulting from multiple strings (section 7.3), and the “phantom partials” associated with the nonlinear excitation of longitudinal motion in the strings (section 7.4). Both effects are known to be clearly audible, and to contribute to the sound of a piano. We will not try to add these effects in here, because the purpose of this section is to explore the physics of bouncing hammers, rather than to produce the most realistic piano synthesis. But this would certainly make an interesting project for the future.

F. The clavichord

Figure 27. A German clavichord of 1763, in the Metropolitan Museum of Art, New York. Image: Christian Kintzing, CC0, via Wikimedia Commons

The final topic in this section concerns the clavichord: a stringed keyboard instrument that involves impact, but not bouncing. An example is shown in Fig. 27. The essential design of a clavichord is very simple: it is sketched in Fig. 28. Each string (or, often, pair of strings) passes over a bridge on the soundboard at one end, just as in a piano. The other end of the string has felt woven through to damp out vibration. When a key is pressed, a metal spike called the “tangent” moves up into contact with the string, between the felt and the bridge. The impact of the tangent starts the string vibrating. The player holds the key down for as long as they want the note to last. The section of string between tangent and bridge continues to vibrate, but in the other section the vibration is quickly damped out by the felt. When the player releases the key the two sections of string are once more joined, the felt damps out all the vibration, and the note stops.

Figure 28. Sketch of the mechanism of a clavichord, showing a single string and a single key

The clavichord shares one important characteristic with the piano: the speed of the key press directly affects the striking speed on the string, and thus the loudness of the note. The player can even give some vibrato to the note: modulating the finger force on the key changes the tension of the string slightly, and thus modulates the pitch. But one could not describe the sound of a clavichord as offering “piano — forte”: it is more a matter of “pianissimo — piano”. The clavichord is notoriously quiet, which is why you rarely come across clavichord recitals (although recordings are another matter: the recording engineer can amplify the quiet sound).

The first challenge in constructing a simulation model for a clavichord is how to describe the excitation by the tangent, which is neither a pluck (like a harpsichord) nor a bouncing hammer (like a piano). There is very little scientific literature devoted to the clavichord, but there is a paper by Thwaites and Fletcher [6] which gives us a strong lead. This is not the first time that we have looked at an unusual instrument, and found a seminal reference involving Neville Fletcher: he was a giant of the field of musical acoustics.

Thwaites and Fletcher measured the motion of key and tangent during normal playing of a clavichord. The tangent strikes the string, and normally it does not bounce but remains in contact. The string is displaced a short distance away from its resting position, to provide sufficient reaction force to ensure no bouncing. But this displacement cannot be too big, because it results in an increase of tension in the string which runs the risk of the note playing unacceptably sharp. Figure 29 shows the predicted frequency increase for a particular clavichord string, as a function of the displacement. The frequency shift is expressed in cents (100ths of a semitone), and we saw back in section 6.4 that a shift by more than about 5 cents is likely to be unacceptable. So we deduce that for this particular string, the maximum displacement is probably about 2 mm. Reassuringly, this distance fits neatly with the range observed by Thwaites and Fletcher.

Figure 29. Predicted frequency deviation as a function of the displacement imposed by the tangent, for a clavichord string made of brass with vibrating length 0.99 m and diameter 0.48 mm. The length of string on the other side of the tangent is 0.2 m.

What the player has to do, therefore, is bring the string to rest within that distance, whatever the initial striking speed may have been. Thwaites and Fletcher suggested a simple model for this, which we follow here. We assume that the tangent speed decreases exponentially, at a rate sufficient to ensure that the total displacement is not too large. The faster the initial striking speed, the more rapid the exponential decay needs to be. Figure 30 shows three examples which all produce 2 mm of final displacement following three very different striking speeds. The player would need considerable delicacy of touch to control the tangent trajectory over this kind of range.

Figure 30. Three possible trajectories of tangent motion, all reaching a final displacement of 2 mm. The three initial striking speeds are 0.1 m/s (black), 0.5 m/s (blue) and 1 m/s (red).

To construct a simulation model, we need to impose tangent motion like one of these curves. The string is initially at rest, then one end of it is forced to move in this particular way. This is enough to determine the resulting string vibration, and the bridge force waveform exerted by the string on the soundboard. The details are described in the next link.


But before we show simulation results, there is another issue to be considered. When a note on the clavichord is analysed, it reveals a pattern of damping that is rather different from other stringed instruments we have looked at. The red stars in Fig. 31 show measured Q-factors from a clavichord note ($D_2$, 73.4 Hz), and the blue circles show the corresponding Q-factors for the same note played on a harpsichord with a somewhat similar string specification (material, length and diameter) [7]. There is inevitably some scatter in the points, probably caused mainly by the frequency dependence of the soundboard vibration for the two different instruments. But some trends are clear through this scatter. Above about 500 Hz the results look rather similar for both instruments, but for the first few overtones of the string sound, the clavichord note has significantly lower Q-factors, or in other words higher damping and a faster decay.

Figure 31. Measured and predicted Q-factors for the first 25 overtones of a clavichord note (red) and the same note played on a harpsichord (blue). Stars and circles show the measured results, curves show the fitted model.

The major difference between these two trends is caused by an additional damping mechanism that operates in the clavichord. Recall that during a note, the player has to hold the key down so that the tangent remains in contact with the string. The tangent can then act as a reflector for waves on the string, but it is not a perfect reflector: the tangent/key/finger combination is not rigid, and a small amount of movement of the tangent allows some energy to get past the tangent into the other section of the string, where it is damped out by the felt. This energy leakage is stronger at low frequencies: the mass of the key and tangent automatically mean that it becomes a better reflector at high frequency.

A simple model of this process is described in the next link. We can combine this new effect with the other damping mechanisms we already talked about in Chapter 5: energy loss into the soundboard, energy loss due to air damping, and energy loss within the string itself. The resulting model gives the predictions shown in the two curves in Fig. 31. The blue curve, relevant to the harpsichord string, only includes the effects we already knew about. The red curve adds in the new effect of energy leakage past the tangent. Both curves track the trend of the respective measurements quite convincingly.


We can use the damping model represented by the red curve to complete our simulation model. Figure 32 shows a typical animation of the result: it shows how the string moves in the early stages after the tangent makes contact. This particular example corresponds to a striking speed of 0.5 m/s, and the left-hand end of the string moves according to the trajectory shown in the blue curve in Fig. 30.

Figure 32. Animation (with exaggerated vertical scale) of the string motion during the first 25 ms of a simulated clavichord note ($D_2$, 73.4 Hz). The tangent striking speed is 0.5 m/s.

The resulting bridge force waveform is shown in the blue curve of Fig. 33. That plot also shows the corresponding waveforms for two other striking speeds: the two shown in the black and red curves of Fig. 30, which represent the extremes of what a player might use as reported by Thwaites and Fletcher [6]. All three bridge force waveforms show a kind of growing wiggle before the bridge force takes its big jump upwards. This is the effect of bending stiffness in the string, which means that waves at high frequency travel a little faster than ones at lower frequency. The high-frequency components of the initial jump in string velocity (as the tangent hits) arrive at the far end of the string a little sooner than the main pulse.

Figure 33. Simulated bridge force waveforms for three strikes on the same string using different striking speeds: 0.1 m/s (black); 0.5 m/s (blue); 1 m/s (red). The blue curve corresponds to the animation shown in Fig. 32. The three tangent trajectories were shown, with the same colours, in Fig. 30.

It is clear from Fig. 33 that the striking speed of the tangent certainly does make a difference to the amplitude of the bridge force waveform, and thus to the loudness of the sound. But recall that the final displacement of the tangent is the same in all three cases, 2 mm. The big difference between the cases is the time-scale of the tangent motion, revealed in Fig. 30. At the highest speed, the motion of the tangent is more or less complete in 5 ms. You can see from Fig. 33 that this is a small fraction of the period of the string vibration. For the slowest striking speed, in contrast, Fig. 30 shows that the tangent motion is still not complete after 50 ms, about 3 period-lengths of the vibration. Because the motion of the tangent is so slow, it does not excite very much string vibration. It is intuitively obvious that if the tangent was brought very slowly up to its 2 mm displacement taking several seconds, there would be virtually no string vibration.

So now we can see why the clavichord is so quiet. To make a loud sound, the player would have to use a very fast striking speed. But they would still need to limit the motion to not much more than 2 mm, or the note will play unacceptably sharp. So the deceleration of the key after string contact needs to be very rapid, and of course there is a limit to the capability of human player to control such a fast trajectory. This translates directly into a limit on loudness.

Figure 34 shows a comparison between our simulated clavichord notes and a plucked harpsichord note played on the same string, starting with a plectrum displacement of 2 mm so that it is exactly the same as our clavichord tangent displacement. The pluck is applied at a typical harpsichord position, 1/7.8 of the string length away from the end of the string. The resulting bridge force signal is plotted in green, compared to the black and blue clavichord waveforms which are repeated from Fig. 33. The damping model used for the harpsichord note is the one shown as the blue curve in Fig. 31.

Figure 34. Bridge force waveforms comparing the simulated clavichord with a simulated harpsichord. The black and blue curves are for the clavichord, and are the same as the corresponding coloured curves in Fig. 33. The green curve is for an ideal pluck of the same string, at a typical harpsichord plucking position 1/7.8 of the string length from the end. The initial pluck displacement was 2 mm, the same as the total tangent displacement for the clavichord notes.

An alternative view of the difference in amplitude between these waveforms is given by the corresponding frequency spectra, shown in Fig. 35. The colour coding is the same as Fig. 34, to make it easy to compare. The harpsichord, in green, has peaks that are over 20 dB higher than those of the clavichord played with striking speed 0.5 m/s (in blue), and the peaks for the quietly-played clavichord (in black) are a further 10—15 dB lower. The synthesised clavichord really is quiet!

Figure 35. The frequency spectra of the three bridge force waveforms from Fig. 34, plotted in the same colours.

You can listen to the three bridge force waveforms from Fig. 34 in Sound 12. The three are combined in one sequence, so that the relative loudness is preserved. First you hear the very quiet sound corresponding to the black curve in Fig. 34; then the blue curve, significantly louder; then finally the harpsichord note from the green curve. You can compare these with recordings of the actual sound from a clavichord and a harpsichord, in Sounds 13 and 14 respectively. These sound recordings do not preserve the loudness difference, but you can make comparisons of the quality of sound with the synthesised efforts in Sound 12.

Sound 12. Synthesised notes from a clavichord and harpsichord. The clavichord note is first heard with striking speed 0.1 m/s, then again with striking speed 0.5 m/s. Finally, the same string is plucked with an ideal pluck at a typical harpsichord position, 1/7.8 of the string length from one end.
Sound 13. Recorded sound from a real clavichord playing the same note as the synthesised versions (although perhaps the tuning was slightly flat).
Sound 14. Recorded sound of the same note played on a harpsichord.

Of course, in the synthesised notes we are listening to bridge force rather than the actual radiated sound from the soundboard. Nevertheless, to my ear the comparison of the harpsichord sound with the synthesised version is quite persuasive. The clavichord notes do not sound so good, though. Many factors may contribute to this disparity, but one important factor takes us back to Fig. 29. The pressure of the player’s finger has a very sensitive effect on the string tension, and hence on the pitch. For a long note like the recorded Sound 13, the player had to hold the key steady while the note rang on. But, being human, and indeed not being a real clavichord player, my finger was not in fact perfectly steady. There were small fluctuations in pressure, which produced fluctuations in the sound of the note. This contributes a “roughness” to the sound which is quite distinctive when compared to the sound of the harpsichord. The synthesised clavichord falls somewhere between the two.


[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).

[6] Suszanne Thwaites and N. H. Fletcher,“Some notes on the clavichord”; Journal of the Acoustical Society of America 69, 1476–1483 (1981).

[7] The measurements on a clavichord and harpsichord were done as part of a student project: William Layzell-Smith, “An investigation of the acoustic vibrations in different keyboard instruments”, MEng dissertation, Cambridge University Engineering Department (2021). The clavichord and harpsichord were made available for measurement by Dan Tidhar.