If a vibrating object is too close to something solid, it may come into contact intermittently during the vibration. The result is usually described as a “buzz”. If you hear a buzz from a musical instrument, it often suggests bad news. A buzz from a violin body may mean that a joint is coming unglued. A buzz from a guitar string usually means that the fret heights are not perfectly adjusted: it will annoy the player and send them to their repair shop to have it sorted out. But over the centuries the innovative folk who invent and enhance musical instruments have found ways to harness many physical phenomena for positive musical purposes, and buzzes are no exception. We will look at some of these in this section.
Perhaps the most familiar example is the snare drum, heard in every military marching band and a key part of every drum kit. Figure 1 shows a typical snare drum, but this view does not reveal the important feature. This kind of drum has two drum-heads: one on the top which the player hits (the “batter head”) and a second on the underside (the “resonant head”). Figure 2 shows a view of a snare drum from underneath, and you can see a band of coiled wires running across the width of the lower head (although you can’t see the head itself because it is transparent).
These “snares” or “rattles” are lightly touching the surface of the lower head, unless the drummer chooses to loosen them so that they go out of contact. With the snares loosened, the drum sounds much like any other drum, but with them in contact the sound has the crisp rat-a-tat quality of a military drum. When the player hits the batter head, sound waves travel through the air in the drum cavity and make the lower head vibrate as well. This then buzzes against the snares. You can hear the effect in Sound 1: the same 14 inch snare drum is played with and without its snares in contact, first with single taps and then with a brief but typical snare drum pattern.
This characteristic snare drum sound has the same underlying physical explanation as any other buzz. In the absence of a buzz or rattle, the vibration of an object like a drum involves a mixture of the vibration resonances. Almost invariably, the resonances at higher frequency die away more rapidly than the ones at low frequency. The result is that the long-lasting part of the sound is dominated by low frequency: in the context of drums, this is the “thud” of a bass drum, for example. But when buzzing takes place, each individual impact generates a very short pulse of force, and as we know, the frequency spectrum of a short pulse must extend to high frequency. So repeated buzzing impacts are a way to feed energy into high-frequency vibration of the drum or whatever, changing the balance across the frequency spectrum so that the sound has a bright, penetrating, “buzzy” quality.
Back in section 7.4 we met another example of the positive use of a kind of buzz, when we looked at the sound of a lute compared to a guitar. Because a lutenist usually plucks the strings with the soft flesh of their fingertips, the sound would tend to be rather mellow and dull — in other words, to be dominated by low frequency. But the sound of a lute can be much brighter than this argument would lead us to expect. We tracked the effect down to a kind of fret buzz, exciting higher frequencies in the string vibration by the mechanism just described. The lutenist does not want to hear too much “buzz”, though, and it seems likely that in normal playing there might be just a single impact with a fret, shortly after the note is plucked. But this is enough to change the frequency spectrum, and the perceived quality of the sound.
We will come back to other examples of deliberate use of this kind of string buzzing, but first we will look at a very different use of a buzz in an unusual stringed instrument, the hurdy-gurdy. Figure 3 shows a hurdy-gurdy player with his instrument. By turning the crank he rotates a rosin-coated wheel, which then “bows” a number of strings. Some of these strings act as drones, with a constant pitch, but there is also a set of melody strings which are controlled by a set of keys operated by the left hand of the player.
Figure 4 shows how the instrument works. Each key presses a set of “tangents” against the melody strings: three of them in this image. These act to shorten the vibrating length of the strings. They remain in contact with the strings throughout the note, so the player can also modify the string tension a little by pressing harder: just as in the clavichord, this allows the possibility of vibrato and other ornaments. These melody strings pass over a bridge similar to a violin bridge, so that the soundboard is set into vibration just as in other stringed instruments.
The drone strings can be seen running outside the keybox on both sides of it. One of these, on the left in the picture, passes over a component labelled as the “buzzing bridge”, and this is what we are interested in. Figure 5 shows two close-ups. In most stringed instruments the bridge is either glued to the soundboard (as in a classical guitar or a piano), or it is held firmly in place against the soundboard by the string tension (as in a violin or cello).
The main bridge of the hurdy-gurdy is like that, but the buzzing bridge (or “dog”) is different. It is rather loosely held, and it can pivot around a tongue held in a slot (visible in the right-hand image of Fig. 5). The string (labelled as the “trompette”) is pulled to the side by a control wire held by a tuning peg, the “tirant”. This puts a sideways force on the dog, and you can probably imagine that if that sideways force was too big, the dog would be unstable — it would jump upwards and probably fly off the instrument.
The trick is to adjust the sideways force to be a little less than this instability threshold. The dog stays in place, but it is almost unstable and it doesn’t take much extra perturbation to make it jump a little. This perturbation is provided by the force exerted by the string, which like any other bowed string is set into stick-slip vibration by the “bow”, the turning wheel. When the control wire adjustment is just right, if the player turns the wheel slowly the dog remains in contact with the soundboard and a normal drone note is heard. But if the wheel is turned faster, the amplitude of the perturbing force from the string increases, and the dog starts to jump. It doesn’t jump right off the instrument, but it bounces up and down and buzzes against the soundboard, making a characteristic sound.
Now the player can do something clever. They continue to turn the crank, so that all the strings play their notes, but they modulate the turning speed in a rhythmical way so that it is sometimes slower than the buzzing threshold, and sometimes faster. The result is a conspicuous rhythmical buzz, adding accents to the music being played. In essence, a hurdy-gurdy player can be their own percussionist. You can hear an example in this YouTube video: about 37 s into the tune, the player starts to engage the buzzing bridge with an obvious audible effect.
Our final topic is a different kind of “buzzing bridge”, which can be found in one form or another in several instruments from around the world. We will look at three of them: one European and two from India. The European example is an early form of harp, usually called the “gothic harp” or, tellingly, the “bray harp”. The Indian examples are the sitar and the tanpura. The sitar is a well-known soloist’s instrument, usually accompanied by a tabla (see section 3.6) and a tanpura, a stringed instrument which is used to provide a kind of background “sound cloud”. Figures 6 and 7 show a sitar and a tanpura.
All three of these instruments have a characteristic sound quality which is the result of a kind of deliberate buzz. The bray harp is the easiest to describe. As usual in all harps, each string emerges from a hole in the soundboard. But in a bray harp there is an L-shaped “bray pin” also in the hole, as sketched in Fig. 8. The bray pins can be rotated out of the way of the strings, whereupon the harp sounds like any other. But when the pins are in the position shown in the sketch, the point of the L is more or less in contact with the string, so that when the string vibrates it can buzz against the pin. Figure 9 shows a bray harp, with a close-up showing bray pins in the two positions. You can see and hear a selection of other bray harps in this YouTube playlist.
In the sitar and the tanpura, a different arrangement is used to achieve a similar effect. Figures 10 and 11 show close-ups of the bridges of the instruments in Figs. 6 and 7, from the side. The sitar has two bridges, because it has a set of sympathetic strings running underneath the main playing strings. All three bridges in these figures are wide, and at first glance they may look flat. But actually they all have gently curving tops, so that the exact point where the string lifts off might vary during the vibration of the string.
In the case of the tanpura bridge, an extra feature can be seen. Short lengths of cotton thread have been pulled beneath the strings, and then very carefully adjusted by the player to make the best sound. The result is shown in sketch form in the upper plot of Fig. 12. The thread, shown as a red dot, has lifted the string, so that it is almost touching the curved bridge a few millimeters away from the thread. The resulting buzzing sound is called “jivari” in the Indian tradition (the word can be transcribed into English in various different spellings). The threads used to adjust to get the best sound can be called jivari threads.
We want to choose the simplest mechanical model in order to create some simulations of these buzzing bridges. An idealised configuration that applies, approximately, to both the tanpura and the bray harp is shown in the lower sketch of Fig. 12. A short distance from the string’s termination, there is a rigid stop with which the vibrating string can collide at a single point. This means that during its vibration the string alternates between two states: in contact with the stop, or not. This is a fairly good model for the bray harp. It also works quite well for the tanpura, although it probably does not capture the full complexity of the tanpura bridge. It cannot be applied directly to the sitar bridge, which does not have jivari threads. However, the sounds of a sitar string and a tanpura string have enough family resemblance that we may hope to capture at least some aspects of what is going on with the simplified model. The next link explains how the resulting simulation model is put together.
Sounds 2 and 3 give an example of what the model can produce. The parameter values for these simulations are given in the previous link. They are loosely based on a tanpura string about 1 m long, with the rigid “stop” 5 mm from one end. In Sound 2, the string is driven by an ideal pluck, at a position 30% of the string length from the bridge. Sound 3 is for the same string, initialised with just the first mode shape. A real finger pluck will be somewhere between these extremes: not as sharp as the ideal pluck, but with more than the first mode initially excited. Both sounds capture enough of the characteristic “Indian” sound to be very encouraging.
The simulations show three noteworthy features, all first pointed out by Valette and Cuesta  and shown by them to correspond quite well to measured behaviour of a tanpura. The first feature is shown by the animation in Fig. 13. This shows the initial stages of the simulation from Sound 3, starting with sinusoidal displacement of the string in the shape of the first vibration mode. The string (shown with an exaggerated vertical scale, of course) is plotted in black when it is not in contact with the stop, changing to red when it is in contact. Note that the stop is very close to the right-hand end of the string, so close that the direct effects of contact are barely visible here. By the end of the animation, the string motion has changed from the initial sinusoidal shape: rather unexpectedly it has developed a circulating “corner”, very reminiscent of the Helmholtz motion of a bowed string (see section 9.1).
The second feature concerns the nature of the sound we heard in Sounds 2 and 3. Figure 14 shows a spectrogram of Sound 3. At the start (bottom of the plot) only the fundamental component of the bridge force is present, because we chose to initialise the motion with the lowest string mode. Figure 14 then shows that other overtones of the string motion are quite rapidly excited, by the effects of the repeated contacts between string and stop. The amplitudes of these higher overtones show an interesting pattern: there is a leftward-sloping patch of bright colour in the frequency range 500—700 Hz. This corresponds to a kind of “formant” in the sound, which falls in frequency as the note develops. So Sound 3 shows a progressively increasing harmonic richness at early times, combined with the falling frequency of the formant at somewhat later times. Both are clearly audible in the sound example.
The third feature is illustrated by Sound 4 and Fig. 15. These correspond to Sound 3 and Fig. 14, except for one change: the bending stiffness of the string has been set to zero, so that the associated inharmonicity of natural frequencies disappears. As Valette first noted , the typical “Indian” sound that we heard in Sound 3 depends critically on the presence of bending stiffness and inharmonicity: it is essentially absent from Sound 4.
The spectrogram in Fig. 15 tells the story. Soon after the start of the note, at the bottom of the plot, the picture is very similar to Fig. 14. Starting from the first mode only, energy is rapidly transferred to higher overtones of the string by the nonlinearity implicit in the collisions between string and stop. But we do not see the time-varying formant feature, confirming that this feature is a key ingredient of “Indianness”.
We can get an inkling of what is going on from Fig. 16. This shows a series of short snapshots of the bridge force waveform from the simulations used for Sound 3 and Sound 4. The top pair of waveforms, plotted on top of each other in blue and red, show the pattern after 0.2 s. The blue curve shows the case without bending stiffness, Sound 4. Notice the feature marked by the green oval: the force waveform has developed a sharp upward jump, corresponding to a “corner” travelling on the string.
But in the corresponding red curve, with bending stiffness, this jump has become more gentle, with a slight “precursor wiggle”. This is an effect of bending stiffness which we have seen before: look back, for example, at Fig. 33 of section 12.2. The sharp jump in the blue curve involves a wide range of frequencies, and for the string without bending stiffness, those frequency components all travel at the same speed along the string so the jump can remain sharp.
When bending stiffness is added in, these frequency components do not all travel along the string at the same speed: the higher frequencies travel a little faster, making the “precursor”. The other red curves in Fig. 16 show further snapshots from Sound 3, with bending stiffness, at successive intervals of 0.4 s. The precursor grows more and more pronounced, and its shape gradually changes. This is the feature responsible for the formant in Fig. 14.
Another thing to note in Fig. 16 is that already by the second red curve, the shape is becoming rather like the “sawtooth” shape we saw with bowed strings back in Chapter 9. This is a signature of the Helmholtz motion, so it gives another view of what we saw in the animation, Fig. 12.
 C. Valette and C. Cuesta, “Mécanique de la corde vibrante”, Hermés, Paris (1993). A summary in English is given by C. Valette, “The mechanics of vibrating strings”, Chapter 4 of “Mechanics of musical instruments”, ed. A. Hirschberg, J. Kergomard and G. Weinreich; CISM Courses and Lectures no. 355, Springer-Verlag (1995).