13b: Drums

The following diagrams are the first eight modes of a circular membrane, clamped along the outer edge (seen in the previous simulation). The mode numbers and frequencies are also given for each one. The figures are shown in the order of increasing frequencies rather than increasing mode numbers. In the figure the green area is moving towards you while the red area moves away; then these areas switch and go in the opposite direction.

If you go back to the Chladni plate on YouTube and listen to the frequencies you will notice that, unlike strings or tubes, the resonances are not harmonics; they aren't multiples of the fundamental. Likewise in the square plate simulation, the circular membrane simulation and the above diagrams you will notice that the modes of vibration of a circular surface also are not multiples of the fundamental. The second frequency shown, f₂ (the first overtone) is 1.59 times the fundamental, f₁. This is an essential difference between instruments that make a perceptible pitch and those that do not. If you perceive a pitch from an instrument at least some of the overtones are harmonic (multiples of the fundamental) but this is not the case for many percussion instruments.

A second feature of percussion instruments is the short duration of the sound. As mentioned in Chapter 9 on Fourier analysis, the uncertainty principle comes into play for short pulses of sound. A short pulse requires many frequencies and few of these additional frequencies are multiples of a fundamental frequency as can be verified by a Fourier analysis of a cymbal or drum strike. Combinations of frequencies which are not multiples of some fundamental frequency usually sound like noise to our ears. Examples of sound sources that have lots of frequencies which are not harmonics include slaps, cracks, crashes, hand claps and thunder claps.

Thunder is formed when there is an electrical discharge from a thundercloud that suddenly heats air to nearly 30,000 ^oC along the path of the discharge. The air can reach a pressure ten times that of normal atmospheric pressure in a few microseconds. The sudden expansion causes a shock wave to propagate away from the region of discharge. Due to irregularities in the path and heating of the air, a large range of frequencies centered around 100 Hz are produced. Variations in air density in the storm and local topographical features also affect the range of frequencies and what someone on the ground will hear. Some frequencies are below 20 Hz and cannot be heard by humans but can be felt. As a result we do not hear thunder as a discernible tone.

Just like in the case of a string, the location of the initial displacement of a membrane activates different modes. Below are some images from a simulation of a circular membrane. The modes which are activated are shown in the columns below the membrane; the first column is n = 0 and m equal to one, two, three, etc.; the second column is n = 1 and m equal to one, two, three, etc. In the first picture the mouse was used to "strike" the membrane in the center. Notice that mostly modes in the first column are activated (modes (0,1), (0,2), (0,3) etc. as you go from top to bottom) with a few in the third column (modes (3,1), (3,2), etc. This is the case of a drumstick with a relatively large end (a mallet) hitting the center (note that the displacement is broad). The second picture is the case where a drumstick with a smaller profile made an indention on the drum head at the center (the "poke" selection in the simulation). For this case the fundamental modes are almost exclusively activated (only the first column is lit up). In the third picture the drum head is struck off center. Now many higher modes are activated (as indicated by the colors in the squares at the bottom). This tells us that, even though we still may not hear a pitch (because the overtones are not harmonic) the drum will sound different because there are different overtones. This is very similar to the case of a string that is plucked or bowed; the initial shape determines which specific modes of vibration (overtones) are activated.

Most drum membranes are stretched over a hollow body, often cylindrical in shape. This configuration allows for two different types of resonances. The air cavity inside the drum will have a set of resonance frequencies determined by its shape and size. Like the air resonances in a guitar body or in a tube, this will emphasize some frequencies at the expense of others and this effect may even be strong enough to give the drum a pitch (see the next section). The body of the drum also has its own modes of vibration as shown in the following diagram of the cylindrical sides of a drum. Here, green is an area moving towards the center of the drum while the red area moves away from the center. This means, again, that some of the membrane's overtones will be reinforced by resonance while others are not, much like the body of a guitar or violin reinforce some (but not all) of the string harmonics.

The body of the drum has one further function in that it controls how sound coming from the top surface of the membrane interacts with sound from the bottom of the surface. As shown in the following figure, with no body these sounds would tend to cancel due to destructive interference (as the surface moves it creates a pressure wave from the top that is exactly out of phase with the pressure wave from the bottom). If, instead, the sound from the underside has to travel through the drum body so that it comes out in phase (half a wavelength behind) then there is constructive interference. This will depend on the wavelength produced so a body of a given size will emphasize some frequency modes and tend to cancel others. This is why tall drums and drums with irregular shaped sides (like conga drums) sound different than shorter drums, even if they have the same drum head.

Here is a YouTube of a cymbal crash in slow motion. The modes are not the same as for a drum head because the edges of the cymbal are free instead of clamped and the cymbal is not flat. Because they are made of metal, which is stiffer than a drum head, the frequencies produced are higher. But the same sort of thing happens; the overtones are not harmonics so the cymbal does not have a fixed pitch.

Video/audio examples:

13c: Harmonic Percussion Instruments

If we want a percussion instrument to have a perceptible pitch there are two choices. We can activate only one single mode so that there is just one frequency present or we can modify the instrument so that at least some of the overtones are harmonic. The tabla is an Indian drum where the head changes thickness from the outer edge inwards which can be done by rubbing material into the drum head. This shifts the overtones to be harmonic so that the tabla has a specific pitch, as can be heard in this tabla YouTube video.

The tympani or kettle drum used in symphony orchestras also has a distinctive pitch. These drums have a closed, bowl shaped body which may be hemispherical or parabolic in shape. The air trapped inside tends to lower the overtones because it cannot move freely. This plus the shape of the body ends up causing the drum to have harmonics at a fifth (ratio of three to two), a major seventh (a ratio of 15 to eight) and an octave (ratio of two to one) above the fundamental. These harmonics are enough to cause our ears to hear a pitch instead of noise.

Many percussion instruments are composed of a series of bars of different lengths that are struck. The xylophone, vibraphone, glockenspiel, triangle, and marimba are examples. As a first approximation of these instruments we consider the modes of vibration of a rod. In general the modes do not produce harmonics but if the main fundamental has a larger amplitude than the other modes it will be perceived as a pitch.

The following pictures are the first four modes of a free rod. The displacement is exaggerated; real xylophone bars, for example, are much too stiff to bend that much. The first mode has a node at the center so the left side will go down while the right side moves up and the middle stays fixed. The second mode has a two nodes at one quarter (0.25) and three quarters (0.75) of the way along the length. Mode three has nodes at about 0.16, 0.50 and 0.84 (meters if the rod is one meter long). You can see higher modes and the vibrational motion of these modes with the simulation of vibrations of simple systems from Wolfram (you may need to download their plug in to play with this demonstration). Click on the + sign under time in the simulation to view the controls for simulating the mode vibrations over time. Here is a simulation of vibrations of a bar.

The xylophone, vibraphone, glockenspiel and marimba have rectangular bars that have the similar modes as the ones shown above for a rod. The bars are of different lengths and are usually suspended at the nodes of the second vibrational mode. This helps damp out other modes so that each bar has a predominant frequency, usually the first overtone (second vibrational mode). The rates of decay of different modes are not the same for these instruments. This means that the initial tone may sound harsh because it has a large number of anharmonic overtones (not multiples of the fundamental) but the tone may improve in the next few milliseconds as some overtones fade away more quickly than the first overtone. Like strings and other instruments, striking the bar in the middle is more likely to excite the lower overtones. Marimbas and vibraphones also have resonating tubes of different lengths beneath each bar which acts as a resonator.

Marimbas originally were made with gourds as resonators but these have been replaced by tubes in modern instruments. Without these tubes the sound from the top of the bar would be exactly out of phase with sound from the bottom and tend to cancel since the top and bottom move in the same direction when struck (as was the case for a drum head without a body, mentioned above). The tubes create a resonance sound wave exactly out of phase with the sound from the bottom of the marimba bar, canceling it. This accentuates the fundamental frequency from the top of the bar and also produces harmonics for that frequency. This is part of the reason why marimbas and vibraphones are more melodic sounding but the xylophone, which does not have resonators, has a harsh sound. A final characteristic of the bars on a marimba that give it its distinctive sound is they are not precisely rectangular. The under surface of the bar is carved and tuned by ear to produce higher harmonics. The smaller mass of the bar where it is being struck also decreases the impedance mismatch between the bar and the mallet, allowing more energy to go into the bar from the mallet. The cross sectional shape of a marimba bar is shown below.

Video/audio examples:

Tympani solo.
Marimba solo.
Xylophone solo.
Glockenspiel solo.

The orchestral triangle is, in effect, a rod fixed at one end and bent into three sections. Here are the first four vibrational modes for a rod fixed at one end. Notice that the left end is fixed and stays does not move. The vibrations of these and higher modes can also be seen using the Wolfram simulation of vibrations of simple systems or with this simulation of a vibrating bar.

Orchestral chimes (sometimes called tubular bells) are tubes of different lengths that are struck at the top with a light, soft hammer. An interesting property of these instruments is that the fundamental frequency does not (and in fact none of the vibrational frequencies) correspond to the perceived pitch (called the strike tone). The vibrational modes are similar to a free rod and it turns out that modes four, five and six are approximately in the ratio two to three to four (there are frequencies at 2f, 2.9f and 3.9f of the perceived strike tone, with frequency f). This is an example of virtual pitch (or missing fundamental which was discussed in Chapter 10). Our ear/brain system interprets the harmonics as belonging to a fundamental frequency that is not actually present.

Bells have been around as musical instruments almost as long as flutes. The vibrational modes depend on the diameter of the bell, the thickness and the exact shape. These modes are tuned by trimming material from the inside of the bell so that there are harmonics of the fundamental frequency, giving the bell a distinct tone. Even so the modes of vibration are quite complicated as shown in these animations of bell modes. Many higher bell modes can be seen in these holograms of ancient Chinese bells and in this animation of ancient bell modes. In the hologram pictures the center of the circular regions are anti-nodes. Notice that some centers are dark at the same time other centers are light. This indicates that they are moving in opposite directions (are 180 degrees out of phase). Some of the bells pictured here were made as early as 1766 BCE. The different modes can be excited by striking the bell in different locations so that one bell can play more than one note.

Steel drums or steelpans are traditionally made by bending and stretching the top part of a large, 55 gallon oil drum to play different notes at different locations on the head. They became popular in the immigrant, former slave communities of the Caribbean in the 1930s. Each region of the surface of the drum is stretched until it has the harmonics of the desired note.

The gamelan is an orchestra of instruments from Indonesia that consists of a large number of mostly percussion instruments (occasionally stringed instruments and voice are included). Each player in the orchestra strikes one or more different instruments in turn to produce the melody. The tuning has five notes to a scale and so sounds odd to western ears (western music uses an eight tone scale).

Video/audio examples:

Steel drum solo.
Gamelan examples from YouTube: one, two, three.

13d: Other Musical Instruments

There are many older instruments which only exist in museums today such as the shawm, viol, cornophone and crumhorn. Others, like the Hurdy Gurdy, have been around since the 11th century and still exist but are rarely heard. The following web sites describe various antique instruments and have sound samples for some of them: Renaissance Instruments, Guide to Medieval and Renaissance Instruments, Edinburgh University Collection of Musical Instruments, Zzounds renaissance instruments.

Musicians continue to make new instruments and modify older ones.

Video/audio examples:

YouTube performance of a Hurdy Gurdy.
YouTube of the Stroh violin.
The Tromba-marina is an interesting instrument popular in the 18th century.
YouTube of the wine glass harmonica.
YouTube of several unusual instruments and a web page of a few more unusual instruments.
Wikipedia on the Jew's harp.
Chinese singing fountain bowl YouTube.
Tibetan singing bowl YouTube. One with water.
Here is a web site that discusses how to create and play a pitch-bending accordion.
Aeolian percussion instruments at the Harmonic Fields.
The Hang is a relatively new instrument similar to a steel drum but played by striking it with the hand. YouTube of a Hang being played.
The Theremin is an interesting instrument that uses electrical capacitance to change an electronic circuit to make sound. The performer does not actually touch the instrument as seen in these YouTube videos: one, two.
Here are a series of Ted talks about unusual instruments and performances.
If you ever thought of making a unique instrument that never existed before you might want to take a look at Wikipedia's Experimental Instruments Page or this New York Times Article on Experimental Instruments.

Summary

Percussion instruments tend to have overtones that are not harmonic (not multiples of the fundamental). If the anharmonicities are strong enough the sound is perceived as noise, not a pitch. Under certain circumstances the anharmonic overtones can be damped or tuned so that there are enough harmonics to cause the instrument to have a perceived pitch. This can happen with harmonics of a real fundamental frequency and also when the harmonics are multiples of a missing fundamental that isn't a vibrational mode of the instruments.

End of chapter exercises: Percussion Instruments.