13a: Instruments with Non-harmonic Overtones

As we have seen, string and tube modes of vibration can be labeled by a single mode number, n, and the modes are harmonic. Each subsequent mode (or overtone) produces a frequency that is a multiple of the fundamental. (Note: There are slight variations to this rule when the string amplitude gets very large or if the string is very stiff but we won’t worry about that here.) In the simulation of the square surface in Chapter 11 we saw that two mode numbers, n and m, are needed to specify a mode on a two dimensional surface. We also discovered that some of the modes were degenerate meaning two different combinations of n and m lead to the same frequency but in general the frequencies were not harmonic. This is generally true of most vibrating surfaces and membranes; they do not have harmonic overtones. As a result our ear-brain system does not detect a distinct pitch from most drums.

Simulation Exercise 13A (turn in answers on a separate sheet of paper): Simulation of a circular membrane

Nice simulation but does this happen in real membranes? Take a look at the following web pages, play the videos and decide if real drums are like the simulation you just looked at.

Membrane driven by a speaker.

Slow motion video of a snare drum being stuck.

The amplitude of the vibration on a string can be given as the function of the location, x along the string which makes it a one dimensional system. This means harmonics can be numbered with one number, n (n = 1 is the fundamental, n = 2 is the second harmonic, etc.) as we did in Chapter 11 on strings. Because drum (and other) surfaces need two variables to locate a position on the surface, we also need two numbers to define the modes of vibration. The following diagrams are the first eight modes of a circular membrane, clamped along the outer edge. The mode numbers are also given above each one. The figures are given 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 in every case. This is a generic feature of surfaces (square, rectangular, round, etc.). Likewise in 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 a fundamental 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.

As mentioned in Chapter 9 on Fourier analysis, the uncertainty principle comes into play for sounds from percussion instruments. Because the sound is formed as a short pulse there has to be many frequencies present, as can be verified by a Fourier analysis of a cymbal strike. Because there are many frequencies present, few of them are multiples of a fundamental frequency. Combinations of frequencies which are not multiples of some fundamental frequency often 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 C 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 in the case of a string, the location of the initial displacement of a membrane activates different modes. Below are some images from the simulation of a circular membrane that you played with previously (Simulation 13A). 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.


The following is a video 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.

Although there are no musical instruments based on a rectangular membrane, the behavior is very similar. Here is a simulation of a rectangular membrane from Wolfram (you may need to download their plug in to play with this demonstration). Also, an applet simulation of a rectangular membrane. Note than you can change the size of the rectangle and look at different modes. How are these modes similar to those of the circular membrane? How are they different?

Video/audio examples:

Snare drum solo.

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