3.1 Harmonics and non-harmonics

Armed with an understanding of vibration modes and natural frequencies, there is one interesting musical question we can tackle straight away. Most small children get their first glimmerings of practical music-making by banging on saucepan lids and tables, perhaps graduating after a bit to a toy xylophone. (Strictly, these toys are glockenspiels rather than xylophones because they have metal rather than wooden bars for the notes, but I will continue to call them toy xylophones for the sake of familiarity.) At least in principle, a child can play tunes on the “xylophone”, but not really on the saucepan lids. Why is that? In this chapter we will explore what is needed to create a tuned percussion instrument, with examples from around the world. This will reveal a story of ingenuity by people from many different cultures, which in itself illustrates how widespread and deep is the human commitment to music-making.

There is one type of vibrating object that everyone agrees you can use to play music: a stretched string, as on a guitar or piano. When you pluck or strike a stretched string, you always hear a sound with a definite musical pitch. The pitch can be changed by varying the length, tension or weight of the string, all familiar to musicians. What is different about strings, compared to the sound of banging a saucepan or a table?

The simplest mathematical description of the vibration of a stretched string, described in the next link, reveals a very special pattern in the set of natural frequencies. Once the lowest (or fundamental) frequency has been fixed by choosing the weight, tension and length of the string, then all the other frequencies are whole-number multiples: if the first is $f_1$, then the second is $f_2=2f_1$, the third $f_3=3f_1$ and the $n$th is $f_n=nf_1$. This simple numerical pattern was already encountered in section 2.2: it is called a harmonic series. The idealised stretched string has natural frequencies that are harmonics, quite different from the irregularly-spaced natural frequencies of the drum described in section 2.2.


Just as we saw for the drum, each natural frequency of the vibrating string is associated with a particular mode shape. The first few of these shapes are shown in Fig. 1: they consist simply of sine waves, fitted into the length of the string in all the ways that allow the ends of the string to be fixed. The simplest way, with the longest wavelength and the lowest frequency, has one half-wavelength trapped in the length of the string. The next mode has two half-wavelengths, and then three, and so on.

Figure 1. The first three modes of an ideal stretched string. The movie shows one period of the fundamental mode, alongside the second and third modes at the correct relative frequency.

All modes except the lowest have one or more points where the string does not move: these are called nodal points or just nodes. These will be important later. If you want to support a structure in a way that doesn’t interfere with the vibration of a particular mode shape, these nodal points are the place to use. For the particular case of a vibrating string, the nodal points have another significance that is familiar to musicians. If you touch a string lightly with a finger at the exact mid-point, then pluck or bow the string somewhere else, your finger will rapidly damp out any motion which does not have a node at the mid-point. You are left with the second mode, fourth mode and so on. This is, of course, what a violinist or guitarist calls “playing a harmonic”, in this case at the octave. Touching the string at 1/3 or 1/4 of its length will produce higher “harmonics”.

Many aspects of this simple model of string vibration are not quite right if examined carefully, and these details are very important for some musical questions. For example, a musician’s “harmonics” are not truly harmonics on a real string, although they would be on our idealised model string. We will return to this in section 5.4, with some details in section 5.4.3. But for now, the simple model gives a strong hint about the most important thing we need to know for investigating tuned percussion instruments. When a structure has some or all of its natural frequencies in something close to a harmonic series, any sound you can make on it is likely to be perceived as having a definite musical pitch. Most structures, like the toy drum, do not have this special property and they do not give rise to a definite pitch.

The art and science of creating a tuned percussion instrument, starting with a structure that does not naturally produce harmonics, involves playing around in some way with the design details in order to coax at least a few of the lower natural frequencies into approximate harmonic relations. The more modes that can be acoustically engineered like this, and the more accurate the harmonic relations, the more clear will be the musical pitch. That, at least, is my claim. Of course you should not believe it without some evidence.

We will look at tuned percussion instruments of several different types, and each case will be accompanied by sound files. By the time you have listened to these you should become convinced that there is some truth in the claim about the benefits of harmonically-related natural frequencies. That is not to say that this is the only important issue for the sound quality of these instruments, but it is definitely a big part of the story.

However, as we explore various sound examples we will rather rapidly find ourselves amid the “smoke and mirrors” of perception. In case you think that pitch is a simple thing, a famous auditory illusion serves to demonstrate that the perception of pitch can be rather slippery. Listen to the sound example below. You should hear a rising scale, with each note seeming to be clearly higher than the one before. But after a bit you realise that the pattern is somehow circular and repetitive. This particular illusion takes advantage of the fact that our pitch perception is particularly shaky when it comes to deciding which octave a given note is in: the pattern “sneaks” down an octave, somewhere during the ever-rising pattern. Try pausing the demo after each note and humming it: that will force you to pay attention to the octave. This demo is a scale based on so-called “Shepard tones”: look them up in Wikipedia if you want to know how it is done.

Ever-rising scale based on Shepard tones