Up to now, we have been concentrating on linear systems. In the previous section we saw examples of musical instruments that were “almost linear”, but with nonlinear effects beginning to show, and sometimes proving important for the sound. Shortly, we will look at some seriously nonlinear instruments. But before that it is useful to step back a little to give a brief survey of what we might expect to see, and what methods we might bring to bear in order to make sense of nonlinear phenomena.
This will not be an easy task, and you should not expect a complete, organised account of everything nonlinear systems can do. “Nonlinear” is one of those terms that is defined purely by its negative. “Linear” means something definite: linear systems are amenable to systematic discussion, as we illustrated in Chapter 2. But “nonlinear” simply means “not one of those…”. It is a bit like the term “agnostic”, which only tells you something about what a person does not believe: it tells you nothing about what they do believe.
When nonlinearity becomes important, some of the concepts we have been using regularly up to this point become useless; or at the least, questionable and potentially misleading. The prime example is the idea of a frequency response function. We have been showing various plots of frequency response, for example using bridge admittance to characterise the behaviour of the soundbox of a stringed instrument. Well, think right back to Chapter 2. That was where we first met the idea of frequency response plots, which relied on the idea of linearity in two critical ways.
First, we learned that sine waves are special waveforms. For any linear system, if you feed a sine wave in you get another sine wave out, at the same frequency. Its amplitude and phase may have been changed, and those changes may vary with the frequency of the particular sine wave, but the waveform is always sinusoidal. The second crucial property of linear systems is superposition: if you can express the input as a sum of separate terms, and if you know how each of those terms separately is affected by the system, then you simply add the separate outputs together to find the total response. We applied this idea to Fourier analysis: express the input as a combination of sinusoidal terms, work out how each term is affected by knowing the frequency response function, then add these output sinusoids together to find the actual output.
Both these properties go out of the window with a nonlinear system. We can give very simple illustrations to demonstrate that claim. Let’s start with the “sine wave in, sine wave out” property. Think about an electrical amplifier, like the ones in home hi-fi systems. An ideal amplifier simply takes an input electrical signal, and boosts it by a certain factor so that it can drive a loudspeaker. An amplifier that behaves like this certainly satisfies the sine wave property: every input waveform, sine waves included, should produce output with the identical waveform except for the amplitude being increased.
Well, there is a problem. Any amplifier is likely to have a maximum output voltage that it is capable of producing — often related directly to the voltage of the power supply which provides it with electrical power. Now suppose you feed in a sine wave, with an amplitude big enough that a correctly amplified version would exceed this limit. What is likely to happen is a type of nonlinear response known as clipping. The peaks and troughs of the output sine wave will be chopped off at the amplifier’s maximum voltage. The result, in idealised form, will be an output waveform like the red curve in Fig. 1, instead of the dotted black curve which represents a correctly amplified sine wave.
What can we say about the clipped output waveform? Well, it certainly isn’t a sine wave. But it is still a repetitive, or periodic, waveform. We know from section 2.2.1 that any periodic waveform can be expressed as a Fourier series, which will consist of a sine wave at the fundamental frequency (the frequency of our input sine wave), plus a series of sine waves at exactly harmonic frequencies. So here we meet one of the characteristic effects of a nonlinear system: it might generate harmonics of a frequency you feed in. In the case of the hi-fi amplifier, this would be a bad thing. You don’t want your amplifier to change the sound of what you are listening to, just make it louder. But the sound of the clipped waveform will be very different from the sound of a sine wave: much brighter, because of all the high-frequency components appearing in the Fourier series. You can hear in in Sound 1: first the sine wave, then the clipped version.
Clipping isn’t always bad, though. A less harsh version of clipping, called “soft clipping”, seems to be part of the reason for the enduring preference of electric guitarists for valve (vacuum tube) amplifiers. Modern solid-state amplifiers may be more likely to produce “hard” clipping of the kind illustrated in Fig. 1 and demonstrated in Sound 1. The soft clipping effect enters more gradually as the input level approaches the amplifier’s limit: a schematic example is shown in Fig. 2. Guitarists can make constructive use of the effect, whereas hard clipping comes in more abruptly and simply sounds harsh. This is a first, tiny, glimpse that a musician may make constructive use of nonlinear effects. That will be a major theme of this chapter: we will meet examples in many different guises.
The idea of superposition also goes out of the window with any nonlinear system. We have already seen an example of this in the discussion of “phantom partials” in the previous section. The nonlinearity we looked at in section 7.4.1 was the simple example of a square law. What we found was that if you take a single sine wave and square it, the result is a sine wave at double the frequency you started with. This is a rather extreme example of “harmonic generation”: the nonlinearity gives an output containing only second harmonic, with none of the original fundamental frequency.
So what happens if we take two sine waves at different frequencies and put them through the square-law nonlinearity? Taking each one separately and squaring it would give two sine waves at the two doubled frequencies. But we saw in section 7.4.1 that when you add the two sine waves together and then square them, the result includes extra sinusoidal terms at the sum and difference of the two original frequencies. This is definitely not a superposition of the two separate outputs.
There is another idea that we have made much use of, but which becomes questionable in the presence of nonlinearity: the idea of modes of vibration. Modes give a very powerful and general tool for understanding the vibration of linear structures. Lord Rayleigh, back in the 19th century, was intimately familiar with that fact, and it has been a cornerstone of vibration theory ever since. Well, people do sometimes talk about “nonlinear modes”, but this is a somewhat contentious idea which is still a topic of current research. As a rule of thumb we should not expect discussions of nonlinear effects to centre around modes in the same way as we have been using for linear vibration. At the very least, we will exercise caution in any such discussions.
All this seems very negative. Should we regard nonlinearity simply as a nuisance, to be avoided wherever possible? In the context of a lot of industrial problems involving noise and vibration control, the answer would be “yes”. But for musical problems, the answer is an emphatic “no!”. At the risk of offending percussionists, isn’t there something a bit limiting about a lot of percussion instruments? Of course you can make interesting music on a marimba, but the range of possibilities is much less wide than with the human voice, or a violin. Nonlinearity is the key to this difference.
To give an impression of why that might be, it is useful to list some of the possible consequences of nonlinearity: we will go into these in more detail in subsequent sections. So far, we have only met some rather basic nonlinear phenomena: harmonic generation (as in the clipping example);combination frequencies (such as phantom partials); the pitch glide effect in which vibration amplitude affects the frequency; and (in the lute string example) the fact that some kind of impact might change the bandwidth of excitation, and thus change the brightness of the sound.
But these examples give little hint of the richness of behaviour that can stem from nonlinearity. The next phenomenon to mention gives the key to entire families of musical instruments: the possibility of self-excited vibration. Up to now, we have been discussing vibration and sound caused by some obvious source of excitation: a drumstick, the clapper of a bell, the finger or plectrum of a guitarist, the hammer of a piano. But what about the wind instruments? A player blows steadily into the mouthpiece of a recorder, for example, and a musical note may or may not emerge depending on whether they get the technique just right. In some way, the player’s lungs must be providing the energy to sustain the vibration and sound. But where does the vibration come from? The same question could be asked about a violin: the player pushes the bow across a string, and vibration appears “from nowhere”.
Linear theory can give no more than a hint of the answer to that question. We will see in section 8.3 that it is sometimes possible to predict the threshold for self-excited vibration using a linearised theory. Reed woodwind instruments can illustrate the effect. If you blow very gently into a clarinet or saxophone, you hear nothing except a bit of “breath noise” from your blowing. But if you gradually increase your blowing pressure, at some stage a musical note will “light up” from nowhere. That critical blowing pressure is an example of a threshold of instability: we will see how that works in section ?. But the only thing linear theory can predict is that once you are beyond the threshold, the volume of the note will continue to grow in an exponential way, louder and louder without limit. The real note doesn’t do that: it settles to a steady amplitude. That kind of sustained self-excited vibration always requires nonlinearity, whether we are talking about wind instruments or bowed string instruments, or about the disastrous wind-driven vibration of the Tacoma Narrows bridge.
We can return to the recorder or clarinet examples to illustrate the next nonlinear phenomenon. You have gradually increased blowing pressure to find the threshold, and achieved a decent musical note. If you carry on blowing gradually harder, you may find that at some stage the instrument spontaneously switches to playing a different note: a musician would call this “overblowing”. This tells us that the recorder or clarinet is capable of more than one kind of self-excited note, or regime of vibration, and also that when you vary a parameter like the blowing pressure you may hit critical values where a switch occurs. In mathematical accounts of the underlying theory, such switches are called bifurcations. Regime switches like this are crucial to many musical instruments: indeed, with a valveless brass instrument like a bugle or natural horn this is the only way that a player can change the note they are playing. Of course, in practice the player will not only vary the blowing pressure: they will also do things with their lips, varying their “embouchure” to allow them to control the switches reliably.
Now think about doing a similar experiment with a violin. Place your bow on a string, move it at a steady speed, and gradually increase how hard you press down. You don’t find a clear-cut threshold this time, it all seems more complicated than the wind instruments. When you are pressing very lightly, you may create a variety of whistles and squeaks. Then you will find a value of the bow force when the violin starts to make the kind of sound you (or your violin teacher) are aiming for. But if you carry on increasing the force, this nice musical note will eventually give way to some kind of raucous “crunch”. What has happened is that regular periodic motion of the string, responsible for the acceptable musical note, ceases to be possible. Instead, something non-periodic (and non-musical) occurs instead. This is an example of another important nonlinear phenomenon you will probably have heard of: “chaos”.
The hallmark of chaotic response is something called “sensitive dependence”. If you repeat the experiment with a new bow-stroke, you will produce a similar-sounding “crunch” at the end, but if the two notes are recorded and then compared carefully, the details will be different. Even a very good player will not be able to take a bowed string deep into the raucous regime, and achieve exactly the same motion twice. Tiny differences in the two bow gestures will be amplified in the chaotic regime to produce results that are completely different.
Good violinists probably never stray into this raucous, chaotic regime, but they regularly encounter another manifestation of “sensitive dependence”. During the transient at the start of each note, it is extremely hard to control the bow gesture well enough that two identical waveforms can be produced. By dint of many hours of practice, a good violinist will learn to control their transients well enough that most of the time they can produce something that sounds pretty much the same. But for some kinds of bow gesture it is hard to do even that: we will go into some details on all this in Chapter?. Such behaviour can only occur in a nonlinear system: linear systems simply do not allow anything resembling sensitive dependence or chaos.
This example of bowed-string transients raises the final issue to be mentioned in this introductory discussion. Let me remind you of something I said right back in Chapter 1. I ventured a definition: “A musical instrument is a contrivance which allows a performer to use gestures they are physically capable of performing to make sounds that they like”. Every musical instrument represents a balance between two competing desires. On the one hand, the player wants an instrument with a wide repertoire of possible sounds to give a good “palette” for their performance. But they have to be capable of exercising rather precise control over these sounds, so they want an instrument with good “playability”. The richer the potential palette, the trickier the control problem is likely to become. We will investigate all this and try to pin down at least some aspects of “playability” in Chapter ?.
In summary, in this chapter and the ones that follow we will explore some of the wonderful, but often frustrating, possibilities that arise from different aspects of nonlinear behaviour in musical instruments. Profoundly nonlinear instruments like the violin require a lot of practice before you can even produce a single note with satisfying musical qualities, but in the hands of an expert a violin can produce a whole world of interesting musical sounds. Its motto might be “no pain, no gain”. But none of this musical potential would be possible without the nonlinearity: as the software engineers would put it, nonlinearity is not a bug, it’s a feature.