6.4 Pitch, timbre and excitation patterns

In the loudness test just described, the listener only ever heard one sine wave at a time. But most sounds contain many frequency components simultaneously: the separate harmonics of a complex tone, or the sounds of two instrument playing together, or the sound of someone talking in the presence of background noise. Naturally, this makes the business of understanding perception more complicated — but it is not quite as complicated as one might have feared.

A very important idea comes into play, based around the description we gave in section 6.2 of the behaviour of the basilar membrane within our inner ear. With a sine wave at some particular frequency, the response of the basilar membrane is strongest in a particular region — but of course this region has a finite spread. So a range of hair cells will respond to some extent to this sine wave, and conversely a given hair cell will respond to some extent to a range of frequencies, spread around its frequency of maximum sensitivity. This leads to the idea of an auditory filter.

The mechanical response at a certain point on the basilar membrane will have a frequency response function, similar to the ones we have looked at for musical instruments. We would expect it to have the shape of a rather simple band-pass filter, showing a peak of response at the “resonant” frequency, and lower response for frequencies progressively further away on either side. This might be expected to translate into some kind of frequency response related to the firing rate of the neuron attached to a hair cell at that point on the basilar membrane.

There are various ways to map out the characteristics of these auditory filters. It is possible to insert a small electrode into a single neuron in the auditory nerve, and record directly the firing rate: if the threshold sound level to give detectable firing is measured for different frequencies of sine wave, the result is called a neural tuning curve. Such curves do indeed have the form we have anticipated, with a peak sensitivity at a frequency corresponding to the placement on the basilar membrane of the hair cell attached to the particular neuron.

Alternatively, psychoacoustical methods of testing can be used to find a filter shape directly from listening tests. Most methods are based around the phenomenon of masking. If a loud sound and a quiet sound happen simultaneously, to what extent is the quiet sound covered up by the loud sound? This can be investigated by a rather similar experiment to the one just described for assessing loudness. A test subject, with their headphones, is exposed to a fixed sine wave at some chosen frequency, and a second sine wave at a different frequency. This time, the two sounds are simultaneous rather than being presented alternately. The goal is to measure the threshold for detecting the quiet sound.

What is discovered is that if the frequencies of the two sounds are sufficiently different, the threshold for detecting the quiet sound is exactly the same as it would have been in the absence of the loud sound: in other words, at the level given by the threshold curve in Fig. 1 of the previous section 6.3. But if the two frequencies are close together, then the quiet sound is significantly masked by the loud sound.

However, in practice using a sine wave for a masking experiment like this does not give the most useful results. The main problem is that when two sine waves at slightly different frequencies occur simultaneously, the phenomenon of beats arises. The effect is most striking for sine waves of the same amplitude, as illustrated in Fig. 1. The strong modulation of the amplitude is at a frequency which is the difference of the two separate sine waves. In this example the sine waves are at 200 Hz and 203 Hz, so the beat frequency is 3 Hz. You can hear the waveform in Sound 1.

Figure 1. A sine wave at 200 Hz added to another sine wave at 203 Hz with the same amplitude, leading to a pattern of beats. The upper trace shows a 3 s segment, which you can listen to in Sound 1. The lower trace shows a zoomed view, to show the underlying sinusoidal variation.
Sound 1. The sound of the beating waveform in the upper trace of Fig. 1.

If the two amplitudes are not equal, a weaker version of the same phenomenon occurs. An example is shown in Fig. 2, where the second sine wave has 1/10 the amplitude of the first one. You can hear the waveform in Sound 2. The modulation is still clearly visible in the plot. It is subtle but audible in the sound. This is the effect which can interfere with an experiment to probe masking. Basically, we might be using a different feature detector to notice the beats, not the one the experimenter is trying to exercise.

Figure 2. A sine wave at 200 Hz added to another sine wave at 203 Hz with an amplitude 1/10 of the first one, leading to a modulation pattern that is less strong than in Fig. 1, but still clearly visible. The upper trace shows a segment 3 s long, which you can listen to in Sound 2. The lower trace shows a zoomed view, to show the underlying sinusoidal variation.
Sound 2. The sound of the beating waveform in the upper trace of Fig. 2.

It is better to use random noise of some kind for the masking signal, to avoid this issue of the subject detecting the test sound via beats. In this context “noise” has a technical meaning that is more specific than the colloquial usage. One thing we might do is replace the sinusoidal masking tone with narrow-band noise, which would mean adding together sine waves at a range of closely-spaced frequencies, all with the same amplitude but with random phases. There is also an ingenious technique due to Patterson [1] which uses the opposite pattern, called notched noise: a combination of sine waves at all frequencies except for a range around the test tone.

By carrying out experiments by these various methods, auditory filter shapes can be mapped out. Some examples are shown in Fig. 3. For very low-level sound these filters behave approximately linearly, but for louder sounds the filter characteristics change progressively with sound level because of nonlinear effects. The filter shapes become more asymmetric, and the degree of “tuning” changes. Tuning is sharpest for low-level sounds: it is strongly influenced by the action of the active outer hair cells. As with so much of the subject, it is all rather complicated: for some details, see Moore [2]. Figure 3 is plotted with a logarithmic frequency scale, which reveals that the filters centred on higher frequencies tend to have very similar shapes. This means that their bandwidth is approximately proportional to their centre frequency. But at lower frequency this relative bandwidth gets wider.

Figure 3. Five examples of auditory filters.

The bandwidth of each auditory filter defines an important quantity known as the critical bandwidth. Roughly, any two frequency components falling within a critical bandwidth of each other are too close to be resolved on the basilar membrane. We will see shortly that this has a number of perceptual consequences. The precise numerical definition of the critical band depends on exactly what definition of bandwidth is used. Historically, different authors basing their conclusions on different test procedures have used different definitions. But we need not go into these details.

We will use one particular definition, the equivalent rectangular bandwidth or ERB. This is defined as the width of an idealised rectangular filter shape which would pass the same total power as the actual auditory filter. If, instead of the decibel plot of Fig. 3, the filter shapes were plotted on a linear vertical scale of squared amplitude, the ERB would simply be the width of a rectangle with the same area underneath it as the filter curve, and the same maximum height. What this means in terms of the decibel plot in Fig. 3 is that the ERB is quite close to the 3 dB bandwidth of each peak, the same measure we used earlier for the bandwidth of resonances of mechanical systems. A plot of the ERB as a function of centre frequency is shown in Fig. 4. It is believed that a frequency scale based on ERBs corresponds to a scale of distance along the basilar membrane: 1 ERB corresponds to about 0.9 mm.

Figure 4. The bandwidth of auditory filters, represented by the ERB, as a function of centre frequency.

Armed with the set of auditory filters, we can construct something important. For any given sound input, be it a sine wave or a Beethoven symphony, we can use the set of filters to map out the pattern of motion of the basilar membrane, known as the excitation pattern. If the input sound is steady this will be a fixed pattern, but for normal sounds entering your ears the pattern will vary in time.

We will start by illustrating a steady example. The left-hand plot in Fig. 5 shows a computed excitation pattern for a steady sawtooth wave at 440 Hz, at sufficiently low amplitude that we can use the linear approximation to the auditory filters. The first few of the successive harmonics, at multiples of 440 Hz, are clearly separated, but at higher frequency the pattern blurs into a continuum. The right-hand plot shows why. The ERB at each successive harmonic frequency is plotted, normalised by the spacing between harmonics. The value reaches 1 around the 8th harmonic: above that, successive harmonics are within 1 ERB of each other, so that they are not clearly resolved in the excitation pattern.

Next we look at a simple time-varying example. Back in section 2.4, we saw a spectrogram of a violin note, being played with vibrato. The spectrogram is repeated as Fig. 6, and the associated sound appears as Sound 3.

Figure 6: A spectrogram of a note on a violin, played with vibrato
Sound 3. The sound of a violin note with vibrato, corresponding to the spectrogram in Fig. 6.

We can analyse the same sound as a time-varying excitation pattern. To achieve this, we can take advantage of an approximate form of the auditory filters, called gammatone filters [3]. These have a neat mathematical expression, convenient for computing, and they have been shown to give a good match to the measured auditory filters. The result is shown in Fig. 7. It looks rather like a spectrogram, and it has been plotted in a similar way to Fig. 6. The horizontal axis shows an ERB-based frequency scale which is not very different from a logarithmic scale. Time runs vertically upwards. The excitation magnitude is indicated by colour, on a decibel scale shown in the sidebar.

Figure 7. A time-varying excitation pattern plotted in a similar format to Fig. 6. The frequency scale is based on ERBs, and thus on distance along the basilar membrane. Time runs vertically upwards. The magnitude of the excitation pattern is represented by colour, on a decibel scale given in the sidebar.

The harmonics of the violin note can be seen, but they are much more blurred than the version in Fig. 6. The “wobble” caused by vibrato is still clearly visible in the first few harmonics. However, by about the 7th harmonic it is getting hard to see the separate stripes in the plot, whereas they remained clearly visible in the spectrogram. The reason is exactly the same as the example in Fig. 5: this frequency is around about where the spacing between successive harmonics is comparable with the critical bandwidth so that they are not resolved in the excitation pattern. The spectrogram representation shows the underlying physics more clearly, but the “auditory spectrogram” of Fig. 7 relates much more closely to the way a human listener will perceive the sound.

For the next example, it would be a good idea to listen (rather carefully) to Sound 4 before you know what it contains.

Sound 4. See text for description

What you were listening to was a pair of sine waves, with the same amplitude and gradually changing frequencies. They are always centred on 800 Hz. They start 160 Hz apart, and the two frequencies move symmetrically inwards at a steady rate until they just come together at the end of the file. How would you describe what you hear? To my ears, the two separate tones, gradually converging, are clear at the beginning. But somewhere around the middle of the file the perception changes. It is no longer clear that there are two tones, but there is gradually increasing “roughness” in the sound, which resolves into slowing beats right at the end.

Figure 8 shows the excitation pattern plot generated from this sound, using the gammatone filters again. In the lower part of the diagram, two nearby but separate stripes can be seen, converging as time goes on and looking rather like a pair of trousers. Somewhere near the middle of the plot, these two stripes merge: this is not surprising, because the ERB at 800 Hz is 111 Hz. At the top of the plot, a pattern of modulation in time can be seen. It is quite fast when first visible, and gradually slows down. These are beats, exactly as shown earlier in Fig. 1, except that the beat frequency is changing with time. The beats are at the difference of the two frequencies, so the beat rate slows down towards zero at the end. This example illustrates two things. First, an apparently simple combination of two sine waves can produce perceptions of different kinds: from changing tones to something involving “roughness”, and eventually clear pulsation in time. Second, we get an indication that the excitation pattern plot reveals at least some aspects of these different perceptions; although we can see nothing very obvious associated with the sense of “roughness”.

Figure 8. Auditory spectrogram of Sound 4.

What we have seen is that the excitation pattern captures some aspects of sound perception, although by no means all. There is a striking success story related to loudness. In section 6.3 we saw a bit of the complexity of loudness relating only to single sine waves. One might have expected many more layers of complexity with more complicated sounds, but in fact there is a very successful loudness model by Moore and Glasberg [4], which builds on the excitation pattern. For this purpose, the linear gammatone approximation is not good enough, though: the nonlinear variation of auditory filter characteristics with level must be included. The Moore/Glasberg model first calculates the excitation pattern, steady or time-varying as appropriate, then aggregates that pattern into a single measure of loudness.

For the rather elusive concept of “timbre”, the excitation pattern is useful, but certainly does not tell the whole story. We have already seen an example, in Sound 4 and Fig. 8. The impression of “roughness” did not correspond to anything obvious in the excitation pattern plot. But on the other hand we will see examples in the next section where analysis of excitation patterns gives strong clues about whether a small change to a sound will be audible. A summary of current understanding seems to be that if two excitation patterns are sufficiently different, the two sounds are almost certainly audibly different, but the converse is not true: two sounds can be distinct but have very similar excitation patterns.

A clear example of this comes from something that you may have been wondering about. A typical ERB is a few semitones, when expressed in musical jargon (remember that a semitone is a frequency ratio of about 6%). Music as we understand it would hardly be possible if we could not distinguish two sounds a semitone apart! In fact, our ability to discriminate pitch between two steady periodic sounds with different periods is far more acute than the ERB.

This acuity of pitch perception is usually expressed in terms of cents: a cent is a hundredth of an equal-tempered semitone, so 1 cent corresponds to a frequency ratio of about 0.06%. Well, we cannot discern a difference as small as 1 cent, but under the best conditions people can discern pitches about 5 cents different. (Strictly, I am speaking rather loosely here: what I mean is “people can discern a pitch difference between two frequencies differing by about 5 cents”.) This value is (approximately) the threshold for pitch discrimination when two sounds are heard one after another. If the two sounds are heard simultaneously, we may be even more acute to “out-of-tuneness”, because the phenomenon of beats comes into play again. So when you are tuning your violin, or playing or singing in an ensemble, you need to get pitches remarkably accurate, as all musicians (and parents of budding musicians) know only too well.

How do we manage to distinguish two pitches as close as 5 cents (i.e. a frequency ratio of about 0.3%)? There has been much debate and controversy about that question over the years. The answer almost certainly involves the interplay of two factors. One is the one we have already been talking about: there is relatively coarse segregation in frequency already present on the basilar membrane, and there are subtle features of the exact pattern of neural excitation by different hair cells that may encode a more precise estimate than the ERB would suggest. But there is a second factor. At least at lowish frequencies (up to the mid-kHz range), the firing of individual nerve fibres can be synchronised with the phase of the input sound, presumably by being phase-synchronised in some way with local motion of the basilar membrane. This means that information about the frequency is reaching your brain in this form, as well as the information coded into which hair cells are being activated. Somehow, somewhere, these two sources of information are probably being combined in your brain to allow you to discriminate pitches with the precision that is observed.

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[1] Roy D. Patterson; “Auditory filter shapes derived from noise stimuli”, Journal of the Acoustical Society of America 59, 640–654 (1976).

[2] Brian C. J. Moore; “An Introduction to the Psychology of Hearing”, Academic Press (6th edition 2013).

[3] The gammatone software used here is by Malcolm Slaney: “Auditory Toolbox Version 2”, Technical Report #1998-010, Interval Research Corporation (1998), http://cobweb.ecn.purdue.edu/~malcolm/interval/1998-010/

[4] The Moore/Glasberg loudness model has developed over time, and there are many references. A key one is Brian R. Glasberg and Brian C. J. Moore; “A model of loudness applicable to time-varying sounds”, Journal of the Audio Engineering Society 50, 331–342 (2002).