As a child, I was oddly fascinated by how ears work. I used to ask “How can we hear two things at once and tell them apart?” I still think this is a much better question than it is given credit for, one that people rarely stop to think about. And it is what this post is about.
The more I have learned since, the more I think it’s a miracle (in the secular sense) that we can do this. I’m not a biologist, and there might be a simple biological explanation. But even if there is, I still think it’s a miracle from a physics point of view. I’ll show you what I mean.
Sound as a pressure wave
Sound is caused by vibrating objects. Those vibrations cause disturbances to the particles of the surrounding medium (air, for example). Those particles disturb their neighbours, which disturb their own neighbours. The disturbance propagates outwards as a wave of regions of compression (higher particle density) and rarefaction (lower particle density).
The following animation of that process comes from Dan Russell at Penn State University, and is reproduced under the CC BY-NC-ND 4.0 licence.
You can imagine the vibrating object at the left hand side of the picture, and your ear at the right hand side. Look at how the wave propagates from left to right, even though each particle vibrates backwards and forwards about its mean position (a few are coloured red to highlight that fact). This picture describes a ‘longitudinal wave’, which means that the vibrations are in the same direction as the wave motion. (‘Transverse waves’ behave differently, and the disturbance of each particle is perpendicular to the wave motion, but sound does not routinely work like this, so we won’t dwell on that too much here.)
The compressions have a higher pressure than the rarefactions, due to the higher particle density. As the wave travels from left to right and hits your ear drum, your ear drum experiences those fluctuations in pressure, and it vibrates accordingly.
This is all cool, but it is not the miracle. Please be patient.
Displaying graphs of sound
If we had to draw the diagram above, and animate it, every time we wanted to describe a sound wave, then either we would get very bored, or Dan Russell would start claiming royalties and get very rich…
Luckily for us, there is an easier way to show what is happening. Look at the following two confusingly-similar graphs of the sound wave in the animation.
We’ll only need the second one, but I’ve put both in to meet the potential confusion head-on… In the left-hand graph, the pressure (or number density) is plotted for each point in space at a given time. It’s like a snapshot of the pressure information. Actually, we aren’t plotting the value of air pressure itself, but the pressure difference from the ‘normal’ air pressure. The only effect on the graph is to centre it on zero on the pressure axis, rather than have it high up the vertical axis.
In the second graph, in contrast, we concentrate on a single point in space, and see how the pressure difference varies over time.
In relation to the animation, the first graph is like freezing the motion and taking in the whole picture, whereas the second is like staring at a tiny point of the picture and watching how it changes as you play the video.
The reason that the second is more relevant to this post is that we will be interested in how the pressure at the eardrum (a single place) varies with time as we hear a sound. Watch the animation again – when a compression hits the end of the screen (your eardrum), that corresponds to a peak in the right-hand graph.
The human ear
The less biology I write, the better for everyone. So here is a picture of the inside of the human ear, by nature.com.
Vibrations from the ear drum are transmitted to the three tiny bones in the middle ear. The last of these bumps against the window of the snail-shaped cochlea, and causes the fluid inside the cochlea to move. This fluid movement stimulates a response in the auditory nerve. And then it’s the brain’s problem!
Now, I admit that’s a miracle, but it’s not the miracle. Stay with me!
Mixing two sounds
This post is supposed to be about hearing two sounds at the same time. So now we have introduced sound as a pressure wave, let’s imagine mixing the two pure tones (like the kind you get from a tuning fork) shown individually here.
The amplitude, or maximum pressure difference from equilibrium, is greater for the dark blue tone (A) than the light blue one (B). We perceive this difference as tone A being louder than B. Similarly, tone B is perceived as higher-pitched, because the sensation of pitch is related to the frequency of the vibrations. In fact, tone B is three octaves above tone A, because a doubling of frequency corresponds to a change of 1 octave, and tone B has a frequency eight times that of A (x8 = x2 x2 x2 = 3 octaves).
What’s the ‘input’ to your ear drum?
When the two sounds A and B occupy the same space at the same time, the pressure wave formed comes from the addition of the two separate vibrations.
If we add the two sounds A and B, we get the red pressure wave below.
And this is where the miracle really starts. What your eardrum experiences is a single complicated pressure wave corresponding to the red pattern, formed from the superposition of the two individual waves. Your ear drum has no way of ‘knowing’ that two vibrating objects have made two separate sounds – it just responds to the vibration of the air particles near it. But you don’t hear a ‘mushy single sound’ combining the two – you can still hear the two sounds distinctly! Just to reiterate, when your ear drum receives the red pressure variation, your brain can hear the two blue tones!
I find that amazing. I’ll give a partial description below, but then I’ll show why it’s more miraculous even than that…
The frequency domain
The pure tones A and B in the graphs above are called ‘sine waves’ or ‘sinusoidal variations’. We call them pure tones because they consist each of a single frequency.
The graphs we have seen so far have been ‘in the time domain’ – that is the pressure variations are plotted as a function of time. We can also plot the same information in the ‘frequency-domain’, by plotting the amplitude against frequency. Such a plot is called the ‘frequency spectrum’ of the sound – and because we are looking at pure tones, the frequency spectra are very simple. Here they are.
Sensing and signal processing
Even if we don’t know yet how the ear/brain system accomplishes the separation of the single pressure wave into its two components, the waves seem ‘more reasonably separable’ when you look at them in the frequency domain than the complicated-looking pattern in the time domain. Perhaps your body is ‘operating in the frequency domain’ – to do this it would need a means of detecting the different frequencies present, and recognising them as such. In other words, the ear needs a physical means of being affected differently by different frequencies; its different responses to different frequencies could then be communicated to the brain for ‘signal processing’ and recognising for what they are.
Inside the cochlea
And indeed, that seems to be what happens. If we were to ‘unwind the snail’ of the cochlea and look inside it, we would find that the thin sheet called the ‘basilar membrane’ varies in width along its length. Different widths of membrane are stimulated by different frequencies of vibration. Each part of the membrane is connected to the brain via a nerve. The brain receives electrical signals along this nerve, can tell which part of the basilar membrane sent the signal, and can therefore decode the frequency information. That’s how you hear sounds as high or low.
(By the way, there’s a whole branch of maths that does a similar job – relating a complicated pattern to its frequency components. It’s called Fourier analysis. If you have never learned to do this maths, you needn’t bother – your brain does it automatically anyway when you hear stuff! J)
So I still think that decoding a single pressure wave into frequency components is amazing, but here, at least, is a partial mechanism for how it happens. So far, so common, on websites about hearing. But the miracle doesn’t stop there.
‘Real’ sounds – it really is a miracle!
To go further we need a small digression. Our discussion has focused on ‘pure tones’. But the vast majority of everyday sounds are nothing like pure tones – they aren’t nice, clean spikes in the frequency-domain. ‘Impure tones’ (i.e. basically everything you hear) are not perfectly sinusoidal. Non-sinusoidal waveforms which repeat at a given frequency contain energy at other frequencies as well. In fact, ANY shape of repeating waveform (of frequency f) can be generated by adding a suitable set of sine waves. These are:
- The fundamental (or first harmonic), which has the same frequency, f, as the funny-shaped waveform
- Harmonics, whose frequencies are multiples of the fundamental frequency (for example, the fourth harmonic has a frequency of 4f)
The relative amounts of the different harmonics then determine the quality, or ‘timbre’ of the sound. An ‘A’ on a piano and an ‘A’ on a guitar both have an underlying fundamental vibration at 440 Hz (that’s what makes it an ‘A’), but with different amounts of harmonics – that’s why a piano sounds different from a guitar.
Here’s a fictional time-domain and frequency-domain representation of an ‘A’ on two different musical instruments – notice the different relative amplitudes of the harmonics, and the effect this has on the time-domain representation.
Now the point is, that if both of these instruments play an A at the same time, your ear drum will receive the superposition of the two. That superposition is reproduced below in red.
With the pure tones, the spikes on the frequency spectrum were well-separated. We said “OK, so the basilar membrane reacts differently to two different frequencies, and the brain sorts that out”. But these two sounds occupy the same frequencies! How does the ear/brain separate out the frequency components? When we hear a guitar and piano play an ‘A’ at the same time, how do we distinguish the two sounds?
I have no idea. That is why I think hearing is a miracle…
Postscript
Feel free to stop reading. But if you are still with me, consider these additional points…
The spectra I have drawn are idealised; realistic spectra are not perfectly sharp spikes, and have baseline noise between the harmonics, like this.
Also, that’s just for a single note. Real life involves rapidly changing sounds. That’s not going to make the brain’s task any easier. Whatever method it uses to separate the sounds, it has to do the processing fast enough that we can keep up with our changing soundscape.
And if that weren’t enough…
The ear/brain combination is capable of making sense of sounds over an enormous range of volumes. Such an enormous range, that the decibel scale of volumes has to be arranged in a nifty way to make it manageable. That ‘way’ is called a logarithmic scale, and that’s what our next post will be on…