How the ear hears frequency

Today I'm going to give you an insight into how the human ear hears frequency, and tell you a secret about the magic frequency of 632 Hz (actually 632.45553).

In my recent post about linear phase and minimum phase filters, I used a frequency sweep from 100 Hz to 1600 Hz and talked about the centre frequency 400 Hz. So in what sense is 400 Hz the centre frequency between 100 and 1600? It certainly isn't the average.

Let's start by listening to the sweep...

That was exciting wasn't it? It's the kind of thing that pleases me. I invite your comments.

So let's listen to 100 Hz. You'll need to be listening on proper speakers or headphones. Laptop speakers or eBay earbuds probably won't do much for you.

And now 1600 Hz.

So how can we find the centre frequency between 100 and 1600? Let's take an average.

100 + 1600 = 1700

Divide by 2 gives 850 Hz. Here it is...

Let me play 100, 850, 1600 in sequence so you can judge whether its bang in the middle.

Hmm, I don't really hear it. To me, 850 seems a lot closer to 1600 than it does to 100 subjectively, but it's the same 750 Hz away from both. So this tells us something about the way the human ear works. We hear frequency logarithmically rather than arithmetically. You can learn more about logarithmic scales here...

So how do we find the centre frequency logarithmically? Well I'm sure mathematical geniuses could suggest plenty of ways, but I'm going to use what's called the geometric mean. To get this I don't add 100 and 1600, I multiply them.

So 100 x 1600 = 160,000

Then I don't divide by two, I take the square root.

The square root of 160,000 = 400

So 400 Hz is the centre frequency using this method. Let's listen to 100, 400, 1600 in sequence.

I'm convinced. It sounds halfway to me. If it doesn't to you, let me know in the comments what you think. As I said, it's subjective.

You might, by the way, have noticed that the jumps are two octaves. That's just a coincidence and you can try out this test for yourself with different pairs of frequencies.

So this brings me to the magic frequency of 632 Hz, actually 632.45553. What does it mean? Well, it's the centre frequency of human hearing. Take a moment to absorb that. So how do I work this out? Simple, it's that geometric mean again.

The frequency range of human hearing is normally stated as 20 Hz to 20 kHz. So if I multiply these...

20 x 20,000 = 400,000

Take the square root - 632.45553

Now, I don't expect you to believe me without a demonstration. I can't do it the same way as before since it's unlikely your speakers or headphones go as low as 20 Hz, so you won't be able to hear it. Likewise, although when you're young you can probably hear 20 kHz, with age that limit decreases. So again, probably you can't hear it. So what I'm going to do instead is to sweep the tone upwards and downwards from the centre of 632 Hz. Rather than try to explain, let's just listen.

Now, bearing in mind that your speakers or headphones are probably a limiting factor in the low frequencies, does 632 sound central to you? Let's try it another way...

And maybe try it in stereo...

Well, it's subjective, and you could ask whether it matters. I think it matters because the more you understand about audio, and in particular how the human ear reacts to sound, the better the engineer and producer you're going to be.

I'm David Mellor, Course Director of Audio Masterclass. Thank you for reading.

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