Difference tone

Otokun

You recently explained that a beat occurs when two sounds with slightly different frequencies overlap, but what happens if the difference in those frequencies widens?

father

The difference in frequency is the beat period, so in the example we talked about before, the difference between 205 Hz and 200 Hz was a fluctuation of 5 Hz. This meant the sound got louder and softer 5 times per second, but you can see that the period gets shorter as the difference in frequency between the two sounds increases.

Otokun

Yeah, I understand that, and I can understand a beat if it happens a few times a second, but when the frequency difference is several tens of Hz, it would fluctuate dozens of times a second, so I was wondering what it would feel like.

father

That's right. You mentioned this before, didn't you? It's about the fluctuating sound we talked about when we were discussing the crying dragon. Let's listen again to the sound we heard before, "the sound when the fluctuating period is changed from 1 Hz to 200 Hz."

Figure 1 Click

If you listen carefully, you'll notice that the pitch gradually gets lower starting from around 30 Hz. For example, if you compare a pitch of 4 Hz and 200 Hz, you'll hear that they sound like they're at different pitches.

Otokun

Indeed, I was so focused on the roughness of the sound at the time that I didn't notice, but now that I'm comparing the two, the pitch is different. What does that mean?

father

I think this might be a little difficult to understand, but let me explain. When you amplitude modulate a 1000 Hz carrier wave with 200 Hz, you get a graph of time (Figure 2) and frequency (Figure 3) as shown in the diagram. This is the frequency graph of the 200 Hz sound we just heard, and you can see that it has components of sound that is 200 Hz higher (1200 Hz) and lower (800 Hz) than 1000 Hz.

Otokun

But why did the 200 Hz sound lower than the 4 Hz sound when we compared them? They both started at the same 1000 Hz...

father

Human hearing is fundamentally designed to perceive periods. Pitch is nothing more than a period. However, as we approach the lower limit of the audible frequency range and the period lengthens, it no longer sounds like pitch, but rather like a pattern with a certain rhythm.

Otokun

"Beat" follows the same pattern, doesn't it? Humans can't hear a 4 Hz sound as a pitch, so it must be that the original waveform has a frequency of 1000 Hz, and the 4 Hz period is what we hear as a pattern. But that's unrelated to why a 200 Hz sound sounds low, right?

father

It was quite a while ago, but do you remember when we talked about "frequency masking"?

Otokun

I vaguely remember that when two sounds with similar frequencies are present, one sound masks the other.

father

That's right. The lower note masks the higher note, doesn't it?

Otokun

I see. I understand. In this graph, the 1200 Hz sound is masked by the 1000 Hz sound, but the 800 Hz sound is not masked, so does that mean that sounds lower than 1000 Hz were heard?

father

Correct! Masking occurs when the difference between each frequency is within the critical bandwidth (the bandwidth in which human hearing can distinguish between different pitches), and in this case, it can be considered to be just within that range, so the masking effect is at work.

Otokun

When Dad said that the fluctuating frequency starts at around 30 Hz and gradually gets lower, it means that we are hearing a peak sound that is lower than 1000 Hz by the fluctuating frequency, and the difference in that sound gradually increases downwards from 1000 Hz.

father

That's right. In fact, when the difference between the two sounds that Oto initially asked about widens, the resulting superposition of the two sounds takes on almost the same shape as a waveform with amplitude modulation. This diagram (Figure 4) shows the superposition of the waveforms of 1000 Hz and 1200 Hz sounds.

father

There's another interesting phenomenon when two sounds are superimposed. If you observe this waveform, how many times longer is the period of the same shape repeating compared to, for example, a 1000 Hz waveform?

Otokun

It's five times stronger! I see, the 200 Hz difference between 1000 Hz and 1200 Hz is the period of this waveform with the same shape.

father

That's right. And you can hear the 200 Hz difference. Earlier, I said that human hearing basically perceives periodicity, and that periodicity is the pitch of the sound. Beats are a pattern of time-varying frequency differences, but as the frequency difference widens, you can hear the sound at the frequency of that difference.

Otokun

I see. If you listen carefully, you can definitely hear a low sound.