差音

Oto-kun

The other day, you told me that a beat occurs when two sounds with slightly different frequencies overlap, but what happens if the difference in frequency keeps widening?

father

The difference in frequency is the period of the beats, so in the example I mentioned earlier, with 205 Hz and 200 Hz, there was a fluctuation frequency of 5 Hz. This means that the sound alternated between loud and soft five times per second, but it's clear that as the difference in frequency between the two sounds increases, the period becomes shorter.

Oto-kun

Yeah, I understand that, but while I can grasp a growl happening a few times per second, I wonder what it would feel like when the frequency difference becomes several tens of Hz, resulting in fluctuations occurring several times within a second.

father

That's right. You mentioned this before, didn't you? It's about the sound that varies when we talked about the crying dragon. Shall we listen again to the sound that changes in frequency from 1 Hz to 200 Hz?

Figure 1: Click

If you listen closely, you can notice that the fluctuating frequency gradually becomes a lower sound starting from around 30 Hz. For example, if you compare the fluctuating frequencies of 4 Hz and 200 Hz, you can hear the difference in pitch.

Oto-kun

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

father

It may be a bit difficult, but let me explain. When a 1000 Hz carrier wave is amplitude modulated at 200 Hz, it results in graphs of time (Figure 2) and frequency (Figure 3) as shown. This is the frequency graph of the varying frequency sound at 200 Hz that we just heard, and you can see that there are components of a higher sound (1200 Hz) and a lower sound (800 Hz) compared to the 1000 Hz.

Oto-kun

But why did the sound at 200 Hz seem lower when compared to the fluctuating frequency of 4 Hz, even though the original 1000 Hz is the same...?

father

Human hearing is fundamentally designed to capture cycles. The pitch of a sound is nothing but a cycle. However, as we approach the lower limit of audible frequencies and the cycles become longer, what we hear is not a pitch but a pattern with a certain rhythm.

Oto-kun

"The 'hum' follows that pattern too. A sound at 4 Hz, as a pitch, is inaudible to humans, so the sound at 1000 Hz, with a cycle of 4 Hz, is perceived as a pattern, right? But that has nothing to do with the fact that a sound at 200 Hz is heard as low, does it?"

father

It's been a while, but do you remember when we talked about "frequency masking"?

Oto-kun

Somehow, when there are two sounds with close frequencies, one sound masks the other.

father

That's right. The lower sound masks the higher sound.

Oto-kun

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

father

Correct! If the difference in frequencies is within the critical bandwidth (the range within which human hearing can separate pitch), masking occurs. In this case, it can be considered that the frequencies are just within that range, so the masking effect is thought to be in play.

Oto-kun

Dad said that the frequency is gradually becoming lower from around 30 Hz, and that the peak sound we hear is lower than 1000 Hz by the amount of the frequency variation, which is why the difference becomes larger as it moves down from 1000 Hz.

father

Exactly. In fact, when the difference between the two sounds that Oto first asked about expands, the superposition of the two sounds takes on a shape almost identical to that of an amplitude-modulated waveform. This diagram (Figure 4) shows the overlapping waveforms of the sounds at 1000 Hz and 1200 Hz.

father

There is another interesting phenomenon in the superposition of two sounds. When observing this waveform, how many times longer is the period of the repeating shape compared to a 1000 Hz waveform?

Oto-kun

It's increased by 5 times! I see, the difference of 200 Hz between 1000 Hz and 1200 Hz corresponds to the period of this waveform with the same shape.

father

That's right. And you can hear the sound of a 200 Hz difference. Earlier, I mentioned that human hearing basically captures cycles, and that the cycle corresponds to the pitch of the sound. A beat is a pattern of time variation in frequency difference, but as the frequency difference expands, you start to hear the sound of that frequency difference.

Oto-kun

I see. When I listen carefully, I can indeed hear low sounds.