The last topic that we will consider is difference tones. In Section 7.3 we discussed the perception of two pure tones played together. How the brain perceives the pair of notes is controlled by the difference frequency: |f₁ - f₂|. If the difference is greater than a Critical Bandwidth, the pair is perceived as consonant. However, there is still a beat frequency given by |f₁ - f₂|. If this beat frequency is high enough, it can be heard as a pitch of its own. For example, if you start at a pitch f₁ and play another note a Perfect Fifth higher at f₂ = (3/2)f₁, the difference frequency between these two notes will be at (1/2)f₁ or a pitch one octave lower than f₁. This is actually useful for tuning violin strings: the strings on a violin, viola or cello are all separated by a Perfect Fifth. If you play two adjacent strings at the same time and they are tuned perfectly, you can hear a difference tone at a pitch one octave below the lower string.

There is not too much more to say about difference frequencies besides the fact that they are real and you can hear them, although they can be rather hard to hear. However, while playing around with difference tones, I heard a curious effect. When two notes are tuned exactly a Perfect Fifth apart, I heard a difference tone an octave lower, as discussed above. But, when I detuned the upper note slightly, I heard the difference tone start beating! The question is why did I hear beats in the difference tone – what two frequencies were beating together?

It is often interesting to think about what happens when a note is tuned away from the ideal case – this happens more often than we would like in playing music! For example, if you have two notes a Perfect Fifth apart, they will have some common harmonics that are exactly equal. In terms of perception, this will not lead to any dissonance. However, if you slightly detune one of the notes, the harmonics will not quite be perfectly equal and this can quickly lead to a strong sense of dissonance!

To analyze the effect of detuning a note, we will say that the frequency is changed by a small amount, called d . For example, let’s consider the Perfect Fifth with complex tones:

From this table, we can see that the common harmonic at 3f will beat with a frequency of |3f – (3f+2d )| = 2d . In other words, if the upper note is 4 Hz out of tune, you will hear a beat frequency of 8 Hz in the harmonics. But, what about the difference tones? Between the two fundamentals, 1f and 1.5f + d , there will be a difference frequency of 0.5f + d . However, between 2f and 1.5f + d , there will be a difference frequency of 0.5f - d . So, there are two ways to get a difference frequency an octave lower: one is at 0.5f + d and the other at 0.5f - d . What frequency will these two notes beat at? We have

|(0.5f + d ) - (0.5f - d )| = 2d .

So, in fact, if I play two notes separated by a Perfect Fifth, I will hear a difference tone. If I play one note out of tune by say 5 Hz, the difference tone will beat with a frequency of 10 Hz.

This example is here to demonstrate some of the consequences of difference tones and of playing notes slightly out of tune. However, it is also meant to show that how we perceive sound is quite complicated and subtle. In fact, Chapter 7 has only given the most basic elements needed to begin to understand the perception of sound. Indeed, much more is going on inside the ear and brain and this continues to be an area of active research and controversy. Nevertheless, all of these effects are rooted in fundamental physics and waves and ultimately how we perceive sound cannot be separated from the physics of waves.