# Combination tone

One way a difference tone can be heard is when two tones with fairly complete sets of harmonics make a just fifth. This can be explained as an example of the missing fundamental phenomena (Beament 2001). If $f$ is the missing fundamental frequency, then $2f$ would be the frequency of the lower tone, and its harmonics would be $4f,6f,8f,$ etc. Since a fifth corresponds to a frequency ratio of 2:3, the higher tone and its harmonics would then be $3f,6f,9f,$ etc. When both tones are sounded, there are components with frequencies of $2f,3f,4f,6f,8f,9f,$ etc. The missing fundamental is heard because so many of these components refer to it.
The specific phenomenon that Tartini discovered was physical. Sum and difference tones are thought to be caused sometimes by the non-linearity of the inner ear. This causes intermodulation distortion of the various frequencies which enter the ear. They are combined linearly, generating relatively faint components with frequencies equal to the sums and differences of whole multiples of the original frequencies. Any components which are heard are usually lower, with the most commonly heard frequency being just the difference tone, $f_{2}-f_{1}$ , though this may be a consequence of the other phenomena. Although much less common, the following frequencies may also be heard:
$2f_{1}-f_{2},3f_{1}-2f_{2},...,f_{1}-k(f_{2}-f_{1})$  