Combination tone

There are two types of combination tones: sum tones whose frequencies are found by adding the frequencies of the real tones, and difference tones whose frequencies are the difference between the frequencies of the real tones. "Combination tones are heard when two pure tones (i.e., tones produced by simple harmonic sound waves having no overtones), differing in frequency by about 50 cycles per second [Hertz] or more, sound together at sufficient intensity."

Combination tones can also be produced electronically by combining two signals in a circuit that has nonlinear distortion, such as an amplifier subject to clipping or a ring modulator.

Explanation

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 phenomenon. If ${\displaystyle f}$ is the missing fundamental frequency, then ${\displaystyle 2f}$ would be the frequency of the lower tone, and its harmonics would be ${\displaystyle 4f,6f,8f,}$ etc. Since a fifth corresponds to a frequency ratio of 2:3, the higher tone and its harmonics would then be ${\displaystyle 3f,6f,9f,}$ etc. When both tones are sounded, there are components with frequencies of ${\displaystyle 2f,3f,4f,6f,8f,9f,}$ etc. The missing fundamental is heard because so many of these components refer to it.

For some time, during the 19th century it was explained by Lagrange, Chladni, and Young as a form of acoustic beats. This was incorrect.

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, ${\displaystyle f_2-f_1}$, though this may be a consequence of the other phenomena. Although much less common, the following frequencies may also be heard:

${\displaystyle 2f_1-f_2,3f_1-2f_2,\dots ,f_1-k(f_2-f_1)}$

For a time it was thought that the inner ear was solely responsible whenever a sum or difference tone was heard. However, experiments show evidence that even when using headphones providing a single pure tone to each ear separately, listeners may still hear a difference tone. Since the peculiar non-linear physics of the ear doesn't come into play in this case, it is thought that this must be a separate, neural phenomenon. Compare binaural beats.

Heinz Bohlen proposed what is now known as the Bohlen–Pierce scale on the basis of combination tones, as well as the 833 cents scale.