Editing Combination tone (section)

==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.<ref>Beament, James (2001). ''How We Hear Music'', p.81-2. The Boydell Press. {{ISBN|0-85115-813-7}}.</ref> If <math>f</math> is the missing [[fundamental frequency]], then <math>2f</math> would be the frequency of the lower tone, and its harmonics would be <math>4f, 6f, 8f,</math> etc. Since a fifth corresponds to a frequency ratio of 2:3, the higher tone and its harmonics would then be <math>3f, 6f, 9f,</math> etc. When both tones are sounded, there are components with frequencies of <math>2f, 3f, 4f, 6f, 8f, 9f,</math> 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 [[linear combination|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, <math>f_2-f_1</math>, though this may be a consequence of the other phenomena. Although much less common, the following frequencies may also be heard:
:<math>2f_1 - f_2, 3f_1 - 2f_2, \ldots, f_1 - k(f_2 - f_1)</math>

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{{Citation needed|date=October 2007}}. 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,<ref>[[Max V. Mathews]] and [[John R. Pierce]] (1989). "The Bohlen–Pierce Scale", p.167. ''Current Directions in Computer Music Research'', Max V. Mathews and John R. Pierce, eds. MIT Press.</ref> as well as the [[833 cents scale]].