Most people who know anything about music are familiar with the concept of

beat frequencies, which is very useful for tuning instruments. To synopsize, if two tones that are close together (440Hz and 441Hz, eg) are played simultaneously, audible beats at the frequency of the difference of the two tones (once per second here) will give away the fact that the tones are shifting

in and

out of phase. This happens because of

superposition.

Beat frequencies are one type of difference tone, but for the most part the latter term is used when the secondary tone produced takes on musical properties as a result of its mathematical relation to the primary tones being played. For example, suppose you sit down at a piano and play the 440Hz A note, and the 880Hz A note an octave above. The difference tone is 440Hz, **the same as the lower note.** This is one of the reasons why the octave is the most stable interval in music: playing the octave reinforces the lower note (I believe the other reason is covered in music theory). Similarly, playing the 440 A with the 660 E produces a difference tone of 220Hz, exactly one octave below the lower note of the pair. And playing the 440 Hz A with its major third at the 550 Hz C# produces a difference tone of 110Hz, two octaves below the A.

Of course, if you really want to go hog wild, you can compute secondary and tertiary difference tones based on the differences between the primary tones played and the implicit ones created by their intervals. In the case of A versus C#, the primary difference tone of 110Hz creates secondary tones of 330Hz (A's perfect fifth) and 440Hz (the A again) respectively. I'm no expert on psychoacoustics, so I don't know how deep the layers of underlying tones can go before they're completely inaudible, but just looking at the results theoretically we can see the natural stability that lies in the major third interval: every tone that might possibly be audible is one of: 110Hz (A), 220Hz (A), 330Hz (E), 440Hz (A) and 550Hz (C#).

Intervals can be categorized based on how stable their difference tones are. The
minor third, for example, produces a primary difference tone that is a major third lower than the root note,
so it is less stable than the perfect fifth and major third. The tritone produces tones that have nothing to do with either note played, creating immense dischord. I could try to list the order of the intervals by descending stability, but odds are that I'll mess up, so you'll have to wait until I get access to my chromaticism text.

The shortcut for figuring out the difference tones for various intervals is to memorize the pitch ratios that define each of them. Rather than finding out the exact frequencies for A and C# and subtracting them, just know in advance that the ratio between the pitches in any major third is 4:5 (440:550, in this case), so the difference will be 5-4 = 1 = 1/4 the frequency of the lower tone = the A two octaves lower. The ratio for a major sixth is 3:5, so the difference between A and F# is 5-3 = 2 = 2/3 of the frequency of A, and since 2:3 is the ratio for a fifth, so the difference tone will be the D below the A. Luckily, I'm thoughtful enough to provide you with a table of all the pitch ratios:

`8ve 5th 4th 3rd b3 6th b6 b7 7th 2nd b2 TT
---------------------------------------------------------------
1:2 2:3 3:4 4:5 5:6 3:5 16:25 9:16 8:15 8:9 15:16 1:sqrt(2)
`

The ratios for the octave, fifth and major third can be discovered by examining the earlier examples. The rest I computed on the spot based on their interrelations: the major sixth, for example, is just a fourth plus a major third, so we multiply 3:4 by 4:5... remember that frequencies work on a

logarithmic scale. The more challenging interval of a 2nd is found by squaring the ratio for the the fifth, then "dividing" by an octave (A + fifth + fifth - octave = B). The astute observer will notice that this system is very appromixatory in nature: the ratio for a 2nd squared, which should be equal to the ratio for a major third, turns out to be 64:81. I've never heard an adequate explanation for why music works in spite of this mathematical inconsistency, so all I can say is, allow yourself a small

fudge factor when doing this type of music math. (oh yeah, go read about the

Pythagorean comma if you're interested in that type of thing)