Why do Tempered Intervals Beat?

Why do tempered intervals such as major thirds beat even though the difference between the frequencies of the two notes is too large for beats to occur? For example, consider the equally tempered major third, C—E. The frequency of the C below middle C is 130.81 Hz, and the frequency of the E above that C is 164.81. The difference between these two frequencies is 164.81 – 130.81 = 34 Hz—much too large for beats to be heard.

Consider the example of a perfect major third, C—E. The ratio of the frequency of E to that of C is exactly 5:4 = 1.25 (note that the ratio of the frequencies of the equally tempered third above is 164.81 ÷ 130.81 = 1.2599, not 1.25). Let fC be the frequency of C. Then the first five harmonics of C will be:

fC, 2fC, 3fC, 4fC, 5fC.

The frequency fE of E is 5fC/4, so the first five harmonics of E will be:

5fC/4, 10fC/4, 15fC/4, 20fC/4.

Simplifying, we get:

5fC/4, 5fC/2, 15fC/4, 5fC.

Note that the frequency of the fifth harmonic of C equals that of the fourth harmonic of E, and both of these harmonics will be E’s two octaves above the root E.

Now consider the first five harmonics of the equally tempered third.

The harmonics of the C below middle C are:

130.81, 261.62, 392.43, 533.24, 654.05.

Hear the above five harmonics added in succession.

The first four harmonics of E are:

164.81, 329.62, 494.43, 659.24.

Note that the frequencies of all the above harmonics are too far apart to beat except for the fourth harmonic of E and the fifth harmonic of C (both are written in boldface). The difference between these frequencies is about 5.2 beats per second.

If you click this link, a new window will open. You will hear the C below middle C and the E above it in a perfect (just) 5:4 relationship.

If you click this link, a new window will open. You will hear, in equal temperament, the C below middle C (130.81 Hz), the E above (164.81 Hz), the fifth harmonic the of the C (654.05 Hz), and the fourth harmonic of the E (659.24 Hz). These latter harmonics are both an E two octaves above the root E, and they beat at 659,24 – 654.05 = 5.2 per second. The sound lasts 5 seconds, so there will be a total of 26 beats.