When two pitches of different frequencies are sustained simultaneously, a listener perceives a new pitch whose frequency is the difference between the two. The resultant pitch is known as a difference tone, which, together with the original sounding pitches, yields additional orders of difference tones. When an interval corresponds precisely to its size in the harmonic series, its difference tones reinforce frequencies already sounding in the series, and the interval becomes more resonant. The more it departs from its theoretically “perfect” size, the more its difference tones deviate from the harmonic series, making the interval, somewhat literally, more “dissonant.” Like an image coming into focus through the lens of a camera, the sound of an interval becomes clearer and less distorted when its difference tones are brought into alignment with the harmonic series. Musicians can use difference tones to increase their awareness of pitch and interval sizes. A simple but effective exercise involves playing an ascending or descending major scale against a drone on the tonic. Each scale degree is adjusted to the drone by listening for the difference tones shown in the examples below. Not all difference tones are easy to hear: in theory, every notated pitch should arise, but some may be too faint to perceive. The difference tones change pitch as the interval between the scale degree and the drone changes size. When the difference tones match the notated pitches, the interval is in tune. However, listening for the broader harmonic structure is more effective than focusing on individual pitches. ![[example difference tones with drone above.png]] **Example 1.** Difference tones with drone above. ![[example difference tones with drone above.mp3]] ![[example difference tones with drone below.png]] **Example 2.** Difference tones with drone below. ![[example difference tones with drone below.mp3]] In each example, the top staff shows the exercise: a tonic drone and the major scale played against it, with scale degrees above. The numbers beneath the top staff show how many cents (¢) each scale degree needs to be adjusted from equal temperament. Difference tones are notated on the grand staff below, indicating what should be heard when the interval is in tune. Diamond noteheads mark the first two orders, the easiest to hear; filled noteheads mark additional orders, which are fainter. The Roman numerals show the harmony the difference tones imply. Contrary to the conventional wisdom of listening down for pitch, I find difference tones easier to hear with the drone above, as in **Example 1**. > [!note] NB > The recordings use pure sine waves, tuned to the just ratios shown in the examples. A sine wave has no overtones, so apart from the drone and the scale, any other pitch heard is produced by the ear itself: mostly difference tones, along with much fainter summation tones (at the sums of the frequencies). This makes difference tones easier to pick out than on acoustic instruments, whose overtones often coincide with them. It also means the recordings cannot demonstrate the reinforcement of overtones described above; that requires real instruments. Difference tones grow stronger with volume, so the recordings are best heard at a comfortable but not quiet level; at very high volumes, speakers and headphones can add distortion of their own.