Any two pitches will have a certain distance, or difference in highness and lowness, between them. Musicians call this distance an interval. Of the many different intervals used in music, one called the octave has a special character that makes it particularly important.

If successive pitches are sounded one after another — say, running from low to high up the white keys on a piano — there comes a point at which a pitch seems in some sense to “duplicate” an earlier pitch, but at a higher level. This new pitch does not sound identical to the old one, but somehow the two sounds are very similar. They blend extremely well; they almost seem to melt into each other. This is the octave.

What causes the phenomenon of octaves? Recall from Chapter 2 that when strings vibrate to produce sound, they vibrate not only along their full length but also in halves and other fractions, creating overtones (page 12). A vibrating string that is exactly half as long as another will reinforce the longer string’s strongest overtone. This reinforcement causes the duplication effect of octaves, and pitches that are an octave apart have frequencies related in a 2:1 ratio.

As strings go, so go vocal cords: When men and women sing along together, they automatically sing in octaves, duplicating each other’s singing an octave or two apart. If you ask them, they will say they are singing “the same song” — not many will think of adding “at different octave levels.”

As a result of the phenomenon of octaves, the full continuous range of pitches that we can hear seems to fall into a series of “duplicating” segments. We divide these octave segments into smaller intervals, thereby creating scales.