Pianos and Continued Fractions
Edward G. Dunne, Mark McConnell · Mathematics Magazine · 1999
Sound is produced when an object vibrates in air. Let's assume that the vibration is at v cycles per second. The object will also naturally vibrate at all positive integer multiples nv of v, with the intensity generally decreasing with n. These are the overtones of the basic vibration. They give the tone its distinctive color. For instance, we distinguish a violin from an oboe by unconsciously recognizing the different overtones they produce. (In addition, we distinguish instruments by their attack-how they sound at the start of each note.) Thus, given one tone, Nature, in her guise as physics, picks out tones to go with it. The interval from v to 2 v is an octave. The interval from 2 v to 3v is a perfect fifth. These two intervals, plus a few others, have become the foundation of Western music. Also, notice that intervals correspond to ratios of frequencies: the octave is a 2:1 ratio, and the perfect fifth a 3: 2 ratio. We will expand on this in what follows.