Frequency was originally considered to be the province of musicians. The pitches or frequencies of the notes in a musical scale are defined by ratios—octaves, for example, where the frequency of the higher note is twice the frequency of the lower one.

The 12 notes between octaves in Western music are also defined by ratios—in principle the ratios of small integers, because notes that have this relationship are generally regarded as pleasing when sounded together, and because these relationships arise naturally in the resonant frequencies of vibrating strings and pipes.

In the “equally tempered” musical scale, the most common tuning system in Western music, the ratio of the frequencies between consecutive notes is 21/12 or about 1.0594. An advantage of this scale is that an octave can consist of any 12 consecutive notes, but a disadvantage is that the ratio of the frequencies of notes are not exactly small integers. Pianos are often tuned to this scale, so that a piano can be played equally well (or equally poorly, depending on your perspective) in any key, that is, with an octave that begins on any note.

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