The Pitch Ratio of A Fifth
The justly intoned pitch ratio of a perfect fifth is 3:2 (also known, in early music theory, as a hemiola), meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. In the cent system of pitch measurement, the 3:2 ratio corresponds to approximately 702 cents, or 2% of a semitone wider than seven semitones. The just perfect fifth can be heard when a violin is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune. Just perfect fifths are employed in just intonation. The 3:2 just perfect fifth arises in the C major scale between C and G. Play
Kepler explored musical tuning in terms of integer ratios, and defined a "lower imperfect fifth" as a 40:27 pitch ratio, and a "greater imperfect fifth" as a 243:160 pitch ratio. His lower perfect fifth ratio of 1.4815 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.498 (relative to the ideal 1.50). Helmholtz uses the ratio 301:200 (708 cents) as an example of an imperfect fifth; he contrasts the ratio of a fifth in equal temperament (700 cents) with a "perfect fifth" (3:2), and discusses the audibility of the beats that result from such an "imperfect" tuning.
In keyboard instruments such as the piano, a slightly different version of the perfect fifth is normally used: in accordance with the principle of equal temperament, the perfect fifth is slightly narrowed to exactly 700 cents (seven semitones). (The narrowing is necessary to enable the instrument to play in all keys.) Many people can hear the slight deviation from the idealized perfect fifth when they play the interval on a piano.
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