Cent (music)

The cent is a logarithmic measure of a musical interval introduced by Alexander Ellis. A cent is the logarithmic division of the equitempered semitone into 100 equal parts.

Formula

In terms of a formula, the separation or interval between two frequencies ƒ₁ and ƒ₂ in cents is determined as:

$c=1200\log _{2}\left({\frac {f_{1}}{f_{2}}}\right)\ .$

Consequently, two frequencies ƒ₁ and ƒ₂ separated by an interval of 1 cent are in the ratio:

${\frac {f_{1}}{f_{2}}}=2^{1/1200}\approx 1.005777895\ ,$

that is, by a ratio given by the 1200th root of 2.

Background

The cent appears in an article Alexander Ellis published in 1885 and also in an appendix he added to his translation of Herman von Helmholtz's Die Lehre von den Tonempfindungen published in translation as On the Sensation of Tone As a Physiological Basis for the Theory of Music, and also as On the sensations of tone.

(PD) Image: John R Brews
Average error of nine flutists in playing a note of specified pitch after listening to a pitch A4 = 442 Hz from a piano.

Sensitivity of the ear

According to Ellis, when two notes are played together, a difference of 2 cents is noticeable, and a difference of 5 cents is heard as out of tune.

Recent observations suggest errors of 5-15 cents when playing a specific pitch are common, with errors of 20-50 cents for pitches above A7 (the 7th octave, 3 octaves above the octave containing middle C). (See figure at right.) The increased error at higher pitch was traced to a systematic error in the response of auditory nerves in the ear.