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. 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.

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

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. 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. However, more recent estimates suggest errors of 5-15 cents in pitch estimates are common, with errors of 20-50 cents above A7 (the 7th octave, 3 octaves above the octave containing middle C). This error was traced to the response of auditory nerves in the ear, which exhibit a systematic error.