# 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 ƒ_{1} and ƒ_{2} in *cents* is determined as:

Consequently, two frequencies ƒ_{1} and ƒ_{2} separated by an interval of 1 cent are in the ratio:

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

The *cent* appears in an article Alexander Ellis published in 1885^{[1]} 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*,^{[2]} also published as *Die Lehre von den Tonempfindungen*, translated as *On the sensations of tone*.^{[3]}

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.^{[1]} Recent observations suggest errors of 5-15 cents in pitch estimates are common, with errors of 20-50 cents for pitches above *A*7 (the 7th octave, 3 octaves above the octave containing middle *C*). The increased error at higher pitch was traced to a systematic error in the response of auditory nerves in the ear.^{[4]}

## References

- ↑
^{1.0}^{1.1}Alexander J Ellis (March 25, 1885). "On the musical scales of various nations; §III.–Cents".*Journal of the Society of Arts***33**: p. 487. - ↑
Herman von Helmholtz (1912). “Footnote, p. 41 and Appendix XX, Section C”,
*On the Sensation of Tone As a Physiological Basis for the Theory of Music*, Alexander Ellis translation of 4th German ed. Longmans, Green. - ↑
Herman von Helmholtz (1954).
*On the sensations of tone*, Reprint of 1885 translation by Alexander Ellis. Courier Dover Publications. ISBN 0486607534. - ↑
Ohgushi, K and Ano, Y (2005). "The Relationship between Musical Pitch and Temporal Responses of the Auditory Nerve Fibers".
*Journal of Physiological Anthropology and Applied Human Science***24**(1): pp. 99-101.