# Cent (music)

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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:

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

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

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

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

## Background

The cent appears in an article Alexander Ellis published in 1885[1] 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,[2] and also as On the sensations of tone.[3]

(PD) Image: John R Brews
Audible difference in frequency Δƒ/ƒ at two sound levels heard in rapid succession vs. frequency ƒ.[4]
(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.[5]

## 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.[1]

The figure at right indicates a smaller sound level difference is audible when the sounds are louder, and smaller differences also are audible at higher frequencies.[4] The ability to distinguish pitches is extremely variable among listeners, increases with intensity, increases with the abruptness in change of tone, improves with the richness of timbre,[4] and varies with the shape of the envelope of the waveform that turns the tones on and off.[6]

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.[5]

## References

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.
2. 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 1877 German ed. Longmans, Green.
3. Herman von Helmholtz (1954). On the sensations of tone, Reprint of 1885 translation by Alexander Ellis. Courier Dover Publications. ISBN 0486607534.
4. Carl Emil Seashore (1967). “Figure 1”, Psychology of Music, Reprint of McGraw-Hill 1938 ed. Courier Dover Publications, p. 60. ISBN 0-486-21851-1.
5. 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.
6. William M. Hartmann (1997). Signals, Sound, and Sensation. Springer, p. 443. ISBN 1563962837.