Cent (music)

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This editable Main Article is under development and subject to a disclaimer.

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## Contents

* 1 Formula
* 2 Background
* 3 Sensitivity of the ear
* 4 References

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[edit]

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

c=1200log2(f1f2) .

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

f1f2=21/1200≈1.005777895 ,

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

## Background[edit]

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[edit]

See also: Music perception

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[edit]

1. ↑ 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.
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. ↑ 4.0 4.1 4.2 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. ↑ 5.0 5.1 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.