Cent (music)

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The cent is a logarithmic unit of measure used for musical intervals. Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive notes.

1200 cents are equal to one octave — a frequency ratio of 2:1 — and an equally tempered semitone (the interval between two adjacent piano keys) is equal to 100 cents. This means that a cent is precisely equal to 2^1/1200, the 1200th root of 2, which is approximately 1.0005777895.

If you know the frequencies a and b of two notes, the number of cents measuring the interval between them may be calculated by the following formula:

[n = 1200 \log_2 \left( \frac \right) \approx 3986 \log_ \left( \frac \right)]

Likewise, if you know a note b and the number n of cents in the interval, then the other note a may be calculated by:

[a = b \times 2 ^ \frac]

To compare different tuning systems, convert the various interval sizes into cents. For example, in just intonation the major third is represented by the frequency ratio 5:4. Applying the formula at the top shows this to be about 386 cents. The equivalent interval on the equal-tempered piano would be 400 cents. The difference, 14 cents, is about a seventh of a half step, easily audible. The just noticeable difference for this unit is about 6 cents.

A. J. Ellis based the measure on the acoustic logarithms semitone system developed by de Prony, on Bosanquet's suggestion, and introduced it in his edition of Hermann von Helmholtz's On the Sensations of Tone. It has since become the standard way of measuring intervals in equal temperament systems or for comparison with equal temperament systems.

References

Ellis, A. (1884) "Tonometrical Observations on some existing Non harmonic Musical Scales". Proceedings of the Royal Society, v.37, nr.234. p.368-385

See also

• Interval (music)
• Musical tuning

External links

• [Cent conversion: Frequency ratio to cents and cents to frequency ratio]
• [Tonalsoft Encyclopaedia of Tuning]
• [How to convert a ratio to cents]

 

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