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
( Play (help·info)).
Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed^{[1]}, and further described and employed the system in his edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals with relative accuracy.
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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 the ratio of one 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 (similar to the definition of decibel both formally as well as in its purpose to linearize a physical unit which is exponential but perceived logarithmically by humans):
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:
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 equaltempered 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.
It is difficult to establish how many cents are perceptible to humans; this accuracy varies greatly from person to person. One author stated that humans can distinguish a difference in pitch of about 56 cents.^{[2]} The threshold of what is perceptible also varies as a function of the timbre of the pitch: in one study, changes in tone quality reduced student musicians' ability to recognize as outoftune pitches that deviated from their appropriate values by ±12 cents.^{[3]} It has also been established that increased tonal context enables listeners to judge pitch more accurately.^{[4]}
When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch.^{[5]} One study of vibrato in western vocal music found a variation in cents of vibrato typically ranged between ±34 cents and ±123 cents, with a mean variation of ±71 cents; the variation was much higher on Verdi opera arias.^{[6]}
Normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents and sometimes have trouble with these or larger intervals.^{[7]}
The following .ogg files play various cents intervals. In each case the first note played is middle C. The next note a C which is sharper by the assigned cents value. Finally the interval is played.
Played separately, the notes may not show an audible difference, but when they are played together, a beat may be heard. At any particular instant, the two waveforms reinforce or cancel each other more or less, depending on their instantaneous phase relationship. A piano tuner may verify tuning accuracy by timing the beats when two strings are sounded at once.
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