The cent is a logarithmic unit of measure used for musical intervals. Definition and base Logarithmic scales are either defined for ratios of the underlying quantity or one has to agree to measure In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical 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. In Music, there are two common meanings for tuning: Tuning practice, the act of tuning an instrument or voice
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, and introduced it in his edition of Hermann von Helmholtz's On the Sensations of Tone. Alexander John Ellis ( 14 June, 1814 - 28 October, 1890) was an English mathematician and philologist. Gaspard Clair François Marie Riche de Prony ( July 22, 1755 - July 29, 1839) was a French Mathematician and Engineer Robert Holford Macdowall Bosanquet ( 31 July 1841 &ndash 7 August 1912) was an English scientist and music theorist and brother of Admiral Sir It has become the standard method of representing and comparing musical pitches and intervals with relative accuracy.
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. In Music, an octave ( is the the use of which is "common in most musical systems Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the This means that a cent is precisely equal to 21/1200, the 1200th root of 2, which is approximately 1. 0005777895065548592967925757932, or of one percent.
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. The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to For example, in just intonation the major third is represented by the frequency ratio 5:4. In music just intonation is any Musical tuning in which the frequencies of Notes are related by Ratios of Whole numbers Any interval 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. In Psychophysics, a just noticeable difference, customarily abbreviated with lowercase letters as jnd, is the smallest difference in a specified modality of sensory
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 5-6 cents.  The threshold of what is perceptible also varies as a function of the timbre of the pitch: in one study, changes in tone quality negatively impacted student musicians' ability to recognize as out-of-tune pitches that deviated from their appropriate values by +/- 12 cents. In Music, timbre (ˈtæm-bər' like timber, or, from Fr timbre tɛ̃bʁ is the quality of a Musical note or sound that distinguishes different  It has also been established that increased tonal context enables listeners to judge pitch more accurately. 
When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch. Vibrato is a musical effect produced in singing and on musical instruments by a regular pulsating change of pitch, and is used to add expression and vocal-like qualities to  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. 
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. Amusia refers to a number of disorders which are indicated by the inability to recognize musical tones or Rhythms or to reproduce them 
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.
One Cent Interval
Six Cents Interval
Ten Cents Interval
The notes may not have had a perceivable difference, but when played together, they are comparably out of phase. The sum of the two waveforms either adds to or diminishes from the loudness of the sound wave; there are instances when the waves are of equal magnitude in opposite directions, canceling each other out and producing moments of "silence". A given note on the piano is tuned not only to pitch, but also so that the strings are in phase with one another. A piano tuner may verify this by playing that note against a common interval (4th, 5th, octave).