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| [[Image:Cent versus semitone monochord ET.png|thumb|350px|One cent compared to a semitone on a truncated [[monochord]].]]
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| [[Image:4Octaves.and.Frequencies.svg|thumb|right|350px|Octaves increase exponentially when measured on a linear frequency scale (Hz).]]
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| [[Image:4Octaves.and.Frequencies.Ears.svg|right|thumb|350px|Octaves are equally spaced when measured on a logarithmic scale (cents).]]
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| The '''cent''' is a [[logarithmic scale|logarithmic]] unit of measure used for musical [[interval (music)|intervals]]. Twelve-tone [[equal temperament]] divides the [[octave]] into 12 [[semitones]] of 100 cents each. Typically, cents are used to measure extremely small finite intervals, or to compare the sizes of comparable intervals in different [[tuning system]]s, and in fact the interval of one cent is much too small to be heard between successive notes.
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| [[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,<ref>[[Alexander John Ellis|Alexander Ellis]]: [http://stuart.sfa.googlepages.com/MSVN00.html On the Musical Scales of Various Nations] HTML transcription of the 1885 article in the Journal of the Society of Arts (Accessed September 2008){{deadlink|date=February 2013}}</ref> 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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| ==Use==
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| [[Image:Music intervals frequency ratio equal tempered pythagorean comparison.svg|thumb|550px|right|Comparison of [[equal-tempered]] (red) and [[Pythagorean tuning|Pythagorean]] (blue) intervals showing the relationship between frequency ratio and the intervals' values, in cents. The curve shown is the plot of the equation, at left.]]
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| Like a [[decibel]]'s relation to intensity, a cent is a ratio between two close frequencies. For the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An [[equal temperament|equally tempered]] [[semitone]] (the interval between two adjacent piano keys) decidedly spans 100 cents. An [[octave]] -- two notes that have a frequency ratio of 2:1 -- spans twelve semitones and therefore 1200 cents. Since a frequency raised by one cent is simply multiplied by this constant cent value, and 1200 cents doubles a frequency, the ratio of frequencies one cent apart is precisely equal to 2<sup>1/1200</sup>, the [[twelfth root of two|1200th root of 2]], which is approximately 1.0005777895.
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| If one knows the frequencies ''a'' and ''b'' of two notes, the number of cents measuring the interval from ''a'' to ''b'' may be calculated by the following formula (similar to the definition of decibel):
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| :<math>n = 1200 \cdot \log_2 \left( \frac{b}{a} \right) \approx 3986 \cdot \log_{10} \left( \frac{b}{a} \right)</math>
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| Likewise, if one knows a note ''a'' and the number ''n'' of cents in the interval from ''a'' to ''b'', then ''b'' may be calculated by:
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| :<math>b = a \times 2 ^ {n/1200}</math>
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| 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.
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| ==Piecewise linear approximation== | |
| As ''x'' increases from 0 to 1/12, the function 2<sup>''x''</sup> increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a [[piecewise linear function]] which is numerically correct at semitones. That is, ''n'' cents for ''n'' from 0 to 100 may be approximated as 1+0.0005946''n'' instead of 2<sup>''n''/1200</sup>. The rounded error is zero when ''n'' is 0 or 100, and is about 0.72 cents high when ''n'' is 50, where the correct value of 2<sup>1/24</sup> = 1.02930 is approximated by 1 + 0.0005946*50 = 1.02973. This error is well below anything humanly audible, making this piecewise linear approximation suitable for most practical purposes.
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| ==Human perception==
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| 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.<ref>D.B. Loeffler, "[http://etd.gatech.edu/theses/available/etd-04102006-142310/ Instrument Timbres and Pitch Estimation in Polyphonic Music]". Master's Thesis, Department of Electrical and Computer Engineering, Georgia Tech. April (2006)</ref> The threshold of what is perceptible, technically known as the [[just noticeable difference]] (JND), also varies as a function of the frequency, the amplitude and the [[timbre]]. In one study, changes in tone quality reduced student musicians' ability to recognize as out-of-tune pitches that deviated from their appropriate values by ±12 cents.<ref>J. M. Geringer; M.D. Worthy, "[http://links.jstor.org/sici?sici=0022-4294%28199922%2947%3A2%3C135%3AEOTCOI%3E2.0.CO%3B2-9 Effects of Tone-Quality Changes on Intonation and Tone-Quality Ratings of High School and College Instrumentalists]", Journal of Research in Music Education, Vol. 47, No. 2. (Summer, 1999), pp. 135-149.</ref> It has also been established that increased tonal context enables listeners to judge pitch more accurately.<ref>{{Citation |url=http://www.zlab.mcgill.ca/docs/Warrier_and_Zatorre_2002.pdf |format=PDF|author1=C.M. Warrier |author2=R.J. Zatorre |title=Influence of tonal context and timbral variation on perception of pitch |journal=Perception & Psychophysics |volume=64 |issue=2 |date=February 2002 |pages=198–207 |accessdate=2008-09-27 |doi=10.3758/BF03195786}}</ref> Free, online web sites for self testing are available.<ref>"[http://tonometric.com/adaptivepitch/ Adaptive pitch test]", ''Tonometric.com''</ref>
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| When listening to pitches with [[vibrato]], there is evidence that humans perceive the mean frequency as the center of the pitch.<ref>{{Citation |url=http://www.wellesley.edu/Physics/brown/pubs/vibPerF100P1728-P1735.pdf |format=PDF|author1=J.C. Brown |author2=K.V. Vaughn |title=Pitch Center of Stringed Instrument Vibrato Tones |journal=Journal of the Acoustical Society of America |volume=100 |issue=3 |date=September 1996 |pages=1728–1735 |accessdate=2008-09-28 |doi=10.1121/1.416070 |pmid=8817899|bibcode = 1996ASAJ..100.1728B }}</ref> One study of modern performances of [[Franz Schubert|Schubert's]] [[Ellens dritter Gesang|Ave Maria]] found that vibrato span typically ranged between ±34 cents and ±123 cents with a mean of ±71 cents and noted higher variation in [[Verdi]]'s opera arias.<ref>{{Citation |url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000102000001000616000001&idtype=cvips&gifs=yes |author=E. Prame |title=Vibrato extent and intonation in professional Western lyric singing |journal=The Journal of the Acoustical Society of America |volume=102 |issue=1 |date=July 1997 |pages=616–621 |doi=10.1121/1.419735|bibcode = 1997ASAJ..102..616P }}</ref>
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| 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.<ref>{{Citation |url=http://www.brams.umontreal.ca/plab/downloads/PeretzHyde03.pdf |format=PDF|author1=I. Peretz |author2=K.L. Hyde |title=What is specific to music processing? Insights from congenital amusia |journal=Trends in Cognitive Sciences |volume=7 |issue=8 |date=August 2003 |pages=362–367 |accessdate=2008-09-27 |doi=10.1016/S1364-6613(03)00150-5 |pmid=12907232}}</ref>
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| ==Sound files==
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| The following audio files play various intervals. In each case the first note played is middle C. The next note is sharper than C by the assigned value in cents. Finally, the two notes are played simultaneously.
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| Note that the JND for pitch difference is 5-6 cents. Played separately, the notes may not show an audible difference, but when they are played together, [[beat (acoustics)|beating]] may be heard (for example if middle C and a note 10 cents higher are played). At any particular instant, the two waveforms reinforce or cancel each other more or less, depending on their instantaneous [[Phase (waves)|phase]] relationship. A piano tuner may verify tuning accuracy by timing the beats when two strings are sounded at once.
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| {{audio|Cent on C.mid|Play middle C & 1 cent above}}, beat frequency = .16 [[Hertz|Hz]]<br/>
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| {{audio|Semicomma on C.mid|Play middle C & 10.06 cents above}}, beat frequency = 1.53 Hz<br/>
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| {{audio|Eighth-tone on C.mid|Play middle C & 25 cents above}}, beat frequency = 3.81 Hz
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| {{Listen|filename=One_Cent_Interval.ogg|title=One Cent Interval|description=The file plays [[middle C]], a tone 1 cent sharper, and both tones together.|pos=left}}
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| {{Listen|filename=Six_Cents_Interval.ogg|title=Six Cents Interval|description=The file plays [[middle C]], a tone 6 cents sharper, and both tones together.|style=clear:none|pos=left}}
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| {{Listen|filename=Ten_Cents_Interval.ogg|title=Ten Cents Interval|description=The file plays [[middle C]], a tone 10 cents sharper, and both tones together.|style=clear:none|pos=left}}
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| {{-}}
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| ==See also==
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| {{Portal|Music}}
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| *[[Millioctave]]
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| *[[Savart]]
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| ==References==
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| ===Footnotes===
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| {{Reflist}}
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| ===Notations===
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| * {{Citation | last = Ellis | first = Alexander J. | authorlink = Alexander John Ellis | coauthors = Alfred J. Hipkins | year = 1884 | title = Tonometrical Observations on Some Existing Non-Harmonic Musical Scales | journal = Proceedings of the Royal Society of London | volume = 37 | pages = 368–385 | doi = 10.1098/rspl.1884.0041 | postscript = . | jstor=114325 | issue = 232–234}}
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| ==External links==
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| *[http://in.music.sc.edu/fs/bain/atmi02/cents/default.html Cent conversion: Whole number ratio to cent]
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| *[http://www.sengpielaudio.com/calculator-centsratio.htm Cent conversion: Online utility with several functions]
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| {{Musical tuning|state=auto}}
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| {{Intervals|state=auto}}
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| {{DEFAULTSORT:Cent (Music)}}
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| [[Category:Equal temperaments]]
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| [[Category:Intervals]]
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| [[Category:Units of measurement]]
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