View source for Cent (music)

{{Short description|Musical interval unit}}
[[File:Cent versus semitone monochord ET.png|thumb|350px|One cent compared to a semitone on a truncated [[monochord]].]]
{{multiple image|caption_align=center|header_align=center
 | align     = right
 | direction = vertical
 | width     = 350
 | header    = 
 | image1    = 4Octaves.and.Frequencies.svg
 | alt1      = 
 | caption1  = Octaves increase exponentially when measured on a linear frequency scale (Hz).
 | image2    = 4Octaves.and.Frequencies.Ears.svg
 | alt2      = 
 | caption2  = Octaves are equally spaced when measured on a logarithmic scale (cents).
}}

The '''cent''' is a [[logarithmic scale|logarithmic]] unit of measure used for [[musical intervals]]. [[Twelve-tone equal temperament]] divides the [[octave]] into 12 [[semitone]]s of 100 cents each. Typically, cents are used to express small intervals, to [[Electronic tuner|check intonation]], or to compare the sizes of comparable intervals in different [[tuning system]]s. For humans, a single cent is too small to be perceived between successive notes.

Cents, as described by [[Alexander John Ellis]], follow a tradition of measuring intervals by [[logarithm]]s that began with [[Juan Caramuel y Lobkowitz]] in the 17th century.{{efn|name=Caramuel}} Ellis chose to base his measures on the hundredth part of a semitone, {{radic|2|1200}}, at [[Robert Holford Macdowell Bosanquet]]'s suggestion. Making extensive measurements of musical instruments from around the world, Ellis used cents to report and compare the scales employed,{{sfn|Ellis|1885|p=485-527}} and further described and utilized the system in his 1875 edition of [[Hermann von&nbsp;Helmholtz]]'s ''On the Sensations of Tone''. It has become the standard method of representing and comparing musical pitches and intervals.{{sfn|Benson|2007|p=166|ps=:The system most often employed in the modern literature.}}{{sfn|Renold|2004|p=138}}

==History==
[[Alexander John Ellis]]' paper ''On the Musical Scales of Various Nations'',{{sfn|Ellis|1885|p=485-527}} published by the ''Journal of the Society of Arts'' in 1885, officially introduced the cent system to be used in exploring, by comparing and contrasting, musical scales of various nations. The cent system had already been defined in his ''History of Musical Pitch'', where Ellis writes: "If we supposed that, between each pair of adjacent notes, forming an equal [[semitone]] [...], 99 other notes were interposed, making exactly equal intervals with each other, we should divide the octave into 1200 equal {{sic|hundr|ecths}} of an equal semitone, or ''cents'' as they may be briefly called."{{sfn|Ellis|1880|p= 295}}

Ellis defined the pitch of a musical note in his 1880 work ''History of Musical Pitch''{{sfn|Ellis|1880|p= 293-336}} to be "the number of double or complete vibrations, backwards and forwards, made in each second by a particle of air while the note is heard".{{sfn|Ellis|1880|p= 293-294}} He later defined musical pitch to be "the pitch, or V [for "double vibrations"] of any named musical note which determines the pitch of all the other notes in a particular system of tunings."{{sfn|Ellis|1880|p= 294}} He notes that these notes, when sounded in succession, form the [[Scale (music)|scale]] of the instrument, and an interval between any two notes is measured by "the ratio of the smaller pitch number to the larger, or by the fraction formed by dividing the larger by the smaller".{{sfn|Ellis|1885|p=487}} [[Absolute pitch|Absolute]] and [[relative pitch]]es were also defined based on these ratios.{{sfn|Ellis|1885|p=487}}

Ellis noted that "the object of the tuner is to make the interval [...] between any two notes answering to any two adjacent finger keys throughout the instrument precisely the same. The result is called [[equal temperament]] or tuning, and is the system at present used throughout Europe.{{sfn|Ellis|1885|p=491-}} He further gives calculations to approximate the measure of a ratio in cents, adding that "it is, as a general rule, unnecessary to go beyond the nearest whole number of cents."{{sfn|Ellis|1885|p=488}}

Ellis presents applications of the cent system in this paper on musical scales of various nations, which include: (I. Heptatonic scales) Ancient Greece and Modern Europe,{{sfn|Ellis|1885|p=491-492}} Persia, Arabia, Syria and Scottish Highlands,{{sfn|Ellis|1885|p=492-500}} India,{{sfn|Ellis|1885|p=500-505}} Singapore,{{sfn|Ellis|1885|p=505-506}} Burmah{{sfn|Ellis|1885|p=506}} and Siam,;{{sfn|Ellis|1885|p=506-507}} (II. Pentatonic scales) South Pacific, {{sfn|Ellis|1885|p=507}} Western Africa,{{sfn|Ellis|1885|p=507-508}} Java,{{sfn|Ellis|1885|p=508-514}} China{{sfn|Ellis|1885|p=514-520}} and Japan.{{sfn|Ellis|1885|p=520-525}} And he reaches the conclusion that "the Musical Scale is not one, not 'natural,' nor even founded necessarily on the laws of the constitution of musical sound, so beautifully worked out by Helmholtz, but very diverse, very artificial, and very capricious".{{sfn|Ellis|1885|p=526}}

==Use==
[[File:Music intervals frequency ratio equal tempered pythagorean comparison.svg|thumb|550px|right|Comparison of [[equal-tempered]] (black) and [[Pythagorean tuning|Pythagorean]] (green) intervals showing the relationship between frequency ratio and the intervals' values, in cents.]]

A cent is a unit of measure for the ratio between two frequencies. An [[equally tempered]] semitone (the interval between two adjacent piano keys) spans 100 cents by definition. An [[octave]]—two notes that have a frequency ratio of 2:1—spans twelve semitones and therefore 1200 cents. The ratio of frequencies one cent apart is precisely equal to {{nowrap|2{{sup|{{frac|1200}}}} {{=}} {{radic|2|1200}}}}, the 1200th root of 2, which is approximately {{val|1.0005777895}}. Thus, raising a frequency by one cent corresponds to multiplying the original frequency by this constant value. Raising a frequency by 1200 cents doubles the frequency, resulting in its octave.

If one knows the frequencies <math>f_1</math> and <math>f_2</math> of two notes, the number of cents <math>c</math> measuring the interval from <math>f_1</math> to <math>f_2</math> is:

:<math>c = 1200 \cdot \log_2 \left( \frac{f_2}{f_1} \right)</math>

Likewise, if one knows <math>f_1</math> and the number of cents <math>c</math> in the interval from <math>f_1</math> to <math>f_2</math>, then <math>f_2</math> equals:

:<math>f_2 = f_1 \times 2 ^ \frac{c}{1200} \, .</math>

=== Comparison of major third in just and equal temperament ===
The major third in [[just intonation]] has a frequency ratio 5:4 or ~386 cents, but in equal temperament is 400 cents. This 14 cent difference is about a seventh of a half step and large enough to be audible.

==Piecewise linear approximation==
<!--This section would be severed very well by a graph that draws this linear approximation with the true values over a range around -100 to +100 cents-->
As ''x'' increases from 0 to {{frac|1|12}}, the function 2<sup>''x''</sup> increases ''almost linearly'' from {{val|1.00000}} to {{val|1.05946}}, allowing for a [[Piecewise linear function|piecewise linear approximation]]. Thus, although cents represent a logarithmic scale, small intervals (under 100&nbsp;cents) can be ''loosely approximated'' with the linear relation 1&nbsp;+&nbsp;{{val|0.0005946}}&nbsp;''<math>c</math>'' instead of the true exponential relation 2{{sup|{{frac|''c''|1200}}}}. The rounded error is zero when ''<math>c</math>'' is 0 or 100, and is only about 0.72&nbsp;cents high at ''<math>c</math>=''50 (whose correct value of 2{{sup|{{frac|1|24}}}}&nbsp;≅&nbsp;{{val|1.02930}} is approximated by 1&nbsp;+&nbsp;{{val|0.0005946}}&nbsp;×&nbsp;50&nbsp;≅&nbsp;1.02973). This error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes.

==Human perception==
[[File:Unison versus cent waveform.png|thumb|350px|The waveforms of a unison (blue) vis-à-vis a cent (red) are practically indistinguishable.]]

It is difficult to establish how many cents are perceptible to humans; this precision varies greatly from person to person.  One author stated that humans can distinguish a difference in pitch of about 5–6 cents.{{sfn|Loeffler|2006}}  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.{{sfn|Geringer| Worthy|1999|pp=135-149}}  It has also been established that increased tonal context enables listeners to judge pitch more accurately.{{sfn|Warrier |Zatorre |2002|pp=198-207}} "While intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords."{{sfn|Benson|2007|p= 368}}

When listening to pitches with [[vibrato]], there is evidence that humans perceive the mean frequency as the center of the pitch.{{sfn|Brown |Vaughn |1996|pp=1728–1735}} One study of modern performances of [[Ellens dritter Gesang|Schubert's 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.{{sfn|Prame |1997|pp=616-621}}

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.{{sfn|Peretz |Hyde |2003|pp=362–367}}

==Other representations of intervals by logarithms==
===Octave===
The representation of musical intervals by logarithms is almost as old as logarithms themselves. Logarithms had been invented by Lord Napier in 1614.<ref>Ernest William Hobson (1914), ''John Napier and the invention of logarithms'', 1614, Cambridge, The University Press</ref> As early as 1647, Juan Caramuel y Lobkowitz (1606-1682) in a letter to Athanasius Kircher described the usage of base-2 logarithms in music.<ref>Ramon Ceñal, "Juan Caramuel, su epistolario con Athanasio Kircher, S.J.", ''Revista de Filosofia'' XII/44, Madrid 1954, p. 134 ss.</ref> In this base, the octave is represented by 1, the semitone by 1/12, etc.

===Heptamerides===
[[Joseph Sauveur]], in his ''Principes d'acoustique et de musique'' of 1701, proposed the usage of base-10 logarithms, probably because tables were available. He made use of logarithms computed with three decimals. The base-10 logarithm of 2 is equal to approximately 0.301, which Sauveur multiplies by 1000 to obtain 301 units in the octave. In order to work on more manageable units, he suggests to take 7/301 to obtain units of 1/43 octave.{{efn|name=Heptameride}} The octave therefore is divided in 43 parts, named "merides", themselves divided in 7 parts, the "heptamerides". Sauveur also imagined the possibility to further divide each heptameride in 10, but does not really make use of such microscopic units.<ref>Joseph Sauveur, ''Principes d'acoustique et de musique ou Système général des intervalles des sons'', Minkoff Reprint, Geneva, 1973; see online ''Mémoires de l'Académie royale des sciences'', 1700, [https://gallica.bnf.fr/ark:/12148/bpt6k3502d/f140 Acoustique]; 1701 [https://gallica.bnf.fr/ark:/12148/bpt6k3503q/f128 Acoustique].</ref>

===Savart===
Félix Savart (1791-1841) took over Sauveur's system, without limiting the number of decimals of the logarithm of 2, so that the value of his unit varies according to sources. With five decimals, the base-10 logarithm of 2 is 0.30103, giving 301.03 savarts in the octave.<ref>Émile Leipp, ''Acoustique et musique : Données physiques et technologiques, problèmes de l'audition des sons musicaux, principes de fonctionnement et signification acoustique des principaux archétypes d'instruments de musique, les musiques expérimentales, l'acoustique des salles'', Masson, 1989, 4th edition, p. 16.</ref> This value often is rounded to 1/301 or to 1/300 octave.<ref>"Ordinary savart", 1/301 octave, and "modified savart", 1/300 octave. Herbert Arthur Klein, ''The Science of Measurement. A Historical Survey'', New York, 1974, p. 605</ref><ref>Alexander Wood, ''The Physics of Music'', London, 1944, ²2007, p. 53-54.</ref>

===Prony===
Early in the 19th century, [[Gaspard de Prony]] proposed a logarithmic unit of base <math>\sqrt[12]{2}</math>, where the unit corresponds to a semitone in equal temperament.<ref>Gaspard de Prony, ''Instruction élémentaire sur les moyens de calculer les intervalles musicaux'', Paris, 1832. Online: [https://gallica.bnf.fr/ark:/12148/bpt6k1083022].</ref> Alexander John Ellis in 1880 describes a large number of pitch standards that he noted or calculated, indicating in pronys with two decimals, i.e. with a precision to the 1/100 of a semitone,<ref>The precision is the same as with cents, but Ellis had not yet devised this unit.</ref> the interval that separated them from a theoretical pitch of 370&nbsp;Hz, taken as point of reference.<ref>Alexander John Ellis, "On the History of Musical Pitch," ''Journal of the Society of Arts'', 1880, reprinted in ''Studies in the History of Musical Pitch'', Frits Knuf, Amsterdam, 1968, p. 11-62.</ref>

===Centitones===
<!--[[Centitone]], [[Iring]], [[decitone]], and [[millitone]] redirect directly here.-->
A '''centitone''' (also '''Iring''') is a [[musical interval]] (2{{sup|{{frac|1|600}}}}, <math>\sqrt[600]{2}</math>) equal to two cents (2{{sup|{{frac|2|1200}}}}){{sfn|Randel|1999|p=123}}{{sfn|Randel|2003|pp=154, 416}} proposed as a unit of measurement ({{audio|Centitone on C.mid|Play}}) by Widogast Iring in ''Die reine Stimmung in der Musik'' (1898) as 600 steps per [[octave]] and later by [[Joseph Yasser]] in ''A Theory of Evolving Tonality'' (1932) as 100 steps per equal tempered [[major second|whole tone]].

Iring noticed that the Grad/Werckmeister (1.96 cents, 12 per [[Pythagorean comma]]) and the [[schisma]] (1.95 cents) are nearly the same (≈ 614 steps per octave) and both may be approximated by 600 steps per octave (2 cents).<ref name=huygens-fokker /> Yasser promoted the '''decitone''', centitone, and '''millitone''' (10, 100, and 1000 steps per whole tone = 60, 600, and 6000 steps per octave = 20, 2, and 0.2 cents).{{sfn|Yasser|1932|p=14}}{{sfn|Farnsworth|1969|p=24}}

For example: Equal tempered perfect fifth = 700 cents = 175.6 [[savart]]s = 583.3 [[millioctave]]s = 350 centitones.{{sfn|Apel|1970|p=363}}

{| class="wikitable"
|-
! scope="col" | Centitones
! scope="col" | Cents
|-
! scope="row" | 1 centitone
| 2 cents
|-
! scope="row" | 0.5 centitone
| 1 cent
|-
! scope="row" | 2{{sup|{{frac|1|600}}}}
| 2{{sup|{{frac|2|1200}}}}
|-
! scope="row" | 50 per semitone
| 100 per semitone
|-
! scope="row" | 100 per whole tone
| 200 per whole tone
|}

==Sound files==
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.

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.

{{audio|Cent on C.mid|Play middle C & 1 cent above}}, beat frequency = 0.16 [[Hz]]<br/>
{{audio|Semicomma on C.mid|Play middle C & 10.06 cents above}}, beat frequency = 1.53&nbsp;Hz<br/>
{{audio|Eighth-tone on C.mid|Play middle C & 25 cents above}}, beat frequency = 3.81&nbsp;Hz

{{Listen|filename=Sines 1 cent difference.wav|title=One Cent Interval|description=Sine wave plays at [[middle C]], middle C 1 cent sharp, then both at the same time.|pos=left}}
{{Listen|filename=Sines 12 cent difference.wav|title=Twelve Cent Interval|description=Sine wave plays at [[middle C]], middle C 12 cents sharp, then both at the same time.|style=clear:none|pos=left}}
{{Listen|filename=Sines 24 cent difference.wav|title=Twenty-four Cent Interval|description=Sine wave plays at [[middle C]], middle C 24 cents sharp, then both at the same time.|style=clear:none|pos=left}}
{{Clear}}

==See also==
{{Portal|Music}}

*[[Degree (angle)|Degree]]
*[[Gradian]]
*[[Microtonal music]]
*[[Radian]]

==References==
===Footnotes===
{{notelist|refs=

{{efn|name=Caramuel|Caramuel mentioned the possible use of binary logarithms for music in a letter to [[Athanasius Kircher]] in 1647; this usage often is attributed to [[Leonhard Euler]] in 1739 (see [[Binary logarithm]]). [[Isaac Newton]] described musical logarithms using the semitone ({{radic|2|12}}) as base in 1665; [[Gaspard de Prony]] did the same in 1832. [[Joseph Sauveur]] in 1701, and [[Félix Savart]] in the first half of the 19th century, divided the octave in 301 or 301,03 units. See {{harvnb|Barbieri|1987|pp=145–168}} and also [[Stigler's law of eponymy]].}}

{{efn|name=Heptameride|301 can be divided only by 7 or by 43.}}

}}

===Citations===
{{Reflist|refs=

<ref name=huygens-fokker>{{cite web|url=http://www.huygens-fokker.org/docs/measures.html |title=Logarithmic Interval Measures|website=Huygens-Fokker.org|access-date=2021-06-25}}</ref>

}}

===Sources===
{{refbegin|2|indent=yes}}

*{{cite book|last=Apel|first= Willi|date=1970|title=Harvard Dictionary of Music|publisher=Taylor & Francis}}
*{{cite journal|last=Barbieri|first= Patrizio |date=1987|title=Juan Caramuel Lobkowitz (1606–1682): über die musikalischen Logarithmen und das Problem der musikalischen Temperatur|journal=Musiktheorie|volume=2|issue=2|pages= 145–168}}
*{{cite book|last=Benson|first= Dave |date=2007|title=Music: A Mathematical Offering|location= Cambridge|isbn=9780521853873|url=https://books.google.com/books?id=Ko1NsIq4qLIC}}
*{{cite journal|url=http://www.wellesley.edu/Physics/brown/pubs/vibPerF100P1728-P1735.pdf |first1=J.C. |last1=Brown |first2=K.V. |last2=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 |access-date=2008-09-28 |doi=10.1121/1.416070 |pmid=8817899 |bibcode=1996ASAJ..100.1728B}}
* {{Citation |last1=Ellis |first1=Alexander J. |author-link=Alexander John Ellis |first2=Alfred J. |last2=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|id={{zenodo|1432077}} |doi-access=free }}
* {{Citation |last = Ellis |first = Alexander J. |author-link= Alexander John Ellis| url=https://soundandscience.de/text/history-musical-pitch| title=History of Musical Pitch| date=1880| journal= Journal of the Society of Arts|volume = 21 |issue = 545 |pages= 293–337|doi = 10.1038/021550a0 |bibcode = 1880Natur..21..550E |s2cid = 4107831 |doi-access = free }}
*{{Citation |last=Ellis |first=Alexander J. |author-link=Alexander John Ellis|url=https://acoustics.mpiwg-berlin.mpg.de/text/musical-scales-various-nations |title=On the Musical Scales of Various Nations|date=1885|journal= Journal of the Society of Arts|pages=485–527|access-date= 1 January 2020}}
*{{cite book|last=Farnsworth|first= Paul Randolph |date=1969|title=The Social Psychology of Music|publisher= Iowa State University Press |isbn=9780813815473}}
*{{cite journal|first1=J. M.|last1= Geringer|first2= M.D.|last2= Worthy|jstor=3345719 |title=Effects of Tone-Quality Changes on Intonation and Tone-Quality Ratings of High School and College Instrumentalists|journal= Journal of Research in Music Education|volume=47|issue= 2|date= 1999|pages=135–149|doi= 10.2307/3345719|s2cid= 144918272}}
*{{cite thesis|type=Master's|first=D.B.|last= Loeffler|url=http://etd.gatech.edu/theses/available/etd-04102006-142310/ |title=Instrument Timbres and Pitch Estimation in Polyphonic Music|archive-url=https://web.archive.org/web/20071218232401/http://etd.gatech.edu/theses/available/etd-04102006-142310/ |archive-date=2007-12-18 |publisher= Department of Electrical and Computer Engineering, Georgia Tech|date=  April 2006}}
*{{cite journal |first1=I. |last1=Peretz |first2=K.L. |last2=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  |doi=10.1016/S1364-6613(03)00150-5 |pmid=12907232  |citeseerx=10.1.1.585.2171 |s2cid=3224978 }}
*{{cite journal|first=E.|last= 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|doi-access=free }}
*{{cite book|last=Randel|first=Don Michael |title=The Harvard Concise Dictionary of Music and Musicians|url=https://books.google.com/books?id=7iuZ6HaEMmoC&pg=123|year=1999|publisher=Harvard University Press|isbn=978-0-674-00084-1}}
*{{cite book|last=Randel |first=Don Michael |title=The Harvard Dictionary of Music|url=https://books.google.com/books?id=02rFSecPhEsC&pg=PA154|date= 2003|publisher=Harvard University Press|isbn=978-0-674-01163-2|edition=4th}}
*{{citation|last=Renold|first= Maria |date=2004|title=Intervals, Scales, Tones and the Concert Pitch C = 128 Hz|translator= Bevis Stevens| editor= Anna Meuss |orig-year=1998|location= Temple Lodge|isbn=9781902636467|quote=Interval proportions can be converted to the cent values which are in common use today}}
*{{cite journal |first1=C.M.|last1= Warrier |first2=R.J.|last2= 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 |doi=10.3758/BF03195786 |pmid= 12013375 |s2cid= 15094971 |doi-access=free }}
*{{cite book|last=Yasser|first= Joseph|author-link=Joseph Yasser|date=1932|title=A Theory of Evolving Tonality|publisher=American Library of Musicology}}

{{refend}}

==External links==
*[http://in.music.sc.edu/fs/bain/atmi02/cents/default.html Cent conversion: Whole number ratio to cent] {{Webarchive|url=https://web.archive.org/web/20170422080739/http://in.music.sc.edu/fs/bain/atmi02/cents/default.html |date=2017-04-22 }} [rounded to whole number]
*[http://www.sengpielaudio.com/calculator-centsratio.htm Cent conversion: Online utility with several functions]

{{Musical tuning|state=auto}}
{{Intervals|state=auto}}

{{DEFAULTSORT:Cent (Music)}}
[[Category:Equal temperaments]]
[[Category:Intervals (music)]]
[[Category:100 (number)]]
[[Category:Units of level]]