
The cent is a
logarithmic Logarithmic can refer to:
* Logarithm, a transcendental function in mathematics
* Logarithmic scale, the use of the logarithmic function to describe measurements
* Logarithmic spiral,
* Logarithmic growth
* Logarithmic distribution, a discrete pr ...
unit of measure used for
musical intervals
In music theory, an interval is a difference in pitch between two sounds.
An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or h ...
.
Twelve-tone equal temperament
Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resulting ...
divides the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
into 12
semitone
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 most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
s of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different
tuning system
In music, there are two common meanings for tuning:
* Tuning practice, the act of tuning an instrument or voice.
* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.
Tuning practice
Tun ...
s, and in fact the interval of one 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
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s that began with
Juan Caramuel y Lobkowitz
Juan Caramuel y Lobkowitz (Juan Caramuel de Lobkowitz, 23 May 1606 in Madrid — 7 or 8 September 1682 in Vigevano) was a Spanish Catholic scholastic philosopher, ecclesiastic, mathematician and writer. He is believed to be a great-grandson of J ...
in the 17th century. Ellis chose to base his measures on the hundredth part of a semitone, , at
Robert Holford Macdowell Bosanquet's suggestion. He made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed, and further described and employed the system in his 1875 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.
History
Alexander John Ellis' paper ''On the Musical Scales of Various Nations'', 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
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 most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
.. 99 other notes were interposed, making exactly equal intervals with each other, we should divide the octave into 1200 equal of an equal semitone, or ''cents'' as they may be briefly called."
Ellis defined the pitch of a musical note in his 1880 work ''History of Musical Pitch'' 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". He later defined musical pitch to be "the pitch, or V
or "double vibrations"of any named musical note which determines the pitch of all the other notes in a particular system of tunings." He notes that these notes, when sounded in succession, form the
scale
Scale or scales may refer to:
Mathematics
* Scale (descriptive set theory), an object defined on a set of points
* Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original
* Scale factor, a number ...
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".
Absolute Absolute may refer to:
Companies
* Absolute Entertainment, a video game publisher
* Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK
* Absolute Software Corporation, specializes in security and data risk manag ...
and
relative pitch
Relative pitch is the ability of a person to identify or re-create a given musical note by comparing it to a reference note and identifying the interval between those two notes. For example, if the note ''Do'' and ''Fa'' is played on a piano, a per ...
es were also defined based on these ratios.
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
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...
or tuning, and is the system at present used throughout Europe. 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."
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, Persia, Arabia, Syria and Scottish Highlands, India, Singapore, Burmah and Siam,; (II. Pentatonic scales) South Pacific, Western Africa, Java, China and Japan. 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"..
Use

A cent is a unit of measure for the ratio between two frequencies. An
equally tempered
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...
semitone (the interval between two adjacent piano keys) spans 100 cents by definition. An
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
—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 , the 1200th root of 2, which is approximately .
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 a
decibel):
:
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:
:
To compare different tuning systems, convert the various interval sizes into cents. For example, in
just intonation
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...
, the major third is represented by the frequency ratio 5:4. Applying the formula at the top shows that this is 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.
Piecewise linear approximation
As ''x'' increases from 0 to , the function 2
''x'' increases almost linearly from to . The exponential cent scale can therefore be accurately approximated as a
piecewise linear function that is numerically correct at semitones. That is, ''n'' cents for ''n'' from 0 to 100 may be approximated as 1 + ''n'' instead of 2. 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 = is approximated by 1 + × 50 = 1.02973. This error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes.
Human perception

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. The threshold of what is perceptible, technically known as the
just noticeable difference
In the branch of experimental psychology focused on sense, sensation, and perception, which is called psychophysics, a just-noticeable difference or JND is the amount something must be changed in order for a difference to be noticeable, detectable ...
(JND), also varies as a function of the frequency, the amplitude and the
timbre
In music, timbre ( ), also known as tone color or tone quality (from psychoacoustics), is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and music ...
. 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. It has also been established that increased tonal context enables listeners to judge pitch more accurately. "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."
When listening to pitches with
vibrato
Vibrato (Italian, from past participle of " vibrare", to vibrate) is a musical effect consisting of a regular, pulsating change of pitch. It is used to add expression to vocal and instrumental music. Vibrato is typically characterised in terms o ...
, there is evidence that humans perceive the mean frequency as the center of the pitch. One study of modern performances of
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
Giuseppe Fortunino Francesco Verdi (; 9 or 10 October 1813 – 27 January 1901) was an Italian composer best known for his operas. He was born near Busseto to a provincial family of moderate means, receiving a musical education with the h ...
's opera arias.
Normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with
amusia
Amusia is a musical disorder that appears mainly as a defect in processing pitch but also encompasses musical memory and recognition. Two main classifications of amusia exist: acquired amusia, which occurs as a result of brain damage, and c ...
, however, have trouble recognizing differences of less than 100 cents and sometimes have trouble with these or larger intervals.
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. 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. In this base, the octave is represented by 1, the semitone by 1/12, etc.
Heptamerides
Joseph Sauveur
Joseph Sauveur (24 March 1653 – 9 July 1716) was a French mathematician and physicist. He was a professor of mathematics and in 1696 became a member of the French Academy of Sciences.
Life
Joseph Sauveur was born in La Flèche, the son of a ...
, 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. 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.
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. This value often is rounded to 1/301 or to 1/300 octave.
Prony
Early in the 19th century,
Gaspard de Prony
Baron Gaspard Clair François Marie Riche de Prony (22 July 1755 – 29 July 1839) was a French mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France and died in Asnières-sur-Seine, France.
Educ ...
proposed a logarithmic unit of base