Equal temperament

# Equal temperament

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An equal temperament is a musical temperament
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

, or a system of tuning
Musical tuning
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:...

, in which every pair of adjacent notes has an identical frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ratio. As pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

is perceived roughly as the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for every note in the system.

In equal temperament tunings, an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

— usually the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

— is divided into a series of equal steps (equal frequency ratios between successive notes). For classical music
Classical music
Classical music is the art music produced in, or rooted in, the traditions of Western liturgical and secular music, encompassing a broad period from roughly the 11th century to present times...

, the most common tuning system is twelve-tone equal temperament (also known as 12 equal temperament), inconsistently abbreviated as 12-TET, 12TET, 12tET, 12tet, 12-ET, 12ET, or 12et, which divides the octave into 12 parts, all of which are equal on a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

. It is usually tuned relative to a standard pitch of 440 Hz, called A 440.

Other equal temperaments exist (some music has been written in 19-TET
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

and 31-TET
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

for example, and 24-TET
Arab tone system
The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament , the distance between each successive note being a quarter tone . Each tone has its own name not repeated in different...

) 24-TET is used in Arabic music, but in western countries when people use the term equal temperament without qualification, it is usually understood that they are talking about 12-TET.

Equal temperaments may also divide some interval other than the octave, a pseudo-octave
Pseudo-octave
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...

, into a whole number of equal steps. An example is an equal-tempered Bohlen–Pierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is sometimes preferred. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on.

String ensembles and vocal groups, who have no mechanical tuning limitations, often use a tuning much closer to just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

, as it is naturally more consonant
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

. Other instruments, such as some wind, keyboard, and fret
Fret
A fret is a raised portion on the neck of a stringed instrument, that extends generally across the full width of the neck. On most modern western instruments, frets are metal strips inserted into the fingerboard...

ted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Other wind instruments, that can easily and spontaneously bend their tone, most notably double-reeds
Double reed
A double reed is a type of reed used to produce sound in various wind instruments. The term double reed comes from the fact that there are two pieces of cane vibrating against each other. A single reed consists of one piece of cane which vibrates against a mouthpiece made of metal, hardened...

, use tuning similar to string ensembles and vocal groups.

## Equal temperament in Europe

### Early history

One of the earliest discussions of equal temperament occurs in the writiting of Aristoxenus
Aristoxenus
Aristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and...

in the 4th century B.C.

Vincenzo Galilei
Vincenzo Galilei
Vincenzo Galilei was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicist Galileo Galilei and of the lute virtuoso and composer Michelagnolo Galilei...

(father of Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

) was one of the first practical advocates of twelve-tone equal temperament. He composed a sets of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "Fronimo", 24 +1 ricercar
Ricercar
A ricercar is a type of late Renaissance and mostly early Baroque instrumental composition. The term means to search out, and many ricercars serve a preludial function to "search out" the key or mode of a following piece...

s. He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567. Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino
Francesco Spinacino
Francesco Spinacino was an Italian lutenist and composer. His surviving output comprises the first two volumes of Ottaviano Petrucci's influential series of lute music publications: Intabolatura de lauto libro primo and Intabolatura de lauto libro secondo...

wrote a "Recercare de tutti li Toni" as early as 1507. In the 17th century lutenist-composer John Wilson
John Wilson (composer)
John Wilson , was an English composer, lutenist and teacher. Born in Faversham, Kent, he moved to London by 1614, where he succeeded Robert Johnson as principal composer for the King's Men, and entered the King's Musick in 1635 as a lutenist. He received the degree of D.Mus from Oxford in 1644,...

wrote a set of 26 preludes including 24 in all the major/minor keys.

Henricus Grammateus
Henricus Grammateus
Henricus Grammateus was a German mathematician. In 1518 he published details of a new musical temperament, which is now named after him, for the harpsichord...

drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".
Simon Stevin
The first mention of equal temperament related to Twelfth root of two in the West appeared in Simon Stevin
Simon Stevin
Simon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...

's unfinished manuscript Van De Spiegheling der signconst (ca 1605) published posthumously nearly three centuries later in 1884. However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values. As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is incorrect.

The following were Simon Stevin's chord length from Vande Spiegheling der signconst:
TONE CHORD 10000 from Simon Stevin RATIO CORRECTED CHORD
semitone 9438 1.0595465 9438.7
whole tone 8909 1.0593781
1.5 tone 8404 1.0600904 8409
ditone 7936 1.0594758 7937
ditone and a half 7491 1.0594046 7491.5
tritone 7071 1.0593975 7071.1
tritone and a half 6674 1.0594845 6674.2
four-tone 6298 1.0597014 6299
four-tone-and-half 5944 1.0595558 5946
five-tone 5611 1.0593477 5612.3
five-tone-and-half 5296 1.0594788 5297.2
full tone 1.0592000

A generation later, French mathematician Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

presented several equal tempered
chord lengths obtained by Jean Beaugrand, Ismael Bouillaud and Jean Galle.

In 1630 Johann Faulhaber
Johann Faulhaber
Johann Faulhaber was a German mathematician.Born in Ulm, Faulhaber trained as a weaver and later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen...

published a 100 cent monochord table, with the exception of several errors due to his use of logarithmic tables . He did not explain how he obtained his results.

Zarlino in his polemic
Polemic
A polemic is a variety of arguments or controversies made against one opinion, doctrine, or person. Other variations of argument are debate and discussion...

with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute
Lute
Lute can refer generally to any plucked string instrument with a neck and a deep round back, or more specifically to an instrument from the family of European lutes....

in his "Sopplimenti musicali" in 1588.

### Equal temperament in the Baroque era

From 1450 to about 1800 plucked instrument players (lutenists and guitarists) generally favored equal temperament, and the Brossard lute Manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Bocquet written in a all keys, including the last prelude, entitled "Prelude sur tous les tons", which enharmonically modulates through all of the keys. Wind and keyboard musicians expected much less mistuning (than that of equal temperament) in the most common keys, such as C major. They used approximations that emphasized the tuning of third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

s or fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

s in these keys, such as meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

. Among the 17th century keyboard composers Girolamo Frescobaldi
Girolamo Frescobaldi
Girolamo Frescobaldi was a musician from Ferrara, one of the most important composers of keyboard music in the late Renaissance and early Baroque periods. A child prodigy, Frescobaldi studied under Luzzasco Luzzaschi in Ferrara, but was influenced by a large number of composers, including Ascanio...

advocated equal temperament. Some theorists, such as Giuseppe Tartini
Giuseppe Tartini
Giuseppe Tartini was an Italian baroque composer and violinist.-Biography:Tartini was born in Piran, a town on the peninsula of Istria, in the Republic of Venice to Gianantonio – native of Florence – and Caterina Zangrando, a descendant of one of the oldest aristocratic Piranian families.It...

, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although Andreas Werckmeister
Andreas Werckmeister
Andreas Werckmeister was an organist, music theorist, and composer of the Baroque era.-Life:Born in Benneckenstein, Germany, Werckmeister attended schools in Nordhausen and Quedlinburg. He received his musical training from his uncles Heinrich Christian Werckmeister and Heinrich Victor Werckmeister...

emphatically advocated equal temperament in his 1707 treatise published posthumously.

J. S. Bach wrote The Well-Tempered Clavier
The Well-Tempered Clavier
The Well-Tempered Clavier , BWV 846–893, is a collection of solo keyboard music composed by Johann Sebastian Bach...

to demonstrate the musical possibilities of well temperament
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...

, where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they each described the subtly different dissonances made available within a particular tuning method. However, it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about the moods and colors of particular keys.)

Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and permitted total harmonic freedom at the expense of just a little impurity in every interval. This allowed greater expression through enharmonic modulation
Modulation (music)
In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

, which became extremely important in the 18th century in music of such composers as Francesco Geminiani
Francesco Geminiani
thumb|230px|Francesco Geminiani.Francesco Saverio Geminiani was an Italian violinist, composer, and music theorist.-Biography:...

, Wilhelm Friedemann Bach
Wilhelm Friedemann Bach
Wilhelm Friedemann Bach , the second child and eldest son of Johann Sebastian Bach and Maria Barbara Bach, was a German composer and performer...

, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel
Johann Gottfried Müthel
Johann Gottfried Müthel was a German composer and noted keyboard virtuoso. Along with C.P.E. Bach, he represented the Sturm und Drang style of composition....

.

The progress of Equal Temperament from mid-18th century on is described with detail in quite a few modern scholarly publications: it was already the temperament of choice during the Classical era (second half of the 18th century), and it became standard during the Early Romantic era (first decade of the 19th century), except for organs that switched to it more gradually, completing only in the second decade of the 19th century. (In England, some cathedral organists and choirmasters held out against it even after that date; Samuel Sebastian Wesley
Samuel Sebastian Wesley
Samuel Sebastian Wesley was an English organist and composer.-Biography:Born in London, he was the eldest child in the composer Samuel Wesley's second family, which he formed with Sarah Suter having separated from his wife Charlotte. Samuel Sebastian was the grandson of Charles Wesley...

, for instance, opposed it all along. He died in 1876.)

A precise equal temperament is possible using the 17th-century Sabbatini method of splitting the octave first into three tempered major thirds. This was also proposed by several writers during the Classical era. Tuning with several checks, thus attaining virtually modern accuracy, was already done in the 1st decades of the 19th century. Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century. The ultimate precision was available with 2-decimal tables published by White in 1917.

It is in the environment of equal temperament that the new styles of symmetrical tonality and polytonality
Polytonality
The musical use of more than one key simultaneously is polytonality . Bitonality is the use of only two different keys at the same time...

, atonal music
Atonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...

such as that written with the twelve tone technique or serialism
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

, and jazz
Jazz
Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...

(at least its piano component) developed and flourished.

## Equal temperament in China

The earliest discussion of the equal division of the scale was in the writings of Ling Lun during the 27th century BC.

Jing Fang
Jing Fang
Jing Fang , born Li Fang , courtesy name Junming , was a Chinese music theorist, mathematician and astrologer. Born in present-day Puyang, Henan during the Han Dynasty , he was the first to notice how closely a succession of 53 just fifths approximates 31 octaves...

(78–37 BC) observed that using the Pythagorean comma of 53 just fifths approximates to 31 octaves. This would later lead to the discovery of 53 equal temperament.

An approximation for equal temperament was given by He Chengtian, a mathematician of Southern and Northern Dynasties
Southern and Northern Dynasties
The Southern and Northern Dynasties was a period in the history of China that lasted from 420 to 589 AD. Though an age of civil war and political chaos, it was also a time of flourishing arts and culture, advancement in technology, and the spreading of Mahayana Buddhism and Daoism...

Historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese
Music of China
Chinese Music has been made since the dawn of Chinese civilization with documents and artifacts providing evidence of a well-developed musical culture as early as the Zhou Dynasty...

Zhu Zaiyu
Zhu Zaiyu
Zhu Zaiyu , a prince of the Ming dynasty of China. In 1584 Prince Zhu innovatively described the equal temperament via accurate mathematical calculation...

(朱載堉), a prince of the Ming
Ming Dynasty
The Ming Dynasty, also Empire of the Great Ming, was the ruling dynasty of China from 1368 to 1644, following the collapse of the Mongol-led Yuan Dynasty. The Ming, "one of the greatest eras of orderly government and social stability in human history", was the last dynasty in China ruled by ethnic...

court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-TET in his five thousand pages work Complete Compendium of Music and Pitch(Yuelü quan shu 乐律全书) in 1584.

He obtained his result mathematically by dividing the length of string and pipe successively by

=1.059463094359295264561825, and for pipe diameter by

(after 8 octave still in tune)

According to Gene Cho, Zhu Zaiyu was the first person to solve equal temperament problem mathematically. Murray Bardour said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyii's brilliant solution remains an enigma." The 19th-century German physicist Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.

### Zhu Zaiyu tuning instrument

Zhu Zaiyu illustrated his equal temperament theory by construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon
Victor-Charles Mahillon
Victor-Charles Mahillon was a Belgian musician and writer on musical topics. He built, collected, and described more than 1500 musical instruments....

, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the diameter of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory..

## General properties

In an equal temperament, the distance between each step of the scale is the same interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. Because the perceived identity of an interval depends on its ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly-spaced, and would not permit transposition to different keys.) Specifically, the smallest interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

in an equal-tempered scale is the ratio:

where the ratio r divides the ratio p (typically the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

, which is 2/1) into n equal parts. (See Twelve-tone equal temperament below.)

Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in Ethnomusicology
Ethnomusicology
Ethnomusicology is defined as "the study of social and cultural aspects of music and dance in local and global contexts."Coined by the musician Jaap Kunst from the Greek words ἔθνος ethnos and μουσική mousike , it is often considered the anthropology or ethnography of music...

. The basic step in cents for any equal temperament can be found by taking the width of p above in cents (usually the octave, which is 1200 cents wide), called below w, and dividing it into n parts:

In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

where the modulo is the number of divisions of the octave (usually 12), these integers can be reduced to pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

es, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g. 'C' is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.

## Twelve-tone equal temperament

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...

of the interval between two adjacent notes, is the twelfth root of two:
(98.9545922303676 CENT) (99.9066043792227 CENT) (100.0000094845790 CENT) (100.0000006728490 CENT)

This interval is divided into 100 cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

s. (The cent is sometimes for this reason defined as one hundredth of a semitone.)

### Calculating absolute frequencies

To find the frequency, Pn, of a note in 12-TET, the following definition may be used:

In this formula Pn refers to the pitch, or frequency (usually in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

), you are trying to find. Pa refers to the frequency of a reference pitch (usually 440Hz). n and a refer to numbers assigned to the desired pitch and the reference pitch, respectively. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A4 (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and C4 (middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

) is the 40th key. These numbers can be used to find the frequency of C4:

### Comparison of various equal temperament in history

YEAR NAME RATIO CENTS
400 He Chengtian 1.060070671 101.0
1580 Vincenzo Galilei 18:17 99.0
1581 Zhu Zaiyu 1.059463094 100.0
1585 Simon Stevin 1.059546514 100.1
1630 Marin Mersenne 1.059322034 99.8
1630 Johann Faulhaber 1.059490385 100.0

### Comparison to just intonation

The intervals of 12-TET closely approximate some intervals in just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

. The fifths and fourths are almost indistinguishably close to just.

In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

.
Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation Error
Unison (C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

)
1.000000 0 = 1.000000 0.00 0
Minor second (C
An equal temperament is a musical temperament
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

, or a system of tuning
Musical tuning
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:...

, in which every pair of adjacent notes has an identical frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ratio. As pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

is perceived roughly as the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for every note in the system.

In equal temperament tunings, an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

— usually the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

— is divided into a series of equal steps (equal frequency ratios between successive notes). For classical music
Classical music
Classical music is the art music produced in, or rooted in, the traditions of Western liturgical and secular music, encompassing a broad period from roughly the 11th century to present times...

, the most common tuning system is twelve-tone equal temperament (also known as 12 equal temperament), inconsistently abbreviated as 12-TET, 12TET, 12tET, 12tet, 12-ET, 12ET, or 12et, which divides the octave into 12 parts, all of which are equal on a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

. It is usually tuned relative to a standard pitch of 440 Hz, called A 440.

Other equal temperaments exist (some music has been written in 19-TET
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

and 31-TET
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

for example, and 24-TET
Arab tone system
The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament , the distance between each successive note being a quarter tone . Each tone has its own name not repeated in different...

) 24-TET is used in Arabic music, but in western countries when people use the term equal temperament without qualification, it is usually understood that they are talking about 12-TET.

Equal temperaments may also divide some interval other than the octave, a pseudo-octave
Pseudo-octave
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...

, into a whole number of equal steps. An example is an equal-tempered Bohlen–Pierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is sometimes preferred. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on.

String ensembles and vocal groups, who have no mechanical tuning limitations, often use a tuning much closer to just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

, as it is naturally more consonant
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

. Other instruments, such as some wind, keyboard, and fret
Fret
A fret is a raised portion on the neck of a stringed instrument, that extends generally across the full width of the neck. On most modern western instruments, frets are metal strips inserted into the fingerboard...

ted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Other wind instruments, that can easily and spontaneously bend their tone, most notably double-reeds
Double reed
A double reed is a type of reed used to produce sound in various wind instruments. The term double reed comes from the fact that there are two pieces of cane vibrating against each other. A single reed consists of one piece of cane which vibrates against a mouthpiece made of metal, hardened...

, use tuning similar to string ensembles and vocal groups.

## Equal temperament in Europe

### Early history

One of the earliest discussions of equal temperament occurs in the writiting of Aristoxenus
Aristoxenus
Aristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and...

in the 4th century B.C.

Vincenzo Galilei
Vincenzo Galilei
Vincenzo Galilei was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicist Galileo Galilei and of the lute virtuoso and composer Michelagnolo Galilei...

(father of Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

) was one of the first practical advocates of twelve-tone equal temperament. He composed a sets of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "Fronimo", 24 +1 ricercar
Ricercar
A ricercar is a type of late Renaissance and mostly early Baroque instrumental composition. The term means to search out, and many ricercars serve a preludial function to "search out" the key or mode of a following piece...

s. He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567. Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino
Francesco Spinacino
Francesco Spinacino was an Italian lutenist and composer. His surviving output comprises the first two volumes of Ottaviano Petrucci's influential series of lute music publications: Intabolatura de lauto libro primo and Intabolatura de lauto libro secondo...

wrote a "Recercare de tutti li Toni" as early as 1507. In the 17th century lutenist-composer John Wilson
John Wilson (composer)
John Wilson , was an English composer, lutenist and teacher. Born in Faversham, Kent, he moved to London by 1614, where he succeeded Robert Johnson as principal composer for the King's Men, and entered the King's Musick in 1635 as a lutenist. He received the degree of D.Mus from Oxford in 1644,...

wrote a set of 26 preludes including 24 in all the major/minor keys.

Henricus Grammateus
Henricus Grammateus
Henricus Grammateus was a German mathematician. In 1518 he published details of a new musical temperament, which is now named after him, for the harpsichord...

drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".
Simon Stevin
The first mention of equal temperament related to Twelfth root of two in the West appeared in Simon Stevin
Simon Stevin
Simon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...

's unfinished manuscript Van De Spiegheling der signconst (ca 1605) published posthumously nearly three centuries later in 1884. However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values. As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is incorrect.

The following were Simon Stevin's chord length from Vande Spiegheling der signconst:
TONE CHORD 10000 from Simon Stevin RATIO CORRECTED CHORD
semitone 9438 1.0595465 9438.7
whole tone 8909 1.0593781
1.5 tone 8404 1.0600904 8409
ditone 7936 1.0594758 7937
ditone and a half 7491 1.0594046 7491.5
tritone 7071 1.0593975 7071.1
tritone and a half 6674 1.0594845 6674.2
four-tone 6298 1.0597014 6299
four-tone-and-half 5944 1.0595558 5946
five-tone 5611 1.0593477 5612.3
five-tone-and-half 5296 1.0594788 5297.2
full tone 1.0592000

A generation later, French mathematician Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

presented several equal tempered
chord lengths obtained by Jean Beaugrand, Ismael Bouillaud and Jean Galle.

In 1630 Johann Faulhaber
Johann Faulhaber
Johann Faulhaber was a German mathematician.Born in Ulm, Faulhaber trained as a weaver and later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen...

published a 100 cent monochord table, with the exception of several errors due to his use of logarithmic tables . He did not explain how he obtained his results.

Zarlino in his polemic
Polemic
A polemic is a variety of arguments or controversies made against one opinion, doctrine, or person. Other variations of argument are debate and discussion...

with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute
Lute
Lute can refer generally to any plucked string instrument with a neck and a deep round back, or more specifically to an instrument from the family of European lutes....

in his "Sopplimenti musicali" in 1588.

### Equal temperament in the Baroque era

From 1450 to about 1800 plucked instrument players (lutenists and guitarists) generally favored equal temperament, and the Brossard lute Manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Bocquet written in a all keys, including the last prelude, entitled "Prelude sur tous les tons", which enharmonically modulates through all of the keys. Wind and keyboard musicians expected much less mistuning (than that of equal temperament) in the most common keys, such as C major. They used approximations that emphasized the tuning of third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

s or fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

s in these keys, such as meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

. Among the 17th century keyboard composers Girolamo Frescobaldi
Girolamo Frescobaldi
Girolamo Frescobaldi was a musician from Ferrara, one of the most important composers of keyboard music in the late Renaissance and early Baroque periods. A child prodigy, Frescobaldi studied under Luzzasco Luzzaschi in Ferrara, but was influenced by a large number of composers, including Ascanio...

advocated equal temperament. Some theorists, such as Giuseppe Tartini
Giuseppe Tartini
Giuseppe Tartini was an Italian baroque composer and violinist.-Biography:Tartini was born in Piran, a town on the peninsula of Istria, in the Republic of Venice to Gianantonio – native of Florence – and Caterina Zangrando, a descendant of one of the oldest aristocratic Piranian families.It...

, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although Andreas Werckmeister
Andreas Werckmeister
Andreas Werckmeister was an organist, music theorist, and composer of the Baroque era.-Life:Born in Benneckenstein, Germany, Werckmeister attended schools in Nordhausen and Quedlinburg. He received his musical training from his uncles Heinrich Christian Werckmeister and Heinrich Victor Werckmeister...

emphatically advocated equal temperament in his 1707 treatise published posthumously.

J. S. Bach wrote The Well-Tempered Clavier
The Well-Tempered Clavier
The Well-Tempered Clavier , BWV 846–893, is a collection of solo keyboard music composed by Johann Sebastian Bach...

to demonstrate the musical possibilities of well temperament
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...

, where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe{{Weasel-inline|date=June 2011}} that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they each described the subtly different dissonances made available within a particular tuning method. However, it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about the moods and colors of particular keys.){{Citation needed|date=June 2011}}

Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and permitted total harmonic freedom at the expense of just a little impurity in every interval. This allowed greater expression through enharmonic modulation
Modulation (music)
In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

, which became extremely important in the 18th century in music of such composers as Francesco Geminiani
Francesco Geminiani
thumb|230px|Francesco Geminiani.Francesco Saverio Geminiani was an Italian violinist, composer, and music theorist.-Biography:...

, Wilhelm Friedemann Bach
Wilhelm Friedemann Bach
Wilhelm Friedemann Bach , the second child and eldest son of Johann Sebastian Bach and Maria Barbara Bach, was a German composer and performer...

, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel
Johann Gottfried Müthel
Johann Gottfried Müthel was a German composer and noted keyboard virtuoso. Along with C.P.E. Bach, he represented the Sturm und Drang style of composition....

.

The progress of Equal Temperament from mid-18th century on is described with detail in quite a few modern scholarly publications: it was already the temperament of choice during the Classical era (second half of the 18th century), and it became standard during the Early Romantic era (first decade of the 19th century), except for organs that switched to it more gradually, completing only in the second decade of the 19th century. (In England, some cathedral organists and choirmasters held out against it even after that date; Samuel Sebastian Wesley
Samuel Sebastian Wesley
Samuel Sebastian Wesley was an English organist and composer.-Biography:Born in London, he was the eldest child in the composer Samuel Wesley's second family, which he formed with Sarah Suter having separated from his wife Charlotte. Samuel Sebastian was the grandson of Charles Wesley...

, for instance, opposed it all along. He died in 1876.){{Citation needed|date=June 2011}}

A precise equal temperament is possible using the 17th-century Sabbatini method of splitting the octave first into three tempered major thirds{{Citation needed|date=June 2011}}. This was also proposed by several writers during the Classical era. Tuning with several checks, thus attaining virtually modern accuracy, was already done in the 1st decades of the 19th century{{Citation needed|date=June 2011}}. Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century{{Citation needed|date=June 2011}}. The ultimate precision was available with 2-decimal tables published by White in 1917{{Citation needed|date=June 2011}}.

It is in the environment of equal temperament that the new styles of symmetrical tonality and polytonality
Polytonality
The musical use of more than one key simultaneously is polytonality . Bitonality is the use of only two different keys at the same time...

, atonal music
Atonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...

such as that written with the twelve tone technique or serialism
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

, and jazz
Jazz
Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...

(at least its piano component) developed and flourished.

## Equal temperament in China

The earliest discussion of the equal division of the scale was in the writings of Ling Lun during the 27th century BC.

Jing Fang
Jing Fang
Jing Fang , born Li Fang , courtesy name Junming , was a Chinese music theorist, mathematician and astrologer. Born in present-day Puyang, Henan during the Han Dynasty , he was the first to notice how closely a succession of 53 just fifths approximates 31 octaves...

(78–37 BC) observed that using the Pythagorean comma of 53 just fifths approximates to 31 octaves. This would later lead to the discovery of 53 equal temperament{{fact|date=November 2011}}.

An approximation for equal temperament was given by He Chengtian, a mathematician of Southern and Northern Dynasties
Southern and Northern Dynasties
The Southern and Northern Dynasties was a period in the history of China that lasted from 420 to 589 AD. Though an age of civil war and political chaos, it was also a time of flourishing arts and culture, advancement in technology, and the spreading of Mahayana Buddhism and Daoism...

Historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese
Music of China
Chinese Music has been made since the dawn of Chinese civilization with documents and artifacts providing evidence of a well-developed musical culture as early as the Zhou Dynasty...

Zhu Zaiyu
Zhu Zaiyu
Zhu Zaiyu , a prince of the Ming dynasty of China. In 1584 Prince Zhu innovatively described the equal temperament via accurate mathematical calculation...

(朱載堉), a prince of the Ming
Ming Dynasty
The Ming Dynasty, also Empire of the Great Ming, was the ruling dynasty of China from 1368 to 1644, following the collapse of the Mongol-led Yuan Dynasty. The Ming, "one of the greatest eras of orderly government and social stability in human history", was the last dynasty in China ruled by ethnic...

court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-TET in his five thousand pages work Complete Compendium of Music and Pitch(Yuelü quan shu 乐律全书) in 1584.

He obtained his result mathematically by dividing the length of string and pipe successively by

=1.059463094359295264561825, and for pipe diameter by

(after 8 octave still in tune)

According to Gene Cho, Zhu Zaiyu was the first person to solve equal temperament problem mathematically. Murray Bardour said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyii's brilliant solution remains an enigma." The 19th-century German physicist Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.

### Zhu Zaiyu tuning instrument

Zhu Zaiyu illustrated his equal temperament theory by construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon
Victor-Charles Mahillon
Victor-Charles Mahillon was a Belgian musician and writer on musical topics. He built, collected, and described more than 1500 musical instruments....

, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the diameter of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory..

## General properties

{{Unreferenced section|date=June 2011}}

In an equal temperament, the distance between each step of the scale is the same interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. Because the perceived identity of an interval depends on its ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly-spaced, and would not permit transposition to different keys.) Specifically, the smallest interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

in an equal-tempered scale is the ratio:

where the ratio r divides the ratio p (typically the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

, which is 2/1) into n equal parts. (See Twelve-tone equal temperament below.)

Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in Ethnomusicology
Ethnomusicology
Ethnomusicology is defined as "the study of social and cultural aspects of music and dance in local and global contexts."Coined by the musician Jaap Kunst from the Greek words ἔθνος ethnos and μουσική mousike , it is often considered the anthropology or ethnography of music...

. The basic step in cents for any equal temperament can be found by taking the width of p above in cents (usually the octave, which is 1200 cents wide), called below w, and dividing it into n parts:

In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

where the modulo is the number of divisions of the octave (usually 12), these integers can be reduced to pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

es, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g. 'C' is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.

## Twelve-tone equal temperament

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...

of the interval between two adjacent notes, is the twelfth root of two:
(98.9545922303676 CENT) (99.9066043792227 CENT) (100.0000094845790 CENT) (100.0000006728490 CENT)

This interval is divided into 100 cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

s. (The cent is sometimes for this reason defined as one hundredth of a semitone.)

### Calculating absolute frequencies

To find the frequency, Pn, of a note in 12-TET, the following definition may be used:

In this formula Pn refers to the pitch, or frequency (usually in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

), you are trying to find. Pa refers to the frequency of a reference pitch (usually 440Hz). n and a refer to numbers assigned to the desired pitch and the reference pitch, respectively. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A4 (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and C4 (middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

) is the 40th key. These numbers can be used to find the frequency of C4:

### Comparison of various equal temperament in history

YEAR NAME RATIO CENTS
400 He Chengtian 1.060070671 101.0
1580 Vincenzo Galilei 18:17 99.0
1581 Zhu Zaiyu 1.059463094 100.0
1585 Simon Stevin 1.059546514 100.1
1630 Marin Mersenne 1.059322034 99.8
1630 Johann Faulhaber 1.059490385 100.0

### Comparison to just intonation

{{Unreferenced section|date=June 2011}}

The intervals of 12-TET closely approximate some intervals in just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

. The fifths and fourths are almost indistinguishably close to just.

In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

.
Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation Error
Unison (C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

)
1.000000 0 = 1.000000 0.00 0
Minor second (C
An equal temperament is a musical temperament
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

, or a system of tuning
Musical tuning
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:...

, in which every pair of adjacent notes has an identical frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ratio. As pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

is perceived roughly as the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for every note in the system.

In equal temperament tunings, an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

— usually the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

— is divided into a series of equal steps (equal frequency ratios between successive notes). For classical music
Classical music
Classical music is the art music produced in, or rooted in, the traditions of Western liturgical and secular music, encompassing a broad period from roughly the 11th century to present times...

, the most common tuning system is twelve-tone equal temperament (also known as 12 equal temperament), inconsistently abbreviated as 12-TET, 12TET, 12tET, 12tet, 12-ET, 12ET, or 12et, which divides the octave into 12 parts, all of which are equal on a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

. It is usually tuned relative to a standard pitch of 440 Hz, called A 440.

Other equal temperaments exist (some music has been written in 19-TET
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

and 31-TET
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

for example, and 24-TET
Arab tone system
The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament , the distance between each successive note being a quarter tone . Each tone has its own name not repeated in different...

) 24-TET is used in Arabic music, but in western countries when people use the term equal temperament without qualification, it is usually understood that they are talking about 12-TET.

Equal temperaments may also divide some interval other than the octave, a pseudo-octave
Pseudo-octave
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...

, into a whole number of equal steps. An example is an equal-tempered Bohlen–Pierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is sometimes preferred. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on.

String ensembles and vocal groups, who have no mechanical tuning limitations, often use a tuning much closer to just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

, as it is naturally more consonant
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

. Other instruments, such as some wind, keyboard, and fret
Fret
A fret is a raised portion on the neck of a stringed instrument, that extends generally across the full width of the neck. On most modern western instruments, frets are metal strips inserted into the fingerboard...

ted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Other wind instruments, that can easily and spontaneously bend their tone, most notably double-reeds
Double reed
A double reed is a type of reed used to produce sound in various wind instruments. The term double reed comes from the fact that there are two pieces of cane vibrating against each other. A single reed consists of one piece of cane which vibrates against a mouthpiece made of metal, hardened...

, use tuning similar to string ensembles and vocal groups.

## Equal temperament in Europe

### Early history

One of the earliest discussions of equal temperament occurs in the writiting of Aristoxenus
Aristoxenus
Aristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and...

in the 4th century B.C.

Vincenzo Galilei
Vincenzo Galilei
Vincenzo Galilei was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicist Galileo Galilei and of the lute virtuoso and composer Michelagnolo Galilei...

(father of Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

) was one of the first practical advocates of twelve-tone equal temperament. He composed a sets of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "Fronimo", 24 +1 ricercar
Ricercar
A ricercar is a type of late Renaissance and mostly early Baroque instrumental composition. The term means to search out, and many ricercars serve a preludial function to "search out" the key or mode of a following piece...

s. He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567. Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino
Francesco Spinacino
Francesco Spinacino was an Italian lutenist and composer. His surviving output comprises the first two volumes of Ottaviano Petrucci's influential series of lute music publications: Intabolatura de lauto libro primo and Intabolatura de lauto libro secondo...

wrote a "Recercare de tutti li Toni" as early as 1507. In the 17th century lutenist-composer John Wilson
John Wilson (composer)
John Wilson , was an English composer, lutenist and teacher. Born in Faversham, Kent, he moved to London by 1614, where he succeeded Robert Johnson as principal composer for the King's Men, and entered the King's Musick in 1635 as a lutenist. He received the degree of D.Mus from Oxford in 1644,...

wrote a set of 26 preludes including 24 in all the major/minor keys.

Henricus Grammateus
Henricus Grammateus
Henricus Grammateus was a German mathematician. In 1518 he published details of a new musical temperament, which is now named after him, for the harpsichord...

drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".
Simon Stevin
The first mention of equal temperament related to Twelfth root of two in the West appeared in Simon Stevin
Simon Stevin
Simon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...

's unfinished manuscript Van De Spiegheling der signconst (ca 1605) published posthumously nearly three centuries later in 1884. However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values. As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is incorrect.

The following were Simon Stevin's chord length from Vande Spiegheling der signconst:
TONE CHORD 10000 from Simon Stevin RATIO CORRECTED CHORD
semitone 9438 1.0595465 9438.7
whole tone 8909 1.0593781
1.5 tone 8404 1.0600904 8409
ditone 7936 1.0594758 7937
ditone and a half 7491 1.0594046 7491.5
tritone 7071 1.0593975 7071.1
tritone and a half 6674 1.0594845 6674.2
four-tone 6298 1.0597014 6299
four-tone-and-half 5944 1.0595558 5946
five-tone 5611 1.0593477 5612.3
five-tone-and-half 5296 1.0594788 5297.2
full tone 1.0592000

A generation later, French mathematician Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

presented several equal tempered
chord lengths obtained by Jean Beaugrand, Ismael Bouillaud and Jean Galle.

In 1630 Johann Faulhaber
Johann Faulhaber
Johann Faulhaber was a German mathematician.Born in Ulm, Faulhaber trained as a weaver and later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen...

published a 100 cent monochord table, with the exception of several errors due to his use of logarithmic tables . He did not explain how he obtained his results.

Zarlino in his polemic
Polemic
A polemic is a variety of arguments or controversies made against one opinion, doctrine, or person. Other variations of argument are debate and discussion...

with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute
Lute
Lute can refer generally to any plucked string instrument with a neck and a deep round back, or more specifically to an instrument from the family of European lutes....

in his "Sopplimenti musicali" in 1588.

### Equal temperament in the Baroque era

From 1450 to about 1800 plucked instrument players (lutenists and guitarists) generally favored equal temperament, and the Brossard lute Manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Bocquet written in a all keys, including the last prelude, entitled "Prelude sur tous les tons", which enharmonically modulates through all of the keys. Wind and keyboard musicians expected much less mistuning (than that of equal temperament) in the most common keys, such as C major. They used approximations that emphasized the tuning of third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

s or fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

s in these keys, such as meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

. Among the 17th century keyboard composers Girolamo Frescobaldi
Girolamo Frescobaldi
Girolamo Frescobaldi was a musician from Ferrara, one of the most important composers of keyboard music in the late Renaissance and early Baroque periods. A child prodigy, Frescobaldi studied under Luzzasco Luzzaschi in Ferrara, but was influenced by a large number of composers, including Ascanio...

advocated equal temperament. Some theorists, such as Giuseppe Tartini
Giuseppe Tartini
Giuseppe Tartini was an Italian baroque composer and violinist.-Biography:Tartini was born in Piran, a town on the peninsula of Istria, in the Republic of Venice to Gianantonio – native of Florence – and Caterina Zangrando, a descendant of one of the oldest aristocratic Piranian families.It...

, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although Andreas Werckmeister
Andreas Werckmeister
Andreas Werckmeister was an organist, music theorist, and composer of the Baroque era.-Life:Born in Benneckenstein, Germany, Werckmeister attended schools in Nordhausen and Quedlinburg. He received his musical training from his uncles Heinrich Christian Werckmeister and Heinrich Victor Werckmeister...

emphatically advocated equal temperament in his 1707 treatise published posthumously.

J. S. Bach wrote The Well-Tempered Clavier
The Well-Tempered Clavier
The Well-Tempered Clavier , BWV 846–893, is a collection of solo keyboard music composed by Johann Sebastian Bach...

to demonstrate the musical possibilities of well temperament
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...

, where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe{{Weasel-inline|date=June 2011}} that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they each described the subtly different dissonances made available within a particular tuning method. However, it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about the moods and colors of particular keys.){{Citation needed|date=June 2011}}

Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and permitted total harmonic freedom at the expense of just a little impurity in every interval. This allowed greater expression through enharmonic modulation
Modulation (music)
In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

, which became extremely important in the 18th century in music of such composers as Francesco Geminiani
Francesco Geminiani
thumb|230px|Francesco Geminiani.Francesco Saverio Geminiani was an Italian violinist, composer, and music theorist.-Biography:...

, Wilhelm Friedemann Bach
Wilhelm Friedemann Bach
Wilhelm Friedemann Bach , the second child and eldest son of Johann Sebastian Bach and Maria Barbara Bach, was a German composer and performer...

, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel
Johann Gottfried Müthel
Johann Gottfried Müthel was a German composer and noted keyboard virtuoso. Along with C.P.E. Bach, he represented the Sturm und Drang style of composition....

.

The progress of Equal Temperament from mid-18th century on is described with detail in quite a few modern scholarly publications: it was already the temperament of choice during the Classical era (second half of the 18th century), and it became standard during the Early Romantic era (first decade of the 19th century), except for organs that switched to it more gradually, completing only in the second decade of the 19th century. (In England, some cathedral organists and choirmasters held out against it even after that date; Samuel Sebastian Wesley
Samuel Sebastian Wesley
Samuel Sebastian Wesley was an English organist and composer.-Biography:Born in London, he was the eldest child in the composer Samuel Wesley's second family, which he formed with Sarah Suter having separated from his wife Charlotte. Samuel Sebastian was the grandson of Charles Wesley...

, for instance, opposed it all along. He died in 1876.){{Citation needed|date=June 2011}}

A precise equal temperament is possible using the 17th-century Sabbatini method of splitting the octave first into three tempered major thirds{{Citation needed|date=June 2011}}. This was also proposed by several writers during the Classical era. Tuning with several checks, thus attaining virtually modern accuracy, was already done in the 1st decades of the 19th century{{Citation needed|date=June 2011}}. Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century{{Citation needed|date=June 2011}}. The ultimate precision was available with 2-decimal tables published by White in 1917{{Citation needed|date=June 2011}}.

It is in the environment of equal temperament that the new styles of symmetrical tonality and polytonality
Polytonality
The musical use of more than one key simultaneously is polytonality . Bitonality is the use of only two different keys at the same time...

, atonal music
Atonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...

such as that written with the twelve tone technique or serialism
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

, and jazz
Jazz
Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...

(at least its piano component) developed and flourished.

## Equal temperament in China

The earliest discussion of the equal division of the scale was in the writings of Ling Lun during the 27th century BC.

Jing Fang
Jing Fang
Jing Fang , born Li Fang , courtesy name Junming , was a Chinese music theorist, mathematician and astrologer. Born in present-day Puyang, Henan during the Han Dynasty , he was the first to notice how closely a succession of 53 just fifths approximates 31 octaves...

(78–37 BC) observed that using the Pythagorean comma of 53 just fifths approximates to 31 octaves. This would later lead to the discovery of 53 equal temperament{{fact|date=November 2011}}.

An approximation for equal temperament was given by He Chengtian, a mathematician of Southern and Northern Dynasties
Southern and Northern Dynasties
The Southern and Northern Dynasties was a period in the history of China that lasted from 420 to 589 AD. Though an age of civil war and political chaos, it was also a time of flourishing arts and culture, advancement in technology, and the spreading of Mahayana Buddhism and Daoism...

Historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese
Music of China
Chinese Music has been made since the dawn of Chinese civilization with documents and artifacts providing evidence of a well-developed musical culture as early as the Zhou Dynasty...

Zhu Zaiyu
Zhu Zaiyu
Zhu Zaiyu , a prince of the Ming dynasty of China. In 1584 Prince Zhu innovatively described the equal temperament via accurate mathematical calculation...

(朱載堉), a prince of the Ming
Ming Dynasty
The Ming Dynasty, also Empire of the Great Ming, was the ruling dynasty of China from 1368 to 1644, following the collapse of the Mongol-led Yuan Dynasty. The Ming, "one of the greatest eras of orderly government and social stability in human history", was the last dynasty in China ruled by ethnic...

court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-TET in his five thousand pages work Complete Compendium of Music and Pitch(Yuelü quan shu 乐律全书) in 1584.

He obtained his result mathematically by dividing the length of string and pipe successively by

=1.059463094359295264561825, and for pipe diameter by

(after 8 octave still in tune)

According to Gene Cho, Zhu Zaiyu was the first person to solve equal temperament problem mathematically. Murray Bardour said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyii's brilliant solution remains an enigma." The 19th-century German physicist Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.

### Zhu Zaiyu tuning instrument

Zhu Zaiyu illustrated his equal temperament theory by construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon
Victor-Charles Mahillon
Victor-Charles Mahillon was a Belgian musician and writer on musical topics. He built, collected, and described more than 1500 musical instruments....

, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the diameter of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory..

## General properties

{{Unreferenced section|date=June 2011}}

In an equal temperament, the distance between each step of the scale is the same interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. Because the perceived identity of an interval depends on its ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly-spaced, and would not permit transposition to different keys.) Specifically, the smallest interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

in an equal-tempered scale is the ratio:

where the ratio r divides the ratio p (typically the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

, which is 2/1) into n equal parts. (See Twelve-tone equal temperament below.)

Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in Ethnomusicology
Ethnomusicology
Ethnomusicology is defined as "the study of social and cultural aspects of music and dance in local and global contexts."Coined by the musician Jaap Kunst from the Greek words ἔθνος ethnos and μουσική mousike , it is often considered the anthropology or ethnography of music...

. The basic step in cents for any equal temperament can be found by taking the width of p above in cents (usually the octave, which is 1200 cents wide), called below w, and dividing it into n parts:

In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

where the modulo is the number of divisions of the octave (usually 12), these integers can be reduced to pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

es, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g. 'C' is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.

## Twelve-tone equal temperament

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...

of the interval between two adjacent notes, is the twelfth root of two:
(98.9545922303676 CENT) (99.9066043792227 CENT) (100.0000094845790 CENT) (100.0000006728490 CENT)

This interval is divided into 100 cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

s. (The cent is sometimes for this reason defined as one hundredth of a semitone.)

### Calculating absolute frequencies

To find the frequency, Pn, of a note in 12-TET, the following definition may be used:

In this formula Pn refers to the pitch, or frequency (usually in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

), you are trying to find. Pa refers to the frequency of a reference pitch (usually 440Hz). n and a refer to numbers assigned to the desired pitch and the reference pitch, respectively. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A4 (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and C4 (middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

) is the 40th key. These numbers can be used to find the frequency of C4:

### Comparison of various equal temperament in history

YEAR NAME RATIO CENTS
400 He Chengtian 1.060070671 101.0
1580 Vincenzo Galilei 18:17 99.0
1581 Zhu Zaiyu 1.059463094 100.0
1585 Simon Stevin 1.059546514 100.1
1630 Marin Mersenne 1.059322034 99.8
1630 Johann Faulhaber 1.059490385 100.0

### Comparison to just intonation

{{Unreferenced section|date=June 2011}}

The intervals of 12-TET closely approximate some intervals in just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

. The fifths and fourths are almost indistinguishably close to just.

In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

.
Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation Error
Unison (C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

)
1.000000 0 = 1.000000 0.00 0
Minor second (C{{music/D{{music) 1.059463 100 = 1.066667 111.73 −11.73
Major second (D
D (musical note)
D is a musical note a whole tone above C, and is known as Re within the solfege system.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of middle D is approximately 293.665 Hz. See pitch for a discussion of historical variations in...

)
1.122462 200 = 1.125000 203.91 −3.91
Minor third (D{{music/E{{music) 1.189207 300 = 1.200000 315.64 −15.64
Major third (E
E (musical note)
E or mi is the third note of the solfège.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle E is approximately 329.628 Hz. See pitch for a discussion of historical variations in frequency.-Designation by octave:...

)
1.259921 400 = 1.250000 386.31 +13.69
Perfect fourth (F
F (musical note)
F is a musical note, the fourth above C. It is also known as fa in fixed-do solfège.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle F is approximately 349.228 Hz. See pitch for a discussion of historical variations in...

)
1.334840 500 = 1.333333 498.04 +1.96
Augmented fourth (F{{music/G{{music) 1.414214 600 = 1.400000 582.51 +17.49
Perfect fifth (G
G (musical note)
Sol, So, or G is the fifth note of the solfège starting on C. As such it is the dominant, a perfect fifth above C.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle G note is approximately 391.995 Hz...

)
1.498307 700 = 1.500000 701.96 −1.96
Minor sixth (G{{music/A{{music) 1.587401 800 = 1.600000 813.69 −13.69
Major sixth (A
A (musical note)
La or A is the sixth note of the solfège. "A" is generally used as a standard for tuning. When the orchestra tunes, the oboe plays an "A" and the rest of the instruments tune to match that pitch. Every string instrument in the orchestra has an A string, from which each player can tune the rest of...

)
1.681793 900 = 1.666667 884.36 +15.64
Minor seventh (A{{music/B{{music) 1.781797 1000 = 1.750000 968.83 +31.17
Major seventh (B
B (musical note)
B, also known as H, Si or Ti, is the seventh note of the solfège. It lies a chromatic semitone below C and is thus the enharmonic equivalent of C-flat....

)
1.887749 1100 = 1.875000 1088.27 +11.73
Octave (C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

)
2.000000 1200 = 2.000000 1200.00 0

## Seven-tone equal division of the fifth

Violins, violas and cellos are tuned in perfect fifths (G - D - A - E), which suggests that their semi-tone ratio will be slightly higher than in the conventional Twelve-tone Equal Temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval is covered in 7 steps, each tone is in the ratio of to the next, which provides for a perfect fifth with ratio of 3:2 but a slightly widened octave with ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1 ratio. During actual play, however, the violinist chooses his pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio.

## Other equal temperaments

### 5 and 7 tone temperaments in ethnomusicology

Five and seven tone equal temperament (5-TET {{audio|5-tet scale on C.mid|Play}} and 7-TET), with 240 {{Audio|1 step in 5-et on C.mid|Play}} and 171 {{Audio|1 step in 7-et on C.mid|Play}} cent steps respectively, are fairly common. A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system. Indonesian gamelan
Gamelan
A gamelan is a musical ensemble from Indonesia, typically from the islands of Bali or Java, featuring a variety of instruments such as metallophones, xylophones, drums and gongs; bamboo flutes, bowed and plucked strings. Vocalists may also be included....

s are tuned to 5-TET according to Kunst
Jaap Kunst
Jaap Kunst was a Dutch ethnomusicologist, particularly associated with the study of gamelan music of Indonesia...

(1949), but according to Hood
Mantle Hood
Mantle Hood was an American ethnomusicologist. Among other areas, he specialized in studying gamelan music from Indonesia. Hood pioneered, in the 1950s and 1960s, a new approach to the study of music, and the creation at UCLA of the first American university program devoted to ethnomusicology...

(1966) and McPhee
Colin McPhee
Colin McPhee was a Canadian composer and musicologist. He is primarily known for being the first Western composer to make an ethnomusicological study of Bali, and for the quality of that work...

(1966) their tuning varies widely, and according to Tenzer
Michael Tenzer
Michael Tenzer is a composer, performer, educator and scholar.Tenzer was born in New York City and studied music at Yale University and University of California, Berkeley...

(2000) they contain stretched octaves
Pseudo-octave
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...

. It is now well-accepted that of the two primary tuning systems in gamelan music, slendro
Slendro
Slendro is a pentatonic scale, one of the two most common scales used in Indonesian gamelan music, the other being pélog.-Tuning:...

and pelog
Pelog
Pelog is one of the two essential scales of gamelan music native to Bali and Java, in Indonesia. The other scale commonly used is called slendro. Pelog has seven notes, but many gamelan ensembles only have keys for five of the pitches...

, only slendro somewhat resembles five-tone equal temperament while pelog is highly unequal; however, Surjodiningrat et al. (1972) has analyzed pelog as a seven-note subset of nine-tone equal temperament (133 cent steps {{Audio|Semitone_Maximus_on_C.mid|Play}}). A South American Indian scale from a preinstrumental culture measured by Boiles (1969) featured 175 cent seven tone equal temperament, which stretches the octave slightly as with instrumental gamelan music.

### Various Western equal temperaments

31 tone equal temperament there advocated by Christiaan Huygens and Adriaan Fokker
Adriaan Daniël Fokker , was a Dutch physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the aeronautical engineer Anthony Fokker...

. 31-TET has a slightly less accurate fifth than 12-TET, but provides near-just major thirds, and provides decent matches for harmonics up to at least 13, of which the seventh harmonic is particularly accurate.

In the 20th century, standardized Western pitch and notation practices having been placed on a 12-TET foundation made the quarter tone scale (or 24-TET) a popular microtonal tuning.

29-TET is the lowest number of equal divisions of the octave which produces a better perfect fifth than 12-TET. Its major third is roughly as inaccurate as 12-TET, however it is tuned 14 cents flat rather than 14 cents sharp.

41-TET
41 equal temperament
In music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally-sized steps . Each step represents a frequency ratio of 21/41, or 29.27 cents , an interval close in size to the septimal comma. 41-ET can be seen as a...

is the second lowest number of equal divisions which produces a better perfect fifth than 12-TET.

53-TET is better at approximating the traditional just
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

consonances than 12, 19 or 31-TET, but has had only occasional use. Its extremely good perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

s make it interchangeable with an extended Pythagorean tuning
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...

, but it also accommodates schismatic temperament
Schismatic temperament
The schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz temperament.-Comparison with other tunings:...

, and is sometimes used in Turkish music theory. It does not, however, fit the requirements of meantone temperaments which put good thirds within easy reach via the cycle of fifths. In 53-TET the very consonant thirds would be reached instead by strange enharmonic relationships. A consequence of this is that chord progressions like I-vi-ii-V-I won't land you back where you started in 53-TET, but rather one 53-tone step flat (unless the motion by I-vi wasn't by the 5-limit minor third).

Another extension of 12-TET is 72-TET (dividing the semitone into 6 equal parts), which though not a meantone tuning, approximates well most just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

intervals, even less traditional ones such as 7/4, 9/7, 11/5, 11/6 and 11/7. 72-TET has been taught, written and performed in practice by Joe Maneri
Joe Maneri
Joseph Gabriel Esther "Joe" Maneri , was an American jazz composer, saxophone and clarinet player. Violinist Mat Maneri is his son....

and his students (whose atonal inclinations interestingly typically avoid any reference to just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

whatsoever).

Other equal divisions of the octave that have found occasional use include 14-TET, 15-TET, 16-TET, 17-TET, 19-TET, 22-TET, 34-TET, 46-TET, 48-TET, 99-TET, and 171-TET.

### Equal temperaments of non-octave intervals

The equal-tempered version of the Bohlen–Pierce scale consists of the ratio 3:1, 1902 cents, conventionally a perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

wider than an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

, called in this theory a tritave ({{Audio|Tritave on C.mid|play}}), and split into a thirteen equal parts. This provides a very close match to justly tuned
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

ratios consisting only of odd numbers. Each step is 146.3 cents ({{Audio|BP scale et.mid|play}}), or .

Wendy Carlos
Wendy Carlos
Wendy Carlos is an American composer and electronic musician. Carlos first came to notice in the late 1960s with recordings made on the Moog synthesizer, then a relatively new and unknown instrument; most notable were LPs of synthesized Bach and the soundtrack for Stanley Kubrick's film A...

created three unusual equal temperaments after a thorough study of the properties of possible temperaments having a step size between 30 and 120 cents. These were called alpha, beta, and gamma. They can be considered as equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes:
• alpha: (78.0 cents)
• beta: (63.8 cents)
• gamma: (35.1 cents)

Alpha and Beta may be heard on the title track of her 1986 album Beauty in the Beast
Beauty in the Beast
Beauty in the Beast is an album by Wendy Carlos using alternate tunings and scales and influenced by jazz and world music. On the back she includes a quote by Van Gogh: "I am always doing what I cannot do yet, in order to learn how to do it."...

.

• Musical acoustics
Musical acoustics
Musical acoustics or music acoustics is the branch of acoustics concerned with researching and describing the physics of music – how sounds employed as music work...

(the physics of music)
• Music and mathematics
Music and mathematics
Music theorists often use mathematics to understand music. Indeed, mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical"...

• Microtuner
Microtuner
A microtuner or microtonal tuner is an electronic device or software program designed to modify and test the tuning of musical instruments with microtonal precision, allowing for the design and construction of microtonal scales and just intonation scales, and for tuning intervals that differ from...

• Microtonal music
Microtonal music
Microtonal music is music using microtones—intervals of less than an equally spaced semitone. Microtonal music can also refer to music which uses intervals not found in the Western system of 12 equal intervals to the octave.-Terminology:...

• Piano key frequencies
Piano key frequencies
This is a virtual keyboard showing the absolute frequencies in hertz of the notes on a modern piano in twelve-tone equal temperament, with the 49th key, the fifth A , tuned to 440 Hz...

• Piano tuning
Piano tuning
Piano tuning is the act of making minute adjustments to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is in tune. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches...

• Semitone
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....

• List of meantone intervals
• Diatonic and chromatic
Diatonic and chromatic
Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to intervals, chords, notes, musical styles, and kinds of harmony...

• Electronic tuner
Electronic tuner
The term electronic tuner can refer to a number of different things, depending which discipline you wish to study.In the Discipline of radio frequency electronics an electronic tuner is a device which tunes across a part of the radio frequency spectrum by the application of a voltage or appropriate...