Equal temperament
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Equal temperament is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings an interval — usually the octave — is divided into a series of equal steps (equal frequency ratios). For modern Western music, the most common tuning system is twelvetone equal temperament, inconsistently abbreviated as 12TET, 12TET, 12tET, or 12tet, which divides the octave into 12 (logarithmically) equal parts. 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 19TET and 31TET for example, and Arabian music is based on 24TET), but in western countries when people use the term equal temperament without qualification, it is usually understood that they are talking about 12TET.
Equal temperaments may also divide some interval other than the octave, a pseudooctave, into a whole number of equal steps. An example is an equallytempered BohlenPierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is sometimes preferred. According to this naming system, 12TET is called 12EDO, 31TET is called 31EDO, and so on; however, when composers and musictheorists use "EDO" their intention is generally that a temperament (i.e., a reference to just intonation intervals) is not implied.
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[edit] History
Vincenzo Galilei (father of Galileo Galilei) was one of the first advocates of twelvetone equal temperament in a 1581 treatise, along two sets of dance suites on each of the 12 notes of the chromatic scale, and 24 ricercars in all the "major/minor keys". His countryman and fellow lutenist Giacomo Gorzanis had written music based on this temperament by 1567. Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino wrote a "Recercare de tutti li Toni" as early as 1507. In the 17th century lutenistcomposer John Wilson wrote a set of 26 preludes including 24 in all the major/minor keys.
Historically, there was a sevenequal temperament or heptaequal temperament practice in ancient Chinese tradition.^{[1]}^{[2]} The first person known to have attempted a numerical specification for 12TET is probably Zhu Zaiyu (朱載堉) a prince of the Ming court, who published a theory of the temperament in 1584. It is possible that this idea was spread to Europe by way of trade, which intensified just at the moment when Zhu Zaiyu published his calculations. Within fiftytwo years of Zhu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin.
From 1450 to about 1800 plucked instrument players (lutenists and guitarists) generally favored equal temperament. 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 thirds or fifths in these keys, such as meantone temperament. Among the 17th century keyboard composers Girolamo Frescobaldi advocated equal temperament. Some theorists, such as Giuseppe Tartini, 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 emphatically advocated equal temperament in his 1707 treatise published posthumously.
String ensembles and vocal groups, who have no mechanical tuning limitations, often use a tuning much closer to just intonation, as it is naturally more consonant. Other instruments, such as some wind, keyboard, and fretted 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 doublereeds, use tuning similar to string ensembles and vocal groups.
J. S. Bach wrote The WellTempered Clavier to demonstrate the musical possibilities of well temperament, 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 was a better approximation to just intonation than the nearby alternative equal temperaments. It permitted total harmonic freedom at the expense of just a little purity in every interval. This allowed greater expression through enharmonic modulation, which became extremely important in the 18th century in music of such composers as Francesco Geminiani, Wilhelm Friedemann Bach, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel. Equal temperament became the standard gradually during the Romantic era.
A precise equal temperament was not attainable until Johann Heinrich Scheibler developed a tuning fork tonometer in 1834 to accurately measure pitches. The use of this device was not widespread, and it was not until 1917 that William Braid White published a practical aural method of tuning the piano to equal temperament.^{[citation needed]}
It is in the environment of equal temperament that the new styles of symmetrical tonality and polytonality, atonal music such as that written with the twelve tone technique or serialism, and jazz (at least its piano component) developed and flourished.
[edit] General properties
In an equal temperament, the distance between each step of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenlyspaced, and would not permit transposition to different keys.) Specifically, the smallest interval in an equal tempered scale is the ratio:
where the ratio r divides the ratio p (often the octave, which is 2/1) into n equal parts. (See Twelvetone equal temperament below.)
Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in Ethnomusicology. 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 of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulo is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes, 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.
[edit] Twelvetone equal temperament
In twelvetone equal temperament, which divides the octave into 12 equal parts, the ratio of frequencies between two adjacent semitones is the twelfth root of two:
This interval is equal to 100 cents. (The cent is sometimes for this reason defined as one hundredth of a semitone.)
[edit] Calculating absolute frequencies
To find the frequency, P_{n}, of a note in 12TET, the following definition may be used:
In this formula P_{n} refers to the pitch, or frequency (usually in hertz), you are trying to find. P_{a} 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) is the 40th key. These numbers can be used to find the frequency of C4:
[edit] Comparison to just intonation
The intervals of 12TET closely approximate some intervals in Just intonation. In particular, it approximates just fourths, fifths, thirds, and sixths better than any equal temperament with fewer divisions of the octave. Its fifths and fourths in particular are almost indistinguishably close to just. In general the next lowest viable equal temperament (as an approximation to just) is 19TET, which has better thirds and sixths, but weaker fourths and fifths than 12TET.
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.
Name  Exact value in 12TET  Decimal value in 12TET  Cents  Just intonation interval  Cents in just intonation  Difference 

Unison (C)  1.000000  0  = 1.000000  0.0000  0  
Minor second (C♯)  1.059463  100  = 1.066667  111.73  11.73  
Major second (D)  1.122462  200  = 1.125000  203.91  3.91  
Minor third (D♯)  1.189207  300  = 1.200000  315.64  15.64  
Major third (E)  1.259921  400  = 1.250000  386.31  −13.69  
Perfect fourth (F)  1.334840  500  = 1.333333  498.04  −1.96  
Augmented fourth (F♯)  1.414214  600  = 1.400000  582.51  −17.49  
Perfect fifth (G)  1.498307  700  = 1.500000  701.96  1.96  
Minor sixth (G♯)  1.587401  800  = 1.600000  813.69  13.69  
Major sixth (A)  1.681793  900  = 1.666667  884.36  −15.64  
Minor seventh (A♯)  1.781797  1000  = 1.750000  968.826  −31.91  
Major seventh (B)  1.887749  1100  = 1.875000  1088.27  −11.73  
Octave (C)  2.000000  1200  = 2.000000  1200.0  0 
(These mappings from equal temperament to just intonation are by no means unique. The minor seventh, for example, can be meaningfully said to approximate 9/5, 7/4, or 16/9 depending on context. The 7/4 ratio is used to emphasize this tuning's poor fit to the 7th partial in the harmonic series.)
[edit] Other equal temperaments
[edit] 5 and 7 tone temperaments in ethnomusicology
Five and seven tone equal temperament (5TET and 7TET), with 240 and 171 cent steps respectively, are fairly common. A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system. Indonesian gamelans are tuned to 5TET according to Kunst (1949), but according to Hood (1966) and McPhee (1966) their tuning varies widely, and according to Tenzer (2000) they contain stretched octaves. It is now wellaccepted that of the two primary tuning systems in gamelan music, slendro and pelog, only slendro somewhat resembles fivetone equal temperament while pelog is highly unequal; however, Surjodiningrat et al. (1972) has analyzed pelog as a sevennote subset of ninetone equal temperament. A South American Indian scale from a preinstrumental culture measured by Boiles (1969) featured 175 cent equal temperament, which stretches the octave slightly as with instrumental gamelan music.
[edit] Various Western equal temperaments
Many systems that divide the octave equally can be considered relative to other systems of temperament. 19TET and especially 31TET are extended varieties of Meantone temperament and approximate most just intonation intervals considerably better than 12TET. They have been used sporadically since the 16th century, with 31TET particularly popular in the Netherlands, there advocated by Christiaan Huygens and Adriaan Fokker. 31TET, like most Meantone temperaments, has a less accurate fifth than 12TET.
There are in fact five numbers by which the octave can be equally divided to give progressively smaller total mistuning of thirds, fifths and sixths (and hence minor sixths, fourths and minor thirds): 12, 19, 31, 34 and 53. The sequence continues with 118, 441, 612..., but these finer divisions produce improvements that are not audible.
In the 20th century, standardized Western pitch and notation practices having been placed on a 12TET foundation made the quarter tone scale (or 24TET) a popular microtonal tuning. Though it only improved nontraditional consonances, such as 11/4, 24TET can be easily constructed by superimposing two 12TET systems tuned half a semitone apart. It is based on steps of 50 cents, or .
29TET is the lowest number of equal divisions of the octave which produces a better perfect fifth than 12TET; however, it does not contain a good approximation of the pure major third, and so it is not widely used.
41TET is the second lowest number of equal divisions which produces a better perfect fifth than 12TET. It is not often used, however. (One of the reasons 12TET is so widely favoured among the equal temperaments is that it is very practical in that with an economical number of keys it achieves better consonance than the other systems with a comparable number of tones.)
53TET is better at approximating the traditional just consonances than 12, 19 or 31TET, but has had only occasional use. Its extremely good perfect fifths make it interchangeable with an extended Pythagorean tuning, but it also accommodates schismatic temperament, 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 53TET the very consonant thirds would be reached instead by strange enharmonic relationships. (Another tuning which has seen some use in practice and is not a meantone system is 22TET.)
Another extension of 12TET is 72TET (dividing the semitone into 6 equal parts), which though not a meantone tuning, approximates well most just intonation intervals, even less traditional ones such as 7/4, 9/7, 11/5, 11/6 and 11/7. 72TET has been taught, written and performed in practice by Joe Maneri and his students (whose atonal inclinations interestingly typically avoid any reference to just intonation whatsoever).
Other equal divisions of the octave that have found occasional use include 15TET, 34TET, 41TET, 46TET, 48TET, 99TET, and 171TET.
[edit] Equal temperaments of nonoctave intervals
The equal tempered version of the BohlenPierce scale consists of the ratio 3:1, 1902 cents, conventionally a perfect fifth wider than an octave, called in this theory a tritave ( play (help·info)), and split into a thirteen equal parts. This provides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents ( play (help·info)), or .
Wendy Carlos discovered 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.^{[4]} 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.
[edit] See also
 Physics of music
 Mathematics of musical scales
 Microtuner
 Microtonal music
 Piano key frequencies
 Piano tuning
 Semitone
 List of meantone intervals
 Diatonic and chromatic
 Electronic tuner
[edit] Notes
 ^ Findings of new literatures concerning the hepta  equal temperament (Chinese)
 ^ "七平均律"琐谈兼及旧式均孔曲笛制作与转调 (Chinese) (abstract of About "Seven equal tuning System")
 ^ Sethares compares several equal temperament scales in a graph with axes reversed from the axes here. (fig. 4.6, p. 58)
 ^ THREE ASYMMETRIC DIVISIONS OF THE OCTAVE by Wendy Carlos
[edit] References
 Burns, Edward M. (1999). "Intervals, Scales, and Tuning", The Psychology of Music second edition. Deutsch, Diana, ed. San Diego: Academic Press. ISBN 0122135644. Cited:

 Ellis, C. (1965). "Preinstrumental scales", Journal of the Acoustical Society of America, 9, 126144. Cited in Burns (1999).
 Morton, D. (1974). "Vocal tones in traditional Thai music", Selected reports in ethnomusicology (Vol. 2, p.8899). Los Angeles: Institute for Ethnomusicology, UCLA. Cited in Burns (1999).
 Haddon, E. (1952). "Possible origin of the Chopi Timbila xylophone", African Music Society Newsletter, 1, 6167. Cited in Burns (1999).
 Kunst, J. (1949). Music in Java (Vol. II). The Hague: Marinus Nijhoff. Cited in Burns (1999).
 Hood, M. (1966). "Slendro and Pelog redefined", Selected Reports in Ethnomusicology, Institute of Ethnomusicology, UCLA, 1, 3648. Cited in Burns (1999).
 Temple, Robert K. G. (1986)."The Genius of China". ISBN 0671620282. Cited in Burns (1999).
 Tenzer, (2000). Gamelan Gong Kebyar: The Art of TwentiethCentury Balinese Music. ISBN 0226792811 and ISBN 0226792838. Cited in Burns (1999).
 Boiles, J. (1969). "Terpehua thoughsong", Ethnomusicology, 13, 4247. Cited in Burns (1999).
 Wachsmann, K. (1950). "An equalstepped tuning in a Ganda harp", Nature (Longdon), 165, 40. Cited in Burns (1999).
 Cho, Gene Jinsiong. (2003). The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston, NY: Edwin Mellen Press.
 Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0870132903
 Sethares, William A. (2005). Timbre, Spectrum, Scale (2nd ed. ed.). London: SpringerVerlag. ISBN 1852337974.
 Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972) Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta, Gadjah Mada University Press, Jogjakarta 1972. Cited on http://web.telia.com/~u57011259/pelog_main.htm, accessed May 19, 2006.
 Stewart, P. J. (2006) "From Galaxy to Galaxy: Music of the Spheres" [1]
 Khramov, Mykhaylo. "Approximation of 5limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave", Proceedings of the International Conference SIGMAP2008, 2629 July 2008, Porto, pp. 181184, ISBN 9789898111609
[edit] External links
 Explaining the Equal Temperament
 An Introduction to Historical Tunings
 HuygensFokker Foundation Centre for Microtonal Music
 Telemann's New Musical System
 A.D. Fokker: Simon Stevin's views on music
 Music of Sacred Temperament
 A.Orlandini: Music Acoustics
 "Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)
 Tuned Piano  a web piano tuned to equal temperament.
 12TET Frequency Table Maker  Creates a frequency table (Hz.) for all 12TET pitches
 Brief description of Scheibler's tonometer
 Barbieri, Patrizio. Enharmonic instruments and music, 14701900. (2008) Latina, Il Levante Libreria Editrice
