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Other equal temperaments exist (some music has been written in 19-TET and 31-TET for example, and Arabian music is based on 24-TET ), 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 , into a whole number of equal steps. An example is an equally-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; however, when composers and music-theorists use "EDO" their intention is generally that a temperament (i.e., a reference to just intonation intervals) is not implied. HISTORY Historically, there was Seven-equal temperament or Hepta-equal temperament practice in Ancient Music Of China tradition,http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm Findings of new literatures concerning the hepta - equal temperament http://scholar.ilib.cn/Abstract.aspx?A=xhyyxyxb200102005 "七平均律"琐谈--兼及旧式均孔曲笛制作与转调, but whether it is real equal temperament or not is a controversial topic in academic circles. Vincenzo Galilei (father of Galileo Galilei ) may have been the first person to advocate equal temperament (in a 1581 treatise), although his countryman and fellow Lutenist Giacomo Gorzanis had written music based on equal temperament by 1567. The first person known to have attempted a numerical specification for equal temperament is probably Zhu Zaiyu (朱載堉) a prince of 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 new theory. Within fifty-two years of Chu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin . From 1450 to about 1800 there is evidence that musicians expected much less mistuning (than that of Equal Temperament) in the most common keys, such as C major. Instead, they used approximations that emphasized the tuning of Third s or Fifth s in these keys, such as Meantone 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. Others take issue with dissonance in the higher register, where Beating between harmonics of mistuned consonances is faster, and Combinational Tones are more pronounced. 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 Fret ted-instruments, often only approximate equal temperament, where technical limitations prevent exact tunings, other wind instruments, who can easily and spontaneously bend their tone, most notably Double-reeds , use tuning similar to string ensembles and vocal groups. J. S. Bach wrote The Well-Tempered 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 Modulation , which became extremely important in the 19th century music of composers such as Chopin , Schumann , Liszt , and others. 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 developed a practical aural method of Tuning The Piano to equal temperament. 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. GENERAL PROPERTIES OF EQUAL TEMPERAMENT 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 evenly-spaced, 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 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 ic 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 Class 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 ratio of Frequencies between two adjacent semitones is the Twelfth Root Of Two : : This interval is equal to 100 Cent s. (The cent is sometimes for this reason defined as one hundredth of a semitone.) Calculating absolute frequencies To find the frequency, , of a note in 12-TET, the following definition may be used: : In this formula refers to the pitch, or frequency (usually in Hertz ), you are trying to find. 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''': : ''See Piano Key Frequencies for a list of 12-TET frequencies tuned to A-440.'' Comparison to just intonation The intervals of 12-TET closely approximate some intervals in Just Intonation . 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 . (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.) OTHER EQUAL TEMPERAMENTS 5 and 7 tone temperaments in ethnomusicology Five and seven tone equal temperament (5-TET and 7-TET), 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 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system. Indonesian Gamelan s are tuned to 5-TET 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 well-accepted that of the two primary tuning systems in gamelan music, Slendro and Pelog , 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. 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. Various Western equal temperaments Many systems that divide the octave equally can be considered relative to other systems of temperament. 19-TET and especially 31-TET are extended varieties of Meantone Temperament and approximate most Just Intonation intervals considerably better than 12-TET. They have been used sporadically since the 16th century, with 31-TET particularly popular in Holland, there advocated by Christiaan Huygens and Adriaan Fokker . 31-TET, like most Meantone temperaments, has a less accurate fifth than 12-TET. It has been used in Indonesian music. 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. The explanation for this curious series of numbers lies in the denominators of fractions that approximate the logarithm to base 2 of the frequency ratios of the consonant intervals. 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. Though it only improved non-traditional consonances, such as 11/4, 24-TET can be easily constructed by superimposing two 12-TET systems tuned half a semitone apart. It is based on steps of 50 cents, or . 41-TET is the lowest number of equal divisions which produces a better perfect fifth than 12-TET. It is not often used, however. (One of the reasons 12-TET 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.) 53-TET is better at approximating the traditional Just consonances than 12, 19 or 31-TET, but has had only occasional use. Its extremely good Perfect Fifth s 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 53-TET 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 22-TET .) 55-TET, not as close as 53 to Just Intonation , was a bit closer to common practice. As an excellent representative of the variety of Meantone Temperament popular in the 18th century, 55-TET it was considered ideal by Georg Philipp Telemann and other prominent musicians. Wolfgang Amadeus Mozart 's surviving theory lessons conform closely to such a model. Based on orchestral recordings, it is evident that this intonation survived as a standard practice until about 1930. 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 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 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 15-TET, 34-TET, 41-TET, 46-TET, 48-TET, 99-TET, and 171-TET. Equal temperaments of non-octave intervals The equal tempered version of the wider than an Octave , called in this theory a Tritave , 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, 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. {Link without Title} Their step sizes:
Alpha and Beta may be heard on the title track of her 1986 album ''Beauty in the Beast''. THE PURPOSE OF TWELVE-TONE EQUAL TEMPERAMENT The twelve-tone scale, upon which practically all Western music is based, is far from an arbitrary system. In any scale (whether it has five or fifty notes), some intervals between the notes will be consonant, and others will be dissonant. Consonant intervals include the following:
Scales were once constructed based on these ratios. But while many of the intervals were perfect, as music became more complex, dissonant intervals were more common. In the 1500s, equal temperament was invented to solve this problem. Equal temperament divides the octave into twelve identical intervals. The smallest interval is thus the twelfth root of two. The nature of acoustics means none of the intervals are exactly perfect, varying by perhaps a percent or so from the ideal ratios shown above (except for the octave, which is still perfect.) What makes the twelve-tone equal-tempered scale special is that it approximates the six consonant intervals shown above to an excellent degree - within one percent - and it is the smallest scale to do so. Some larger scales may approximate the consonant intervals better, but at the expense of creating more dissonant intervals. They're also much too large to make a piano out of - a keyboard based on the 53-tone scale would have at least 371 keys. What equal temperament obtains is perfect transposability, that is, key changes, at the expense of imperfect intervals. Just temperament obtains perfect intervals in one key, at the expense of poor transposability. Some meantone temperaments compromise these two qualities, having better intervals than 12-TET but better transposability than Just intonation. Perhaps the present popularity of key changes inclines most composers to remain within equal-tempered scales. SEE ALSO
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