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Encyclopedia > Equal temperament

An equal temperament is a musical temperament — that is, a system of tuning intended to approximate some form of just intonation — in which 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 twelve-tone equal temperament, sometimes abbreviated as 12-TET, which divides the octave into 12 equal parts. This system is usually tuned relative to a standard pitch of 440 Hz. 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. ... In music, there are two common meanings for tuning: Tuning practice, the act of tuning an instrument or voice. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In music theory, the term interval describes the difference in pitch between two notes. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ... STEP has several meanings: Sixth Term Examination Paper The Society of Trust and Estate Practitioners. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... In algebra, a ratio is the relationship between two quantities. ... Western music is the genres of music originating in the Western world (Europe and its former colonies) including Western classical music, American Jazz, Country and Western, pop music and rock and roll. ... A440 is the 440 Hz tone that serves as the standard for musical pitch. ...

Other equal temperaments do exist (some music has been written in 19-TET and 31-TET for example, and Arabian music is based on 24-TET), but in the Western world when people use the term equal temperament without qualification, it is usually understood that they are talking about 12-TET. In music, 19 equal temperament, called 19-tet, 19-edo, or 19-et, is the scale derived by dividing the octave into 19 equally large steps. ... In music, 31 equal temperament, called 31-tet, 31-edo, or 31-et, is the scale derived by dividing the octave into 31 equally large steps. ... 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 (50 cents). ...

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. A pseudo-octave is an interval whose frequency ratio is not 2:1, the definition of an octave, but is treated in some way or ways equivalent to this ratio. ... The Bohlen-Pierce scale (BP scale) is a musical scale that offers an alternative to the octave-repeating scales typical in Western music. ...

Historically,there was Seven-equal temperament or Hepta-equal temperament practice in Ancient Music of China tradition,[1][2],but whether it is real equal temperament or not is a controversial topic in academic circle. 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 wrote 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 had been intensified just at the moment when Zhu Zaiyu went into print with his new theory. Within fifty-two years of Chu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin. Music of China appears to date back to the dawn of Chinese civilization, and documents and artifacts provide evidence of a well-developed musical culture as early as the Zhou Dynasty (1122 BC - 256 BC). ... Vincenzo Galilei (1520 â€“ July 2, 1591) was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer Galileo Galilei. ... Galileo Galilei (15 February 1564 â€“ 8 January 1642) was an Italian physicist, mathematician, astronomer, and philosopher who is closely associated with the scientific revolution. ... The lute is a plucked string instrument with a fretted neck and a deep round back. ... Zhu Zaiyu (Chinese: ; Wade-Giles: Chu Tsai-Yu) (1536-1610), a prince of the Ming dynasty of China was a musician and one of the first people to discover equal temperament in music in 1584. ... Ming China under the Yongle Emperor Capital Nanjing (1368-1421) Beijing (1421-1644) Language(s) Chinese Government Monarchy Emperor  - 1368-1398 Hongwu Emperor  - 1627-1644 Chongzhen Emperor History  - Established in Nanjing January 23, 1368  - Fall of Beijing 1644  - End of the Southern Ming April, 1662 Population  - 1393 est. ... Marin Mersenne, Marin Mersennus or le PÃ¨re Mersenne (September 8, 1588 â€“ September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ... Simon Stevin Simon Stevin (1548/49 â€“ 1620) was a Flemish mathematician and engineer. ...

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 thirds or fifths 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. A major third is the larger of two commonly occuring musical intervals that span three diatonic scale degrees. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... Meantone temperament is a system of musical tuning. ... Giuseppe Tartini. ... In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies. ... Also called a Tartini tone, a combination tone is a usually lower pitch produced inside the inner ear by the presence of two external pitches. ...

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 wind, keyboard, and fretted-instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In music, a consonance (Latin consonare, sounding together) is a harmony, chord, or interval considered stable, as opposed to a dissonance, which is considered unstable. ... A wind instrument consists of a tube containing a column of air which is set into vibration by the player blowing into (or over) a mouthpiece set into the end of the tube. ... A keyboard instrument is a musical instrument played with a musical keyboard. ... The neck of a guitar showing the first four frets. ...

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.) For other people named Bach and other meanings of the word, see Bach (disambiguation). ... Title-page of Das wohtemperierte Klavier A flat major (As-dur) fugue from the second part of Das wohtemperierte Klavier (manuscript) The Well-Tempered Clavier (Das wohltemperierte Klavier in German -- Klavier means piano, but the English word clavier (which means keyboard) looks more like the German title) consists of two... Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ...

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. In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another. ... The only known photograph of FrÃ©dÃ©ric Chopin, believed to have been taken by Louis-Auguste Bisson in 1849. ... For others with the same name see Robert Schumann (disambiguation). ... Portrait by Henri Lehmann, 1839 Franz Liszt (Hungarian: Liszt Ferenc; pronounced , in English: list) (October 22, 1811 â€“ July 31, 1886) was a Hungarian [1] virtuoso pianist and composer of the Romantic period. ...

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. A tuning fork is a simple metal two-pronged fork with the tines formed from a U-shaped bar of elastic material (usually steel). ... Piano tuner Piano tuner redirects here. ...

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. The musical use of more than one key simultaneously is polytonality. ... Atonality describes music not conforming to the system of tonal hierarchies, which characterizes the sound of classical European music between the seventeenth and nineteenth centuries. ... Twelve-tone technique is a system of musical composition devised by Arnold Schoenberg. ... Serialism is a technique for composing music that uses sets to describe musical elements, and allows the composer manipulations of those sets to create music. ... This article needs additional references or sources for verification. ...

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: In music theory, the term interval describes the difference in pitch between two notes. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... This is a page about mathematics. ... In music theory, the term interval describes the difference in pitch between two notes. ...

$r^n_{}=p$
$r=sqrt[n]{p}$

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.) In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

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: The cent is a logarithmic unit of measure used for musical intervals. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Ethnomusicology (from the Greek ethnos = nation and mousike = music), formerly comparative musicology, is the study of music in its cultural context, cultural musicology. ...

$c = frac{w}{n}$

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. This article or section does not adequately cite its references or sources. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ...

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: FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... The Twelfth root of two is a quantity representing the frequency ratio between any two consecutive notes of a modern chromatic scale in equal temperament. ...

$r = sqrt[12]{2} approx 1.05946309$

This interval is equal to 100 cents. (The cent is sometimes for this reason defined as one hundredth of a semitone.) The cent is a logarithmic unit of measure used for musical intervals. ...

Calculating absolute frequencies

To find the frequency, $P^{}_n$, of a note in 12-TET, the following definition may be used:

$P_n=P_a times 2^frac{n-a}{12}$

In this formula $P^{}_n$ 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: MHZ redirects here. ... A440 is the 440 Hz tone that serves as the standard for musical pitch. ... In Western music, the expression middle C refers to the note C or Do located exactly between the two staves of the grand staff, quoted as C4 in note-octave notation (also known as scientific pitch notation). ...

$P_{40} = 440_{Hz} times 2^frac{40-49}{12} approx 261.626_{Hz}$

See Piano key frequencies for a list of 12-TET frequencies tuned to A-440. This is a virtual piano with 88 keys tuned to A440, showing the frequencies, in cycles per second (Hz), of each note (i. ...

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. In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... The cent is a logarithmic unit of measure used for musical intervals. ...

Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation
Unison $2^frac{0}{12} = 1$ 1.000000 0 $begin{matrix} frac{1}{1} end{matrix}$ = 1.000000 0.0000
Minor second $2^frac{1}{12} = sqrt[12]{2}$ 1.059463 100 $begin{matrix} frac{16}{15} end{matrix}$ = 1.066667 111.73
Major second $2^frac{2}{12} = sqrt[6]{2}$ 1.122462 200 $begin{matrix} frac{9}{8} end{matrix}$ = 1.125000 203.91
Minor third $2^frac{3}{12} = sqrt[4]{2}$ 1.189207 300 $begin{matrix} frac{6}{5} end{matrix}$ = 1.200000 315.64
Major third $2^frac{4}{12} = sqrt[3]{2}$ 1.259921 400 $begin{matrix} frac{5}{4} end{matrix}$ = 1.250000 386.31
Perfect fourth $2^frac{5}{12} = sqrt[12]{32}$ 1.334840 500 $begin{matrix} frac{4}{3} end{matrix}$ = 1.333333 498.04
Diminished fifth $2^frac{6}{12} = sqrt{2}$ 1.414214 600 $begin{matrix} frac{7}{5} end{matrix}$ = 1.400000 582.15
Perfect fifth $2^frac{7}{12} = sqrt[12]{128}$ 1.498307 700 $begin{matrix} frac{3}{2} end{matrix}$ = 1.500000 701.96
Minor sixth $2^frac{8}{12} = sqrt[3]{4}$ 1.587401 800 $begin{matrix} frac{8}{5} end{matrix}$ = 1.600000 813.69
Major sixth $2^frac{9}{12} = sqrt[4]{8}$ 1.681793 900 $begin{matrix} frac{5}{3} end{matrix}$ = 1.666667 884.36
Minor seventh $2^frac{10}{12} = sqrt[6]{32}$ 1.781797 1000 $begin{matrix} frac{16}{9} end{matrix}$ = 1.777778 996.09
Major seventh $2^frac{11}{12} = sqrt[12]{2048}$ 1.887749 1100 $begin{matrix} frac{15}{8} end{matrix}$ = 1.875000 1088.3
Octave $2^frac{12}{12} = {2}$ 2.000000 1200 $begin{matrix} frac{2}{1} end{matrix}$ = 2.000000 1200.0

(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.) In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ...

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 gamelans 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. Gamelan - Indonesian Embassy in Canberra A gamelan is a kind of musical ensemble of Indonesian origin typically featuring a variety of instruments such as metallophones, xylophones, drums, and gongs; bamboo flutes, bowed and plucked strings, and vocalists may also be included. ... Jaap Kunst (or Jakob) (b. ... Mantle Hood (? - July 31, 2005) was an American ethnomusicologist who specialized in tn studying gamelan music from Indonesia. ... Colin McPhee photo taken by Carl Van Vechten, 1935 Colin McPhee (February 15, 1900 in Montreal or Toronto, Canada - January 7, 1964 in Los Angeles, CA) was a Canadian composer and musicologist. ... Michael Tenzer (born 1957) is a composer, performer, educator and scholar. ... A pseudo-octave is an interval whose frequency ratio is not 2:1, the definition of an octave, but is treated in some way or ways equivalent to this ratio. ... Slendro (called salendro by the Sundanese) is a pentatonic (five tone) scale, one of the two most common scales used in Indonesian gamelan music. ... Pelog is one of the two essential scales of gamelan music native to Bali and Java, in Indonesia. ...

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[citation needed]. In music, 19 equal temperament, called 19-tet, 19-edo, or 19-et, is the scale derived by dividing the octave into 19 equally large steps. ... In music, 31 equal temperament, called 31-tet, 31-edo, or 31-et, is the scale derived by dividing the octave into 31 equally large steps. ... Meantone temperament is a system of musical tuning. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 â€“ July 8, 1695), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... Adriaan DaniÃ«l Fokker (Buitenzorg, Dutch East Indies (now Bogor, Indonesia), August 17, 1887â€“Beekbergen (near Apeldoorn), September 24, 1972) was a Dutch physicist and musician. ...

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 $sqrt[24]{2}$. A quarter tone is an interval half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone. ...

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 fifths make it interchangeable with an extended Pythagorean tuning, but it also accomodates 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.) In music, 53 equal temperament, called 53-tet, 53-edo, or 53-et, is the scale derived by dividing the octave into fifty-three equally large steps. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... In music, schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ... Turkish music includes the music of modern Turkey, together with related musics in neighbouring regions that once lay within the former Ottoman Empire, and closely related ethnic variants in Central Asia stretching as far as the Xinjiang Autonomous Region of China. ... In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. ...

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[citation needed]. Wolfgang Amadeus Mozart's surviving theory lessons conform closely to such a model[citation needed]. Based on orchestral recordings, it is evident that this intonation survived as a standard practice until about 1930. In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... Meantone temperament is a system of musical tuning. ... Georg Philipp Telemann. ... Wolfgang Amadeus Mozart (IPA: , baptized Johannes Chrysostomus Wolfgangus Theophilus Mozart) (January 27, 1756 â€“ December 5, 1791) was a prolific and influential composer of the Classical era. ...

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). In music, 72 equal temperament, called 72-tet, 72-edo, or 72-et, is the scale derived by dividing the octave into twelfth-tones, or in other words 72 equally large steps. ... Meantone temperament is a system of musical tuning. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... Joseph Gabriel Esther Maneri (born February 9, 1927, Brooklyn) is an American jazz composer, saxophone and clarinet player. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ...

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 Bohlen-Pierce scale consists of the ratio 3:1, 1902 cents, conventionally a perfect fifth 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 $sqrt[13]{3}$. The Bohlen-Pierce scale (BP scale) is a musical scale that offers an alternative to the octave-repeating scales typical in Western music. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ... The Bohlen-Pierce scale (BP scale) is a musical scale that offers an alternative to the 12-tone equal temperament typical in western music. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ...

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. None of the three divides any rational interval equally, but each of them provides a very good approximation of several just intervals.[1] Their step sizes: Wendy Carlos (November 14, 1939 in Pawtucket, Rhode Island) is an American composer and electronic musician. ...

• 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.

Sound waves Variations in air pressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the experience called sound. Most sound that people recognize as musical is dominated by periodic or regular vibrations rather than non-periodic ones (called a definite pitch), and... A musical scale is a discrete set of pitches used in making or describing music. ... A Microtuner or microtonal tuner is an electronic device or software program designed to modify and test the tuning of musical instruments (in particular synthesizers) with microtonal precision, allowing for the design and construction of microtonal scales and just intonation scales, and for tuning intervals that differ (or not) from... This is a virtual piano with 88 keys tuned to A440, showing the frequencies, in cycles per second (Hz), of each note (i. ... Piano tuner Piano tuner redirects here. ... A semitone (also known in the USA as a half step) is a musical interval. ... The following is a list of intervals of meantone temperament. ... Diatonic and chromatic are important terms in Western music theory. ...

Sources

• Burns, Edward M. (1999). "Intervals, Scales, and Tuning", The Psychology of Music second edition. Deutsch, Diana, ed. San Diego: Academic Press. ISBN 0-12-213564-4. Cited:
• Ellis, C. (1965). "Pre-instrumental scales", Journal of the Acoustical Society of America, 9, 126-144.
• Morton, D. (1974). "Vocal tones in traditional Thai music", Selected reports in ethnomusicology (Vol. 2, p.88-99). Los Angeles: Institute for Ethnomusicology, UCLA.
• Haddon, E. (1952). "Possible origin of the Chopi Timbila xylophone", African Music Society Newsletter, 1, 61-67.
• Kunst, J. (1949). Music in Java (Vol. II). The Hague: Marinus Nijhoff.
• Hood, M. (1966). "Slendro and Pelog redefined", Selected Reports in Ethnomusicology, Institute of Ethnomusicology, UCLA, 1, 36-48.
• Temple, Robert K. G. (1986)."The Genius of China". ISBN 0-671-62028-2
• Tenzer, (2000). Gamelan Gong Kebyar: The Art of Twentieth-Century Balinese Music. ISBN 0-226-79281-1 and ISBN 0-226-79283-8
• Boiles, J. (1969). "Terpehua though-song", Ethnomusicology, 13, 42-47.
• Wachsmann, K. (1950). "An equal-stepped tuning in a Ganda harp", Nature (Longdon), 165, 40.
• Cho, Gene Jinsiong. (2003). The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston, NY: The Edwin Mellen Press.
• Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3
• 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" [2]

Notes

1. ^ http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm Findings of new literatures concerning the hepta - equal temperament
2. ^ 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

Results from FactBites:

 Equal temperament - Wikipedia, the free encyclopedia (1977 words) Equal temperament is a scheme of musical tuning in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). True equal temperament was not theoretically possible until 1863 when Hermann Helmholtz published the first rigorous scientific study of tone and acoustics. 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.
 Equal temperament (1748 words) Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). Twelve tone equal temperament was introduced in the West to permit the playing of music in all keys with an equal amount of mis-tuning in each, without having to provide more than 12 pitches per octave on instruments, while still roughly approximating just intonation intervals. At the time equal temperament was beginning to take hold in the West, many people perceived the much-increased mis-tuning of the music, relative to meantone temperament, as a disgrace.
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