In 12-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two: tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music.

Chinese theorists had previously come up with approximations for 12-TET, but Zhu was the first person to mathematically solve 12-tone equal temperament, [10] which he described in his Fusion of Music and Calendar ( 律暦融通, Lǜ lì róng tōng) in 1580 and Complete Compendium of Music and Pitch ( 樂律全書, Yuè lǜ quán shū ) in 1584. [11] Joseph Needham also gives an extended account. [12] Zhu obtained his result by dividing the length of string and pipe successively by 12√ 2 ≈ 1.059463, and for pipe length by 24√ 2 ≈ 1.029302, [13] such that after 12 divisions (an octave), the length was halved. Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a " pseudo-octave" in that system, into 13 equal parts.Some of the first Europeans to advocate equal temperament were lutenists Vincenzo Galilei, Giacomo Gorzanis, and Francesco Spinacino, all of whom wrote music in it. [15] [16] [17] [18] The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and Simon Stevin in 1585. According to Fritz A. Kuttner, a critic of the theory, [5] it is known that Zhu "presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently. [6] P 40 = 440 ( 2 12 ) ( 40 − 49 ) ≈ 261.626 H z {\displaystyle P_{40}=440\left({\sqrt[{12}]{2}}\right) For tuning systems that divide the octave equally, but are not approximations of just intervals, the term equal division of the octave, or EDO can be used. In modern times, 12-TET is usually tuned relative to a standard pitch of 440Hz, called A440, meaning one note, A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440Hz; it has varied considerably and generally risen over the past few hundred years. [3]

An equal temperament is a musical temperament or tuning system that approximates just intervals but instead divides an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency. [2]

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In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12-tone equal temperament, 12-TET or 12-ET, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2 ( 12√ 2 ≈ 1.05946). That resulting smallest interval, 1⁄ 12 the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12-TET.

In this formula P n is the pitch, or frequency (usually in hertz), you are trying to find. P a is the frequency of a reference pitch. n and a are 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, A 4 (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and C 4 ( middle C), and F# 4 are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of C 4 and F# 4: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: Simon Stevin was the first to develop 12-TET based on the twelfth root of two, which he described in Van De Spiegheling der singconst ( c. 1605), published posthumously in 1884. [19] Kenneth Robinson attributes the invention of equal temperament to Zhu [7] and provides textual quotations as evidence. [8] In a text dating from 1584, Zhu wrote: "I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations." [8] Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications". [5] Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered an inventor. [9] China [ edit ] Zhu Zaiyu's equal temperament pitch pipes Other equal temperaments divide the octave differently. For example, some music has been written in 19-TET and 31-TET, while the Arab tone system uses 24-TET.