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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] 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 A comparison of some equal temperaments. [1] The graph spans one octave horizontally (open the image to view the full width), and each shaded rectangle is the width of one step in a scale. The just interval ratios are separated in rows by their prime limits. 12-tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. Play ascending and descending ⓘ

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. 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:

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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: 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. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. ( June 2011) ( Learn how and when to remove this template message)

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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] 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.