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] 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: In an equal temperament, the distance between two adjacent steps 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:

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.

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) 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: 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. where the ratio r divides the ratio p (typically the octave, which is 2:1) into n equal parts. ( See Twelve-tone equal temperament below.)

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 ⓘ 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 pipesP 40 = 440 ( 2 12 ) ( 40 − 49 ) ≈ 261.626 H z {\displaystyle P_{40}=440\left({\sqrt[{12}]{2}}\right) 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]