Varieties of equal temperament

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing 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. In classical music, the most common tuning system since the 18th century has been an equal temperament that divides the octave into 12 parts. The system uses a logarithmic scale with a ratio equal to the 12th root of 2, ( 2 12 {\textstyle {\sqrt[{12}]{2}}} ≈ 1.05946). That resulting smallest interval, ⁠1/12⁠ the width of an octave, is called a semitone or half step. In modern times, 12 TET is usually tuned relative to a standard pitch of 440 Hz, called A 440, 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 440 Hz; it has varied considerably and generally risen over the past few hundred years. 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. 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. 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.

Twelve-tone equal temperament

12-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.

History 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 F. A. Kuttner, a critic of giving credit to Zhu, 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. Kenneth Robinson credits the invention of equal temperament to Zhu and provides textual quotations as evidence. In 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. Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications". Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.

China

Chinese theorists had previously come up with approximations for 12 TET, but Zhu was the first person to mathematically solve 12-tone equal temperament, which he described in two books, published in 1580 and 1584. Needham also gives an extended account. Zhu obtained his result by dividing the length of string and pipe successively by 2 12 {\textstyle {\sqrt[{12}]{2}}} ≈ 1.059463, and for pipe length by 2 24 {\displaystyle {\sqrt[{24}]{2}}} ≈ 1.029302, such that after 12 divisions (an octave), the length was halved. Zhu created several instruments tuned to his system, including bamboo pipes.

Europe 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. 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. Plucked instrument players (lutenists and guitarists) generally favored equal temperament, while others were more divided. In the end, 12-tone equal temperament won out. This allowed enharmonic modulation, new styles of symmetrical tonality and polytonality, atonal music such as that written with the 12-tone technique or serialism, and jazz to develop and flourish.

Seven-tone equal division of the fifth Violins, violas, and cellos are tuned in perfect fifths (G D A E for violins and C G D A for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12-tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of 3 / 2 7 {\textstyle {\sqrt[{7}]{3/2}}} to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves. During actual play, however, violinists, violists, and cellists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio.

Other equal temperaments

Five- and seven-tone equal temperament (5 TET and 7 TET ), with 240-cent and 171-cent steps, respectively, are fairly common.

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