Equal temperament
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave 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 and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament, 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,. That resulting smallest interval, the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means '.
In modern times, 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 equal temperament| and 31 equal temperament|, while the Arab tone system uses
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, 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 ' can be used.
Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.
Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.
General properties
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. Specifically, the smallest interval in an equal-tempered scale is the ratio:where the ratio divides the ratio into equal parts.
Scales are often measured in cents, which divide the octave into 1200 equal intervals. 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 above in cents, called below, and dividing it into parts:
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 modulus is the number of divisions of the octave, these integers can be reduced to pitch classes, which removes the distinction between pitches of the same name, e.g., is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.
General formulas for the equal-tempered interval
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 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:
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, 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, and for pipe length by, such that after 12 divisions, 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 based on the twelfth root of two, which he described in van de Spiegheling der singconst, published posthumously in 1884.
Plucked instrument players 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.
Mathematics
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:This interval is divided into 100 cents.
Calculating absolute frequencies
To find the frequency,, of a note in 12 , the following formula may be used:In this formula represents the pitch, or frequency, that is to be calculated. is the frequency of a reference pitch. The index numbers and are the labels assigned to the desired pitch and the reference pitch. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A is the 49th key from the left end of a piano, and C, and F are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of C and F:
Converting frequencies to their equal temperament counterparts
To convert a frequency to its equal 12 counterpart, the following formula can be used:is the frequency of a pitch in equal temperament, and is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that and have the following frequencies, respectively:
Comparison with just intonation
The intervals of 12 closely approximate some intervals in just intonation.The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.
In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents.
Seven-tone equal division of the fifth
Violins, violas, and cellos are tuned in perfect fifths, 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 to the next, which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves. During actual play, however, violinists 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-, seven-, and nine-tone temperaments in ethnomusicology
Five- and seven-tone equal temperament, with 240 cent and 171 cent steps, respectively, are fairly common.and mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure 1.
- In the tempered perfect fifth is 720 cents wide, and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0 cents.
- In the tempered perfect fifth is 686 cents wide, and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second.
5 tone and 9 tone equal temperament
7-tone equal temperament
A Thai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from. According to Morton,A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175 cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.
Chinese music has traditionally used.