An equal temperament is a
musical temperament
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the ...
or
tuning system
In music, there are two common meanings for tuning:
* Tuning practice, the act of tuning an instrument or voice.
* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.
Tuning practice
Tun ...
, which approximates
just intervals by dividing an
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(or other interval) into equal steps. This means the ratio of the
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of any adjacent pair of notes is the same, which gives an equal perceived step size as
pitch is perceived roughly as the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
of frequency.
In
classical music
Classical music generally refers to the art music of the Western world, considered to be distinct from Western folk music or popular music traditions. It is sometimes distinguished as Western classical music, as the term "classical music" also ...
and Western music in general, the most common tuning system since the 18th century has been twelve-tone equal temperament (also known as 12 equal temperament, 12-TET or 12-ET; informally abbreviated to twelve equal), which divides the octave into 12 parts, all of which are equal on a
logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resulting smallest interval, the width of an octave, is called a
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
or half step.
In
Western countries
The Western world, also known as the West, primarily refers to the various nations and states in the regions of Europe, North America, and Oceania. the term ''equal temperament'', without qualification, generally means 12-TET.
In modern times, 12-TET is usually tuned relative to a
standard pitch of 440 Hz, called
A440 A440 or A-440 may refer to:
* A440 (pitch standard)
* A440 highway (Australia), a road in Victoria, Australia
* Quebec Autoroute 440 (Laval)
* Quebec Autoroute 440 (Quebec City)
See also
* Apollo 440
Apollo 440 (also known as Apollo Four Fo ...
, meaning one note,
A, is tuned to 440
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
and all other notes are defined as some multiple of semitones apart from it, either higher or lower in
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. The standard pitch has not always been 440 Hz. It has varied 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
In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...
and
31-TET
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31- EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equ ...
, while the
Arab tone system The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament (24-TET), the distance between each successive note being a quarter tone ( ...
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
The Bohlen–Pierce scale (BP scale) is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the equal-tempered diatonic scale.
The ...
, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "
pseudo-octave
A pseudo-octave, pseudooctave,"Interview with Max Mathews", p.21. Author(s): C. Roads and Max Mathews. Source: ''Computer Music Journal'', Vol. 4, No. 4, (Winter, 1980), pp. 15–22. Published by: The MIT Press. or paradoxical octave"The Parad ...
" 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.
Unfretted
string ensemble
A string orchestra is an orchestra consisting solely of a string section made up of the bowed strings used in Western Classical music. The instruments of such an orchestra are most often the following: the violin, which is divided into first a ...
s, 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
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
for acoustic reasons. Other instruments, such as some
wind
Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few hou ...
,
keyboard
Keyboard may refer to:
Text input
* Keyboard, part of a typewriter
* Computer keyboard
** Keyboard layout, the software control of computer keyboards and their mapping
** Keyboard technology, computer keyboard hardware and firmware
Music
* Musi ...
, and
fret
A fret is any of the thin strips of material, usually metal wire, inserted laterally at specific positions along the neck or fretboard of a stringed instrument. Frets usually extend across the full width of the neck. On some historical instrume ...
ted 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
trombone
The trombone (german: Posaune, Italian, French: ''trombone'') is a musical instrument in the Brass instrument, brass family. As with all brass instruments, sound is produced when the player's vibrating lips cause the Standing wave, air column ...
s, 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
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
, this scale in even steps is a
geometric sequence
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
of multiplications. (An
arithmetic sequence
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
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:
:
: