An equal temperament is a
musical temperament or
tuning system, which approximates
just intervals by dividing an
octave (or other interval) into equal steps. This means the ratio of the
frequencies 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 ...
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" al ...
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, 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 or half step.
In
Western countries the term ''equal temperament'', without qualification, generally means 12-TET.
In modern times, 12-TET is usually tuned relative to a
standard pitch
Concert pitch is the pitch reference to which a group of musical instruments are tuned for a performance. Concert pitch may vary from ensemble to ensemble, and has varied widely over music history. The most common modern tuning standard uses ...
of 440 Hz, called
A440, 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 o ...
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, 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
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 firs ...
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 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 ...
,
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
* Mu ...
, 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
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 ...
of multiplications. (An
arithmetic sequence of intervals would not sound evenly spaced, and would not permit
transposition to different
keys
Key or The Key may refer to:
Common meanings
* Key (cryptography), a piece of information that controls the operation of a cryptography algorithm
* Key (lock), device used to control access to places or facilities restricted by a lock
* Key (m ...
.) Specifically, the smallest
interval in an equal-tempered scale is the ratio:
:
: