SafekipediaEqual temperament
Adapted from Wikipedia · Adventurer experience
An equal temperament is a special way of tuning musical instruments. It divides an octave—the distance between one musical note and the note an octave higher—into equal parts. This makes each step between notes sound the same size. It is easier to play music in different keys this way.
The most common type is called 12 equal temperament, or 12 TET. This system splits the octave into 12 equal parts, called semitones. Each semitone has a precise frequency. In modern music, this system is often tuned to a standard pitch of 440 Hz for the note A, known as A 440. This standard helps musicians play together smoothly, whether they are using wind, keyboard, or other instruments.
While 12 TET is widely used, other equal temperaments exist. Some divide the octave into 19 or 31 parts. Some systems, like the Arab tone system, use 24 parts. There are even systems that divide intervals other than the octave, such as the Bohlen–Pierce scale. This flexibility lets musicians explore many different sounds and styles in their music.
General properties
In an equal temperament, the distance between each note in a scale is always the same. This is done by splitting an interval, such as an octave, into equal parts. Because our ears hear intervals based on their ratio, this creates a smooth and consistent scale that can easily change between different keys.
Musicians use a unit called "cents" to measure these intervals, dividing an octave into 1200 equal parts. This helps them compare different tuning systems. For any equal temperament, you can find the size of each step by dividing the total width in cents by the number of parts in the scale.
Twelve-tone equal temperament
Main article: 12 equal temperament
12-tone equal temperament splits an octave into 12 equal parts. This system is used in most Western music today. Each step, called a "semitone," has the same size. This makes all steps sound the same.
This tuning helps create new music styles like jazz. It was created separately in China and Europe in the late 1500s. The ratio between any two nearby notes is the twelfth root of two, written as 12√2. This number is about 1.05946. Each note is a little higher in sound than the note before it, making a smooth, even scale across the whole octave.
| Interval Name | Exact value in 12 TET | Decimal value in 12 TET | Cents in 12 TET | Just intonation interval | Cents in just intonation | 12 TET cents tuning error |
|---|---|---|---|---|---|---|
| Unison (C) | 20⁄12 = 1 | 1 | 0 | 1/1 = 1 | 0.00 | 0.00 |
| Minor second (D♭) | 21⁄12 = 2 12 {\displaystyle {\sqrt[{12}]{2}}} | 1.059463 | 100 | 16/15 = 1.06666... | 111.73 | -11.73 |
| Major second (D) | 22⁄12 = 2 6 {\displaystyle {\sqrt[{6}]{2}}} | 1.122462 | 200 | 9/8 = 1.125 | 203.91 | -3.91 |
| Minor third (E♭) | 23⁄12 = 2 4 {\displaystyle {\sqrt[{4}]{2}}} | 1.189207 | 300 | 6/5 = 1.2 | 315.64 | -15.64 |
| Major third (E) | 24⁄12 = 2 3 {\displaystyle {\sqrt[{3}]{2}}} | 1.259921 | 400 | 5/4 = 1.25 | 386.31 | +13.69 |
| Perfect fourth (F) | 25⁄12 = 32 12 {\displaystyle {\sqrt[{12}]{32}}} | 1.334840 | 500 | 4/3 = 1.33333... | 498.04 | +1.96 |
| Tritone (G♭) | 26⁄12 = 2 {\displaystyle {\sqrt {2}}} | 1.414214 | 600 | 45/32= 1.40625 | 590.22 | +9.78 |
| Perfect fifth (G) | 27⁄12 = 128 12 {\displaystyle {\sqrt[{12}]{128}}} | 1.498307 | 700 | 3/2 = 1.5 | 701.96 | -1.96 |
| Minor sixth (A♭) | 28⁄12 = 4 3 {\displaystyle {\sqrt[{3}]{4}}} | 1.587401 | 800 | 8/5 = 1.6 | 813.69 | -13.69 |
| Major sixth (A) | 29⁄12 = 8 4 {\displaystyle {\sqrt[{4}]{8}}} | 1.681793 | 900 | 5/3 = 1.66666... | 884.36 | +15.64 |
| Minor seventh (B♭) | 210⁄12 = 32 6 {\displaystyle {\sqrt[{6}]{32}}} | 1.781797 | 1000 | 9/5 = 1.8 | 1017.60 | -17.60 |
| Major seventh (B) | 211⁄12 = 2048 12 {\displaystyle {\sqrt[{12}]{2048}}} | 1.887749 | 1100 | 15/8 = 1.875 | 1088.27 | +11.73 |
| Octave (c) | 212⁄12 = 2 | 2 | 1200 | 2/1 = 2 | 1200.00 | 0.00 |
Other equal temperaments
See also: Sonido 13
Five- and seven-tone equal temperament are often used in world music. These tuning systems have steps of 240 cents and 171 cents.
In five-tone equal temperament, the perfect fifth is 720 cents wide. In seven-tone equal temperament, it is 686 cents wide.
Research shows that Indonesian gamelans are often tuned to five-tone equal temperament. Thai xylophones studied in 1974 were close to seven-tone equal temperament. Chinese music has traditionally used seven-tone equal temperament.
Many instruments use different ways to divide the octave (EDO) for tuning. Examples include 19 EDO, 22 EDO, 23 EDO, 24 EDO, and many more. Each offers new ways to approximate musical notes.
The equal-tempered Bohlen–Pierce scale divides the tritave into 13 equal parts. Wendy Carlos created special tuning systems called alpha, beta, and gamma. These divide the perfect fifth into equal parts with different step sizes.
It is also possible to create tuning systems that divide the octave into parts that are not whole numbers. These can give better approximations of musical intervals than systems using whole numbers.
Related tuning systems
Equal temperament systems are different from just intonation. In just intonation, chords often sound perfectly in tune. But in equal temperament, three main intervals are changed: the greater tone, the lesser tone, and the diatonic semitone. These changes let us create different types of equal temperaments.
Regular diatonic tunings follow a pattern of steps that include tones and semitones. These patterns can be extended into a spiral of fifths, which does not close perfectly like the circle of fifths in 12-tone equal temperament. Various equal temperaments, such as 5 TET, 7 TET, 19 TET, 31 TET, 43 TET, and 53 TET, are made by changing the sizes of these intervals. Each of these systems tries to copy different old tuning methods and has its own special musical properties.
Main article: just intonation
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