Equal temperament

An equal temperament is a special way of tuning musical instruments. It is a system that divides an octave—the distance between one musical note and the note an octave higher—into equal parts. This means that each step between notes sounds the same size, making it easier to play music across different keys.

12 tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. Play ascending and descendingⓘ

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 ratio, calculated as the 12th root of 2. 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, and 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 allows musicians to explore a wide range of sounds and styles in their music.

General properties

In an equal temperament, the distance between each step in the scale stays the same. This is done by dividing an interval, like an octave, into equal parts. Because our ears perceive intervals based on their ratio, this creates a smooth and consistent scale that can easily shift between different keys.

Musicians often use a unit called "cents" to measure these intervals, splitting an octave into 1200 equal parts. This makes it easier to 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

Zhu Zaiyu's equal temperament pitch pipes

12-tone equal temperament divides an octave into 12 equal parts. This system is widely used in Western music today. Each step, or "semitone," has the same ratio of frequencies, making all steps sound the same in size.

This tuning method allows for new musical styles like jazz and modern compositions. It was developed independently in China and Europe during the late 1500s. One key idea is that the ratio between the frequencies of any two adjacent notes is the twelfth root of two, written as ¹²√2, which is approximately 1.05946. This means each note is slightly higher in pitch than the one before it, creating a smooth, even scale across the entire 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)	2^0⁄12 = 1	1	0	⁠1/1⁠ = 1	0.00	0.00
Minor second (D♭)	2^1⁄12 = 2 12 {\displaystyle {\sqrt[{12}]{2}}}	1.059463	100	⁠16/15⁠ = 1.06666...	111.73	-11.73
Major second (D)	2^2⁄12 = 2 6 {\displaystyle {\sqrt[{6}]{2}}}	1.122462	200	⁠9/8⁠ = 1.125	203.91	-3.91
Minor third (E♭)	2^3⁄12 = 2 4 {\displaystyle {\sqrt[{4}]{2}}}	1.189207	300	⁠6/5⁠ = 1.2	315.64	-15.64
Major third (E)	2^4⁄12 = 2 3 {\displaystyle {\sqrt[{3}]{2}}}	1.259921	400	⁠5/4⁠ = 1.25	386.31	+13.69
Perfect fourth (F)	2^5⁄12 = 32 12 {\displaystyle {\sqrt[{12}]{32}}}	1.334840	500	⁠4/3⁠ = 1.33333...	498.04	+1.96
Tritone (G♭)	2^6⁄12 = 2 {\displaystyle {\sqrt {2}}}	1.414214	600	⁠45/32⁠= 1.40625	590.22	+9.78
Perfect fifth (G)	2^7⁄12 = 128 12 {\displaystyle {\sqrt[{12}]{128}}}	1.498307	700	⁠3/2⁠ = 1.5	701.96	-1.96
Minor sixth (A♭)	2^8⁄12 = 4 3 {\displaystyle {\sqrt[{3}]{4}}}	1.587401	800	⁠8/5⁠ = 1.6	813.69	-13.69
Major sixth (A)	2^9⁄12 = 8 4 {\displaystyle {\sqrt[{4}]{8}}}	1.681793	900	⁠5/3⁠ = 1.66666...	884.36	+15.64
Minor seventh (B♭)	2^10⁄12 = 32 6 {\displaystyle {\sqrt[{6}]{32}}}	1.781797	1000	⁠9/5⁠ = 1.8	1017.60	-17.60
Major seventh (B)	2^11⁄12 = 2048 12 {\displaystyle {\sqrt[{12}]{2048}}}	1.887749	1100	⁠15/8⁠ = 1.875	1088.27	+11.73
Octave (c)	2^12⁄12 = 2	2	1200	⁠2/1⁠ = 2	1200.00	0.00

Other equal temperaments

Related tuning systems

Equal temperament systems can be compared to just intonation, where chords often sound perfectly in tune. In contrast, equal temperament changes the spacing of three main intervals found in just intonation: the greater tone, the lesser tone, and the diatonic semitone. By adjusting these intervals, different types of equal temperaments can be created.

Figure 1: The regular diatonic tunings continuum, which includes many notable "equal temperament" tunings

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 created by changing the sizes of these intervals. Each of these systems approximates different historical tuning methods and offers unique musical properties.

Main article: just intonation

General properties

Twelve-tone equal temperament

Other equal temperaments

Related tuning systems

Images