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Equal temperament
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Equal temperament

Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). The best known example of such a system is twelve-tone equal temperament (sometimes abbreviated to 12-TET), which is nowadays used in most Western music. Other equal temperaments do exist (some music has been written in 31-TET for example), but they are so rare that when people use the term equal temperament it is usually understood that they are talking about the twelve tone variety.

The distance between each step and the next is aurally the same for any two adjacent steps, though, because steps form an geometric sequence, the difference in frequency increases from one to the next. A linear sequence of one frequency difference would create ever smaller intervals (ratios), such as the harmonic series. See also logarithmic scale.

Table of contents
1 Twelve-tone equal temperament
2 Non-12 TET
3 See also
4 External links

Twelve-tone equal temperament

The ratio between two adjacent semitones can be found with a few steps:

  1. Let an be the frequency of a semitone n, with a12 an octave above a0. This creates twelve tones for each octave.

  2. Since the frequency ratio of a tone from one octave to the next is 2:1, the ratio of the frequency of one tone (a12) to the frequency of a tone an octave lower (a0) is 2:1 as well, so

Since the tones are in a geometric sequence, the frequency for a tone k (relative to the tone designated zero) will be equal to ska0 where s is the constant ratio between adjacent frequencies. This gives for k = 12,

Since a12 / a0 was found to be two, the formula with constant ratio s is

Therefore, the ratio between two adjacent frequencies is equal to the twelfth root of two or approximately 1.05946 to one.

The interval
is also known as a cent: the ratio between two tone frequencies with an interval of one hundredth of an equal-tempered semitone.

Twelve tone equal temperament was designed to permit the playing of music in all keys with an equal amount of mis-tuning in each, while still approximating just intonation. This allows much more facile harmonic motion, while losing some subtlety of intonation. True equal temperament was not available to musicians before about 1870 because scientific tuning and measurement was not available. And in fact, from about 1450 to about 1800 musicians tolerated even less mistuning in the most common keys, like C major. Instead, they used approximations that emphasized the tuning of thirds or fifths in these keys, such as Meantone temperament. J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament. There is some reason to believe that when composers and theoreticians of this era wrote of the "colors" of the keys, they described the subtly different dissonances of particular tuning methods, though it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer.

12-TET also allows the use of integer notation and modulo 12, and this allows for proofs concerning pitch. This may be used for atonal music, such as that written with the twelve tone technique or serialism, or tonal music. See also: musical set theory

The following table shows the values of the intervals of 12 TET, along with one interval from just intonation that each approximates, and the percentage by which they differ:

Name Exact value in 12-TET Decimal value Just intonation interval Percent difference
Unison 1 1 1 0.00%
Minor second 1.059 16/15 -0.68%
Major second 1.122 9/8 -0.23%
Minor third 1.189 6/5 -0.90%
Major third 1.260 5/4 0.79%
Perfect fourth 1.335 4/3 0.11%
Diminished fifth 1.414 7/5 1.02%
Perfect fifth 1.498 3/2 -0.11%
Minor sixth 1.587 8/5 -0.79%
Major sixth 1.682 5/3 0.91%
Minor seventh 1.782 16/9 0.23%
Major seventh 1.888 15/8 0.68%

These mappings from equal temperament to just intonation are by no means unique. The minor seventh, for example, can be meaningfully said to approximate both 16/9 and 9/5, depending on context or simultaneously in a chord -- and probably even 7/4.

Non-12 TET

The quarter tone scale or 24-tET is, similarly, based on powers of . Other equal divisions of the octave, though, can be better considered temperaments. 19-tET and especially 31-tET are extended varieties of Meantone temperament and approximate most just intonation intervals considerably better than 12-tET. They have been used sporadically since the 16th century, with 31-tET particularly popular in Holland, there advocated by Christiaan Huygens and Adriaan Fokker. 53-tET is much better still at approximating the traditional just intonation consonances, but has had very little use. It doesn't fit the Meantone mold that shaped the development of Western harmony and tonality. In 53-tET, most traditional compositions would necessitate subtle microtonal pitch shifts or a drifting pitch level in order to make use of the tuning's excellent just intonation triads. 55-tET, not as close to just intonation, was a bit closer to common practice. As an excellent representative of the variety of meantone temperament popular in the 18th century, 55-tET it was considered ideal by Georg Philipp Telemann and other prominent musicians. Wolfgang Amadeus Mozart's surviving violin lessons conform closely to such a model.

In the 20th century, standardized Western pitch and notation practices having been placed on a 12-tET foundation made the quarter tone scale a much more popular microtonal tuning. A further extension of 12-tET is 72-tET, which though not a Meantone tuning, approximates most just intonation intervals, including non-traditional ones like 7/4, 9/7, 11/5, 11/6 and 11/7, better. 72-tET has been taught, written and performed in in practice, for example by Joe Maneri and his students -- whose atonal inclinations typically avoid any reference to just intonation intervals whatsoever. Still other equal temperaments occupying more than a few musicians include 5-tET, 7-tET, 15-tET, 22-tET, and 48-tET.

More generally, every step in n tone equal temperament is 1200/n cents. However, if one wishes to create an equal tempered scale that does not repeat at the octave, a scale with n equal steps in a pseudo-octave p is based on the ratio r

.
This still may be easier to calculate in cents, for instance the pseudo-octave of ratio 2.1:1 is an interval of 1284 cents. Equal tempered scales can also be generated simply by picking the number of cents that each step will consist of.

Wendy Carlos created two equal tempered scales for the title track of her album Beauty In The Beast, the Alpha and Beta scales. Beta splits a perfect fourth into two equal parts, which creates a scale where each step is almost 64 cents. Alpha does the same to a minor third to create a scale of 78 cent steps.

The equal tempered version of the Bohlen-Pierce scale consists of the ratio 3:1, 1902 cents, conventionally an octave and a just fifth, used as a tritave, and split into a thirteen tone equal temperament where each step is

or 146.3 cents. This provides a very close match to justly tuned ratios consisting only of odd numbers.

See also

External links