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.

Table of contents

1 12 TET
2 Non-12 TET
3 External links

12 TET

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

1. Let a_n 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

3. Since the tones are in a geometric sequence, the frequency for a tone k (relative to the tone designated zero) will be equal to s^ka₀ where s is the constant ratio between adjacent frequencies. This gives for k = 12,

4. 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 complex 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. Instead, they used approximations that emphasized the tuning of thirds or fifths in certain keys, such as Mean tone 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. 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

Non-12 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 intervals consisting only of odd numbers.