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 19-TET and 31-TET for example), but they are so rare that when people use the term equal temperament without qualification, it is usually understood that they are talking about the twelve tone variety.

Twelve-tone equal temperament

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

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

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: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

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::rac{a_{12}}{a_0} = 2

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:3. Since the frequencies of 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,

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::a_{12} = s^{12} a_0

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::rac{a_{12}}{a_0} = s^{12}

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

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::2 = s^{12}

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::s = sqrt{2}

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Therefore, the ratio between two adjacent frequencies is equal to the twelfth root of two or approximately 1.05946309 to one.

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:s = sqrt{2} pprox 1.05946309

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The half tone interval:

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: 1 : 2^{1/12}

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is also known as 100 cent. 1 cent is therefore the ratio between two tone frequencies with an interval of one hundredth of an equal-tempered semitone.

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The distance between two notes whose frequencies are f1 and f2 is 12 log2(f1/f2) half tones, that is 1200 log2(f1/f2) cents.

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Cent values of equal temperament

12-TET allows the use of integer notation and modulo 12, and this allows for proofs concerning pitch.

Related Topics:
Integer notation - Modulo - Proof

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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:

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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.

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