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As mentioned above , mean-tone tuning was still very popular in the eighteenth century. J. S. Bach wrote his famous "Well-Tempered Klavier" in part as a plea and advertisement to switch to a well temperament system. Various well temperaments did become very popular in the eighteenth and nineteenth centuries, and much of the keyboard-instrument music of those centuries may have been written to take advantage of the tuning characteristics of particular keys in particular well temperaments. Some modern musicians advocate performing such pieces using well temperaments, in order to better understand and appreciate them. It is interesting to note that the different keys in a well temperament tuning were sometimes considered to be aligned with specific colors and emotions. In this way they may have had more in common with various modes and ragas than do keys in equal temperament.

Equal temperament

In modern times, well temperaments have been replaced by equal temperament, so much so in Western music that equal temperament is considered standard tuning even for voice and for instruments that are more likely to play using just intonation when they can (see above ). In equal temperament, only octaves are pure intervals. The octave is divided into twelve equally spaced half steps , and all other intervals are measured in half steps. This gives, for example, a fifth that is a bit smaller than a pure fifth, and a major third that is larger than the pure major third. The differences are smaller than the wolf tones found in other tuning systems, but they are still there.

Equal temperament is well suited to music that changes key often, is very chromatic , or is harmonically complex . It is also the obvious choice for atonal music that steers away from identification with any key or tonality at all. Equal temperament has a clear scientific/mathematical basis, is very straightforward, does not require retuning for key changes, and is unquestioningly accepted by most people. However, because of the lack of pure intervals, some musicians do not find it satisfying. As mentioned above, just intonation is sometimes substituted for equal temperament when practical, and some musicians would also like to reintroduce well temperaments, at least for performances of music which was composed with well temperament in mind.

A comparison of equal temperament with the harmonic series

In a way, equal temperament is also a compromise between the Pythagorean approach and the mean-tone approach. Neither the third nor the fifth is pure, but neither of them is terribly far off, either. Because equal temperament divides the octave into twelve equal semi-tones (half steps), the frequency ratio of each semi-tone is the twelfth root of 2. If you do not understand why it is the twelfth root of 2 rather than, say, one twelfth, please see the explanation below . (There is a review of powers and roots in Powers, Roots, and Equal Temperament if you need it.)

In equal temperament, the ratio of frequencies in a semitone (half step) is the twelfth root of two. Every interval is then simply a certain number of semitones. Only the octave (the twelfth power of the twelfth root) is a pure interval.

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Source:  OpenStax, Frequency and music. OpenStax CNX. Feb 21, 2006 Download for free at http://cnx.org/content/col10338/1.1
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