Musical temperament

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In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system.

In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies. Such just intervals have a stability, or purity to their sound. If one of those pitches is adjusted slightly, that stability decreases, and slow changes in the timbre of the interval's sound begin to appear: an effect known as beating. As the adjustment becomes more severe, the beating becomes faster. To intentionally choose an interval with beating as a substitute for a just interval is the act of tempering that interval. These adjustments can make different musical possibilities available to the musician that would be impractical in just intonation. The actual measure of these adjustments are usually called commas.

1 Meantone temperament
2 Well temperament and Equal temperament
3 See also
4 References

[edit] Meantone temperament

Main article: Meantone temperament

Before Meantone temperament became widely used in the Renaissance, the most commonly used tuning system was Pythagorean tuning. Pythagorean tuning was a system of just intonation which tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the harmonic practice until then (See: Quartal harmony), but in the Renaissance, musicians wished to make much more use of Tertian harmony. The major third of Pythagorean tuning differed from a just major third by an amount known as Syntonic comma, which musicians of the time found annoyingly impure.

Their solution, laid out by Pietro Aron in the early 16th century, was to temper the interval of a perfect fifth slightly narrower than in just intonation, and then proceed much like Pythagorean tuning, but using this tempered fifth instead of the just one. With the correct amount of tempering, the Syntonic comma is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this tuning system with a slight beating. However, because a sequence of four fifths makes up one third, this beating effect on the fifths is only one quarter as strong as the beating effect on the thirds of Pythagorean tuning, which is why it was considered a very acceptable compromise by Renaissance musicians.

Pythagorean tuning also had a second problem, which Meantone temperament does not solve, which is the problem of modulation (see below), which is restricted because both have a broken circle of fifths. A series of 12 just fifths as in Pythagorean tuning does not return to the original pitch, but rather differs by a Pythagorean comma, which makes that tonal area of the system more or less unusable. In meantone temperament, this effect is even more pronounced (the fifth over the break in the circle is known as the Wolf interval, as its intense beating was likened to a "howling".) 53 equal temperament provides a solution for the Pythagorean tuning, and 31 equal temperament for the Meantone.

[edit] Well temperament and Equal temperament

Main articles: Well temperament and Equal temperament

Just intonation has the problem that it cannot modulate to a different key (a very common means of expression throughout the Common practice period of music) without discarding many of the tones used in the previous key, thus for every key the musician wishes to modulate to, his instrument must provide a few more strings, frets, or holes for him to use. When building an instrument, this can be very impractical.

Well temperament is the name given to a variety of different systems of temperament that were employed to solve this problem. 12 tone equal temperament (12-TET) is the modern standard version of it, and it can be seen as another modification of Pythagorean tuning. Unlike Meantone temperament, which alters the fifth to temper out the Syntonic comma, 12-TET tempers out the Pythagorean comma, thus creating a cycle of fifths that repeats itself exactly after 12 steps. This allowed the intervals of Tertian harmony, thirds and fifths, to be fairly close to their just counterpoints (the fifths almost imperceptibly beating, the thirds a little milder than the Syntonic beating of Pythagorean tuning), while permitting the freedom to modulate to any key and by various means (e.g. common-tone and enharmonic modulation, see modulation). This freedom of modulation also allowed substantial use of more distant harmonic relationships, such as the Neapolitan chord, which became very important to Romantic composers in the 19th century.